Word problems. Decision problems and the Burnside problem in group theory

WORD PROBLEMS Decision Problems and the Burnside Problem in Group Theory

Edited by

W. W. BOONE

F. B. CANNONITO

University of Illinois Urbana

University of California Irvine

R.C . LYNDON University of Michigan Ann Arbor

1973

NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM LONDON

@ North-Holland Publishing Company - 1973

AN rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form op by any m e w s , electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner

Library of Congress Catalog Card Number: 70-146190 North-Holland ISBN: 0 7204 2271 X

PRINTED IN THE NETHERLANDS

Dedicated to the memory of our friend and colleague Hanna Neumann (1914-1971)

INTRODUCTION This volume grew o u t of the conference “Decision Problems in Group Theory”, (CODEP) held at the University of California, Irvine, in September 1969, with the support of the United States National Science Foundation, the Army Research Office, Durham, and the Air Force Office of Scientific Research (Air Force Systems Command). These same organizations supported the preparation of this volume. A l t h o u g h c u t t h g broadly across logic, group theory and other fields of algebra, topology and computer science, the mathematics discussed in the conference and this book has, we would argue, an inherent unity of purpose and method. The unifying aspects are the common concern with algorithms and the common use of a certain kind of combinatorial analysis. This analysis had been used in logic for a long time, particularly in the study of proofs, before it then spread - in the very development of the field we are here presenting - t o the study of “word problems” in groups and semigroups. To us it seems unlikely that the argument of Novikov and Adjan for the Burnside Problem, published in 1968 and whose central ideas are explained in the present volume, o r the argument of Britton for the same result in the present volume, could have been discovered by a “pure” group theorist! All this is not t o say that the field of word problems did not exist before this confluence of group theory with logic. Indeed, the origin of the field may be traced back t o Max Dehn’s fundamental problems posed in 191 1. At that time Dehn was studying the fundamental groups of compact manifolds of genus at least 2. His concern was with the existence of algorithms t o determine if given paths were contractible t o a point, if given paths were homotopic, and if given spaces were homotopy equivalent. In group-theoretic terms these decision problems are, respectively, the word (or identity) problem, the conjugacy (or transformation) problem, and the isomorphism problem - now all known t o be unsolvable in general ix

X

Introduction

for finitely presented abstract groups. Dehn did solve the word problem corresponding t o the case of closed 2-manifolds of genus at least 2. Further, Wilhelm Magnus’ proof of the Freiheitssatz, and his related solution of the word problem for groups given by one defining relation, also appeared in the 1930’s before the union with logic. We should mention, too, that early on in the century, as an independent line of development, Axel Thue had formulated the word problem for finitely presented semi-groups - or, as one now says, Thue systems - and solved various special cases of this general problem. But negative results, unsolvability results in group theory, were impossible before the union with logic, since the very idea of an algorithmically unsolvable problem was lacking. This lack was overcome in 1935- 1936 by Alonzo Church and, independently, in 1936 by A.M. Turing when they gave equivalent precise mathematical definitions of the intuitive notion of algorithm. All these definitions, known today as “Church’s Thesis”, led t o Church’s negative solution of the decision problem for first-order arithmetic; and, subsequently, to independent negative solutions by Church and Turing t o Hilbert’s Entscheidungsproblem for pure predicate logic. Hindsight shows that virtually all unsolvability results in mathematics arc, in the final analysis, a translation of such classical results into a new setting.” The propitious event, t o our minds, constructing the bridge from logic to algebra occurred in 1946-1947. F o r then Emil Post and A.A. Markov, independently, showed the word problem for Thue systems unsolvable. This result was the first unsolvability result outside the foundations and it marked off the field of word problems in much its present form. Ultimately this bridging would lead to the situation that, while one still had positive and negative results regarding the existence of algorithms, the logic and algebra are so merged in theorems and proofs that it is impossible to separate one from the other. * Other formalizations of the intuitive notion

of effective process are due t o Gijdel (based on a suggestion of Herbrand), Kleene, Post, and A.A. Markov. That all these definitions have turned out to be equivalent is often taken as evidence for the correctness of Church’s ‘I’hcsis.

Introduction

xi

This book is intended not only as a volume for the expert but also as a vehicle for entry into the field at various levels of knowledge and into various specialties. We have included many survey articles; the Britton proof and the Wos-Robinson proof are entirely self-contained. The foremost reference we can mention t o obtain the necessary background for this volume is the following: Joseph J. Rotman, The theory of groups, an introduction (Allyn and Bacon). The second edition (1973), particularly the last chapter, of Rotman’s text gives virtually all the background material from logic (recursive function theoretic), needed t o get into the present volume. Indeed, the only sizable gap for the reader would then be the FriedbergMuchnik Theorem on the existence of “in between” recursively enumerable degrees - and this can be found in several standard logic texts. Our next most important reference is: Wilhelm Magnus, Abraham Karrass and Donald Solitar, Combinatorial group theory (Interscience, New York, 1966). Referring t o the text, the reader will be able t o follow or look in at a deeper level. An historical account of the field is given in: William W. Boone, The theory of decision problems in group theory: u survey; this is available on tape from the American Mathematical Society and is to appear in the Bulletin of the American Mathematical Society. The following three items are parallel t o parts of this volume: William W. Boone, Wolfgang Haken and Valentin Pognaru, On recursively itnsolvable problems in topology und their classification, in : Con tribi I tions to Math cinu tical Logic ( N orth-H olland , Am sterdam, 1968). Roger C. Lyndon and Paul Schupp, Geometric group theor], (tentative title), Ergebnisse der Mathematik (Springer, Berlin, to appear) . Charles F. Miller, 111, On group-theoretic decision p r o b l e m and their classification, Annals of Mathematic Studies (Princeton Univ. Press, Princeton, N.J., 1971). We d o not attempt to attribute authorship t o the problems listed in this volume and we wish here t o thank all who have so generously helped in the preparation of this volume by their contribution t o this very interesting problem set.

xii

Introduction

The editors also wish to acknowledge their gratitude to the NorthHolland Publishing Company and particularly t o Einar Fredriksson for the considerable help given t o us during the preparation of this work. Similarly Jens Mennicke’s mathematical help in carefully checking Britton’s proof was so important that h e deserves some such title as “honorary editor”, which we hereby confer. Our thanks, too, to Peter X. Sarapuka for his careful translation of Adjan’s article from the original Russian. And, finally, we thank the contributors, without whose efforts and cooperation, this volume, of course, could not exist. To them belongs the lion’s share of credit for whatever merits this volume may have. William W. Boone Frank B. Cannonito Roger C. Lyndon

A PROOF OF HIGMAN'S EMBEDDING THEOREM USING BRITTON EXTENSIONS OF GROUPS S t i l AANDERAA University of Oslo

$1. introduction

The aim of this paper is t o give a new proof of the following theorem due t o Higman (see Higman [ 5, Theorem 1, p. 4561 ):

Theorem 1 . A finitely generated group can be embedded in a finitely presented group i f f it is recursively presented. Shoenfield [ 9, pp. 3 2 1-3261 presents a variant of Higman's proof. As pointed out by Higman [ 51, one half of Theorem 1 is trivial. Hence, from now on we shall concentrate on the non-trivial half of the theorem, which we shall state as a lemma:

Lemma 1 . Every finitely generated and recursively presented group can b e embedded in a finitely presented group. Definition 1 . A positive group presentation is a presentation (SID), where each relation in D is of the form w = 1 for some positive word over S. Note that every group presentation (SID) can be turned into a positive presentation of the same group by replacing S by S' = {s'l s E S ) u S and D by D', where D' is obtained from D by replacing each occurrence of s-l by s' and by adding the relation ss' = 1 for every s in S . 1

4

S. Aanderaa, Proof of Higman s embedding theorem

s i f s E S , n S,, dS) =

I i f s €S1\S2

can be extended t o a homomorphism of G, into G,.

Definition 4. Let W be a word. We shall say that W is trivially (or freely) reduced iff W contains n o subword of the form sl's or ss-'. Definition 5. Let (S*ID*) be a Britton extension of (SID) with stable letters P. A pirich is a word of one of the following forms: (i) p - ' Cp, where C is a word on S , and C E A ( p ) and p E P ; o r (ii) pCj>-', where C is a word on S , and C E B ( p ) and p E P. Definition 6. Given (S*ID*) with basis ( S I D )and stable letters P , then a word on S" is p-redzrced (for a fixed p E P ) iff W contains no pinches of the forms p - ' C p or p C p - l as subwords. Moreover, W is p-reduced iff W is p-reduced for all p E P. Bootie [ 2, p. 571 defines the p-reduction of a word. We shall use p-reduction in a special case.

Definition 7. Let (S*ID*) be a Britton extension of (SID) with stable letters P. Let p E P and suppose all relations in D" involving p are of the form / I - ' X p = X.Let W and W ' be words 011 S". Then W ' is a p-rediictioiz of W iff W contains a pinch p - e C p e ( e = 1 ) and W ' is the result of replacing p - e C p e by C.

*

Definition 8. A set X is called afree set of'geiierators in G if the subgroup F generated by X is free on X. Lemma 2. Let G" be a Brittoi? extension o f ' G with stable letters P. Let F be a f r e e subgroup of' G which is free on the set X . Let H be u siibgroiip of'G siiclz that F n €1 = {I}. Let P , C P and suppose F n A ( p ) = F n R ( p ) = {I} f o r all p E P,. Let F* be the subgroup o f G* geiierated 61. X u P , Then F* is a f i e e group on the set X u P , , arid I;" n H = { I } . M o r e o i ~ suppose r that H' is u group ,

S. Aanderaa, Proof of Higman S embedding theorem

5

generated by Y = { h i p i g i I 1 < i G l} , where h i , gi E G, p i E P, p i $ P, and p i # pi if i # j (i = 1 , 2 , ..., I; j = 1 , 2 , ..., I ) , then H' is a f r e e g r o u p on Y a n d F* n H ' = { 1).

Proof. Let w be a trivially reduced word on P, U X,and suppose w = h in G* for some h E H. We shall prove that this implies w = 1, which implies that F* is free on the set X u P, and that F* n H = = ( 1 ) . Suppose that w is a word which involves at least one stable letter p E P,. Since wh-l = 1 in G*, and since p does not occur in h , we have that w contains a pinch of the form p-'Cp' (E = f l ) , where p E P, and C E A ( p ) n F = { 1) or C f B ( p ) n F = { 1 ) . Hence C = 1 and p-' Cp' = 1 , which contradicts the fact that w is trivially reduced. Hence w involves n o letter p E P,. That is, w E F , since w is a word on X.But F n H = { 1) which shows that w = 1 and h = 1 in G*. To prove the last part of the lemma, let w be trivially reduced word on X U P, and let u be a trivially reduced word on Y , and let w = u in G*. Then w-l u = 1 in G*, and exactly as before we can prove that w is a word on X and if u contains a pinch of the form p ; , C p , then ZL contains a subword of the form gi Pi- ' h :i' h i p i g i , which contradicts the assumption that u is trivially reduced. Moreover, by a similar argument we can prove that u contains n o pinch of the form p i C p ; ' . Hence u = 1 and w = 1 in G*. This proves the last part of the lemma.

-'

$4. The Turing machine and the corresponding semigroups

The following discussion leading to Lemma 4 presupposes familiarity with Turing machines. We shall use the convention that so is the blank square.

Definition 9. T is a Turing machine, its alphabet is the set {so, s, ..., sM-,} of all s-letters x c u r r i n g in its quadruples. If w is a positive word on the non-blank symbols {s,, s2, ..., s M - , } , then T accepts w if there is a computation of T beginning with q 1w.(We use "computation of T" in the sense of Davis [4, Definition 1.9, P. 71.)

6

S.Aanderaa, Proof of Higman S embedding theorem

Informally, we regard the machine T as being started in the state y 1 scanning the leftmost symbol in w , and T accepts w iff it eventually stops. We shall also call w an input t o the Turing machine T i n the case we are studying the (finite or infinite) sequence of instantaneous descriptions q , w + a 2 + a 3 -+ ... . We shall say that a set of positive words is recztrsively enlimeruble iff the set of its Godel numbers is recursively enumerable.

Lemma 3 . Let E be a recursively enziinerable set of positive words o t i the letters { a , , a 2 , .._,a m } . Then there exists a Turing machine and states Z,. with alphabet S'= {so, sl, s 2 , . ., Q' = (41, y 2 , ..., Y N - ~ szich } that ( 1 ) {ci1,u2 ,..., arn}c { ~ ~ , s ~ , . . . ,: s ~ - ~ l ( 2 ) i f ' w is a positive word on the letters { a l ,a 2 , ..., a m } , then the Tiiring machine Z, accepts w i f f 'w E E. Remark. Note that from now on E is a recursively enumerable set of words on { a l , a 2 , _ _a,}, _ , while in Definition 2, E was a recursively enumerable set of positive words on ( i l l , u 2 ,..., urn}. Proof of Lemma 3 . Let w be the set of non-negative integers. Numbers are represented by a sequence of symbols 1 on its tape. An input is numerical iff it consists of a sequence of l's, i.e. non-negative integers. A subset A of w is recursively enumerable (abbreviated r.t'.) iff A = p or A = range ( f ) for some recursive function f: A basic theorem on r.e. sets says that A is r.e. iff A = domain(+) for wine partial recursive function Since Turing computable is the same as rccursive, we have that A is r.e. iff there exists a Turing machine which accepts the numerical input n iff n E A . Since a Turing machine can compute the Godel number of a word w given as input, we obtain Lemma 3 .

+.

By Post's construction (see [ 6 ] ), or better by Davis modification of Post's construction, we obtain a semigroup T = TE which corresponds t o the r.e. set E . T is generated by S U Q, where S = S' u { s M } and Q = Q' u { q o ,q,v-l, y N } . We shall write 4 for 40.

S. Aanderaa, Proof of Higman S embedding theorem

I

A word C on the letters S U Q is special if either C = q o z q or C = wqiw' for some j E (0, 1,2, ..., N } and for some positive words w and w' on S = {so,s l , ..., s M } . Lemma 4. L e t E be a recinrsive1.v enumerable set oj'words o n the letters {al, a,, ..,, a,}. Then there exists a semigroup T = T, generated b y the set S u Q ( S = {so,sl, _..,s M } ,Q = { q o ,q l , ..., q N } ) and dejining relations

C i = ri (i = 1, 2, ..., P),

where each C iand riis a special word, such that f o r each positive word w on the letters { a l , a,, ..., a,} we have that (9)

h q , wh = q in T i f f w

E

E.

Moreover,

(**I

if W = q in T, then W = q or W = hpqip'h, where j E ( 0 , 1 , 2 , ..., N } and /3 arid /3' are positiw words on {S()'S1, ..., SM-1).

$5.Properties of the Boone-Britton group By Britton's modification of Boone's constri~ctionwe obtain a group G = G, such that the following lemma holds.

Lemma 5. Let E be a r.e. set ofpositive words on { a l , a 2 , ..., a,}. Then there exists a finitely presented group G = G, siich that (1) G, is generated by S U Q LJ { r l ,r 2 , .._,r p , 1, , I,, _..,l p , t, k } . (2) { a , , u 2 , ..., am> c { S ' , s,, ..., SM}. ( 3 ) I f C is a special word then IK' C-' t C k = C - l t C in G, iJf C = q in T,

(4) I f w is a positive word on { a l , a 2 , ..., u,,,} the??

S. Aanderaa, Proof of Higman s embedding theorem

8

It turns out that we need t o know more about Boone's group. In order t o prove Lemma 1 I , we start with the following lemma: Lemma 6. The subgroup of G generated by the set S is free o n this set.

U

Q

U

{t}

Proof. The detail of the proof is left t o the reader, since by using Lemma 2 the proof becomes easy, although somewhat lengthy since many details have t o be checked. We first prove that S U { p o , ..., p N } is a free set of generators in G,, by repeated application of Lemma 2. Then we prove that the group generated by S U { p o ,p l , ..., p N , z } in H , is free on this set (see Britton [ 3 , p. 24-25] ). Hence S U Q is a free set of generators in G,. By application of Lemma 2 again, we complete the proof of Lemma 6. Lemma 7 . The sinbgroirp of G generated by the set S free on this set.

U

Q

U

{ k } is

Proof. Let G ; = ( G 2 , / i I l i - ' ~ k = ~k, - ' r i k = r i , i = 1 , 2,..., P ) .

Then we have that

G = ( G ' , t l t - l y t = y , t-'zzt = li, t - ' ( q k y - ' ) t =

Hence G; is a Britton extension of G, with stable letter k and G is a Britton extension of G, with stable letter t . Hence Lemma 7 can be proved i n the same way as Lemma 6. This completes the proof of Lemma 7. By modification of Britton's proof we can prove:

S. Aanderaa, Proof of Higman's embedding theorem

9

Lemma 8. Let A be a word on so,s l , ..., s M , q, q o ,41, ..., qN and let A be trivially reduced. Then k-' A-l t A k = A-' t A implies that A is a special word. Proof. First we shall show that A contains exactly one occurrence of a symbol from the alphabet { q o ,q,, ..., q N } . Exactly as in [ 3 , eq. (4), p. 251, we can prove that

where N o , N , , ..., Nb are words on so,sl, ..., s M ,p o , p , , ..., p N , and where L is a word on I,, I,, ..., l p , y and where R is a word on r l , r2, ..., rp, x. Let K be the group generated by so, s,, ..., s M , p o , p , , ..., p N . Then K n A ( z ) = 1 and K n B ( z ) = 1 by Lemma 2 (see the proof of Lemma 6). Since (4') implies

we have by Britton's Lemma that P - ' L - ~ N ,E A ( z ) . Hence for some C E B ( z ) we have

But since K n A ( z ) = 1 and K n B ( z ) = 1 , since K, A ( z ) and R ( z ) are free groups, and A is freely reduced, we have that b = 1 and i, = 1. Hence we have that

where F and G are words on so, sl, ..., sM . It remains t o prove that F and G are positive words. To prove this we change Lemma 8 of [ 3 , p. 261 t o read as follows:

Lemma 8'. Let F and G be reduced words on so,sl, ..., sM and let A = Fq, G. If u , , ..., vn and e l , ..., en and words L (in li, y ) , R (in ri, x ) exist such that (6) and ( 7 ) hold, then A is a positive word

S. Aanderaa, Proof'of Higman s embedding theorem

10

(and hence a special word), moreover, in the semigroup T, A can be transformed into q by a sequence of at most n elementary transformations. Proof. The proof is by induction on n as before. The first place where we have t o change the proof is in part (s) of [3, p . 271. Change the second paragraph of part (s) t o read as follows: "The reader is reminded that, in (8), F and Hiare trivially reduced words (possibly empty) in the s b , and Hiis even a positive word; and Y is a word in y only. We shall prove that F has the form F UH, ( i x . , the form UH,), for some reduced word U. Now reduced h o r d s W , F",Mi in the sb certainly exist such that w and Hl! are positive and such that F = F' W , 11, = Hl!W , and suppose U' has maximal length, so it is sufficient t o prove that Hf = 1. Assume that Hi! $ 1 ." The rest of part (s) may be left almost unchanged. We shall here quotc some basic facts and add some new facts: Since A = Fqs G' = UIii4,, K i V , the word A can be transformed into the word A* 3 UF,qsiGjV by one elementary transformation, as before. By ( 6 " ) and (7") of [3, p. 291, we obtain that A * = F*q si G" = UFjysiGiV is a special word, and A* can be transformed into q in at most n - 1 steps. Hence U and V are positive words, and since If, and Ki are positive words, we obtain that UH,q,,, KiV = A is a special word; moreover, A can be transformed into q i n at most n steps. When el = - I , only slight changes have t o be made as before. This completes the proof of the revised Lemma 8'.

-

Hence our Lemma 8 is proved. Remark. A shorter proof of our Lemma 8 may be possible by applying [ 3 , Lemma 26, p. 741 and some other results in [ 2 ] . Lemma 9. Lct A , , A,, A 3 , A4 be redirced words on so, sl, ..., s M , y. 4 0 , q l , ..., qN. Sirppose I\-' A i t A , k = A;' tA,, in the group G,.. Tllcvl A , = A , = A 3 = A4 and A , is a special word.

S. Aanderaa, Proofof Higman s embedding theorem

11

Proof. Since the group obtained by adding the relation k = 1 t o G is isomorphic to G,, the group obtained from G by deleting k as a generator and deleting all relations containing k , we have that k-lA;ltA,k=A;ltA, implies A;ltA2 =AjltA,. Hence A , = A , and A2 = A4 since the letters in Ail tA2 and A j l t A , form a free set of generators, by Lemma 7. In the same way, by adding the relation t = 1 to G which is isomorphic t o the group G', obtained by deleting t from the generators and deleting all relations in the presentation of G which contain t , we obtain that Ii-' A,' t A , li = A;' tA, and t = 1 implies k-' A;' A , k = A;' A,. Hence A , = A, and A 3 f A,. Thus - A , = A, and / C I A ; l t A , k = A j l t A l , hence A , is a A =A = special word by Lemma 8. This proves Lemma 9.

,

Lemma 10. Let H , be the subgroup of G = G, geiierated bj. (1)

S u Q u { t ,k } .

Then H has the presentation

where C is a special word and C = y in T ) . Proof. Let H i be a group having the presentation above. Let IV be a word on the letters (1 ). Suppose w = 1 in Zi;, then I.V = 1 in H , by [ 3 , Lemma 7, p. 231. Suppose \v = 1 i n ZZ, and IV # 1 in H i . We shall prove that this leads t o a contradiction. We may choose w to have as few occurrences of k as possible. From Lemmas 6 and 7 it follows that w must contain at least two occurrences of t and two occurrences of k . Since w = 1 in H , we also have w = 1 in G. By Britton's Lemma, we have that w must contain a pinch of the form li-'CkE ( E = 2 l ) , where C E A ( k ) = B(X). Hence IC'CIi' = C in Zi,. Suppose k-'C/i' = C is true in Hi.Then w = w' in H i . wherc IV' is obtained from w by replacing K'Cli' in \V by C. Then \Y' = 1 in H , and w' = \V # 1 in H i . Since IV' contains a smaller number of occurrences of Ic than w, this contradicts the fact that 12' is minimal. Hence IC1 Ck = C in If, but li-' Ch # C in ZZ;.

12

S . Aanderaa, Proof of Higman's embedding theorem

Let \ti" = k-' CkC-'. Then w" = 1 in H , and w" # 1 in H i ; moreover, w" contains just two occurrences of k. The group G of Britton (see [ 3, p. 281 ) is obtained from G2 b y a Britton extension as follows: Gl = (C2 ; t I t-l li t = li , t

-'y t = y , i = 1 , 2 , ..., P I ,

G = ( G 1 ; k I k - ' r i k = r i ,k - ' x k = x , k - l ( q - l t q ) k = q - ' t q ,

i = 1 , 2,..., P ) . Here t is first added as stable letter and then k is added as stable letter in the second extension. But we may as well obtain G from G, by first adding k as stable letter and then t in the next extension: G; = ( ~ ; 2 ; l ~ l l ~ - 1 r i k/ ~- l=x rk i=, x , i = 1 , 2,..., P ) ,

i = 1 , 2,..., P ) . Among the words w on the letters S u Q U { t, q } choose a w such that (1) w = 1 inN1; (2) w f 1 in H i ; (3) w contains just two occurrences of k ; (4) w contains as few occurrences o f t as possible. We can now prove that w contains just two occurrences of t by using the fact that G is a Britton extension of G ; . The proof is analogous t o the proof above. Hence w contains n o more than two occurrences of k and two occurrences of t. By using Britton's lemma and Lemma 9 it is easy t o prove that w = 1 in H i also. Hence we have a contradiction which proves Lemma 10. Lemma 1 I . Let k , = hkh-' and let to = q i ' h - l thq,. Then the sub-

groiip oj'G generated by a l , a,, ..., a,, t Q ,kQ has the presentation:

S. Aanderaa, Proof of Higrnan 's embedding theorem

(a1,a2,..., a,,

13

t o , ~ I k , ' ( W - ' t o w ) k o= w-'t,w,

w a positive word on { a l , ..., a,}

and w E E )

Proof. hql wh = q in T = TE by Lemma 4, and we obtain Lemma 1 1 from Lemma 10, since ki'w-' towk, = w-' tow iff

Note that our Lemma 1 1 corresponds t o [ 5, Lemma 5.1, p. 4731 and to [9, Lemma 18, p. 3351.

96. Completion of the proof of Lemma 1 Given a word w o n the letters ( a l , a,, ..., a,} , let w, be the result of replacing all occurrences of ai in w by ui for all i = 1 , 2 , ..., m simultaneously. Moreover wb is defined t o be a word on the letters { b l ,b,, ..., b,} in the same way. (Example: ( a 1 a 2 a ~ 1 )=, u1zi2u;', and (ulu2a;'), = b,b,b,'.) Let

R = ( ~ 1 ~,

2 ..., , U,

I W , = 1, w

E

E).

Let G, be the free product of G = G, and R , i.e.,

G,=G*R. We shall now consider a sequence of extensions of G , and prove that we have a sequence of Britton extensions:

G, = (G4; b 1 , b 2 ,..., b, I bJ1uibi= tii, b;'uibi = ui, bf' kobi = kouT1, for all i, j = 1 , 2 , ..., m ) , G6 = (G, ; d I d-'

k,d = k,, d-' aibid = ai, for all i = 1 , 2 , ..., m ) ,

for all i = 1, 2, ..., m ) .

14

S. Aandcraa, Proof of Higman's embedding theorem

Lemma 12. The sitbgrozips ( u l , .. . , a, , Ic, ), and ( u1, ... , a, , to ), of' G, are frce on the displayed generating sets. Remark. The subscript 4 indicates that the subgroups are computed i n G,, i.e., only the relations of G, are used. Proof of Lemma 12. The Lemma is an immediate consequence of Lemma 1 1 . Lemma 13. G, is a Britton extension of G4 Proof. By Lemma 12, we have

A ( b , ) = ( u l . ._.,I(,, al , ..., a,,

k,

)4

Hence the mapping pj ( c i j ) = zij , pi ( a j ) = aj and pi (k,) = k , zi;' may be extended t o a homomorphism of A ( b i ) o n t o B(bi). Since R(b,) = A()?,)and the map $ i ( z i j ) given by $ i ( z i j ) = u j , cpi(uj) = aj and $i(/io) = k0ui can be extended t o a homomorphism of B (b i ) into A( b,) , and since the extension of $i is the inverse of the extension pi we have that pi is an isomorphism. This shows that the isomorphism condition holds and Lemma 13 is proved. Lemma 14. G6 is u Uritton exterision o f G,. Proof. Let G; = (G, I hi = iii = 1). Then G; is isomorphic t o G,. Since ( a , , a 2 , .._,a,,*, lc, )4 is a free group on the displayed letters, (a1,u 2 . ..., a,, k,) and ( a l b , , a 2 b 2 ,..., amb,, k,) are also free groups on the displayed letters in G; and then also in G,. This makes it easy t o prove that the isomorphism condition holds. The rest of the proof of Lemma 14 is left t o the reader.

Definition 10. Let G i , G ; , Gk, G; be the result of deleting all the relations of the form wLl= I , w E E , in the presentations of G,, G,, G 6 , G,. respectively.

S. Aanderaa, Proof of Higman's embedding theorem

15

Lemma 15. Let the presentation o f H be an extension of the presentation of Gk. Let w be a word on the letters a l , a 2 , ..., a , . Then w, = 1 in H iff k i l wbko = wb in H.

Proof. Since b f l k,bi = kOui-' is a defining relation in G;, and hence also in Gk and H , we have that k6' bi k , = biui. Since bi and ui commute, we also have k;' wb k , = wb w,. Hence if w , = 1 in H , then k i l w , k , = wb in H . Moreover, if k i l wb k , = wb, then wb = wb w , , hence w, = 1. This proves Lemma 1 5 .

Lemma 16. Let the presentation o f H be an extension o f the presentation of Gk. Let w be a word on the letters a l , a 2 , ..., a, , and suppose k6'w-l t , w k , = w-l t o w in H. Then k6O' w-l t,dwk, = w-l t,dw in H if w , = 1 in H. Proof. Since ai and bi commute in H , and since dai we have that

= aibid

in H ,

Here k , commutes with d , and we have supposed that k , commutes with w-l t o w . Hence

k6' w - l tOdwkO

= w-l t o d w ,

I

by (8)

kO1(w-' t o w ) w b d k o= w - ' t o w w h d , (W-lt,W)k,'Wbk,d

0 =

by the hypothesis (W-'

t,W)Wbd,

f? by cancellation

k;' wb k ,

-

-

h

wb, by Lemma 15

= 1.

S. Aanderaa. Proof of Higman's embedding theorem

16

This proves Lemma 16. Lemma 17. G, is a Britton extension o f G,

Proof. The group A ( p ) has the presentation (9) ( a l , u 2 , ..., a,,

to,ko I k i l w-'towko = w-l tow, w

E

El,

by Lemma 1 1, since G is embedded in G,. In order t o prove that the isomorphism condition holds, it is sufficient to prove that the group B ( p ) = ( a l ,a 2 ,..., a m , t d , ko )6 (where t d = t o d ) has the presentation

B'

=

( a 1 ,a 2 , ..., U r n ,

td,

ko I k6'w-l tdwko = W - l t d W , W

E

E).

Let w E E . Then / c ~ O ' to Wwk0 - ~ = w-l t o w in G6 by Lemma 1 1 and \vb = 1 in (;6 by definition. Since the presentation of G, is an extension of the presentation of Gk, we have by Lemma 16 that k i l w - l td w/co = 1 v - l tdw in G , and hence also in ~ ( p )Let . W be a word on the generators of B'. Suppose W = 1 in B ' . Then W = 1 in B ( p ) since B ( p ) satisfies the defining relation of B'. Suppose W = 1 in R ( p ) . Let

Then G: is a homomorphic image of G,, and G" is isomorphic to G. Hence W = 1 in G i . Let W ' be the result of replacing t d by to in W . Then W ' = 1 in G i , but W ' is a word on the generators of G and G is isomorphic t o G:. Hence W ' = 1 in G and so W ' = 1 in A ( p ) . But A ( p ) and B' are isomorphic by definition. Hence W = 1 in B ' . This proves Lemma 17.

Lemma 18. Gb is isomorphic to G,, G; is finitely presented and R is embedded in G;. Proof. Recall that G; is the result of deleting all the relations of the form w,, = 1 (w E 15') in G,. We shall prove that all the relations

S. Aanderaa, Proof of Higman 's embedding theorem

17

w , = 1 ( w E E ) can be proved in G; . Let w E E. Then (10)

k;'w-ltowko = w - l t 0 w o .

Hence (1 1 )

p-' (k6'w-l tow k o ) p = p-' (w-' tow o ) p ,

(1 2)

k;' w-' todwko = w-' t o d w ,

since p commutes with all letters in (1 0) except to and p-' t o p = t o d . The presentation of G; is an extension of the presentation of GL . Hence by (1 0), (1 2) and Lemma 16 we have w , = 1 in C ; , as desired. Hence G; and G, are isomorphic and G; is obviously finitely presented. Moreover, since G, is obtained from G, by a sequence of Britton extensions, we have that R is embedded in G7 . Hence R is embedded in G; . This completes the proof of Lemma 18. Since R was an arbitrary recursively presented group, this also proves Lemma 1. Acknowledgements

In July 1970, I wrote an outline of this paper. I am very grateful to Professor W.W. Boone, Dr. C.F. Miller, 111, and Dr. D.J. Collins for proposing some simplifications and for encouraging me t o write down my work and get it published. I am especially grateful t o Professor W.W. Boone; without his encouragement this paper would hardly have been written. I would also like t o thank Professor J.J. Rotman. He has given me an early draft of chapter 1 2 of the second edition of his book on group theory. I have benefited from his exposition in certain places.

18

S. Aanderaa, Proof of Higman's embedding theorem

References [ l ] W.W. Boone, The word problem, Ann. Math. 7 0 (1959) 207-265. [ 21 W.W. Boone, Word problems and recursively enumerable degrees of unsolvability. A

sequel on finitely presented groups, Ann. Math. 84 (1966) 49-84. [ 3 ] J.L. Britton, The word problem, Ann. Math. 77 (1963) 16-32. [4 ] M . Davis, Computability and unsolvability (McCraw-Hill, New York, 1958). [ 5 ] G. Iligman. Subgroups of finitely presented groups, Proc. Roy. Soc. London A 262 (1961) 455-475. [ 6 ] E.L. Post, Recursive unsolvability of a problem of Thue, J. Symbolic Logic 1 2 (1947) 1-11. [7 1 H.Rogers, Jr., Theory of recursive functions and effective computability (McCrawHill, New York, 1967), [ X I J.J. Rotman, The theory of groups: An introduction (Allyn and Bacon, 1965; second ed. 1973). [9 1 J .R. Shoenfield, Mathematical logic (Addison-Wesley, Reading, Mass., 1967).

BURNSIDE GROUPS OF ODD EXPONENT AND IRREDUCIBLE SYSTEMS OF GROUP IDENTITIES S.I. ADJAN Steklov Institute of Mathematics, Moscow

In 1902 Burnside [ 1 ] proposed the following problem: “Is every group finite when it has a finite number of generators and satisfies the identity relation

This question was answered in the negative in [ 4 1 . It is shown in [ 4 J that, for arbitrary in > 1 and arbitrary odd 12 2 4381, there is an infinite group r ( m ,n ) on in generators satisfying ( 1 ). Prior t o this, affirmative answers t o the problem proposed by Burnside were obtained for n = 3 [ 1 1 , YZ = 4 [21, and IZ = 6 [ 3 ] . In order t o construct r(m,n ) in [ 4 ] , a certain classification of periodic words on the group alphabet u l , u2,..., a,,

a;! u;’ , ..., u;:

is introduced and a theory for transformations of words, which is consistent with ( 1 ), is constructed with a fixed odd I Z > 4 3 8 1 . Affirmative answers t o the word and conjugacy problems for free Burnside groups with odd tz > 4381 were obtained by the same writers on the basis of this theory in the sequels [ 5 J and [ 61. The existence of infinite groups in which all abelian subgroups are finite is proved in [ 61 . By using a somewhat altered version of the theory constructed in [ 4 ] , the writer of the present paper proved in [7 J and [91 that the infinite system of group identities. 19

S.I. Adjan, Btirnside groups of odd exponent

20

where ti > 438 1 is a fixed odd number and k ranges over all the prime numbers, is irreducible, that is, n o identity in this system can be derived from the others in this system. From this it can be deduced that there is a group which is defined by t w o generators and a recursively enumerable set of identity relations of form ( 3 ) and has an unsolvable word problem. The present paper sets forth the fundamentals of the theory of word transformations constructed in [ 4 J* and the ideas of the proofs of the results obtained in [4,5,6 and 71. In addition to the odd number n 2 438 1 , we also set b y definition some auxiliary numerical parameters which will be used in our determinations:

g2 = g ,

+ 2 p , , g 3 = g , + 2 p , , g = g 3 + 2 p 1 + 1.

We consider words on (2). The empty word is denoted by 1. Then length of the word X is denoted by a(X). The word that is the inverse of X is denoted by X - l . If A is a nonempty word and t a positive integer, A f denotes the result of writing t copies of A one after another. F o r example, A is A A A . Letter-for-letter identity of two words X and Y is denoted by X X Y. We say that word E occurs in word X if there are words P and Q such that X X PEQ. In this case, if P (resp. Q) is empty, we say that E is an end (resp. start) of X . The same word E can occur in different places in a given word X . In order t o distinguish different occurrences of a word E in a given word X , we shall use the symbol I , placing it before the first letter and after the last letter of the occurrence of E that is being considered. If X = PEQ, where X , P, E and Q are words on (21, then the word P * E * Q shows us a particular occurrence of E in X . A word of the form

* The condition

a(X)

o f c h a p t e r I in [ 4 ] .

2 p d ( A ) IS omitted from the statements of Lemmata 36 and 41.1

S.I. Adjan, Burnside groups of odd exponent

21

P * E * Q is called an occurrence in PEQ and E is called the base of this occurrence. Occurrences are denoted by the letters V and W , with various indices or without them. We denote equality by definition by the symbol +. Let V + P * E * Q and W + R * D * S be two occurrences in the same word, that is, PEQ Z RDS. If a(P) < a ( R ) and a(Q) > a(S), we say that V lies t o the left of W and we write V < W . If a ( P ) < a ( R ) and a(Q) < a(S),we say that W is contained in V . If P Z R and a(Q) 2 a(S),we say that W is a start of V or that V starts with W . If a ( P ) < a ( R ) and Q 0 S , we say that W is an end of V or that V ends with W. Let V + P * E * Q and W + R * E * S be respective occurrences of the same word E in words X and Y . If an occurrence V , + P , * C * Q1 is contained in V , then there are wordsA and B such that P I 2 P A , E 0 ACB and Q , Z BQ. Then the occurrence R A * C * BS in Y is denoted by $( V , , V ; W ) . A word X on ( 2 ) is called uncancellable if n o words of the form a i u i 1 or ai-' a j occur in it. Uncancellable words of the form A'A 1 , where A , is a start of A and t > 1 , are called periodic words with period A . An occurrence P * E * Q in a periodic word A f AI is called an occurrence interior relative t o the period A , if a ( P ) 2 8 a ( A ) and a(Q) 2 8a(A).Two occurrences P * E * Q and R * D * S in a periodic word A'A are said t o correspond in phase relative t o the period A if a ( R ) - a(P) = a(Q) - a(S) = k a ( A ) , for some integer k . This means that R * D * S is obtained from P * E * Q as a result of a shift to the left or right by an integral number of periods A . In this case we have E 4 D . The following fundamental concepts of the theory under consideration are introduced by means of joint induction on a natural parameter a , called the rank. (1) Kernel of rank a . Reduced and minimized words of rank a . By II, and M , we mean the set of all reduced words of rank a and the set of all minimized words of rank a , respectively. If X E M a and a word E occurs in X , then E f TI,. (2) T h e relation X L? Y , called an equivalence of rank a , is defined for X and Y both in n,. It is reflexive, symmetric and transitive. The function W = f a ( V , Y ) is uniquely defined for an arbitra-

22

S.I.Adjan, Burnside groups of odd exponents

ry kernel V of rank a of a reduced word X of rank a if X 2 Y and Y E n a , and its value W is a kernel of rank a of the word Y . If W = f a ( V , Y ) , then V = fa(W, X ) . ( 3 ) The operation [X, Y ] = 2, called a close-up of rank a , is defined for arbitrary words X and Y both in n,, and its value 2 also is in n,. We first define these concepts for a = 0. If X is a word on (2), every occurrence P * E * Q in X , where E is one of the letters in (2), is called a kernel of rank 0 of the word X . By a reduced word of rank 0 we mean any uncancellable word on (2). It is called a minimized word of rank 0, that is, M , = II,. Two words X and Y in no are called equivalent in rank 0 if X Z Y . We write X 2 Y in this case. I f X 2 Y and V is a kernel of rank 0 of X , then by definition we set fo ( V , Y ) = V . A close-up of rank 0 for X , Y E no is defined in the fAlowing manner: [ X , Y ] = 2 if and only if there are words X , , Y , and T such that X X X , T, Y E T-' Y , , X , Y , o Z a n d Z E no. Now let us assume that cr > 0 and that all the concepts listed in paragraphs ( 1) - ( 3 ) are defined for a -1. Then we define all of these concepts for a . Together with these concepts, we also define a number of other concepts for a - 1 and a which are n o less important for the theory under consideration than the ones given in paragraphs (1)-(3). We d o not include them in the initial listing only because they are either derived from the enumerated concepts or they are meaningful only for cr > 0. In defining these o r other concepts of our theory, we shall at the same time cite some of their characteristic properties in order, as far as possible, to make the meaning of these properties understandable for the reader. Some of these properties follow easily from the definition, while the proofs of others require painstaking analysis of the concepts introduced. A n occurrence W in X € na- is called a regular occurrence of rank a - I if it starts and ends with supporting kernels of rank a - 1 of X . Let us assume that X,Y E nap X Y and W is a regular occurrence of rank a - 1 in X . If a kernel V , of rank a - 1 is a start of W and a kernel V , of rank cr - 1 is an end of W , then & - l ( W , Y ) denotes an occurrence in Y for whichfa-, ( V , , Y ) is a

,

23

XI. Adjan, Burnside groups of odd exponent

start and f a - ( V , , Y ) is an end. From W , = f a - ( W , Y ) we obtain W = f a - ( W , , X ) since this is true for kernels of rank a - 1 by the induction hypothesis. Let us assume that V + P * E * Q and W =+R * C * S are regular respectively. We say that occurrences of rank a - 1 in X , Y E TIaW and V are mutually normalized i n rank a - 1 if the following where Z conditions are satisfied: for arbitrary 2 E X and f a - ( V ,2)= P I * D * Q , , there is a Z , E nap such that Z, Y and f a _ l ( W , Z , ) = R , * D * S , , for some R , a n d S 1 ; conversely, for any such word 2, there is a 2 E TIa- such that Z a ~ l ( V , Z ) = P*, D * Q l , f o r s o m e P 1 a n d Q , . It is evident from the definitions given that two occurrences P * E * Q and R * D * S respectively in X , Y E II, are mutually normalized in rank 0 if and only if E 4 D . The relation of being mutually normalized in rank or - 1 is reY and W is a regular ocflexive, symmetric and transitive. I f X currence of rank a - 1 in X , than W and the occurrence fa- ( W , Y ) are mutually normalized in rank a - 1. A periodic word X + A f A with a period A is called a periodic word of rank a if X E nap , a(X) 2 2 p d ( A ), and there is a kernel of rank a 1 of X interior relative to the period A . It follows from properties of the concepts introduced by us that if two occurrences V and W in a periodic word X of rank a that are interior relative t o the period A correspond to each other in phase relative to the period A and if one of them is a kernel (resp. regular occurrence) of rank a - 1 , then the other is also a kernel (resp. regular occurrence) of rank a 1, moreover, in this case V and W are mutually normalized in rank or - l . This assertion is obvious for a = l . A word Y E TIa_ is called an integral word of rank a if it 15 equivalent in rank a - 1 t o some periodic word X of rank a . In this case X is called an original periodic word for Y . I n particular, every integral word of rank 1 is letter-by-letter equal t o its original periodic word of rank 1. In the general case an integral word need not be periodic. I n virtue of the reflexivity of the relation X Y, every periodic word of rank a is an integral word of rank a . Every kernel V of rank (Y -- 1 of a periodic word X of rank a thdt is interior relative to a period A is called a supporting hernel: A

n,

,

~

,

~

~

XI. Adjan, Burnside groups of odd exponent

24

kernel V of rank a - 1 of an integral word Y of rank a is called a supporting kernel if for an original periodic word X of rank a t h e kernel f a ( V , X ) is a supporting kernel of X . Supporting kernels V and T1/ of an integral word Y of rank a :ire said t o correspond in phase relative t o a n original periodic X if and only if the supporting kernelsf,( V . X ) and & - ( W , X ) of X correspond t o each other i n phase relative t o ;I period A of X. I f a regular occurrence P * E * Q of rank a -- 1 in an integral word Y of rank a starts and ends with supporting kernels and contains at least three supporting kernels which correspond t o each other in phase relative to ;in original periodic word X , then t h e word E is called a semi-integral word of rank a . Then, P * E * Q is called a generating occurrence of the semi-integral word E of rank Q . By the number of segments of a semi-integral word E of rank a we mean the maximal number of supporting kernels that are contained i n its generating occurrence and which correspond t o each other i n phase relative t o the original periodic word X . We say that A is a minimal period in rank a 1 of a periodic word X =+A ‘ A of rank a if, for any generating occurrence W =+P :i: L * Q in X and for any generating occurrence W , + R * E * S that is mutually normalized with it and is in an arbitrary integral word Y , of rank a that has an original periodic word X I with a period I), the following condition is fulfilled: if two supporting kernels V, and V-, of Y , are contained in W , and correspond t o each other relative t o X, then the occurrences @I(Y , , It’, : 11’) and $( V 2 ,W , ; W ) are supporting kernels of X and correspond t o each other i n phase relative to A . A periodic word X with ;i period A minirnal i n rank Q - 1 is called a11 e1enient:iry periodic word of rank Q if for every occurrence 11’ + P :!: I:‘ * Q in X that is interior relative t o A , where E is ;I semi-integral word of rank a generated by ; Igenerating occurrence M’, + K .4: I < ..ir S in an integral word Y , of rank a with an original periodic word X I which has ;i period R minimal in rank a 1 and contains at least p segments, the following condition is fulfilled: if I V a n d W , are mutually normalized in rank a 1, then any two supporting kernels V , and V , of Y that are contained ~

,

,

,

~

.

25

S.I.Adjan, Burnside groups of odd exponent

in W , correspond t o each other in phase relative to X, if and only if the occurrences @(I/, , W ,; W ) and @(V,, W , ; W )are supporting kernels of X and correspond t o each other in phase relative t o A . If X is an elementary periodic word of rank a , then every integral word Y of rank a obtained with original word X will be called an integral elementary word of rank a , and every semi-integral word E of rank a generated by an occurrence P * E * Q in Y will be called an elementary word of rank a generated by P * E * 0. If the number of segments of elementary word E is not less than the natural number I, we say that E is an elementary fpower of rank a . Let W , + R * E * S be an occurrence in a word Z E IIa- of an elementary word E of rank a generated by an Occurrence W * P * E * Q in an integral word Y of rank a . If V , and V , are phase-correspondent supporting kernels of generating occurrence W , then the occurrences @ ( V , ,W ; W ,) and @( V,, W ; W , ) in Z are called supporting kernels that correspond i n phase relative t o W . If W , and W are mutually normalized in rank a -1, then W , is called a normalized occurrence of the elementary word E in the word Z . If W , is normalized, then every one of its supporting kernels is a kernel of rank a - 1 of Z . Generally speaking, different generating occurrences of the same semi-integral word E of rank a may have different phase-correspondences of their supporting kernels. However, if E is an elementary p-power of rank a , then any two of its generating occurrences W and W' are mutually normalized in rank a - 1 and the phase-correspondence of their supporting kernels are transformed from one t o another by the function @( V , W ; W ' ) . Therefore, the question of whether an Occurrence of an elementary p-power E of rank a in Z E n a p ,is normalized, the phasecorrespondence of its supporting kernels and, consequently, also the number of its segments, are all independent of the choice of W. If R 0 R u and M/, =+R * uE * S is a normalized occurrence of the elementary word uE of rank a , then W , is called a normalized continuation of W , t o the left. Moreover, if the word u is nonempty, we say W , is continuable t o the left in normalized fashion.

,

,

26

S.I. Adjan, Birrnside groups of odd exponent

We also define in a similar fashion the normalized continuation to the right R * Ezt * S and the normalized continuation in both directions R * uEu * S , . A normalized occurrence W , of an elementary word E is called maximal if it is not continuable in normalized fashion either t o the left or t o the right. It is obvious that for every normalized occurrence W , of an elementary word of rank a in Z E TIap, its maximal normalized extension is uniquely defined. Two normalized occurrences W , and W , of the respective elementary words E l and E 2 of rank a in the same word Z are called compatible if there is a common normalized continuation for both of these occurrences. The compatibility relation is symmetric and transitive. T w o compatible occurrences of elementary words have the same maximal normalized continuation. If two normalized occurrences of elementary p-power of rank a in Z E TIa.-, are not compatible, then neither is contained in the other, i.e., one lies t o the left of the other. Every normalized occurrence 14' + R * E * S of an elementary word E of rank a in Z E I I a ~ ~is ~a regular occurrence of rank 0-1. Moreover. if Z a=1 2,and 2,E 11,then the occurrence j & ( W , Z , 1 in Z , (cf. par. ( 2 ) ) is a normalized occurrence of some elementary word of rank a which contains exactly the same number of segments as E. If P * E d: Q is any occ~irrencein a word Z E nap of an elementary word E of rank a that contains I > p segments and L? X L4E1 u , where Pu * E l 4: UQ starts with the supporting kernel that is seventh from the left and corresponds in phase to the starting supporting kernel of P * E * Q and ends with the supporting kernel that is seventh from the right and corresponds in phase t o the ending supporting kernel off' * E 4: Q. then Pi1 * El * UQ is a normalized occurrence of an elementary (I - 12)-power of rank a . Let W + P * B ' B , * Q be a normalized occurrence of an elementary p-power B I B , of rank a with period B in a word X E M a - where I3 X B , B , . If there are words Y E nap D ,P I , Q I , Ti and T -, such that

,

,

S.I.Adjan, Burnside groups of odd exponent

21

then the transition

is called a simple reversal of the occurrence W . Moreover, if the maximal normalized continuation of W contains at least 1 2 p segments and the occurrence P , * D * Q 1 contain a normalized occurrence W , of an elementary I-power of rank a generated with an original periodic word wit11 period B-1, then transition (4) is called a simple I-reversal of rank LY ?f the occurrence W . Moreover, we say that the occurrence W and W , corresponds t o each other in individuality in reversal (4). A regular occurrence V of rank a - 1 contained in an occurrence W 2 =+* P , * T IEQ is called stable in the reversal (4) if V is mutually normalized in rank a - 1 with an occurrence V‘ ++ @( V , W, ; W , ) in the word P , DQ , where W, + * P I * DQ, . Moreover, V’ is called the image of occurrence V in the reversal (4). In a similar fashion, we define the stability of the regular occurrence of rank a contained in PET, * Q * and its image in reversal (4). If V is a normalized occurrence of an elementary p-power of rank a that is stable in (4), then we say that V and its image V’ in reversal (4) correspond to each other in individuality in this reversal. Let W * P * E * Q be a normalized occurrence of an elementary p-power E of rank a in a word X E na- . The transition X Z , where 2 E nor1 , is called an I-reversal of rank a of the occurrence W if there are words X , E M a - and 2, E II, such that X a;1 X 1 ’ 2 a=1 Z, and the transition X , + 2 , is a simple I-reversal of rank a of the occurrence f a - ( W , X , ) in X I . We say that a regular occurrence V of rank a - 1 in the word X is stable in the reversal X -+ Z and that the occurrence V , in the word 2 is its image if the occurrence f a - ( Vl , 2, ) is the image of the occurrence f a - ( V ,X,) in the simple reversal X , + Z , . In this case we write f x + z ( V ) = V , . Normalized occurrences W , and W , of elementary

,

+

,

S.I. Adjan, Burnside groups of o d d exponent

28

p-powers of rank cr in the words X and 2,respectively, are said to correspond in individuality in the reversal X + Z if the occurrences fa- I ( W , , XI ) and fa-. ( W , , 2,) correspond in individuality in the simple reversal X I Z , . Moreover, if W , ' and W,' are normalized occurrences of elementary p-powers of rank Q in the words X and 2 and are compatible with W , and W , , respectively, then they also are said t o correspond in individuality in the reversal X + Z . If I > Y > p , then every I-reversal of rank cr is an r-reversal of rank a . If I > p and if a maximal normalized continuation of an occurrence W of an elementary p-power of rank a in a word X E n,- contains a t least I and at most 12-1-28 segments, then an .!-reversal of occurrence W can be made. I f two normalized occurrences W , and W, of elementary p-power of rank a in a word X E n,- are compatible, then every I-reversal of W , is an I-reversal of W 2 . If a transition X + 2 is an l-reversal of an occurrence W and an occurrence W, in the word Z corresponds t o W in individuality in this reversal, then the transition Z + X is an I-reversal of the occurrence W , . Assume the transition X 2 is a g , -reversal of rank Q of the occurrence W , , while M/ + R -* E * S is a normalized and not compatible with W , Occurrence in X of an elementary I-power E of rank a , where I > 2 p , + p . If W , < W , then the end W' of W that starts with the ( p , + 1)- th supporting kernel from the left which corresponds in phase t o the starting supporting kernel o f W is a normalized Occurrence of an elementary (I- p , )-power o f rank a which is compatible with W and stable in the reversa1.X + Z. A similar statement holds for W < W , . If V , < 1' < W ( o r W < V < V , ), where V and V , are regular occurrences of rank a -~ 1 and V is stable i n a p-reversal of W , then V , is also stable in a reversal of W . Let N, denote the set of all words X E such that X does not contain any normaliLed occurrences of elementary (12 - g + 3 p , )powers of rank a . Let W , and bV2 be two normalized occurrences of elementary g , -powers of rank a: in X E N, that are not mutually compatible. We say that W , and W , adjoin each other if there is n o normalized occurrence of an elementary p-power of rank a between them which is stable in a reversal of either one of them. It is easy t o construct on the three symbols 1, 2 and 3 an infinite sequence

,

--f

,

+

nap,

S.I.Adjan, Burnsidegroups of odd exponent

(5)

c, , c2, ..., ci,

29

...

such that, for arbitrary i, the word c, c2 ... cj cannot be represented in the form REES, with non-empty E (cf. [ 81 ). Let us choose some such sequence ( 5 ) . Let V , + P * A , * Q be a normalized occurrence of an elementary g-power A , of rank a in a word X E N,. The set of occurrences in X

where i = 1 , 2, ..., k and k > 1, is called a right cascade of the occurrence V , if the following conditions are fulfilled: (a) F o r 0 < i < k , each V , is a normalized occurrence of an elementary g, -power of rank Q where c, is the i-th term in ( 5 ) and a maximal n6rmalized right continuation of V , contains less than g + p 1 segments. Occurrence Vk is a normalized occurrence of an c/ elementary (g + p1)-power of rank a . 'k (b) The occurrences V , and V , + , adjoin each other for O < i < k. (c) F o r 0 < i < k , every V , is stable in the reversals of V l - l and V i + l , and n o normalized continuation of V,, differing from it, has this property. We call occurrence Vl the first element of cascade (6). The left cascade of occurrence V o is defined in a similar fashion. An occurrence V o is said t o be real on the right (left) if either V , does not have a right (left) cascade o r V , is stable in the reversal of the first element of its right (left) cascade. I f V , is real both on the right and on the left, we say that V , is a real occurrence of rank 0. Let the transition X Y be a g-reversal of a normalized occurrence V of an elementary p-power of rank 01 in a word X E N , and let Y E N,. We call this reversal X Y a real reversal of rank a if there are real occurrences V , and W , of rank (Y in the words X and Y , respectively, such that V , is compatible with V and W , corresponds to V in individuality in this reversal. An occurrence V is said to be a really active occurrence of rank Q if it has a real reversal of rank a . -+

-+

S.I. Adjan, Burnside groups of odd exponent

30

A normalized occurrence V of an elementary p-power of rank a in a word X E N, is called a kernel of rank a of the word X if it is stable in the reversal of any really active occurraice of rank a in the word X that is not compatible with it and n o normalized continuation of V possesses this property if the continuation differs from V . A kernel V of rank a is called a really active o r really inactive kernel, depending on whether it is a really active occurrence or not. If a kernel V contains less than t segments, its maximal normalized continuation contains less than t + 2 p , segments. Every normalized occurrence of an elementary ( p + 2 p , )-power of rank a in a word X E N, is compatible with some kernel of rank a of X.If two kernels Vl and V , of rank a in the word X are different, then they are not compatible and neither of them is contained in the other, i.e., either V , < V , or V , < V , . If a kernel V of rank a of a word X is really active, its maximal normalized continuation does not intersect with other kernels of X. If X Y is a real reversal of rank a , then each kernel V of rank a of the word X corresponds in individuality in this reversal to a unique kernel V ' of rank a of the word Y , moreover the kernels V and V' of the words X and Y , respectively, also correspond to each other in individuality in the real reversal Y + X.We write this correspondence as V ' = ja(V , Y ) or V = f a ( V ' ,X ) . A word X is ca'lled a reduced word of rank a if X E Naand every kernel o f rank a of X contains less than I I - g - - 7p1 - 103 segments. Moreover. if X E Ma and each kernel of rank a contains less than ( I I + 3117 segments. then the word X is called a word minimized i n rank a . By &(respectively, Ma) is denoted the set of all reduced (respectively, minimized) words of rank a . A word occurriiig in a reduced word of rank a need not itself be reduced. However. n o elementary ( 1 2 + + ',pl + 1 ',)-powers of rank a occur in X if X t Ma. I-Ience, PEQ t M a implies E E II,. The relations Ila C nap and Ma C Ma_ follow immediately from the definitions. Two reduced words X and Y of rank a are called equivalent in rank a i f either X % I Y or there is a sequence of real reversals of rank a +

3

31

S.I.Adjan, Burnside groups of odd exponent

Here the words X i need not be reduced, but they are all in the set N,. The equivalence in rank (11 of the words X and Y is denoted by X 2 Y. Obviously, the relation X 2 Y is reflexive, symmetric and transitive. For every word X E II, there is a Y E Ma such that X 2 Y. Y, then an occurrence V in Let X , Y E TIa and X 2 Y. If X X is a kernel of rank a if and only if the occurrence & - ( V , Y) is a kernel of rank a in Y. In this case the single-valued function mapping all the kernels of rank a of X into kernels of Y is said t o be equal by definition t o the function fa., ( V , Y). In the case when the words X and Y are not equivalent in rank a 1 and a sequence of real reversals (7) exists, the function fa( V , Y) is defined as a composition of the mappingsf,( V , Y) that were described above for one real reversal of rank a . It is obvious that if X 2 Y , then V , = f a ( V , Y ) implies V = f , ( V , , X ) . A word Z E n, is called the result of a close-up of rank 01 of the reduced words X and Y of rank CY if there are words XI E n, and Y, E n a s u c h t h a t X 2 X , , Y 2 Y, a n d Z = [ X , , Y l ] o . I n t l i i s case we write Z = [ X , Y],. This operation is defined for any pair of reduced words X and Y of rank a and is independent of the choice of the intermediate words up to equivalence in rank a , i.e., ifX, 2 X x , andY, 2 Y 2 , t h e n [ X 1 , Y 1 ] , 2 [X2,Y2],.Theoperation [ X,Y 1 is associative. With this, the inductive definition of all the basic concepts of the theory under consideration is completed. Now we can define the desired group r ( m ,n ) . A word X on (2) is called an absolutely reduced word of rank 0 if X E no and n o elementary p-power of rank 1 occurs in X . A word X E n,, if a > 0, is called an absolutely reduced word of rank a if there is a normalized occurrence in X of some elementary p-power of rank a and there are n o normalized occurrences in X of elementary p-powers of rank a + 1 . The set of all absolutely reduced words of rank (Y is denoted by Via. It follows easily from the definitions we have introduced that, for arbitrary i and j , !)I, C TII and if X E Yli, then X - l E Q( i.T w o words X and Y from the set

,

~

,

XI. Adjan, Burnside groups of odd exponent

32

Y for some a . The equiv91 =+ Uy=, 9( are called equivalent if X Y . Let 23 denote alence of the words X and Y is denoted by X the set of all the classes of equivalent words that are formed when Vi is partitioned by the relation X Y . If X E 9 f , then an element of % containing the word X is denoted by {X}. The operation of multiplication in % is introduced in the following manner. By the product of elements u and u in % is meant an element w E % such that for some representatives of these classes, X E 11, Y E u and Z E w, the following relations hold: X E ? f a , Y E !)Lo and Z = [ X , Y ] , , where y = max (a,p). The product of the elements zi and u defined in this way is denoted by u u. The operation ZL u is uniquely defined for any two elements of 'li and is associative. Moreover, the set 23 together with this operation constitute a group for which the elements {al } , { a 2 }, ..., { u,, } form a set of generators, while the element { 1 } is the identity element. The inverse of { X},if X E ?L , is the element { X-l}. This group is denoted by r ( m ,n ) . The identity relations ( I ) are fulfilled in it, where n is a fixed odd number chosen by us. T h e proof of this is quite involved. I t is simple t o show that r ( m , n ) is infinite and we can d o this here. It follows from the definition of a kernel of rank a that, for a > 0, the base of any kernel of rank a is an elementary p-power of rank a + 1 is an elementary p-power of rank a.Hence, for arbitrary a > 0, an elementary p-power of rank a contains an occurrence of a n elementary p-power of rank 1, i.e., a periodic word containing at least p - 1 = 17 complete periods. Assume X E %,. 'Ihen X does not have any kernels of rank a , for arbitrary a > 0. Hence, for arbitrary a > 0, a real reversal X X, of rank a is impossible. This means, for arbitrary a > 0 and Y E Vf , that X 2 Y implies X u ~ Y.l Thus, X Y implies X 3: Y , i.e., the class { X } consists of tlie one word X . In particular, the generators { ui}and the unit e k m c n t { 1 } of our group also are of this kind. Thus, it is sufficient t o show that VL is infinite. I!ut this follows trivially from the fact that m > 1 and the existence of a sequence on two symbols n l and a7 constructed as in IS]

-

0

-

0

+

-

S.I. Adjan, Burnside groups of odd exponent

33

for which, for arbitrary j , the word a . a . ... a . cannot be repre' 1 '2 'j sented in the form PEEEQ, with non-empty E. Let B(m, n ) be a free Burnside group with generators (2) and identity relations ( I ) . For an arbitrary word X on alphabet (2) a word Y can be actually displayed such that Y E 2i and X = Y in B(m, n). T w o words X and Y from '21 are equal in B(m, n ) if and only if X Y . I t follows immediately from these two assertions that the correspondence

-

induces an isomorphism of B(m, n ) o n t o r(m,n). I t follows from X E 91 j , X Y and Y E 91 that d ( Y ) < r(i, d(X)), where r(i,j)is some recursive function. Therefore we have an algorithm that solves the word problem for B(m, n ) . Let B(m, n, a ) , for a > 0, denote a group defined by the generators (2) and all defining relations of the form

-

An = 1, where A n is an elementary word, with period A , of rank < a . If two words X and Y are equal in B(m, n ) , then they are equal in B(m, n, a ) , if a = d(X) + d(Y). If B(m, n ) can be defined by means of a finite number of defining relations, then, for some a > 0, every equality in B(m, n ) is also fulfilled in B(m, n, a ) . However, it is easy t o construct an elementary word C n ,with period C, of rank a + 1 , that is not equal to 1 in B(m, n, a ) . Consequently, identity relation ( I ) can not be replaced by a finite number of defining relations, when we have odd n 2 438 1 . All the results stated in the last two paragraphs are given with complete proofs in [ 51. The following fundamental lemma is proved in [ 61 by using the theory described above and the isomorphism between B(m, n ) and r ( m , n ) . If A B = BC and if A # 1 in B ( m , n ) , then there is a word E such that B = E in B(m, n ) and a ( E ) < n.r(j,r(j,j ) ) , where j = d(A) + d(C). This provides us with an algorithm that solves the conjugacy problem for B(m, n ) , when we have odd n 2 438 1. It follows from this lemma that the centralizer of any non-unit ele-

34

S.I. Adjan. Burnside groups of odd exponent

nient ofU(m, n ) contains a finite number of elements. Thus, for 111 > 1 and odd 11 > 438 1 , B ( m , n ) is an example of an infinite group in which all of the abelian subgroups are finite. We consider in [ 71 and [ 91 a generalization of the theory of word transformation that is described above, in order to prove that tlie system of group identities ( 3 ) is irreducible when n = 438 1 and k ranges over all the prime numbers. In order to construct this new theory, i n addition to using all of the concepts in [ 4 ] , we also define by induction on the rank (Y the concept of an admissible word of rank Q which is associated with a given set of prime numbers K . I f , as a result of replacing the variables s and 1 ' on the left-hand side of (3 1 by reduced words A and B of rank a - 1 and interpreting the operation of multiplication as a close-up of rank a 1, a word is obtained within the brackets which is equivalent in rank Q 1 t o the result of a close-up of rank (Y - 1 of the form [ T, Cs, T-I 1 I , where s > 0, then we say that the word C is generated by the substitution in rank Q - 1 of the pair of words ( A , B ) in (3). Let K be some set of prime numbers. A periodic word C t C , of rank a with period C' minimal in rank a I is said to be admissible 1 relative t o the set K by a periodic word of rank a if in rank a there are words R and Lz such that the word C is generated by the substitution in some rank < a of the pair of words ( A , B ) in some relation ( 3 1, for k t K. In particular, a periodic word CfCI of rank I with period c' is admissible in rank 0 if and only if some word of the form ~

~

~

~

where k E K and where A and B are words on (2), is conjugate t o some word C ' S in the free group, where s > 0. Integral, semi-integral and elementary words of rank (Y obtained with original periodic words of rank a that are admissible in rank (Y -- 1 are also said to be admissible i n rank a - 1. Keversals of occurrences of admissible elementary words of rank cy are called admissible reversals. Only admissible reversals of rank Q are considered i n tlie new theory. It is necessary t o consider only

X I . Adjan, Burnside groups of odd exponent

35

normalized occurrences of admissible elementary words of rank a in defining a cascade of rank a of a real o r really active occurrence of rank a . A normalized occurrence of an elementary word of rank a which is not admissible is not really active, n o matter how many segments it may contain. Hence, the conditions concerning the number of segments that are stated in the definition of the sets of words N,, I I a and M a apply only t o occurrences of admissible elementary words of rank a . In particular, words belonging t o these sets may contain occurrences of kernels of rank a having more than n segments, but their bases will not be admissible elementary words of rank a and these kernels will not be really active. The proofs in the new theory of all the assertions made in [ 41 remain basically the same. O d y some of these assertions need to be reformulated by taking ir.to account the remarks made above. For example, an admissible elementary p-power of rank a + 1 always contains occurrences of elementary p-powers of rank a (the bases of its supporting kernels), but it can not contain occurrences of admissible elementary p-powers of rank a . If an elementary word E of rank a , generated by some original periodic word of rank a with period C, does not contain an occurrence of any admissible elementary 2p-power of rank /3 < a , then E is a periodic word with period C. Hence, if a word X E II, does not contain an occurrence of any admissible periodic 2p-power of rank 1 and if X 2 Y , then X E Y . In a manner similar t o that in which the group r ( m ,n ) was defined above, a group r ( m , n, K ) , which also depends on the choice of the set K , is constructed on the basis of the concepts of the new theory. Moreover, the definition of the set V i a must involve normalized occurrences of admissible elementary p-powers. The assertion that, for k E K , all identities (3) are fulfilled in r ( m , n, K ) is proved in a manner similar to that for Lemma 22 of Chapter VII in [ 41 . Moreover, important use is made of the following basic property of the concept of admissibility: if a word C E nap is generated by the substitution in rank a > 0 of a pair of words ( A , B ) in some relation (3) and if C does not contain an occurrence of any admissible elementary 2p-power of rank a , then C can be generated by the substitution in rank a - 1 of some pair of words

36

S.I. Adjan, Burnside groups of odd exponent

( A B , ) in the same relation ( 3 ) . From this property it follows that if C does not contain an occurrence of any periodic 2p-power of rank 1 that is admissible in rank 0 and if the periodic word C"C, of rank a with period C is admissible in rank a -- 1 , then it is also admissible in rank 0. Since a word

for k $ K , does not contain an occurrence of any periodic p-power of rank 1 that is admissible in rank 0, then (8) is in the set Yi of the new theory and is not equivalent t o any word different from it. This means that (3) is not fulfilled in r(m,a, K ) when k $ K . Thus, identity relation (3) is fulfilled in the group r(m,n, K ) that we constructed if and only if k E K . If we take as K the set of all prime numbers not equal to a given prime number I, then we find that relation ( 3 ) for k = 1 does not follow from the other identities in system ( 3 ) . The fact that the system of group identities ( 3 ) is irreducible implies immediately that a continuum exists of systems of group identities of the form ( 3 ) that are not pairwise equivalent. A proof that the set of all group varieties is a continuum which is based on a completely different idea has been obtained by A.Ju. Ol'shanskii (Ol'shanskii's result is published in the journal lzvestija AN SSSR, Ser. Mat. 1970, V. 34, No2). He defines group varieties not by using systems of identities, but by means of systems of finite groups belonging to these varieties. Since the set of all finite systems of group identities is denumerable, this result implies the existence of a group variety that is not defined by any finite system of identities. However, it has not been possible to actually exhibit such a variety on the basis of 01 shanskips proof. I f an identity has been derived from some infinite system of identities, then only a finite number of identities from this system could have been used in the derivation. Hence, it follows from the fact that the system of identities (3) is irreducible that this system (and every one of its infinite subsystems) is not equivalent t o any finite system of identities, i.e., a group variety defined by ( 3 ) does not have a finite basis.

S.I. Adjan, Burnside groups of odd exponent

37

Let S be some recursively enumerable and undecidable set of prime numbers, while r(S)is a group defined by two generators and the identity relations (3) for k E S. Every relation in r(S)is an identity relation. Moreover, for given n , ( 3 ) is fulfilled in r(S)if and only if k E S. Consequently r(S)has an unsolvable word problem. References [ 11 Burnside, W., On an unsettled question in the theory of discontinuous groups, Quart, J. Pure Appl. Math. 33 (1902) 230-238. [2] Sanov, I.N., (Solution of Burnside's problem for exponent 4) ReSenie problemy Bernsaida dlja pokazatelja 4, Leningrad. Gos. Univ. UEen. Zap. Ser. Mat. Nauk. 10 (1940) 166-170. (Russian) MR2, 212. 131 Hall, Marshall, Jr., Solution of the Burnside problem for exponent 6, Proc. Nat. Acad. Sci. USA 4 3 (1957) 751-753. MR 19, 728. [4] Novikov, P.S. and S.I. Adjan, (Infinite periodic groups) 0 b e s k o n e h y h periodiEeskih gruppah, I , 11,111, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968) 212-244, 251-524, 709-731 = Math. USSR Izv. 2 (1968) 209-236, 241-479, 665-685. MR 39 # 1532 a-b-c. [5] Novikov, P.S. and S.I. Adjan, (Defining relations and the world problem for free periodic groups of odd order) Opredeljajdtie sootnosenija i problema totdestva dlja svobodnyh perioditeskih grupp netetnogo porjadka, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968) 971-979 = Math. USSR Izv. 2 (1968) 935-942. [6] Novikov, P.S. and S.I. Adjan, (On abelian subgroups and the conjugacy problem in free periodic groups of odd order) 0 kommutativnyh podgruppah i probleme s o p r j a h n o s t i v svobodnyh perioditeskih gruppah neEetnogo porjadka, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968) 1176-1190 =Math. USSR Izv. 2 (1968) 1131-1144. MR 38 # 2197. [ 7 ] Adjan, S.I., (Infinite irreducible systems of group identities) BeskoneEnye neprivodimye sistemy gruppovyh tozdestv, Dokl. Akad. Nauk SSSR 190 (1970) 499-501 = Sov. Math. Dokl. 11 (1970), N o l , 114-115. 181 Adon, S.E., (Proof of the existence of n-valued infinite asymmetric sequences) Dokazatel'stvo suSEestvovanija n-znaEnyh b e s k o n e h y h asimmetriEnyh posledovatehoste:, Mat. Sb. (N.S.) 2 (44) (1937) 769-779. (Russian). 191 Adjan, S.I., (Infinite irreducible systems of group identities) Beskonehye neprivodimye sistemy gruppovyh totdestv, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970) 715734 = Math. USSR Izv. 4 (1970)

CONSTRUCTION OF A NON-SOLVABLE GROUP OF EXPONENT 5 S. BACHMUTH, H.Y. MOCHIZUKI and D.W. WALKUP University of California, Santa Barbara, California, Boeing Research Laboratory, Seattle, Washington

1. Introduction

Let R be the free, associative, noncommutative ring of characteristic 5 without unit element generated by the countable set of i n d e t e r m i n a t e s ~x~2, , ... . Let L be the Lie ring imbedded in R generated by x l , x 2 , ..., where addition in L is the same as in R and multiplication in L is commutation [x,y ] = xy - yx in R . Let I be the ideal of R generated by all cubes x 3 ,x E L , and let H be the ideal of R generated by all elements g(x, y ) = x 2 j 3 3 x y x + 3.v.x2, x, y E L . A major part of this paper, $ 3 , is devoted to proving the following theorem. ~

Theorem 1 . The ring RIH is not nilpotent. 111particirlar, x 1 x2’... x, F 0 ( m o d H ) j o r anv n 2 1 . Since g(x, x) = x 3 , it is obvious that H 3 I and hence Theorem 1 implies that the ring R/Z is not nilpotent either. Theorem I also implies:

Corollary 1 . There exists a Lie ring L’ o f churacteristic 5 which satisfies the third Engel condition, i.e., [ [ [x, y ] ,y ] , j.1 = 0 for all .x,y E L’, which is n o t nilpotent as a Lie ring.

39

40

S. Bachmuth et ab, A non-solvable group of exponent 5

Since this corollary is not needed for our group theory applications below, we will not indicate a proof here. A proof that Corollary 1 and the first part of Theorem 1 are actually equivalent has been given by Walkup in his thesis [ 9 ] . In 3 4 we use Theorem 1 and the tools developed in 5 3 to estab lish :

Theorem 2. Tlicre exists a group of exponent 5 which is locally so h a Ole hi 1 t I 1 o t so Ivab le. A theorem of Kostrikin [ 61 states that for any finite cardinal

rn and any prime p there is a largest finite group of exponent p with at most 171 generators. This result and Theorem 2 taken together imply:

Corollary 2. There exists a variety of groups which is a locally solvuble arid lo call^^ finite variety but not a solvable varietv. Other consequences of Theorem 2 relevant t o the restricted Burnside problem may be inferred from the results of G. Hignian [ 5 ] and Kostrikin [ 6 ] . For the sake of completeness we mention the following strengthening of a result of Higgins and Heineken (see Bruck [ 23 o r ~:aclimutli,Mochizuki and Walkup [ 1 1 ). A proof can be found in [91.

Theorem 3. Suppose the ring R' has characteristic prime to 2 , 3 , and 5 (i.e.,30s = 0 implies x = 0 ) . Suppose fiirther that R' is genrrutcd h.v elenimts .Y,, s 2 ,_... Let L' be the Lie ring embedded in R' generated 61,x l , . y 2 , . _ .and let I ' be the ideal of R' generated b y tlic cwhcc. s3,Y E L'. Then R'II' is nilpotent of index at most 9. 2. Preliminaries In this section we introduce some useful elementary concepts, and in Lemma 1 below we apply the familiar process of lineariza-

S. Bachmuth el al., A non-solvable group of exponent 5

41

tion t o obtain a set of elements which span the ideal H as a subspace of R . A natural basis for R , considered as a vector space over the field Z , of integers modulo 5 , is provided by the set of all monomials M = x. xi* ... xi,, m > 1. The integer m is the degree of M . Unless ‘I otherwise indicated, when we write a product of the form PMQ, M a monomial, we include the case in which M is the “empty” monomial of degree 0, that is, we include the product PQ as one of the cases. The analogues of the monomials in R are the commutators in L , which may be defined inductively as follows: Any indeterminate is a commutator of degree 1 , and if C, and C, are commutators of degree d , and d , respectively, then C = [ C , , C, 1 = C , C, - C, C, is a commutator of degree d = d , + d,. It is easy t o see that the commutators span the Lie ring L , considered as a vector space over Z , . Given any monomial M = x i lx i 2... xi, .in R , we denote by R, the subspace of R spanned by all monomials M ‘ (including M ) obtainable from M by rearranging the order of the indeterminates in the product. In the special case M = x1 x2 ... x, we will write R, for R, . Of course the ideal H is a subspace of R , and we write Hm=R,nHandHn=RnnH. We shall say that an ideal J of the ring R is an I-substitution ideal if x E J implies x‘E J , where X’ is obtained from x by substituting any (not necessarily distinct) indeterminates systematically for the indeterminates which figure in x. Similarly, we shall say that J is a Lie substitution ideal if x E J implies x’E J , where x’is obtained from x by substituting any elements of L for the indeterminates of x.Clearly the ideals I and H defined in the introduction are Lie substitution ideals, and any Lie substitution ideal is obviously a substitution ideal. Lemma 1. For any monomial M , the subspace H, R, is spanned b y the polynomials o f the form

where h is the polynomial function defined b y

= R,

n H of

42

S. Bachmuth et al., A non-solvablegroup of exponent 5

h(x,y , z ) = x y z

+ z y x + 2yzx + 2yxz,

M , and M , are (possibly empty) monomials, and C, , C,, C, are commutators such that M , C , C, C,M, is in R, . Proof. By its definition as an ideal of R , H i s spanned by the elements of the form P , g ( L , , L,)P, , where P I ,P, are arbitrary polynomials from R and L , L , are arbitrary elements from the Lie ring L. It is readily computed that 2h ( x ,x , y ) - h ( x , y , x ) = 6 x 2 y + 2xyx

~

2yx2

= x 2 y - 3xyx + 3yx2 (mod = g(x, Y ) (mod 5)

5)

and

id?,,-Y> + g ( z , x) + g(x, z ) + g(y, z ) - g o / + z , x ) - g(x+y,z ) = -4xyz

xyz 5

-

4zyx

+ 2yzx + 2yxz

+ z y x + 2yzx + 2yxz (mod 5 )

h ( x ,y , z ) (mod 5).

It follows that H is also spanned by the elements of the form P , h ( L L , L 2 , L , )P,. Since every element Piof R is a sum of monomials. since every element Li of L is a sum of commutators, and since 11 is linear in each of its arguments, it is clear that If is spanned by elements of the form (2.1 ). The restriction t o HIMis obvious.

Remurk. Since R has characteristic 5 , the above congruences modulo 5 are superfluous statements. However, in the above lemma and in later places, we often add this extra comment to emphasize the role played b y characteristic 5.

S. Bachmuth et al., A non-solvablegroup of exponent 5

43

3.Proof of Theorem 1 In this section we prove the following strengthened version of Theorem 1. Some of the machinery developed here will also be used in the proof of Theorem 2 in 54.

Theorem 1'. For each n 2 1, H , = R , n H is u proper subspace o f R,, and x1 2 x 2 2... x n 2 together with H , span R,. The proof of Theorem 1' may be divided into three main steps. Let H! be the subspace of H , spanned by generators of H , of the form M , h(x,y , z)M2 where x,y , and z are indeterminates. In the first step we define a Collection Algorithm which enables us t o use induction on n. In the second step (Lemma 2 ) , we use this Collection Algorithm to define a linear map 4 from R , into itself with kernel H; and image spanned by x1 _..x, * . In particular, H; and x1 ... .x, span R,, so that Theorem 1' holds with H: in place of H,. In the final step (Lemma 3 ) we show that H , = H:. F o r any (nonempty) monomial M let H& be the subspace of HM = H , n H spanned by the polynomials of the form M , h ( x ,y , z ) M 2 , where M , ,M , are (possibly empty) monomials and x,y , z are indeterminates such that M , x.vzM2 is in R , . The relation h(x,y , z ) = 0 modulo H implies many others. We indicate below six of the simplest ones. Each of the six expressions in (3.1 ) is equivalent t o 0 modulo H for any x,y , z in L . If.u, y, z are restricted t o being indeterminates, each of these expressions, when pre- and post-multiplied by appropriate monomials M , and M 2 , becomes a member HA. h ( x ,y, z ) = xyz

x2y (3.1)

-

y.x2

+ zyx + ' y x z + zyzx

= 2h(x,x,y ) - 2h(x,y, x) (mod 5)

+ 2yx2 = -2h(x, y , x) (mod 5 ) 2x23: + x y s = -h(x, s,y ) - h ( s ,y , x ) (mod 5) @.Y, y , z ) = zyx + xyz + 2xzy + 'zxy

xyx

= -2h(x, y , z j - h ( s , z , .I<)

~

hCr, .Y, z) (mod 5)

44

S.Bachmuth et al., A non-solvablegroup of exponent 5

+ s z y + y x z + zxy + yzx + zyx =5 4 { Iz(x,y , z ) + h(x,y , z ) }.

S ( Y. y , z ) = x y z

Notice that h ( x , I), z ) is the polynomial obtained from h ( x , y , z ) by reversing the order of the factors in each term. I f Theorem 1 ‘ is true, then for each polynomial P in R n there exists a unique polynomial p(P) of the form Ox, x 2 2_..x n 2 (0 not always t e r o ) such that P 5 p(P) (mod H,, ). In the process of proving Theorem 1 ’ we will construct an explicit algorithm for computing p(P) for any P in R,, . A principle ingredient in this algorithm is the fo 1lowing :

Collection algorithm. An integer PZ > 1, and a polynomial P in some R,w* are given. It is assumed that A4* is of degree at most 2 in the indeterminate Y,!. A polynomial O n ( P ) is sought such that P - On(P) (mod ll,ii*) and such that the occurrences of the indeterminate x , in O,, ( P ) ;ire restricted t o the two right-most positions in each term. ( 1 ) Set Q equal t o P. ( 2 ) Choose a n y term OM of 0 ,/3 f 0 (mod 5), in which x , appears other than in the last two positions. I f no such term can be found, go t o step 6. Otherwise, go t o step 3. ( 3 ) Ikterrnine (possibly e m p t y ) mononiialsM1 and M , such that the monomial M chosen in step 2 has the form M , x r , x j x k M 2 , where vJ and Y~ are indeterminates not necessarily distinct from Y,, and M , does not involve x, . ( 4 ) Perform one of the following operations on Q . ( a ) I f / = I I # k , replace PM by / 3 M 1 x k x i M 2 . ( b ) I f j # 17 =I\.replace/3Mby-2/3Mlxkx,2M2. ( c ) l f j # I T f k , replace /3M by

( 5 ) Collect like terms of Q , reduce all coefficients modulo 5, and go t o step 2. (6) Define O , / ( P )t o be Q and stop.

S. Bachmuth et al., A non-solvable group of exponent 5

45

Proposition 1. For any n, any R,*, and any P E R,*, the Collection Algorithm is well-formed, terminates in a finite number of steps, and yields a unique polynomial 8, ( P ) independent o f the order in which terms are chosen in step 2. Moreover en is a linear map of R,* into itseif and P = 8 , ( P ) modulo HA* for all P in R,*. Proof. The proposition is readily derived from the following five observations: (i) Given any M satisfying the conditions in step 2, the representation M = MlxnxixkM2 defined in step 3 is unique. (ii) Given the representation in step 3, exactly one of the conditions listed in step 4 must hold. (iii) Each of the operations given in step 4 preserves the property that Q E R,*. (iv) Each of the operations given in step 4 replaces OM by a polynomial, in each of whose terms the occurrences of x, are farther to the right. (v) Each of the operations given in step 4 is equivalent t o adding elements of the form M l f M 2 to Q, where each f has the form of one of the expressions listed in (3.1 ). Proposition 2. Suppose n and P sati.yfy the restrictions stated in the Collection Algorithm and P can be factored in the form P , P, P 3 , (respectively P, P3 or P , P , 1, where P , does not involve x,. Then each Pi is an element of some RMi,n and Pisatisfy the restrictions of the Collection Algorithm, and

(respectively

or

Lemma 2. For each n 2 1 , H t is a proper subspace of R,, and x12 x 2 2 ... x n 2 together with H; span R,.

S. Bachmutli et al., A non-solvable group of exponent 5

46

Proof. Define a map pt7of R,, into itself by induction on y1 as follows: p1 is the identity map of R , onto itself, and if 17 > 2 then p t I ( P= ) p,7pliP’)x,12, where O , , ( P ) = P ’ x , ~ .Obviously pn is a linear map of R,, o n t o the subspace of dimension 1 spanned by .vl 2 . ~ ~_..2s , , 2 and p,,(P)= Y modulo H:. I n order t o complete the proof of the lemma if suffices t o show that pn takes H! into zero. This is ininiediatc tor IZ = I , since HP is trivially zero by Lemma 1 . T h ~ i sthe p r e x n t lemma will follow by induction on y1 if we show that 0, takes H! into HnOp x, 2 . To d o this it suffices, i n view of the definition of H: to show that

for any monomials M I ,M , and (not necessarily distinct) indeterminates x,y , z such that M I .vyzM2 is in R,, . We distinguish 12 cases depending on the location of the two occurrences of x n in ill, h ( . ~y , z)ill,. In what follows N idenotes a (possibly empty) monomial and .v, y, z , .s, t, etc., denote indeterminates distinct from s,, but not neccssarily distinct from each other. Since h is symmetric i n the first and third argument, it is unnecessary t o list additional cases involving h(x, .v, s,,), etc. I

(1)

(2)

N ,h(.v, J’, Z ) N ~ . Y , N ~ . K , , N ~ N l , ~ , , N ~ . ~ , , / ~ ’ 3J’,I ~z)N4 ~.~,

(3)

N , h(-vtz,.Y,?, z ) N 2

(4)

X I h ( . Y I i , j’,.Yn ) N 2

(5)

N, .~,,.h’~ h ( . ~ ,J,’,, z ) N ,

(0)

/vl.v,,l~’2/?i~v, s,,, z > N 3

(7)

N,h ( ~, J’, , z)s,, N 2

(8)

!V, h ( X , , , .l’, z ) tN2S,,N3

S. Bachmuth et al., A non-solvable group of exponent 5

41

Consider Case 1. By Proposition 2

which is obviously an element of H:_ I x, 2 . This takes care of the first, and easiest case. Consider Case 2. Again using Proposition 2, we have

which is again obviously an element of H!-l .v/, ? . (Here we have made use of the observation that 0, {x, Q} = Qxpl2 . Note that this is an observation about the action of the Collection Algorithm, and not just an observation about squares of indeterminatcs commuting modulo H i !) Now consider one of the most complex cases, Case 12. Again by Proposition 2, we have

-

0, { N , X, N2 k ( x ,,v,Z ) tN3X , N 4 } = 8,,{ N , 8 { s,,N , 1h (x,y , z ) tN3I,N4 } . From the description of the Collection Algorithm it is obvious that

where each si is an indeterminate. (If the degree of N 2 is 0 or 1 , then 8, { x n N 2} = x,, or x,s respectively.) Actually, a close examination of the Collection Algorithm will disclose that a n y s i , if pres-

48

S. Bachmicth et al., A non-solvable group of exponent 5

ent at all, must be the last indeterminate of N,. Thus, after another application of Proposition 2, we have

By the same reasoning

Substituting this back into (3.3) and applying Proposition 2 again, we obtain

where O n { X , N ~ X} ,=~Phxn2 and 8, { x, tN3x,, } = Pi x n 2 .In order to prove (3.2) for Case 12 it will suffice t o prove each Q,, lies in the appropriate H i . Now it may happen that certain of the indeterminates r, x, I’, z,t are not distinct. However, an examination will show that the Collection Algorithm depends in n o way upon the identity of any indeterminate other than x,. Thus the proof of (3.2) for Case 17 will follow from a proof of the propositions:

S. Bachmuth el al., A non-solvable group of exponent 5

49

In simi!ar fashion the proof of the remaining nine cases reduces t o the proof of the following propositions. In some cases it turns out in practice that the application of the Collection Algorithm yields 0 directly. These cases have been indicated.

Each of these is a proposition about an explicit finite space R,,, of dimension at most 5 ! over 2,. A FORTRAN program which the authors have used to verify these propositions is reproduced in the Appendix. The algorithm employed by the program is explained in 5 5 .

Remark. Characteristic 5 is crucial in the proof that tpn maps H: into zero. If division by 5 were possible, (3.2) would n o longer be valid. In fact if we replace characteristic 5 by characteristic 7, and use identity g(x, y ) = 0, then x 2y 2 = 0 (cf. Higgins [ 4 J , Theorem 3).

50

S. Bachmirth et aL, A non-solvable group of exponent 5

Proposition 3. I f M = M , M , is any monomial in R,, then M , M , M ,-M , inodiilo H,O.

=

Proof. The proposition is trivial for n = 1. Assume therefore that 2 2 and t h e pr3position has been proved for smaller values of n . Clearly it will suffice t o prove M = y M , = M,y (mod H:), where y is ; i n indeterminate. Since 11 > 2 there is some indeterminate xIf y sucli that M has the form M = ~ N ~ x , N , x + VApplying ~. Propositions 1 and 2. we have ii

wherc 0,( u,/V, y I ) = P u l l . From thc inductive hypothesis it follows that \*NlPN, = A', P N 3 1' (mod H/- 1. T ~ L JI ISM , = M O y (mod H t ) as rccluired.

,

'Tlie proof o f Lemma 3 will involve repeated applications o f the simple ohservation contained i n the following proposition.

Proof. By the first hypothesis, P ( t , , _ _ t_r l,l )is the sum of polynomials o f thc form 'V, Iz(ti, t i , t,)N,. Hence P ( C , , ..., C m )is the s ~ i mo f polynomials of the form P I h(Ci, Ci, C, )P,, each of which is in /I,(/ by the third hypothesis and the assumption concerning the degree o f the commutators C j. We can now prove the following lemma. which, combined with Lemnla 7 PI-ovecTheorem 1 ' .

S. Bachrnuth et a[., A non-solvable group of exponent 5

51

Lemma 3. Hn = H:

Proof. We must show that every generator (2.1) o f H n lies in H.! In view of Proposition 3, we may restrict attention to generators of the form G = M,h(C', C", C"'). We prove the assertion G E H; by induction on the pair of parameters d 2 3 and M 2 1 , where d = d , + d, + d, is the sum of the degrees of the commutators C', C", C"'. Clearly the assertion is trivially true for any M if d = 3 and vacuously true for any d if IZ = 1. Consider, therefore, any generator G with d > 3 and IZ > 1 and assume the inductive hypothesis that the assertion holds when either d o r 12 are smaller. We distinguish four cases. Case 1. C" has degree at least 2. Then the G has the form G = M , h ( C , , [ C,, C, 1 , C4). Note that no three of the commutators C , , C,, C,, C, and the commutators of degree 1 (indeterminates) in M , have a sum of degrees equal to d . The desired conclusion G = 0 (mod H!) follows from Proposition 4, the inductive hypothesis in d , and the identity

(3.6)

h(w,[x,y ] , z ) = 0 (mod

This identity and the identities (3.7), (3.9), (3.10) and (3.1 1 ) below can all be verified by the method described in § 5 . CUSP 11. Either C' or C"' is of degree at least 3. Then G has the form G = M , h( [ [ C ,, C,] , C, 1 , C4, C, ). Applying Proposition 4, the inductive hypothesis in d , and the identity

(3.7) we get

[[x,y l , z l

-2(.xz + z x ) y + 2(yz + z q ' ) . ~(mod H&)

S.Bachnticth et al., A non-solvable group of exponent 5

52

Applying Proposition 4, the inductive hypothesis in d , and the identity

(3.9)

x(yz + z y )

(.YZ + z y ) x (mod

H&)

repeatedly we may commute (C, C , + C,C,) in the first term of (3.8) t o the left until it is outside the influence of h . Applying the same process t o the second term as well, we have

which is equivalent t o 0 modulo H: by the inductive hypothesis on (1.

Cusc 111. G has one of the f o r m s M , h ( [ x , Z J , y , y ) , M , h ( [ x , y l, y , z ) , orhll/z([x, v l , y , [ z , wl ). This case is dismissed immediately by the identities (3.10)

/I(

[ Y, z ] , v,J , )

h( [ Y, y ]

0 (mod H xov

. 1'. z ) = 0

(mod H o

XY

z

)

).

For 111, /z( [.v, 1.1 , J-, [ z , w ]) we use the second identity, induction on tl. and Proposition 4. Cusr 1V. G = N , i3N,h([w, s ],y , C, ). Of course, by Proposition 2 and the argument preceding (3.3), I

By Proposition 4, the inductive hypothesis in d , and the identities

we have

S. Bachmuth et al., A non-solvable group of exponent 5

53

But by the inductive hypothesis on n, N , P , h( [ w ,X I , s, C,) = 0 (mod H$), where N 1 N 2wxC, E R M ~Thus . G = 0 modulo H:. This completes the proof of the lemma. 4. A non-solvable group of exponent 5

Assume now that R has an identity. Let G be the multiplicative semigroup of R / ( H + U) generated by all elements of the form ei = 1 + xi, where U is the ideal of R spanned by all monomials with at least one indeterminate factor repeated. It is readily seen that e t = 1 - x i (mod u) is an inverse of ej in G. Hence G is a group. We defer until the end of this section a proof that G has exponent 5. We show now that G is not solvable. Consider the group commutator ( e l , e2) = e l - l e2-le , e 2 . We have

( e l , e 2 ) - ( 1 - x l ) ( l -x2)(1 + x , ) ( l + x 2 ) =

1 + [ x 1 , x 2 ]- x 1 2 + x 1 x 2 x , + ...

= 1 + [x1,x2]

(mod U ) .

Moreover

( e l , e2)-l = (e2, e l ) = 1

-

[ x 1 , x 2 1 (mod U ) .

And in general any (compound) group commutator formed exclusively from the generators ei is equivalent modulo U t o 1 plus the corresponding ring commutator in the indeterminates xi. In particular 1 + C, is a member of the s-fold derived group G @ )of G, where C, is any member of the following sequence of ring commutators:

S. Bachmuth e t al., A non-solvable group of exponent 5

54

Now suppose, in order t o reach a contradiction, that G is solvable, i.e., C, = 0 (mod H + U ) for some s. (We may suppose s 2 3 . ) Then obviously C, = 0 (mod H ) , and it follows by substitution of indeterminates that C,* = 0 (mod H ) , where

etc. Using the methods in 8 5 it is easy t o verify the identity

and in general, if s > 3 >

By

ail

application of the identity

S. Bachmuth et al., A non-solvable group of exponent 5

55

it follows that

Since this contradicts Theorem I f ,we conclude that G is not solvable. It remains t o be shown that G has exponent 5. Every element of G has the form g = ( l +Xi,)(]+ X i J

... ( I +xi ) n

= 1 + P ( X i , , x i z , .,., Xi,)

for some n. Now

We shall show that P 3 = 0 modulo H + U. To d o so we need t o introduce a family of polynomials T , described by Bruck in [ 21. F o r each y 1 > 1 the polynomial T , ( x l , ..., x r z )is defined t o be the multilinear part (homogeneous of total degree I? and of degree one in each of x l , ..., x n ) of the polynomial

It is observed on p. S2.1 of [ 2 ] that

(Note that the second term on the right has fewer arguments.) It is readily computed that T , (xl ) and T , (x, , x2)are zero and T , ( x l ,x 2 ,x,) is just the symmetric function S ( x x 2 ,x3 ) already mentioned in (3.1). Let 9 denote the ideal of R spanned by elements of the form

56

S. Bachmirth et al., A non-solvablegroup of exponent 5

for M 2 3 . It is apparent from (4.4) and the definition given in 8 2 that 9 ,as well as N.is a Lie substitution ideal.

Lemma 4. 9 C H. Proof. Since 9 and H are both Lie substitution ideals, it will suffice to show that the following proposition holds for all n : I f y l , ...,_ v n are commutators in x l , ..., x, such that y 1 y 2 ...yn E R

xm,

then T 3 b l ,...,y n ) - 0 ( m o d H ) .

This proposition is true trivially for n < 2. Moreover, since H is a Lie substitution ideal, it suffices to prove the proposition for a fixed but arbitrary 17 2 3 in the special case each of the commutators y i is just the indeterminate xi,assuming the inductive hypothesis that the proposition is true (without restriction on the degrees of the commutators) for smaller values of n . Now by (4.4) and the inductive hypothesis, T 3 ( x 1 ..., , x n ) is equivalent modulo HM t o the same expression with the indeterminates permuted in any way. Thus T 3 ( x I , ..., x n ) 6 T 3 ( ~ 1..., , xn) 5

S * { T 3 ( xl , . . . , ~ n ) } ( m o d H M ) ,

where S*{P } denotes the sum of P and the five other polynomials obtained from P by permuting the three indeterminates x l , x 2 , and x 3 . Clearly, S*{ T3( x l , _..,x,)} is the sum of polynomials of the form Q = S * { N , x , N 2 x , N 3 x 3 N , 1 . From the properties of the Collection Algorithm, including Proposition 2, it is clear that each Q is equivalent modulo HM to a sum of expressions of the form

s * { "x

XsX2XtX3N" }

s * { N'xl X2XtX3"'

}

S * { N ' x 1 ~ ~ ~A"' 2 1x 3 S * { N ' x 1 ~ 2 x N" 3 }.

But each of these is in I f M as a consequence of identities

S. Bachmuth et al., A non-solvable group of exponent 5

51

which can be verified by the methods of § 5. Thus each Q and hence T3(x1, ..., x n ) is in HM.This completes the proof of the lemma. We may now complete the proof that G has exponent 5 . In view of (4.3) and the fact that H + U is an I-substitution ideal, it suffices to show

P3 = ((1 + x l ) ( l + ~ ~ ) . . . ( I + x ~ ) -=l 0} ~(modH+Cr). Certain of the terms in the expansion of P 3 are in U. It is not very difficult to see that the remaining terms may be regrouped in the form

where the sum is taken over all (increasing) subsequences x = (Xi,, xi,, ...

)

Xis)

of ( x ~ ..., , xn), for 3 < s < n. Thus P 3 E 7 + U. By Lemma 4, P 3 E H + U , which is what we proposed to show.

Remarks. We have given a more or less direct proof that G is not solvable. Other proofs are possible. Thus a variation on the proof given above will show that G is not nilpotent. Then by a theorem of Tobin [8] the group G of prime exponent can not be solvable. (Actually Tobin speaks only of finitely generated groups, but the proofs can be adapted to the general case.) Yet another approach, which the authors used originally [ 11, is t o construct a group Q which is the split extension of an abelian normal subgroup A by G. Nilpotency of Q implies x l x 2 . . . x, = 0 (mod H + U), which contradicts Theorem 1 . Thus Q is not nilpotent. By the theorem of Tobin, Q is not solvable, and consequently G is not solvable.

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S. Baditiiirth et al., A lion-solvable group o f exponent 5

5. A reduction algorithm In $ 3 and $ 4 we have made use of more than 20 identities which hold modulo 11° in various spaces R , . I n this section we ?I present an algorithm for automating the verification of these identities. W e do not know if t h e Reduction Algorithm given below is capable of verifying every valid identity in R , modulo for any M, but we do not need so strong a result. A s :I minor convenience, let us make the following definition: ~ . . has . Property A if We say that ;I niononiial M = . Y ~ ~ . Y ~s. m il < il+z for 1 < / < ! t i - - ? . Note that a monomial can not have Property A if i t has degree 3 o r more in any one indeterminate.

Hk

Reduction Algorithm. A polynomial P in some R,, is given. It is desired to produce ;I polynomial $ ( P ) such that P = $(P)modulo fl;;* and such that each niononiial of $ ( P ) has Property A . ( 1 ) Set Q equal to I-’. ( 3 ) Choose any term PM of Q , /3 # 0 (mod S), which does not Iiave Property A . If no such term can be found, go t o step 6. Otherwisc. go t o ctcp 3. ( 3 ) Determine (possibly e m p t y ) inonoriiids M , and M 2 such that the monomial ,ill chosen in step 2 has the form ill, . Y ~ . Y ~ .where Y , ~ Ii ~2. k . but M , xixj has Property A . ( 4 ) Perform one of the following operations on Q. (a) I f i = i = li. delete OM from Q. ( b ) l l ’ i = j > I;, replacepM b y P M l x , x j 2 M 2 . ( c ) I f i > j = k . replace Pizf by P M , . Y , * . Y ~ M ~ . ((1) I f i = li < j . replace OM by -~-2pM, xi2siM2. ( e ) I f i = k > j , replace pM by -?OM, x j x i 2 M 2 . ( f ) I f i > j > h. o r i > k > j , replace fiM by

(g) I f j > i > k . replace OM by

S. Bachmuth el al., A non-solvable group of exponent 5

59

(5) Collect like terms of Q, reduce all coefficients modulo 5 , and go t o step 2. (6) Set $(P) equal to Q and stop.

Proposition 5. For any M * and any Y E R M , , the Reduction Algorithm is well-formed, terminates in a finite number of steps, and yields a unique polynomial $(P) independent o f the order in which terms are chosen in step 2. Moreover $ is a linear map of R M , into itself, P = $ ( P ) modulo H i , , and each term of $ ( P ) has Property A. Proof. The proof is essentially the same as for Proposition 1, except that observation (iv) must be replaced by: (iv') Each of the operations given in step 4 either deletes PM or replaces it by a polynomial, each of whose terms is lexicographically less than OM. (The term P'M' is lexicographically less than PM if M = MlxiM2 , M' = MlxjM;, and i > j . ) As mentioned above, we do not know whether the Reduction Algorithm will reduce every element of H& t o the zero polynomial. However, any element of R, which does reduce to zero must be in H i . We have found in practice that each of the elements of H i corresponding t o the identities used in $ 3 and $4, as well as all others experimented with in the course of finding reasonably direct proofs, were reduced to zero by the Reduction Algorithm. Nevertheless, it is conceivable that the Reduction Algorithm might fail to completely reduce some of the expressions listed below if the indeterminates were renumbered. It is for this reason that we take the trouble to list them explicitly.

60

S. Bachmuth et al., A non-solvable group of exponent 5

We conclude with an observation concerning the Collection Algorithm and Lemma 3 . As is shown in the final example in the Appendix, $ h ( [ ~, x, 2 ] x , 3 ,x4) is not zero. As already noted, this does not immediately prove that h( [ x l ,x2], x 3 ,x4) is not an element of H ~ l x z x 3 xHowever, 4. a tedious but straightforward hand computation shows that the six monomials in R, I x 2 x 3 x 4 with Property A form a basis for RXIX2X3X4 modulo H x0 I x 2 x 3 x 4I t. follows that h([x,,x2],x3,x4) is not an element of Hx 0 x 2 x 3 x 4 and consequently H:,x x 3 x 4 f HxIx?x3x4, in contrast with the result in Lemma 3 . This okservation points up the importance of Case IV and the induction on n in Lemma 3 . Appendix

Reproduced below for the convenience of the reader is a FORTRAN program which the authors have used t o verify all the assertions in (3.4) and (3.5) and the identities listed at the end of 5s. The program output for typical cases (and the more difficult cases) is also included. We remark that the verification of most of the cases in (3.5) and at the end of 5 s can be done routinely by hand. Indeed, the original proof of the authors as outlined in [ I ] did not rely on a computer at all, either directly or indirectly as an experimental tool.

S. Bachmuth e t al., A non-solvable group of exponent 5

61

The proofs presented here achieve a reduction in complexity and overall length at the modest expense of verifying a few complicated identities. COMMON I N O E T l 1 0 ~ 2 0 0 ) ~I C O E F l 2 O O ) . I T E M P l 1 0 ) * I T C O E F COMMON LENGTH, N, ITERM, LASTRM L O G I C A L COLECT, REDUCE C LENGTH = NUMBER OF FACTORS I N EACH TERM. = I N D E T E R M I N A T E TO BE COLLECTED. (IF N = 0 N C ONLY THE REDUCTION ALGORITHM I S TO BE C PERFORMED.) C C = TOTAL NUMBER OF TERMS CURENTLY I N Q. LASTRM C ICOEF(1) = C O E F F I C I E N T OF ITH TERM OF POLYNOMIAL Q. C INOETIL,I) = SUBSCRIPT OF LTH INDETERMINATE FACTOR C OF ITH T E R M , O F T H E POLYNOMIAL a. C I TERM = TERM OF Q CURENTLY B E I N G EXAMINED. ( A L L PREVIOUS TERMS HAVE BEEN C C - . COLL ECT E D/ REOU CEO. I 10 C A L L I N P U T L I M I T = LENGTH 2 COLECT = .FALSE. REDUCE = .TRUE. IF(N.LE.0) GO TO 90 COLECT = .TRUE. REDUCE = .FALSE. 90 I F ( L I M I T . L E . 0 ) GO TO 200 ITERM = 0 100 I T E R M = I T E R M + 1 1FlITERM.GT.LASTRM) GO TO 200 C F I N O F I R S T P O S I T I O N I N TERM WHERE OPERATION I S POSSIBLE. . no 110 L = 1 , L I M l T I = INOETlL,ITERM) J = INOETlL+1.ITERM) K = INOETlL+2.ITERM) 1FICOLECT.ANO.I.EQ.N .OR. 1 REOUCE.ANO.I.GE.K.AN0.J.NE.NE.N) GO TO 1 1 5 . I 1 0 CONTINUE C NO OPERATION P O S S I B L E ON T H I S TERM. GO T O 100 C 115 COPY O R I G I N A L TERM AND DELETE FROM POLYNOMIAL. 115 ITCOEF = ICOEF(1TERM) 00 120 LL = 1,LENGTH __ 1 2 0 ITEMPILL) = INOETlLL,ITERM) I O E L = ITERM CALL OELETEIIOEL) I F ( R E 0 U C E ) GO TO 130 C DETERMINE WHICH C O L L E C T I O N OPERATION TO PERFORM. IF(J.EQ.N.AND.K.NE.NI GO TO 140 1FIJ.NE.N.AND.K.EQ.N) GO T O 170 1FIJ.NE.N.AND.K.NE.NI GO TO 180 STOP 1 C 130 DETERMINE WHICH REDUCTION OPERATION T O PERFORM. GO TO 100 130 1FII.EQ.J.ANO.J.EQ.KI IFlI.EQ.J.ANO.J.GT.Kl GO TO 140 1FII.GT.J.AND.J.EQ.K) GO TO 150 IFlI.EP.K.AND.K.LT.JI GO TO 160 1FlI.EP.K.ANO.K.GT.J) GO TO 170 IF(I.GT.J.AN0.J.GT.K .OR. 1.GT.K.AND.K.GT.J) GO TO 1 8 0 1FlJ.GT.I.ANO.I.GT.K) GO TO 190 STOP 2

-

__ ~

62 C

C

C

C

Cl

S. Bachmuth e t al., A non-solvable group of exponent 5 140 R E P L A C E XXY BY YXX. 140 I T E M P I L ) = K ITEMPlL+2) = I C A L L AOTERM G O T O 100 1 5 0 R E P L A C E XYY BY YYX 150 ITEMPIL) = K ITEMP(L+21 = I C A L L ADTERM GO T O 100 160 R E P L A C E XYX BY -2XXY 160 I T C O E F = M O D ( - 2 * 1 T C O E F ~ 5 1 ITEMPlL+2) = J ITENPlL+ll = K C A L L ADTERH G O TO 100 BY -2YXX 170 R E P L A C E X Y X 170 I T C O E F = H O O I - 2 * I T C O E F ~ S ) ITEMP(L) = J ITEMPlL+l) = I C A L L ADTERH -~ __-___ -GO TO 100 e o REPLACE XYZ BY -ZYX -2YZX -2YXZ . __ 180 I T C O E F = M O O I - I T C O E F ~ 5 1 ITEMPIL) = K _._ ITEMPIL+2l = I C A L L ADTERH ITCOEF = M 0 D I 2 * I T C D E F ~ 5 1 ITEMPlLl = J ITEMPIL+ll = K C A L L ADTERH ITEMPlL+Zl = K ITEMPIL+ll = I C A L L ADTERM GO TO 100 190 R E P L A C E XYZ BY -ZYX -2ZXY -2XZY 190 I T C O E F = M O D I - I T C O E F . 5 ) ITEMP(L1 = K ITEMPlL+2I = I C A L L AOTERM ITCOEF = MOO12*ITCOEF151 ITEHP(L+ZI = J ITEMPIL+l) = I C A L L AOTERM - -- . ITEMP(L+lI = K ITEMPILI = 1 C A L L AOTERM G O T O 100 200 C O L L E C T I O N OR R E D U C T I O N A L G O R I T H M C O M P L E T E D . 200 I F I R E D U C E I G O TO 250 WRITEl6~201) 2 0 1 F O R M A T I l r ' R E S U L T OF C O L L E C T I O N A L G O R I T H M - . . ' * / I C A L L OUTPUT I F P O L Y N O M I A L I S ZERO, GO T O N E X T CASE. GO T O 10 IF(LASTRM.EQ.0) R E D U C E = .TRUE. COLECT = .FALSE. G O TO 90 250 H R I T E l b r 2 5 1 1 251 FORMATI/,' R E S U L T OF R E D U C T I O N A L G O R I T H M . - . ' * / I C A L L OUTPUT

.

.

__

~

~

~

~

~

C

.

-

~

C

C

G O TO 10 EN0

~

S. Bachmuth et al., A non-solvable group of exponent 5 -~

~

100

101

102 10 103

10

101 102

C

SUBROUTINE D E L E T E I / I D E L / ) COMMON I N 0 E T l 1 O 1 2 0 0 ) ~ I C O E F I Z 0 0 1 r COMMON LENGTH. N t I T E R M , L A S T R M D E L E T E T E R M NUMBER IOEL FROM a. 00 10 I = 1OEL.LASTRM ICOEFII) = ICOEFlI+l) 00 10 LL = 1,LENGTH 10 I N O E T I L L v I ) = I N D E T I L L s 1 + 1 1 IF(ITERM.GE.IOEL) ITERM = ITERM LASTRM = LASTRH 1 RETURN EN0

-

ITEMPIlO),

ADJUST

AND

-

C

ITERM.

~

-

S U B R O U T I N E AOTERM ADDS T E R M I N I T E M P TO THE POLYNOMIAL. COMMON I N D E T I l O t 2 0 0 ) ~ I C O E F l 2 0 0 ) r I T E M P l 1 0 ) * I T C O E F COMMON LENGTH. N, I T E R M . L A S T R M _ _ _ 00 2 0 I = 1.LASTRM DO 10 LL = 1,LENGTH 10 I F I I N O E T l L L ~ I ~ . N E . I T E M P O ~GO TO 20 T E R M I N TEMP C O M B I N E S W I T H TERM NUMBER I OF T H E P O L Y N O M I A L . I C O E F I I I = M O O l I C O E F ( I 1 + I T C O E f: , 5 l I F l I C O E F l I ) . N E ~ O l RETURN TERMS CANCEL. DELETE. CALL DELETE1 I ) RETURN 20 C O N T I N U E TERM DOES NOT COMBINE. ADD NEW TERM A LASTRM = LASTRM + 1 IFlLASTRM.GT.200) STOP 3 ICOEFILASTRM) = ITCOEF . DO 30 LL = l t L E N G T H

~

_ _ _ -__

~

C

__ -

-

~

C

__

ITCOEF

LASTRM

1

__

~

-

~~

S U B R O U T I N E OUTPUT COMMON I N O E T ( 1 0 ~ 2 0 0 l I~C O E F l 2 O O ) * I T E M P ( 1 0 ) . ITCOEF-_ COMMON LENGTH. NI I T E R M t L A S T R M DATA X/' X '/ IFILASTRM.GT.0) GO T O 10 WRITE l6r101) - __ RETURN DO 2 0 I = 1,LASTRM - -WRITEl6s102) I C O E F l I ~ ~ l X ~ I N O E T l L L . I ) r L L = I ~ L L ~ l ~ L E ~ G T H ~ RE TURN FORMATI' P O L Y N O M I A L IS ZERO.') FORMATI5X~12ilOlA2t11)) END

-

C

-

SUBROUTINE INPUT COMMON I N O E T l 1 0 ~ 2 0 0 ) I~C O E F l 2 0 0 ) r I T E M P l 1 0 ) t I T C O E F _ _ _ COMMON LENGTH, N. I T E R M , L A S T R M D I M E N S I OY TI T L E ( 2 0 ) READ(5.100) TITLE FORMATl2OA4) WRITEI6plDO) TITLE READl5.101) LENGTHt Nv LASTRM FORMAT1315) WRITEl6.102) LENGTH' N LASTRM ,13,', NO. O F TERMS ='IS,/) FORMATI/,' LENGTH = ' I 3 I N 00 10 I = l r L A S T R M REAOl5.103) ICOEFlI),l NDET(LL,I).LL=l.LENGTH) FORMAT 1 2 x 1 I 2 12x1 1012) C A L L OUTPUT RETURN END --

20

63

~~ ~

~-

S. Bachrnuth el al., A non-solvable group of exponent 5

64

__ .__

3 0 I NDET I L L , L A S T R M I = I T E M P I L L ) RETURN END

XbXlH(X2.X3*X4)X5 L€YGTH =

6 N =

6,

X2 X4 X3 X3

X3 X3 X2 X4

1 X6 X 1 X6 X 2 X6 X 2 X6 X

1 1 l 1

NO. X4 X2 X4 X2

OF

TERMS =

X5 X5 X5 X5

R E S U L T OF C O L L E C T I O N A L G O R I T H M 1 X2 X 1 X2 X 1 -2 X I XZ 1 X4 X1 -2 X4 X 1 -2 X I X 4 -2 X 3 X I -2 X 3 X 1 -1 X I X 3 -2 X 2 X 1 -4 X 2 X 1 1 X I X2 -3 X 1 X 2 -2 X 1 X 2 -2 X 4 X I -4 X 4 X I 1 X1 X4 -3 X 1 X 4 -2 X 1 X 4 -4 X 3 X I -3 X 3 X 1 -2 X 1 X 3 1 X1 X3 -4 X 3 X 1 -3 X 3 X l -2 X I X 3 1 X1 X3 -1 X I X 3

-2

RESULT

X4 X3 X4 X2 X3 X2 X4 X2 X4 X3 X3 X3 X3 X3 X3 X3 X3 X3 X3 X2 X2 X2 X2 X4 X4 X4 X4 X2

X3 X4 X3 X3 X2 X3 X2 X4 X2 X5 X4 X5 X4 X4 X5 X2 X5 X2 X2 X5 X4 X5 X4 X5 X2 X5 X2 X4

X6 X6 X6 X6 X6 X6 X6 X6 X6 X4 X5 X4 X5 X6 X2 X5 X2 X5 X6 X4 X5 X4 X5 X2 X5 X2 X5 X6

X5 X5 X5 X5 X5 X5 X5 X5 X5 X6 X6 X6 X6 X5 X6 X6 X6 X6 X5 X6 X6 X6 X6 X6 X6 X6 X6 X5

,

.

OF R E D U C T I O N A L G O R I T H M

POLYNOMIAL

IS .. .

ZERO.

...

...

4

..

.

S. Bachmuth et al.,A non-solvable group of exponent 5 XIX2H((X3.X41.XliX51 LENGTH =

6

N =

I x1 x2 x3 1 x1 x2 x5 2 X I x2 x 1 2 x1 x2 x 1 - 1 x 1 x2 x4 - 1 x1 x2 x5 -2 X I xz X I - 2 x1 x2 x1 I X I XI x3 1 XI X I x5 2 Xl x1 x2 2 X I X I x2 -1 x1 x1 x4 -1 x1 x 1 x5 -2 x1 x 1 x2 -2 x1 XI x2

XlXlH((X3rX4).X2.X5)

OF TERMS =

0. NO.

X I x3 x4

x5 x4 x5 x3 x5 x3 x5 x4 x4 x5 x3 x4 x2 x5 x4 x3 x3 x5 x4 x 3

x4 x3 x1 x4 x3 x4 x2 x3

,

-

__

~

.

OF R E D U C T I O N A L G O R I T H M

-4 x3 x1 x4 x2

-4 X I x3 1 X I x2 -1 x2 X l 1 xz x1 3 X I x2

x2 x4 x4 x3 x3

-

...

I S ZERO.

1 x4 x3 x1 x2 2 x3 XI x2 x4 2 x3 x4 XI x2 -1 xz x 1 x3 x4 - 1 x4 x3 x2 X I -2 x3 xz x1 x4 -2 x3 x 4 x2 X I RESULT

16

x4 x1 x5

x3 x5 x3 x1 x4 x5 x4 x2 x3 x5 x3 x2 x4 x5

R E S U L T OF R E D U C T I O N A L G O R I T H M POLYNOMIAL

65

x4 x3 x3 x4 x4

.

__

.

66

S. Bachmuth e l al., A non-solvable group of exponent 5

References [ 11 S. Bachmuth, t1.Y. Mochizuki, D.W. Walkup, A non-solvable group of exponent 5 , Bull. Amer. Math. Soc. 76 (1970) 638-640. [2] R.H. Bruck, Engel conditions in groups and related questions, Lecture notes, Australian National Univ., Canberra, Jan. 1963. [3] €1. Heineken, Endomorphismenringe und Engelsche Elemente, Arch. der Math. 1 3 ( 1 962) 29-37. [ 4 ] P.J. Higgins, Lie rings satisfying the Engel condition, Proc. Cambridge Philos. SOC. 5 0 (1954) 8-15. [ 5 ] G. Iligman, On finite groups of exponent 5 , Proc. Cambridge Philos. Soc. 52 (1956) 38 1-390. ( 6 ) A.I. Kostrikin, Solution of the restricted Burnside problem for the exponent 5 , Izvestia Akad. Nauk. SSSR, 19 (1955) 233-244; [ 71 A.I. Kostrikin, The Burnside problem, Izvestia Akad. Nauk. SSSR, Ser. Mat. 23 (1959) 3-34. [ 8 ] S. Tobin, On a theorem of Baer and IIigman, Canad. J. Math. 8 (1956) 263-270. (91 D.W. Walkup, Lie rings satisfying Engel conditions, Thesis, Univ. of Wisconsin, 1963, 154 pages.

THE EXISTENCE OF INFINITE BURNSIDE GROUPS J.L. BRITTON University of Kent at Canterbury

Preface

This paper is a record of research work carried out at the University of Glasgow and the University of Kent at Canterbury: also during visits of about two months each t o the University of Western Australia (1 965) and the University of Illinois, Urbana (1 968). I first reported my ideas in a brief address at the University of Oxford (1964) and gave progress reports later at the International Conference on the Theory of Groups at Canberra ( 1965), at the International Congress of Mathematicians, Moscow (1 966) and more fully at the University of Illinois, Urbana ( 1 968). However I first reported on the final version t o the Edinburgh Mathematical Society in Aberdeen (1 970). It was a personal disappointment that working out the technical details of the method known in principle in 1964, and unchanged since then, should have taken so much time. I wish to thank the North-Holland Publishing Company and the Organizing Committee of the CODEP conference, W.W. Boone, F.B. Cannonito and R.C. Lyndon, for suggesting that 1 should present the full text in this volume rather than a report of my address to the conference. My apologies are due t o the other contributors since the manuscript reached the publishers six months later than the date then promised. During this time I verified “Wright’s Law” that the difficulties of presenting a research paper increase with the cube of its length. I wish t o thank the U.S. Army Research Office-Durham for a travel grant enabling me t o attend the conference. 61

68

J. L,. Britton, The existence of infinite Burnside groups

I owe special thanks to J. Mennicke for his valuable comments and for suggesting many improvements to the manuscript. I acknowledge with thanks the support received from the University of Bielefeld and the Deutsche Forschungsgemeinschaft. Summary

The primary purpose of this paper is t o prove, in Chapter 11, that the Burnside Group B: on d generators with exponent e ( d 2 2, e > 0) is infinite for all sufficiently large odd e. This proof is independent of the proof of Novikov and Adyan [ 51 . The secondary purpose of this paper is t o give in Chapter I a self-contained account of a new class of groups, called Generalized Tartakovskii Groups. We contend that B: is a Generalized Tartakovskii Group, for sufficiently large odd e. N o previous knowledge of the results of Tartakovskii I61 or related results of Greendlinger, Lipschutz, Lyndon, McCool, Schiek, Schupp and the author is assumed. Indeed the entire paper assumes little more that the concepts free product, free group, group given by generators and defining relations.

J. L. Britton, The existence of infinite Burnside Groups

69

CHAPTER I Generalized Tartakovskii Groups

5 1. Introduction The results of this chapter are presented in Theorems 1 , 2, 3, 4, 5, 6. For Chapter 11, only the statements of Theorems 4, 5 , 6 are required together, of course, with their supporting definitions (including 2.1, 2.8B, 3.1-3.5, 6.1-6.3, 6.6).

§ 2.

2.1. Notation. Let ll be a free product of groups G,(y E I‘). An element X of ll is either the identity I or has n’xmal form a l a ,... a,, where n Z 1 is the length L(X)of X,ai E GYi (i = 1, ..., n ) and yi # y i + l (i = 1, ..., n - 1). We use capital letters for elements of 11, and small letters for elements of ll having unit length. The G, are called the constituent groups of ll and = denotes equality in ll. x y means that x, y belong to the same constituent group. We write In X,Fin X for a l , an respectively. The product of X , Y E II is denoted by X . Y but we shall use the “dot convention” that writing W ,W,...W, instead of W,.W,. ... . W, indicates that the length of the element is

-

q=1 L (

wj

1.

Let X E ll: if L(X)2 2 and In X , Fin X are in different constituent groups we say that X is externally reduced (E.R.). If L(X)2 2 and In X,Fin X are not mutually inverse we say that X is externally cancelled (E.C.).

10

J.L. Britton, The existence of infinite Burnside Groups

Let X = a1a2...an be E.R. Then a cyclic arrangement (C.A.) of X is any element ui". ~ , , u ~ . . (. ia= ~1 ,~...) ~ n). By a split of an element x of length I we mean any conjugate i'-l . .x.y where 3: x: a split of an E.C. element X = u1a2...an is defined as follows: put Y 3 a2." a;al if a1 an otherwise put Y = X : thus Y is E.R., say Y = b,b2... b,: then (i) any element

-

-

where bi = bj:.by is a split of X and (ii) any C.A. of Y is a split of X . Let Sp(X) be the subset of n consisting of all splits of X : similarly we may write C.A.(X) if X is E.R. Throughout the paper, we mean by a word the normal form of an element of n: the empty word means I . A subword of W E TI is an ordered triple A , B, C of elements of ll such that W = ABC; the value of the subword is B . Distinct subwords may have the same value, but from now on we shall not distinguish notationally between a subword and its value. If W = CD we call C a left subword of W and D a right subword of W. A left subword of W is sometimes denoted by WL or W1. Similarly W R or Wr denotes a right subword. An arbitrary subword of W I S sometimes denoted by WM or W m . I f X.Y E KI we write p(X, Y ) . E ( X , Y ) for the number of cancellations and amalgamations respectively in the product X . Y . Thus e ( X , Y ) = O o r 1 a n d L ( X , Y ) = L ( X ) + L ( Y ) - 2 P ( X ,Y ) - E ( X , Y ) . We may write X . Y = XLcE Y R where L ( X L ) = L ( X ) - p - E , L( Y K ) = L ( Y ) p - E , E = €(X,Y ) , p = p ( X , Y ) . By the driul of a statement we mean the new statement obtained by interchanging the words "left" and "right". If X $ I then an external cancellation is the operation of deleting I n X and Fin X from X if these are mutually inverse and L ( X ) > 2. Denote by E ( X ) the result of carrying o u t all external cancellations: it will be either E.C. 07 of length 1. ~

2.2. The relations = 0 0. The relation X = Y where X , Y E II means Y is E.C. and X is a split of Y .

J. L. Britton, The existence of infinite Burnside Groups

71

The relation X = Y 0 2 where X , Y , Z E II means Y , Z are E.C. and there exist X ’ , Y‘, Z’ such that Y‘, 2’ are splits of Y, 2, Y’. 2’ I, E( Y’.2 ’ )z X ’ , X is a split of X ‘ . Note that X has length 1 or is E.C. (Informally, X arises from the E.C. words Y, 2 by the following process. Take splits Y’, Z ’ , then externally cancel the word Y‘.Z’ and finally take a split. Similarly, below, X arises from Y,, ..., Y, by repetition of this process.) We consider certain formal expressions involving elements of II, the symbol 0 and brackets. These are defined inductively as follows. (i) Any element of lI is an admissible expression: (ii) if 6 and q are admissible expressions then so is (t 0 q). The relation X = Y , 0 Y, 0 ...o Y , , where X,Y , , ..., Y, E n ( n 2 2) and where Y , 0 Y , 0 ... 0 Y , has been assigned brackets making it an admissible expression, means the following. Where Y , 0 Y, 0 ... 3 Y , has the form (Y, 0 ... 0 Y,) 0 ( Y , 0 ... 0 Yn),there exists A , B such that A = Y, 0 ... 0 Y,., B = Yr+l0 ... 0 Y, and X = A 0 B . Note that the case n = 2 is consistent with the previous definition. C l e a r l y i f X = Y l o . . . o Y,, t h e n X - T ; l . Y , . T,....T,-l. Y;T, for some T,, _ _T, _ ,in II.

+

+

,

2.3. We shall consider a subset s2’ of II satisfying the following conditions and the group G = II/[L!’] where square brackets denote “normal subgroup generated by”. 1 . Each element of C2’ is E.R. and s2’ = L u L u ... . 2. L i is closed with respect t o the operations cyclic arrangement and inverse, (i = 1 , 2, ...). 3. The sets L i are pairwise disjoint, and if Li is empty then so is

, ,

Lj+l.

2.4. Let 52 = Sp(C2’). F o r n = I , 2 , 3 , ..., put XE C, if there exist Y , , ..., Y, in 52 such that X = Y, -..o Y, for some bracketing. Thus

c, = a .

0

12

J. L. Britton, The existence of infinite Burnside Groups

2.5. Definition. Let W be a relator, i.e., W E that there exists a representation

[a'I and let W $ I so

(TiE n, W iE 52'). The preweight of this representation, and of the sequence W , , __., W,, means the sequence ( d , , d,, ...) where d s is the number of W , , ..., W n in 1s. ( d , , d,, ... ) + ( d ; , d ; , ... ) means ( d , + d ; , d , + d ; , ...). Put ( d ; , d i , ...) < ( d , , d,, ...) if there exists j such that d i < d j and k > j 3 d i = d,. This totally orders the preweights of the representation of W : it is a well-odering and the weight of W means the least preweight of any representation of W . It is denoted by w ( W ) . If W E 1, then the weight of W is < (0, ..., 0, 1,0,0, ...) where the 1 occurs in the a t h place: if equality holds we call W irredundant. 2.6. Definition. J E Fl is a natural element if there exist W , , ..., W , in 52' such that (i) J = W , 0 ... 0 W , with some bracketing and (ii) the weight of J equals the preweight of the sequence W , , ..., W , . (Each of W , , ..., W , is then irredundant.) 2.7. Our first aim is t o prove

Theorem 1 . Ij'no nattiral element has length 1 then every nonernptv relator has a natural element as a subword, hence has length ut least 2.

2.7A. Let R , $ I be a relator and take a fixed representation N

of smallest preweight. Consider all representations of the form h

J.L. Britton, The existence of infinite Burnside Groups

13

where

{a(i, l), ..., a(i, q i ) } being a subset Ji of J ={I, 2, ..., N } . I t follows that J = J , U .-.U J h . (1) itself is of this form. Of all representations (2) take those with h least; of these take those with X f = , L ( S ; ' . A i . S i ) least; finally of these choose one with least vector ( l , , ..., l h ) where li = L<S;'. A,. Si)and ( I l , ... , l h ) < ( k l ,.._,k h ) means that for some i, li < ki and j < i * li = ki. Let M 1 means the choice of ( 1 ) and let M2, M3, M 4 be the successive restrictions just made on the choice of (2). Since if (3) holds for A i it holds for any split of A i , we may assume SITIAiSito be written without dots. If Ai is expressed in the form II;=l U,..l. Vi.Ui( ViE st')then the preweight of V , , ..., Vt is at least that of Wa(i,1 ) , ..., Wa(i,qi) 2.8. In the following lemmas we assume that in (2) each A / has length2 2. 2.8A. Lemma. The number of cancellations and the number 6 of amalgamations between the ith and ( i + 1)th terms in ( 2 ) satisfy 1. P < Max(L(Si), L ( S i + l ) ) , 2. 2p + f < l i + l , 3. 2p + E < li. 2.8B. Note. If X , Y are two consecutive terms in (2) and X . Y =XLcEY R (cf. 2.1 ) then 2.8A implies that X L f I and YR $ I.

Proof of 2.8A. In this proof, abbreviate L ( X ) < L ( Y ) to X < Y . Write the terms in question T-lAT, S-lBS E. 1 : Suppose S < T : (there is a dual proof if T < S ) . Suppose p 2 T . Then E = T - l E , say and

J.L. Britton, The existence of infinite Burnside Groups

14

By M 3 , 2 T + A

< 2 8 , + A , i.e., T < E l , hence

T G & ( T +E , ) = ~ - = Es+$B< S+B. Since T - ' E , = S - ' B S we have T - ' = S - ' B , , B Let = B 2 .B , . Then

E

B l B 2 ,El

= B2S.

contradicting M2. 2 : I f p < S then 2 p + E < 2S+ 1 < 2 S + B = I i + , so a s s u m e p > S. Now p < Max (S ,T ) so < T, T = T , X , L ( X ) = 0, E F X - l E l , T - l A T . S - l B S = X - l T ; l A T , . E , - E . M whereM E E ; l . T ; l A T l . E l . ByM3,M> T 1 A T . N o w T 1 f I s o

2E, +2T1+A-2e>,2T+A=2T,+2X+A Hence E - E > X and E = X + El >, 2 X + E = 2p + E . I f 1; = 29 + E then El = 20 + E < 2T + E < 2T + A and also E = X + E hence L(M) = L(T-1AT) contrary t o M4. 3 : This is proved similarly t o 2 except that now equality may ocCUI.

2.8C. Lemma. Let X , Y, Z be three consecutive terms in ( 2 ) . Writing X. Y = X L C E Y R , Y.Z = YLd"'ZR where E , E' are the number oJ'umalgamations in X . Y, Y. Z respectively we have X . Y. Z = XLHZR f b r some H of length 2 1 and I n 11 In(cEYR), Fin H F i n ( Y L d E ' ) Also / I has the jorin cE.JdE'.

-

-

75

J.L. Britton, The existence of infinite Burnside Groups

Case 1. L ( Y R ) + L ( Y L ) > L ( Y ) . Here Y = PQR, Y L = PQ, Y R = QR say, Q f I. Hence

without dots and the result follows. Case2 L ( Y R ) + L ( Y L ) < L ( Y ) Here L( Y ) - - E + L ( Y )- 0'- E'

( L (Y ) - 20 - E )

+ ( L (Y )

-

< L( Y ) - 1 , so 20' - f ' )

< E + E'

-

2

By the previous lemma 20 + E < L( Y ) and 20' + E' < L( Y). Hence 0 < E + E' - 2, a contradiction. Case 3. L( Y R ) + L( Y L ) = L( Y ) . Modifying the argument of Case 2 we find 0 < E + E ' . Now Y L # I : for L ( Y ) - P' - e' = 0 implies 20' + E' < 0'+ E' hence 0'= 0 and L ( Y ) = E' < 1, a contradiction: similarly Y R $ I . Thus 1 <j
X.y . z= XLce. de'ZR

-

We show that the dot can be removed. If E = E' = 1 then c yi + yi+l d . If E = 0, E ' = 1 then FinXL + In Y R = y i + l d . I f € = 1,d=Othenc-yi=FinYL+ InZR. Let H = cede'. We show In H In(c' Y R ) :this is trivial if E = 1 so letE=O. I n H = d - y i + l = I n Y R . S i m i l a r l y F i n H - Fin(YLde').

-

-

-

2.8D. Lemma. Let K = B i . B i + l . ... . Bi be a product o j a t least three consecutive terms o f ( 2 ) . Let

Thus we may write

J. L. Britton, The existence of infinite Burnside Groups

76

B j . B i + l = BLC''B:~

B.

1-1

-Bi=B!

1-1

d'BR:

Proof. This is true for three factors by the previous lemma. Assume it is true for K = B i . '.. . Bi: we prove it is true for B . ... . B . . B . = K . B . I 1+1 .&,+I ' LetBi.Bi+l =Bi f B ~ R + ~ . N oKw= A d ' B ~ a n d B i p l . B i . B i + l = H By, L ( H )2 1 ,

,

In H

- Iii(d'BP),

Fin H - Fin(B)f'"),

H

= d'Jf'"

Heiice K.Bi+l =AdcR;-(B)-ld'BF)-l.BL

-

1-1

HB:+l

=A-HBF+l.

Now Fin A + In(d'By) In H . Since A = BIC''Q we have B,... Bi+l = B ~ c " Q d ' J f E " B > l . 2.9. Lemma. R , coritains some A i as a subword.

Proof. This is so if h = 1, so assume h 2 2. Consider X = SfT1 A i S j where L(S,) is maximal. If i = 1 let Y = S,lA,S,. Then p(X, Y ) < L(S,). Write X . Y = X L C ' YR. Then R, = X L c E D .But XL has the form S f l A j P ,hence A j is a subword of R,. Similarly, if i = h. Now let 1 < i < h . Then

Hence R ,

= QAiSj.(SrlAiSi)-l.S;lAjR = QAiR as required.

2.10. Proof of Theorem 1 . This now follows since in (2) each of A , , ..., A, is natural.

J. L. Britton, The existence of infinite Burnside Groups

I1

§ 3.

In this section we introduce some concepts we shall need.

3.1. Definition. Let W,, W,,..., W, (n 2= 1 ) be a sequence in II of which not all the terms are I . Its component sequence and patches are as follows. First define the component sequence of a non-empty word W = x1x2...x, to be xl, x,, ..., x, and the component sequence of the word I to be the empty sequence. Then the component sequence of w,,..., is the result of writing in succession the component sequences of W ... , W,. Denote it by d , , d,, ..., dp. Then p 2 1. It may be written S , , S,, ..., S4 where S j is a non-empty subsequence all of whose terms belong to the same constituent group G and yi f yi+, ri ( i = 1, ..., q - 1). Call S , , S,, ..., Sq prepatches. If q 2 2 and y1 = y, the patches are S,, S,, _..,S,-, and S,S,. In the opposite case the patches are S , , S,, ..., S4.

w,,

3.2. Definition. A partition is a sequence W , , ..., W , (17 2 1) in II such that either (i) all terms are I or (ii) the product of the terms in each patch is not I. It is proper if no W j is I . (If W = W,.W,-... . W, we say W , , _..,W, is a partition of W) 3.3. Note. Let W be a partition of one term. Then W is I , or L ( W ) = 1 or W is E.C.

3.4. A partition is strong if it has the form (ii) of 3.2 and the product of the terms in any non-empty subsequence of a patch is not I. This is the case if no patch has more than two terms. A onetermed partition is strong.

3.5. Let W E II. Then a division of W is a subsequence W , , ..., W,, whose product is W such that, if the component sequence

d , , ..., dp is non-empty and is factorized into S , , S,, ..., Sr (Sinonempty, all terms in SiE GTi,y j # yi+,) then the product of the terms in each S j is not I.

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.I.L. Britton, The existence of infinite Burnside Groups

Each Wi is called a segment of W .

3.6. Lemma. Let W , , ,.., W , be a division o f W . Let T = W i . W i + l . Then W,,.,., W i P 1 ,T , W i + 2 ,..., W , is a division of W . Similarly for partitioris. Proof. Let W i = x l . . . x,, W i + l = y l . . . y , (the lemma is trivial if W i or W i + ] is I or if x,, y1 does not hold). Cusc~1. Both Wi, W i + l have length > 2. Then x;yl # I. If there is an amalgamation the new component sequence is

-

if not the new component sequence is the same as the old. Cbse 2.m = 1 , ti 2 2 (or dually). Let z , , ..., zy, x, y l (r Z 0 ) be the patch containing x, , y 1 in the original sequence. Then the new patch is either zl, _ _z_, ,if there is cancellation (when x, y , = I and r # 0): z 1 , _ . ,z,, (xlny , ) if there is amalgamation: zl, ..., z,, ,x, y , if there is no amalgamation. Hence the product of terms in the patch is unchanged. Cuse 3 in = 11 = 1 . Again the product is unchanged.

3.7. Corollary. Let W,,_..,W , be u division of W =xI".xq.Then the pvxhtct of the t e r m in SJ is xi and the rzuinber of SJ is y. Proof. By 3.6 W is a division o f W and when we combine two terms the product of terms i n the j t h patch is unchanged.

3.8. Corollary. AYZJdivision of an E.C. word is a partition. A n y divisioii of'u word of length 1 is a partition. 3.9. Lemma. Let W , , _ _W_, ,be a partition (division) o f W and let U , , _ _Uq _ , bc a division o j W i . Then W , , _ _W _ i,- , , U , , ..., Uq , W I + l...,, W,, is a partition (division) o f W .

Proof. Say W i = x 1. . . xp.Then the component sequence of u,....,Uq isx:...?iC;lx:...,~~2. . . xP l. . . xPa P w h e r e x1. = x I! . xJ ? . . - .x?. . I Hence the product of terms in any patch of W , , ..., Wn is preserved.

J.L. Britton, The existence of infinite Biirnside Croups

19

3.10. Lemma. Let R , , ..., R , and S , , ..., So be two divisions of W , W z I. Then there is a common subdivision, i.c., there is a division U , , ..., U p of W and divisions R,!, ... , R f i of R,, Sj, ..., o j S i such that U , , ..., U p is R : , ..., R{1, ..., R i , ..., R P and also is Si, ..., S:', .._,S:, ..., S?. Finally, p < p + 0 - 1 .

Sy

Proof. Let W = x1x2...xp . Consider any xi. Then the j t h prepatch of R , , _..,R , is such that the product of its terms a l , a,, .... a, is xi. Define c by a , = h , . h,. ... . hn - c where b,, b,, ..., b, is the j t h prepatch of S , , S,, ..., S,. Then b, = ca2a3...a, since b , ... b,-l ca,"' a, = xi = b , b,... b,. Consider the product

,'

Note that any a j ( i = 1 , ..., m ) is a subproduct and any b, ( k = I , ..., n ) is a subproduct. Rewrite (2) as x!J x2J ... x!i.J Now consider the product (3)

x1 ... x P l x l ... xPz... 1

1

2

2

x' ... x P p ,

p

each of R , , ..., R , is equal t o the product of terms in a subproduct of (3) and similarly for each S , , ..., So. Thus (3) factorises into p < p + u - 1 part products where if Ui is the product of terms in the jth part product ( j = 1, ..., p ) then each Ri and each Si is a part product of

otherwise by examining (2) it No U , can obtain both x!,J x!"J would follow that u k properly contains the end point of an R , or 'b.

Hence the subsequence of ( 3 ) which forms U, is just the normal form of uk. Hence U , , ..., U, is a division of W. Consider any R,. We have, say, R , = U i . U i + , .... . U p . Now each prepatch P is the component sequence of Ui, ..., U[, is clearly a subsequence of a prepatch P' in the component sequence of U , , ..., U, in fact P is R , n P ' . If we show the product of the terms in P is not

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J.1,. Britton, The existence of infinite Burnside Groups

I then Ui, ..., UP is a division of U i . ... . Up i.e., o f R I . But a typical P' in (2) so a typical R , n P' is either b , , b,, ..., b, c or ak (some k >, 2).

-,,

3.1 I . Note. Trivially if W , , ..., W , is a partition then so is W j + l , Wi+,, _ _W,,, _ , W , , ..., Wi where 1 < i < n. 3.12. Lemma. Let W , , ..., W , be a strong partition of W and let U be a split o f W. Then there is a strong partition of U of the form

where W i , Wi.' is a division o f W i .

Proof. This is trivial if U = W so let U # W . Then

x . Y = a).,

U =S . R,

W=R.S,

L ( Y ) < 1.

Let the component sequence of W , , ..., W , be d , , d,, ..., d p . Since a strong partition is a division, a, = dg. dg+,. ... . dh say, and R = d I . ... . dg- 1 .x,S = y . d h i l . ... . d p . I f for some .f x = dg . dg+ . ... . d f , g < f < h, then

u = d f + . .. . . d

I1

. ... . d . d . ... . d P

1

g-1

. dg . . .. . d f

and the result follows. If x cannot be written in the above form put dg = x.z . Then n o non-trivial subproduct of x . z . dg+l. ... . dh equals I and

Suppose dg lies in W isay: W i = dk ... dg-..dl. Put Wi = d , ... dg-, Wl!' = zd g + l ... d,. The result follows.

x,

3.1 3. Proposition. Let X , Y E II,X . Y # I, X # I, Y # I. Let Z be the

J.L. Britton, The existence of infinite Burnside Groups

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result of cancelling the element X . Y externally. Then X,Y have the form 1. X = X ' Q E , Y=E-lRV-lZVS-lX'-l , or 2. X = T-l BQE, Y = E-l RCT, where Z = BSC, or 3. Y'-?s-~ V - ~ Z V Q E , Y = E - ~ R Y ' . Here either Q, R, S are all I, or all have length 1 and S = Q . R.

x=

Proof.X.Y=X'SY', X = X ' Q E , Y = E - l R Y - ' , whereQ,R,Sare all I or all have length 1 and belong to the same constituent group; S=Q.R. T exists such that T-' ZT = X'SY'. Case I . L(X'S) < L(T-'). T-1 = X'SJ7-1 say, y' = V--'ZT= V-lZVS-lX'-l Case 2. L(T-') < L ( X ' S ) < L(T-'Z). Here L(T-') < L(X') so X' = T-l B say, and BSY' = ZT. Now Z = BSC say, so Y' = CT. Case 3. L(X'S) > L ( T - ~ z ) . HereL(X')> L ( T - ' Z ) s o X ' = T - ' Z V s a y a n d T = VSY'. 3.14. Lemma. Let X, Y be E. C. and X. Y # I. Let Z be the result of cancelling the element X . Y externally. Then there are splits X,, Yo of X, Y respectively such that 1. X i ' , F - ' , Z , F is a division of Yo for some F, or 2. elements L, M , N exist such that L , N-' is a division of X, N , M is a division of Yo L, M is a division of Z, or 3 . F - ' , Z, F, YO' is a division of X, for some F.

Proof. In Case 1 of 3.13 XI-', E-' R ZVS-' is a division say of Yo, hence Yo is a split of Y. Now E-' R V - ' , Z, VS-' is a division of E-'RV-'ZVS-' and E-lQ-', S V - l is a division of E-lRV-'. Hence by 3.9 X'-l, E - l Q-', S V 1 ,2, VS-' is a division of Y o hence so is Xi',S V - l , 2, VS-l where X, = QE. X'. This is a split of X. In Case 2, BQE, T-l is a division of X,, a split of X. Now BQ, E is a division of BQE so BQ, E, T-', hence BQ, (E. T - l ) is a division of X,. Similarly ( T .E-' ), RC is a division of Yo, a split of Y . It remains to show that BQ, RC is a division of 2. This is so.

32

J. L. Britton, The existence of infinite Burnside Groups

Case 3 is now obvious.

3. IS. Definition. A srrizischeme of .k rows on symbols u l , _..,u, is an array

.. .

...

. ..

...

consisting of k rows and a result r1r2 ... r6 where 1. li> I , 4, 2 1 ( i = 1, ..., k ) , b > 0: 2. each xii has the form ur or u;': each ri has the form ur: 3 . for each s = 1, ..., a the number of occurrences of uj' is two: either an occurrence of us in a row and an occurrence of u,' in a row. or an occurrence of 14, in a row and us in the result: in the first case [ I , and uil are of type C while in the second case both occurrences of us are of tj'pe R: (The total number of entries is q l + ... + 4 k + b = 2a.) 4. the order in which the rows are written does not matter: the terms i n any row or in the result may be cyclically permuted. 3.16. Definition. Given a sequence 7 ] , 7 2 , ..., rp each term of which is a non-empty sequence of symbols each of the form zifl (i = 1, ..., a ) an clrrnciitary opcration means any one of the following. E 1 . Replace some 7 i by ;I cyclic arrangement of ri. E2. Cancel two adjacent mutually inverse symbols in some 7 i . E3. For some j , li where j f k delete T k and replace ri by 7 i 7 k . Notation: If after ;i finite sequence of elementary operations r l , r 2 , ..., r p is tranformed into v l ,v 2 , ...,vq say, write 7 1 %7 2 .

..., 7,,

Exaiizple: Let r 1 be d , Then r , , 7 2 , r3 u. --f

+

v ] ,v 2 , ..., vq 6 : r2 be e, b - ' , c : 73 befl

c - l , a:u

be e, cl, f.

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83

3.1 7. Definition. A semischeme is reducible if the rows p l , ..., p k can be transformed into the result u by elementary operations, i.e.,

The following are trivial.

3.1 8. If p1,p2,..., p k -+ v l , v 2 , ..., vq then the sequence p , , ..., p k is the union of q disjoint subsequences e l , ..., 6 , such that O i + vi ( i = 1, ..., 4). 3.1 9 . If cp is a sequence of elements of a group let p(p) denote the product of these elements. If p , , ..., p k + v then p ( v ) = I I f Z 1 Tlrl . p ( p i ) - Ti for some Ti in a free group. 3.20. Let p , , ..., p k v. Let A be a term of p , , say p1 = t , A , q and let A survive the sequence of elementary operations, i.e., assume A is not cancelled in any elementary operation: thus A is a term of v , say v = Er, A , qr. Then +

k

p(q') p(tf) = p ( q ) p ( t )

n T;'

i=2

. p ( p i ) . T~

in a free group.

3.21. Assume p l , ...,pk + v and p1 contains two t e r m s A , B that survive the sequence of elementary operations: thus p 1 = A , t , B , q and v = A , tr,B, q' say. Then p ( t ' ) = p ( t ) II;=, T;lp(pi) Ti and p ( q ' ) = p ( q ) I I f = r + , T L 1 p ( p i )Ti (in a free group) without loss of generality .

3.22. Proposition. In a semisclzeme with rows p l , ..., p k where two distinct rows have the f o r m a , Z and Z-' , /3 and the result is u we have

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.I. I,. Britton, The existence of infinite Burnside Groups

Proof. Z and 2-l cancel in the reduction say at the sth elementary opedtion. Hence there exists r < s such that after r - 1 operations the two rows have the form t 72, and 2, T I , 8, and the rth operation brings them together giving ,$ZqfZ-'O or { Z - l O EZq: we consider the first case only. After the (s - 1 )th operation we have say A, Z', Z ? , p ( E = 2 1) and the next operation changes it t o A,p. Suppose E = -- 1 : the case E = 1 is similar. Then

hence [)(a) = T - ' p ( a ) p ( P ) T ~ T ; ~ T ; ~ T ;where T ~ T ;7 ~ ;is a conjugate of T i (i = 1 , ..., 5).

3.23. Proposition. E2 fhllowed by E l or E 3 is equivalent to E l or E 3 jbllowed b.v E2. Proof. Trivial.

3.24. Proposition. l f ' p l , p 2 , _ _p,_ , + 1 (the empty sequence) and 1 la, p1 then p l , p 2 , ..., pm + u. Converselj, p i , p2, ..., P,,~+ ii implies p l , p2. ..., pm 1.

i f p i is

+

Proof. The converse is trivial. Now p l , p2, _._, p, ul, a2,..., (Tb 1 where u l , a2,..., (Tb arises from p l , p 2 , ..., pm by E l , E3 only and (51, 02, ..., (Tb 1 by E2 only (hence b = 1). Say p l , p 2 , ..., p, +V1. Then p1 is t o , t l, _ _ ~_ ,, a n d ~ ~ i s a C . A . o f ( l ) ~ ~ a ~ ~ ~ ~ ~ ~ a ~ ~ ~ w p 2 , . _ _p,, + a l , ... , a s . Hence --f

+

+

J.L. Britton, The existence of infinite Burnside Groups

85

Now u1 cancels t o the empty word hence so does (1). Hence u toa 1 El a, tscancels to u . 1..

3.25. Proposition. If a, x and x - l , are two distinct rows p l , p2 of a reducible semischeme with rows pl, p2, ..., p k and result u then ( a , 01,P 3 , ..., P k 0. +

i3XEX-E

+

hence a ;

pi

up:, a:

+

+

1 and

pi a :

+

u7

( a ; pi

+

6+u.

1. By 3.24

)pi a;

-+

( a ; p: )

-+

p; a;

+

6.

Similarly if E = - 1.

3.26. Replacing ri by r i , u , , ~in both row and result of a semischeme S gives a semischeme S' on u l , ..., u,, u , + ~ If. S is reducible then so is S ' . 3.27. Note. If a # 1, deleting both occurrences of any u i in S yields a semischeme S ' . If S is reducible then so is S ' .

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J.L. Britton, The existence of infinite Burnside Groups

3.28. Note. In a semischeme S, delete r l , r2 from the result and, in the row containing r l , replace r1 by Y;' : then a semischeme S' is obtained. If S is reducible then so is S ' .

3.29. Definition. Let X , , ..., XuE Il and let S be a semischeme on u l , .._,11,. The array obtained fromS by replacing u: by X: ( E = 1 or -1 i = 1 , ...,a)iscalledascheme. 3.30. Proposition. Consider a reducible scheme such that the rows and the result are partitions. Let the result be W , , ..., W , where W , - , = Xy' W i y and y' y . I f we replace Xy' by A , y-' in the row, where A = X y ' . y , and replace Wi-,, Wi in the result b y A then we obtain a reducible scheme whose rows and result are partitions.

-

Proof. In the underlying semischeme let r l , r2 correspond t o X y ' and y respectively. Replacing r1 by r l , u we obtain a reducible semischeme S' by 3.26. In S ' , delete u, r2 from the result and, in the appropriate row, replace u by ril obtaining a reducible semischeme s'' by 3.28. In s'' replace r , , r2 by A , y and replace all the other symbols as in the original scheme. We obtain a reducible scheme whose rows and result are partitions.

3.30A. Note. I f some row of the original scheme is a division, the corresponding row of the new scheme is a division. 3.31. Proposition. Let W , , ..., W , be a proper partition that is not strong. Then n 2 2 and some W i has length 1 and is in the same constituent group as Fin Wi-l and In Wi+,: (interpret W , to be W , und W n + , to be W , ) . Proof. Some patch contains three or more terms hence contains d i + l , di, di+' say. Then di is W i , di-, = Fin Wi-, and di+, is InWi+,.

§ 4.

The purpose of this section is t o prove Theorem 2 below. We re-

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mark informally that a geometrical interpretation is possible in which each row p becomes a region p' in the plane, each entry A in p corresponds to an edge A' of p' and if X , X - ' are of Type C (cf. 3.15) then the edges X ' , ( X - l ) ' coincide. (cf. Lyndon [31). Theorem 2. Let 52' satisfy 1 , 2 , 3 of 2.3. Let W be a natural element so that W = W,o ... 0 W , ( W i E a'),k 2 1. Then there is a scheme satisfying the following conditions. 1 . (a) It is a reducible scheme of k rows. (b) The total number of entries is < 6 k - 4. 2. The ith row is a partition of a split Wl!o f W i . 3'. The result is a proper strong partition o f W. 4. No row has the form ..., X , X - ' , ... where X and X-' are partners in the underlying semischeme. 5. In the underlying semischeme if one row has the form ..., x ', y ' , ... then no other row has the form ..., y - ' , X-', ... . No row has the form ..., x', y " , . . . , y - " , ~ - ~ , .(.E. = f 1 , E' = ? 1). Note. A supplement to Theorem 2 is given in 6.12.

4.1. Lemma. I f there is a scheme satisfying 1(a), 2 and also: 3. The result is a partition of a split o f W. then we may assume that in 3. the partition is proper and strong. Proof. Of all schemes satisfying the hypothesis choose one whose result R , , ..., R , has the least number of terms. Then no R j is I otherwise we could delete Ri and its partner R j . The result follows if n = 1 so let n 2 2. If R , , ..., Rn were not strong we could obtain a contradiction by using 3.30, 3.3 1.

4.2. Lemma. With the hypothesis o f Lemma 4.1, there is a scheme satisfying 1 (a), 2 and 3'. Proof. Take a scheme S satisfying l(a),2 and with result R,, ..., R , a proper strong partition of W ' , a split of W. This determines a strong partition of W of the form RY, ..., R,, R,, ..., R ; where RI, RY is a division of Ri. If in S we replace Ri in both row and result by R ; , RY we obtain S' satisfying la, 2 and

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3". The result is a strong partition of W. If Ri o r Ri-' is 1we may delete it from row and result to obtain 3'.

4.3. Lemma. A scheme exists satisfying 1 , 2 , 3 : Proof. The proof is by induction on k . k = 1. Here W = W , , i.e., W is a split of W,. For the required scheme take the single row W and result W . For the induction step we have W = X o Y

Since W is natural, so are X and Y . Since W = X o Y , X and Y are E.C. There are splits X ' , Y' of X , Y such that X ' . Y ' # I , Z = E ( X ' . Y ' ) , W is a split of 2.By 3.14 there are splits X,, I', of X',Y ' , h e n c e o f X , Y s u c h t h a t 1 , 2 o r 3 o f 3 . 1 4 h o l d . Case 1. 1 holds, i.e., X i ' , F - l , Z, t;is a division of Yo. Now .I'~ = Wa+l ... W , so by induction hypothesis there is a reducible scheme S , of k - a rows satisfying 2 for i = a + 1, ..., k and whose result Y , , ..., Y m 2is a proper strong partition of Y o ,hence a division of Y o . Hence there are divisions 0

0

d(Yi)

of Yi

( i = 1, 2, ..., m 2 )

such that

.41so there is a reducible scheme S , of a rows satisfying 2 for ..., a and whose result X , , ..., X , is a proper strong partition of X,. Hence there are divisions d ( t i ) df ti(i = 1, ..., a ) d ( X j ) of Xi ( j = 1, ..., m , ) such that

i = 1,

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Also there are divisions d ( p i ) of pi (i = 1, ...,b ) d(Fi) of Fi ..., d ) such that

( j = 1,

Let S ; arise from S , by replacing each Xi by d(Xi). Then S ; has a rows and is reducible and 2 still holds for i = I , ..., a : the result is 4 t , ). * * 4 t , ) (= P , say). Let Si arise from S2 by replacing each Yiby d ( Y i ) . Then Si is reducible, with k - a rows and satisfies 2 for i = a + 1, ..., k . Its result is the right side of ( I ) . Let S;l arise from S; by replacing ti1 by d(&)-', p;' by d ( p i ) - l , Fi by d(Fi). Si is reducible with k - ci rows and satisfies 2 for i = a + 1 , ..., k . Its result, u say, is

Let S , arise by taking the rows of S ; and S i and taking for result Z , , ..., 2,. S , is a reducible scheme since

S , has k rows and satisfies 2 for i = 1, ..., k . We next consider the number of entries in S,. The number of terms on each side of ( 1 ) is m2 - 3, the number for (2) is m l + a - 1 and the number for ( 3 ) is b + d - 1. From ( 1 ) a + b + c + d = m2 + 3 . The increase in the number of entries in the rows (and also the increase in the number of entries in the result) from S , to S ; is a - 1 from S , to Si is 3 from S; to S; is (ml - 1 ) + (2(b + d - 1 ) - b - d ) . Let the total number of entries in the rows of S,, S, be a,, a2 respectively. S , has c terms in the result so its total number of entries is a1 + a2 + (a - 1 ) + 3 + (ml - I + b + d - 2) + (m2 + 3 - a - b - d ) = (a1 + m l ) + (a2 + m2) + 2 < ( 6 -~4) + ( 6 ( k - a)-4) + 2 = 6 k - 6.

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J. L. Britton, The existence of infinite Burnside Groups

The result of S , is a division of Z , which is E.C. or has length 1 since it is a split of W and W = X o Y . Thus the result is a partition of Z . Now 2 is a split of W so as in 4.2 we can obtain from S, a scheme satisfying 1 , 2 , 3 ' with at most 6k - 4 entries. Case 2. 2 of 3.14 holds. We may take L , M t o be a proper strong partition. Schemes S , , S, exist as in Case 1 . By considering X , , ..., X,, and L , N-' we see that there are divisions d ( X , ) , d ( L ) and d ( N - ' ) cucli that d ( ~ , ) . . , d ( ~ ,=, d ( ~ ) d ( ~ -Say l ) .d ( ~ - ' i)s N i 1 ,...,N ; 1 . Let S ; arise from S , by keplacing X i by d ( X , ) . Then S ; has result d ( L )N-' * ... - N i l . There are divisions d( Y , ) ( i = 1, ..., m,) and d ( N ) ,& M ) such that d( Y , ) ... d( Y , ) = d ( N )d ( M ) . Let S; arise from S, by replacing each Y , by d( Then its result is T I . Tq d ( M ) where d ( N ) is T,, ..., Tq. Considering N , , ..., N p and T I ,..., Tq we see that there are divisions such that

qj).

(*I

d ( N , ) ... d ( N p )= d ( T , ) ... d ( T q ) .

Let Sy arise from S ; by replacing Nf' by d ( N i ) - ' . Let Si arise from S; by replacing Tiby d ( T i ) . Now take the rows of S ; and of Si and result d ( L ) d ( M )t o obtain a reducible scheme of k rows satisfying 2 for i = 1, ..., k : it is reducible since by (*)

The increases in the row entries from S,, S,, S ; , S ; t o S ; , S ; , Si, S i respectively are 1,1, 4 - 1, p - 1 respectively. Now d ( L ) , d ( M ) have m l + 1 - p , m2 + 1 - 4 terms respectively, hence the number of entries in the final scheme is a1 + m l + a, + m2 + 2 < 6 k - 6 as before. Since L , M is a division of 2, so is d ( L ) d ( M ) . As in Case 1 we find a scheme satisfying 1 ., 2., 3'. Cuse 3. Part 3 of 3.14 holds. This is similar t o Case 1.

4.4.Lemma. Ij'thertj is u scheme S satisjying 1 , 2 , 3 ' then there is a scheme satisjying 1 , 2 , 3 ' , 4 , s .

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91

Proof. If S does not satisfy 4 consider a row ..., X , X - l , ... . Since this is a partition L ( X ) < 1. If L ( X ) = 1 then X , X-' are two terms of a patch, so deleting them does not affect the product of the terms in this patch: thus if we delete X , X-' from the row it remains a partition: this is also the case if X = I. But deleting X , X - l does not destroy reducibility. If S does not satisfy 5, combine XE, Ye' into a single element (their product): partitions are not destroyed. Now deleting y in the underlying semischeme preserves reducibility so the result follows. 4.5. Note. Theorem 2 now follows.

§ 5.

5.1. Lemma. In a reducible semischeme with k rows El, ..., Ek and result r l , r 2 , ..., r n , i f the first row El has the form r1 ri ... then j = 2 or r2 r3".rI.- 1 is in a free group equal to S

where ql,..., q s form a subset of the rows

E2) ..., Ek

Proof. Consider an elementary operation applied to r l , r 2 ,..., rq where r1 is a cyclic arrangement of r l , a , rj, p (a and p are possibly empty sequences) and none of a , p, r 2 , ..., rq contains r1 or ri. A cyclic arrangement of one of the new terms has the form r1afrip' where either a = a' in a free group or a = a1e2 a' = a1 T, a 2 where 2 < w < q. In the second case a' = a ( a i l T, a 2 ) in a free group. Initially a is empty, and after a finite sequence of elementary operations it becomes r2 rj ... riV1 if j Z 3.

5.2. Definition. Let a , X , be a.partition of a split of an element of a',in L, say. Let L ( X ) Z 2. Let q be an integer Z a. We say X is q-special in a,X , p if

92

J.L. Britton, The existence of infinite Burnside Groups

1 . where X - ' , B is a partition of an element of L b and b < q then w ( A . B )< Z+ 6,where A = 0.a and u = (0, ..., 0, I , 0, ...) with 1 as the ath term, and similarly for b. 2. if a = 1 then if X - ' , B is a partition of a split of A . X we have A.B. =I. We say X is special in a , X , 0 if it is q-special in a , X , 0 for all q >a.

5.3. We now make a further assumption about the subset

a' of

2.3 : 4. Jf X I , X,, ..., Xu is a partition of an element of Sp(S2') where u < 5 then at least one X i is special with respect t o X,,..., X u . 5.4. Lemma. Let S2' satisfy 1 , 2 , 3 , 4 . I f W = W , . . . o W , is natural, so that there is a scheme S satisjying I , 2,3', 4 , 5 then f o r every such S we have: 6'. no row o f S has the form a , X , P , X - l where X , X-' is a cancelling pair in the underlying semischeme and X is special in this row. 0

Note: The proof of this lemma is postponed t o Section 6 (6.12A). Theorem 3. Let S2' satisfy 1 , 2 , 3 , 4 . Every non-empty relator R has a segment A such that there is a partition 2,. ..., 2, o f an element of Sp (a2') where u < 4 and (i) 2,is A (ii) 2 , is special but none o f Z,, ..., 2, are special (with respect to 2,, .. . , 2,). Hence L ( R )2 L ( A ) 2 2. 5.5. Proof that Lemma 5.4 implies Theorem 3. The least possible weight of R is (1,0,0, ...). In this case R = U-' W ' U where W ' E S( L '), and the theorem is true in this case with u = 1. Now let the theorem be true for all non-empty relators R' with weight smaller than that of R . Let k

R =

n T[' .Wi.Ti

i=l

( W i E a')

93

J.L. Britton, The existence of infinite Burnside Croups

where W,, ..., W, has weight equal t o the weight of R . Then h

R

E

n S:'.Ai.Si,

j=,

1

Ai x W a ( i , l ) o . . . o Wff(1,4i) .

If h > 2 then 4i < k for all i so the theorem is true for a11 A j , hence L ( A , ) 2 2. Hence by 2.9, R contains some A i as a subword. Hence the theorem is true for R . Thus, changing the notation, it remains t o prove that if W = W, ...o w k is natural then the theorem is true for W. There is a scheme S satisfying 1 , 2 , 3 ' , 4 , 5 . By 5.4 S satisfies 6 ' . We may assume that the total number of terms is minimal. Let XI, ..., X, be a row consisting entirely of terms of type C. Suppose c < 5. Then some X i is special. By 6 , the partner Xf' of X i does not lie in the row X I , ..., X c . Thus we have two distinct rows a , X i and Xf', 0. By the irredundancy of W,, ..., W , (cf. 7.6) we have 0

w(a . p) < w(a . X i >+ w(Xf1. p) But W = S - ' aflSII;,, 7';'. W i . Ti, contradicting that the weight of W equals the weight of W , , ..., w k . Hence c > 5. Let di rows have i terms (i = 1 , 2 , ..., 5) so that q = k - Cdi rows have >, 6 terms each. Then

r +d,

+ 2d2 + 3d3 + 4d4 + 5d, + 64 < 6 k

-

4.

Hence c < -4 + 5d, + 4 d 2 + 3d3 + 2d4 + d,. Let d{ rows have i terms of which j are of type R. Suppose (i) d! = 0 (i = 1 , 2 , 3 , 4 ) . Hence d , = 0. Every row with two terms has both of type R. The three-termed rows contain together 2d: + 3d; terms of type R. Each row with four terms has at least two terms of type R. Each row with five terms has at least one of type R. Hence 2 d 2 + (2d: + 3 d ; ) + 2d4 + d ,

< r,

O < -4+ 2 4 +d:

J. L. Britton, The existence of in,finite Burnside Groups

94

If (i) is false there is a row Y , , Y,, ..., Y,, u < 4, where Y , has type R and Y,, ..., Y , have type C. None of Y,, ..., Yu are special otherwise a contradiction would follow as before. By 4, Y , is special. Hence the theorem is true. If (i) is true then some row Y, s, 6 has the terms Y, s of type R so the result is either (a) a r p s y or (b) a s p r y . Since d: + d: 2 2 there are at least two such rows so we may assume that if (b) holds then not both a , y are empty. Assume (a). If /3 is non-empty then p ( p ) is a product T;'. q i .T, where t < k and q , , ..., qt is a subset of W , , ..., W , . By induction the hypothesis /3 has a segment as in the theorem hence so does W (cf. 3.6, 3.9). Hence we may suppose 0 is empty. Then the terms Y, s may be amalgamated to form a single term (in both the row and the result), decreasing the total number of terms. Assume (b). Then p ( y )p ( a ) = n1 and p ( p ) = p ( 6 ) 71,

":=,

so that s p ( p ) r is a product of less than k conjugates provided 71, is nonempty, and the theorem follows as before. Now let n, be empty Since the result is a proper strong partition a and y are both empty, a contradiction.

I n this section we introduce Generalized Tartakovskii Groups.

6.1. Definition. Let M be a non-empty subset of II closed to C.A. and inverses. Let e O ,e l , E , be non-negative real numbers. Then M has a size function using e 0 , e l , e2 if to each pair A , B E II such that A B E M there is defined a real number sA5 ( A ) such that 1. 0 < sAB ( A) < 1 . If A = I then sA5 ( A ) = 0. If B = I then SAB ( A ) > 1 €0. 2. L ( A ) = 1 SAB ( A ) < € 2 3 . S A B ( A ) = sA-'5.' (A-1) ~

J. L. Britton, The existence of infinite Burnside Groups

95

6.2. Notation. If X,Y E K then X s Y means that L ( X ) = L ( Y ) and ( a ) X = Y S I o r (b) L ( X ) = 1 and X - Y or ( ~ ) X xl".x, F Y = y 1" . y n and x, y , , x, y n , x i = y i ( i = 2,3, ..., n - I )

-

-

6.3. Definition. A non-empty subset R' of II satisfies the Generalized Turtukovskii Conditions, and G = II/[R'l is a Generalized Tartukovskii Group if subsets L L ,, ... of II and real numbers p , po,pb, e0, e l , € , exist such that the following conditions hold: put Gi = II/[L 1 U ... U L i ] , Go = II. 1. R'= L u L u .-.and each element of R' is E.R. 2. The L i are pairwise disjoint, and if Li is empty then so is L,+, (i = 1 , 2 , ...I 3. Each L i is closed under C.A. and inverses 4. l/pb > 0, l / p 2 l / p o > l/pb + € 1 + € 2 5(l/p+E1)< 1 - € 0 + € 1 5 . There is a size function on R' using e0, e l , e2. 6. IfAX, E L t , B X 2 E L s and X , = X,, say t < s, then (a) and (b) below hold where P is defined by

,,

,

(and L ( x ) = L ( y ) = 1).

a'

6.4. Definition. A non-empty subset o f II satisfies the Turtakovskii Conditions and G = II/[R']is a Turtakovskii Group if a real number p o exists such that

96

.I. I,. Britton, The existence of infinite Burnside Groups

1 . 52' consists of E.R. elements and is closed under C.A. and inverses. 2. 5 < p O < L = Min L ( W ) for W E 52'. 3 . I f AX,, B X , cz 52' and X, E X , and P is defined by A X , = E x , BX,-l
6.5. Proposition. ko vslcii G r o i y .

Aiiji

Tartakovskii Group is a Generalized Tarta-

Proof. Let 52' satisfy the Tartakovskii Conditions. Put E , = 1/ L , E 1 = €0 = 0 , 1 / p = 1 / P o , 1 /pb = 1 /Po - €2. Then S / , B ( n ) = L ( A ) / L ( A B ) is a size function for 52'. Put L 1 and take L2, L 3 , ... empty.

=a'

6.6. Definition. Let X , , ..., X, be a partition. The left (right) semiclosure of X i is the product Yi of those patches of which the last (first) term arises from Xi( i = 1 , ..., m).

6.7. Proposition. Let 52' satisfv the Generalized Tartakovskii Cond i t i o m Let a , X be u partition of a split of an element o f 1 1. Let Y be the l e f t semiclosure of X . Let X-',6 be u partition o f a split of un element of L , . (i) Let s y z ( Y ) > l / p O . Then we have t < s; i f t = s then w(cu .p) < X + farid X is special in a , X. (Here Y Z is the uppinpriatc elcinent o f 1 .) (iii) I j srz( Y ) > 1 / p b and s y z ( Y ) > e2 then t < s or w(a.0) < S+ t. Proof. Since 1 / p > l/po> E , we have s Y z ( Y ) > e2, hence L ( Y ) 2 2, in a11 cases. 1Ience X - x 1 x 2 . . ~ x nn, > 2. Now ( 1 ) Y = X 1 x 2 - . . x n - 1 if x , ~is not a patch (of a . X ) or (2) Y ~ X 1 x 2 ~ ~ ~ x nif -x , xisna xl. Consider P - l , X.This is a split of an element patch, where i1 A .I of L,s which either ( 3 ) contains a subword T = 2 1 x 2 . . . x n - 1 x nif , x l , x , , are i n different patches, or (4) contains a subword A T = x , . . . x , ? if x,, x,, are in the same patch: here x^, x l , i\ x,, s,. Cuse 1. ( 1 ) and ( 3 ) d o not both hold: (2) and (4) d o not both hold.

-

.:,

-

-

J.L. Britton, The existence of infinite Burnside Groups

97

Then Y s T. Let (i) hold. Then syz( Y ) Z 1 /po Z 1 /pb hence t < s. If t = s then “P = I in GtP1”: but P here is conjugate in II to a.0 so w(a.0)< t. If (iii) holds then t < s. Assume (ii) holds. If t<sthenP=U.C.U-’ inGt-l,CE L,,sow(a.p)<S+~.Ift=s then w(a.0)< f < S+ t. Case 2. ( 2 ) and (4) both hold. Since x 1 x, we have n 2 3. Now where Y = Y ’ x , we have Y ’ s T . M o r e o v e r T E L,.Hences(T)Z 1 - e 0 2 5 / p + 4 e 1 Z l / p 2 l/po 2 l/pb. Hence s < t. If s = t then since s ( T )Z l/po, w(a.0)< f < Y + E if s < t then since s ( T ) > l / p , w(a.0) < S + t . Case 3. (1) and (3) both hold. A Here Y = T1. . . x n P 1and T = x 1 * - x , - ~ ~ , .Put T’ = .-.x,-~. The argument of Case 1 now holds with T replaced by T ‘ .

-

6.8. Proposition. If‘ a’ satisfies the Generalized Tartakovskii Conditions it satisfies 1 , 2 , 3 , 4 o f 2.3 and 5.3. Proof. Trivially 1 , 2 , 3 hold. Let X , , ..., X , be a partition of a split of W E L, where c < 5. Let Y j be the left semiclosure of X i . Then Y , Y, ... Y , has no dots and is a C.A. of W . Omitting certain subscripts we have 1 - Eo

< s( Y , ... Y c )< s( Y1) + ... + s( Y , ) + ( c - 1 ) E l

Now c( 1/p + € 1 ) < 5( 1 / p + €1) < 1 - € 0 + € 1 So 1/ p < ( 1 - e0 - (c - l ) e l ) / c hence 1 /p < s( Y i ) for some j (1 < j < c). By 6.7 Xi is special.

6.9. We shall continue the discussion of Generalized Tartakovskii Groups in Section 7. Our aim in this section is to prove Lemma 5.4. 6.9A. Proposition. Let In’ satisfy the Tartakovskii Conditions and assume (*) no constituent group of II has an element ojorder 2. Then 5.4 holds in this case. Note: It will appear later than (*) is unnecessary (cf. 6.12A).

98

J.L. Britton, The existence of infinite Burnside Groups

Proof. Let W = W , ..-o W, be natural and let S be a scheme satisfying 1 , 2 , 3 ' , 4,5. Assume one row has the form a , X , 0,X-' where X is special. Let A = 0 . X - l . a , B = a . X . 0.Then X - ' , B and X , A are partitions so w(A - B )< (1,0,0, ...), i.e., A . B = I. Hence 0

Let C = X . p - X - ' . Then C . a = C - ' - a - ' . Now L ( X ) > 2 and a, X , 0, X-' is a partition hence /3 I; similarly cu I . Hence L ( C ) > 3. Suppose ( i )there is n o amalgamation in C . a and C-' .a-' . Then C = C-' so C 2 = I and by ( * ) C z I, a contradiction. Now let (+) be false. Where a = al ... a,, C = c1 ... c, we have

+

+

= c;' ... cil. I n particular el = c;' so Hence c1 c1 ... c , ~c , =~ c;' ... c;' ci' . Thus C = C-' , yielding C = I as before.

6.1 0. Lemma. Let F be the free group o n sl, s2,... . Let be a subset of F consisting of' elements o f the f o r m S:; S:; . - . S E i(ei= f 1 ) ui where j 2 6 and ui # ui+' ( i = 1 , 2 , ..., j ) where vi+l means u l . Let Lll be closed under cyclic arrangements and inverses. Assume that if U, V E Ll, then either the number p ( U , V )of' cancellations in U . V is at most one or U . V = 1. Then every non-identity element U of [a,]has a sitbword A such that (3B)A B E al,L ( B ) < 3 .

Note: Here we have temporarily adapted our notation and definitions t o the case of free groups rather than free products. Proof. This is a special case of e.g. Greendlinger [ 1 ] but we shall derive it from our present results. Let k 2 15 be an integer and let H be the free group on

I f W is a word in sl, s 2 ,... let @ b e the word obtained from W

J.L. Britton, The existence of infinite Burnside Groups

99

by replacing si by silsi2...sik (i = 1, 2, ...). Let 52, be the set of all w ( W E 52,) and let 52, be the closure of 52, under cyclic arrangements and inverses. Let K = H/[52, ] . Then K = H / [ n 2] . Add new generator si and new defining relations si = sil si2 ... s i k . Then remove this defining relation with the generator sil. This shows that K is the free product of a free group with F/[52, I . Now U = 1 in F/[SZ,] hence U = 1 in K and u=1in K. We show 52, satisfies the Tartakovskii Conditions with po = 6 k / ( k + 3 ) . Now 5 < po < 6 k . If U, V E 52, we find p(U, V )< k or U . V = 1. Let A X , , B X , E 52, and X , z X,, L ( X , ) / L ( A X , ) Z l/po. Then L ( X , ) 2 k + 3 so X , = c l X d l , X , = c 2 X d 2 ,L ( X ) 2 k + 1. NowP(dlAc,X,X-'c~'R-'dI') > k s o d l A c l X - X - ' c Z I B - ' d Z 1 = 1. Since the constituent groups are infinite cyclic, 5.4 hence Theorem 3 are available (cf. 6.9A). Hence Ucontains a segment A such that there is a partition A Z , , ..., 2, of an element J of S(52,) where u < 4 and none of Z,, ..., Z , are special. Hence L ( Z i ) < k + 3 . Some split J , of J is in 52, so has the form (W E nl).Let h = L ( W ) . Then Xk < L ( A . Z , . ... -2")< L ( A ) + 3 ( k + 2), so L ( A ) 2 k ( h - 3 ) - 6 where A = c A ' d , A' is a subword of g a n d of an element Vof 52, and _ _L ( A ' ) Z k ( h - 3 ) - 8. But a cancellation of length a between X , Y implies a cancellation of length b between suitable C . A of where b is the least number such that klb and a < b. Hence Uhas a subword A" of length Z k ( h - 3 ) where A"B" E 52, and L(A"B") = hk. Hence the result.

,,

w

r

6.1 1 . Corollary. Let F be the free group on s, ,s2, _... To each si assign a positive integer p i . If X = sf1 ... (with no cancellations 1 possible), 6 , = f 1, let w ( X ) = p i , + + pill. Let 52, be a subset of F, closed under C.A. and inverses, consisting of elements o f the form

sfi

where i f j 2 2 then ui # ui+, ( i = 1 , ..., j - l), uj # u,. Assume if U, V E a,, U - U ' X , V = X - ' V' then w ( X ) < 1 or U . V = 1. Then every non-identity element U of [al] has a subword A such that ( 3 B ) A B E 52, and w ( B ) < 3 .

.I.l.. Britton, Thc existence of infinite Burnside Groups

100

Proof. Replace

pi

by

tr

t I 2... tipi.

6.1 2. Supplement to Theorem 2. L e t S2' satisfy 1, 2 , 3 , 4 . I f 2= w, ... W, is riutural. so that there is a scheme S' satisfying I , 2, 3'. 4, 5 , theri f o r every such S' we have 6. 1 1 0 row ofS' hus the f o r m a , X , 0,X-'where X , X-' is of T I . ~C C iri J thc i/tidc~rh~irig semischeme.

w

0

0

Proof. If some row of S' has the form a , X 7 / 3 , X - ' where X , X - l is a pair of Type C call X,X - ' a pair of T y p e CS. Assume some Type CS pair exists: we shall obtain a contradiction. Consider any sequence of elementary operations transforming the rows of S' to its result and consider the first elementary operation in this sequence in which a Type CS pair say X,X-' is cancelled:

a.x,p,X - 1 P2

a',X,X-I +

... +

u2

P

a' +

u2

-+

... + u. Hence p 2 -+ ... -+ 1

without loss ofgenerality. Now none of 0,p 2 , . . . , p h contain a term of Type R of Type CS. If some pi ( i = 2 , ..., h ) has c terms where c < 5 then a contradiction would follow as in 5.5. Hence each of p 2 . ..., pI, has at least six terms. Let the symbols in p 2 , ..., ph be u , . .... 11,. Then = 1 in the group ( u l , ..., u,Ip2 = 1 , ..., ph = 1). By 6.1 1 contains at least three consecutive terms from some C.A. of p'' (for some i = 2, ..., h ) . This is a contradiction since S' satisfies 5. 6.1 2A. N o t e : Lemma 5.4 follows immediately from 6.12. Also by 5.5 we have that Theorem 3 holds. Finally we note that in 6.9A, the hypothesis (*) is unnecessary.

J.L. Britton, The existence of infinite Burnside Groups

101

The main result for Generalized Tartakovskii Groups is: 7.1. Theorem 4. Let S2' satisfy the Generalized Tartakovskii Condi. R has a subword A such that there tions and let I f R E [ f i r ] Then is an element A'B of S2' where A E A' and

Proof. By 6.8, Theorem 3 is available. Hence R has a segment A (i.e., A is a term of a division of R ) and there is a partition A , , A , , ..., A , of J E S(S2') where v < 4 and none of A , , ..., A , is special. Let Yi be the left semiclosure of A i . As in 6.8,

,

1 - €0

< s( Y , Y , - . Y , ) <

s( Y , ) + ... + s( Y,) + (v - 1 ) f,.

Now if s( Yi) 2 l / p then A i is special (cf. 6.7): hence 1 - e o < s ( Y 1 ) + 3 ( l / p + e l ) . Therefore L ( Y , ) > 2, otherwise s ( Y 1 )< E , < l / p and

< € o + ( l - € o + € l ) - € , = 1. HenceAl = x 1 x 2 . - - x nn, 2 2. Y , is ( a ) 3 1 x 2 . . . x n - 1or (b) X1x 2 ... Yn and an appropriate E.R. split of J contains a subword (c)A'- X 1 x 2 - - . x n - , X nor (d)A' = 7 1 x 2 . . . x n - 1((d) . is the case if and only if x,, x , lie in the same patch.) Not both (b), (d) can hold. Hence Y , C A'. Now a subword of R has the form ~ 1 ~ 2 - . . ~ n - l of which a left subword A s A ' . We may suppose J = A ' B say. Then

~ n

102

J. L. Britton, The existence of infinite Burnside Groups

We have L ( A ) = L ( A ' ) 2 L ( Y 1 )> 2. To prove the last part of the theorem, first note that if s2" = 1 u ... u 1, then 5-2" satisfies the Generalized Tartakovskii Conditions: moreover if a,X , (3 is a partition of a split of an element of 1, and L ( X ) > 2 then, with respect to a'',

X is q-special in a,X , (3 * X is special in a,X , (3. Thus if s( Y i )2 1 / p o for some i = 2, ..., u then by 6.7 A i is q-special: hence A i is special: but this is a contradiction. Thus

and the required inequality follows.

I n this section we discuss equations of the form A - B = I and A . B .C = I in a Generalized Tartakovskii Group.

8.1. Theorem 5 . Let s2' satisfy the Generalized Tartakovskii Conditions. Let C = D in G n where C, D are not both I. Assume that if S is *-contained in C or D, i.e., S = S' for some subword S' of C or D, and i f S T E L i , 1 < i G n then

Then there are divisions C = P,,..., P,, D = Q , , __., Q,, and words X o , X , , ..., X , where X o = X , = I such that f o r j = 1 , 2 , ..., m. 1 . I # Pi = Qi, Xi = Xi-, = I. (In this case put ti = 0.) or 2. P;', X,?, , Q i , X i is a partition o f ' a split of an element Wi of 1 where 1 < ti < n. 'i

103

J.L. Britton, The existence of infinite Burnside Groups

ti is called the type of Piand o f Qi Moreover in 2 i f A denotes left semiclosure then, where

~(2i-l~) < l/p;,

l/po, l l p according as ti

> , = , < ti-1

< l/p;,

l/po, l / p according as ti

> , = , < ti+l.

A

s(Xi)

Similarly for right semiclosure. A

8.1A. Note. Since 1 - e0 < s(p;') we have

4

A

A

+ s(Xidl) + s(Qi) + s ( X i ) + 3E1

1 - e0

-

2/p - 3e1

1 - e0 - 2/p0 - 3e1

-'

if ti


if ti = n.

Proof of 8. I . We may suppose C & D. Let D F T-' B , C = A T where T is taken maximally. Then A . B = I . Next A . B = A Lc E B R ( E = 0 or l ) , and contains a natural element N . N *-contains a subword of L of size > 1 -e0 - 3 ( l / p + e l ) (if q = n replace p by p o ) . Hence N A L and N (# BR. Thus there is a division of N of the form A R , BL . There is a scheme as in Theorem 2 whose result R , , ..., R , is a proper strong partition of N , and such that no row has the form R,, R i + l ,... ( i = 1, ..., b). By 3.10 there is a division R J ,RI'of some R , and divisions R 1 , ..., R,-', Ri of A R : RI', R,+', .._,R , of B L . In the row containing R , replace R i by RI, RY. Let G ' be the group with generators the symbols of this scheme and defining relations the rows of this scheme. Let 0 consist of all C.A. and inverses of the defining relations. The scheme satisfies 1a, 2,3', 4,s and no row has the form ..., X , ..., X - ' , ... . Also any row consisting of Type C terms has at least six terms. The defining relations satisfy p(U, V )< 1 or U . V - 1 . Hence the result contains a subword E where EF E 0 and L ( F ) < 3.

8

J.L. Britfon, The existence of infinite Burnside Groups

104

We first show that B contains RI or RY. If not then without loss of generality E C R , , .,.,R i - l . E contains some Ri (1 < j < i - 1). Hence EF is a C.A. of the row containing Ri. Hence E does not contain Ri- or Ri+,. Hence E is Ri. F contains n o term of the result since L ( F ) < 3 . This means that A’ “contains more than” 1 - 3 / p - 3e1 - e0 of an element of a’, contrary to hypothesis. Next E contains both RI, RY : for if it contains only one then the same contradiction occurs. Hence R , ...Rip1F-’Ri+,...Rb= 1 in the group obtained from G ‘ by deleting the defining relation E F = 1, T h e left side could be empty but if not, n o cancellation occurs, and it contains C where CD E 0 ,L ( D ) < 3. Now C n F < 1 so C is not contained in any of

,

If C c R , ... R i p ]F-’ then A’ would “contain more than” 1 4/17 -4e1 of an element of 52‘. Hence C - R i - , F - l R i + l . Thus we see that the rows of the scheme are ~

With respect t o 52‘ consider any scheme as in Theorem 2 and let a , X and X - ’ , /3 be two distinct rows, say the ith and j t h rows respectively where X , X-’ is a pair of Type C. Say W iE L t , W iE L,. By the definition of a natural element (2.6) w(a.P) 2 s + t. Hence by 6.7 if Y is the left semiclosure o f X in a . X then (i)s(Y)> I/po * t < s (ii)s(Y)< I/p (iii) s( Y ) > 1 * t < s.

/pi

J.L. Britton, The existence of infinite Burnside Groups

105

Thus t > s implies s( Y ) < 1/ p i , t = s implies s( Y ) < 1/ p O and t < s implies s( Y ) < 1/ p . DefineAl,B1 byA = A , A R , B - BLB1.Then

Either A 1. B, = I or A , - B, contains a natural element N,. The theorem now follows by a simple induction argument.

8.1B. Corollary. In Theorem 5 , i f L ( C ) < 2 then C = D . 8.2. Theorem 6. Let A'. B'. C' = I in Gn where if S is *-contained in A', B' or C' and ST E Li ( i < n ) then 1 - 5 I p - 5 ~ ~- e O i f i < IZ 1 - 5 / p 0 - 5el - e0 if i= n

Then

1. there are divisions A ,, A , , A , of A ' : B,, B,, B, o f B': C, , C,, C, of C'. Also A , . B, = X , B,. C, = Y , C, . A = 2 in Gn for some X,Y , 2. 2. Either ( 2 a ) A , , X,B,, Y , C,, Z is a partition o f a split of an element o f i2' or 42b) A,. X . B, . Y . C,. Z = I. Moreover if (2b) hoids then either ( i ) A , , B, , C, all have length 1 and X = Y = Z =I, or (ii) A , = B , = C, = I. 3. For the equation A , . B, = X we have (unless A , = B, = I ) divisions A ; , = p , , ...,P k , B, = Q , , ..., Qk where for i = 1, ..., k either Pi = Qi and X i E X i = I , or P;' Xi:, Qi X i is a partition of a split o f an element o f a'.Also X, = I, Xi1 = X . Similarly for the other two equations.

,

-,

Proof. Take U maximal so that ( 3 V ) A' = A , U B' = VB, U - V = 1 in G (possibly U = I ) . V is uniquely determined by U ; if B' = V'B;, U . V' = 1 then V = V' = VJ say and J = I . Now B' 3 J : if J were not empty B' *-contains E where EF E i2'

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Thus A , . B , . C1 = 1. Take X maximal such that

(3Y)B,-BX,

C ' = YC,,

X.Y=1.

Then A . B . C, = 1. Take S maximal such that

Then

(2)

A ' = TA U, B ' = VBX,

C ' = YCS.

Note that if A = A"E, B = FB", E . F = 1 then E = F = 1 for A' = TA"E U, B' = VFB"X hence E = I : by the uniqueness F = 1 . Similarly for the pairs B , C and C, A . Cuse 1. A = B = C = I. HereA'=TU,B'= V X , C ' - Y S , U . V = X , Y = S . T = l . Case 2. Exactly one or exactly two of A , B, C are I. If two are I then the other = I in G hence = I. If A is I then B . C = I , hence B = C = I, a contradiction: similarly if B is I and if C is I . Cuse 3. None o f A , B, C is I. Case .?a. All of A , B, C have length 1 A ' = Tau,

B ' r VbX,

C'F YcS,

a.b.c-I

Case 3b. Exactly two have length 1. A .b .c = I,

A = c-l b-'

hence A

= c - l b-'

(no dot).

Thus we have the contradiction that cancellation occurs in A . b. Case 3c. At most one of A , B, C has length 1. Note that if A R . B . C L= 1 then C R . A L= 1. Hence CR = A L = 1 , A R is A and CL is C.

J. L. Britton, The existence of infinite Burnside Groups

107

Also note that A , B, C i s a partition (this is so if all of A , B, C have length Z 2 so there remains the case a, b,,

..., b,, e l , ..., en

(m2 2, y1 2 2).

Now a . b , * I , en . a $ 1 Suppose cn . a . b , = I . Then by 8.1 B = m = 1 a contradiction. Now A . B . C i s a non-empty relator so has a natural element as subword. N is either a subword of A ,B or C or has the form A R .BL or B R . CL or A R .B . CL. Hence N - A . B . C. By Theorem 2 and 3.10 there is a scheme S satisfying 1 a, 2,4,5. The result is a partition of N and has the form y1

where A *.'A,is a division of A :similarly for B and C. No row is ...X-l... and any row consisting of Type C terms has Z 6 terms. Let GO be the group with generators the symbols of this scheme and the rows as defining relations. Let 0 consist of all C.A. and inverses of the defining relations. If U, V E 0 then p( U, V )< 1 or U . V = 1. Consider A .-.A,B , -..Bs. If this contains E where EF E 0 and L ( F ) G 4 then by an earlier argument E -A,B,. Since L ( F ) < 4, F contains no result term. Consider (4) A , ...A , F - l B Z - - B S noting that

,

-,

in G' where G' is obtained from Go by deleting the defining relator EF. If (4) contains C where CD E 0 and L ( D ) < 4, i.e., L ( C ) Z X - 4 then since C n A , . - . A r has the form A j and moreover L ( A j )G X - 6 and since F-' n C < 1 we have C = A , F - ' B 2 . Consider ( 5 ) A . . * A r-2D-1 B , .-B, and note that

,

-,

-,

obtained from G' by deleting CD. Then by induction we have (6) A.C, C2-.-Ct= 1 in G i where A does not contain E where EF E 0 ,

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J.L. Britton, The existence of infinite Burnside Groups

L(I+')< 4 and either i = 0 and A is A , . . . A r B ," . B ,

or A is A , . . . A r _ i H - l B i + l . . . B s

and L ( H ) < 4. Combining these case A = A l . . . A r - i H - 6 i B i + l" - B ,

(ai = 0 o r 1 ). Consider B i + l ... B, C , ... Ct and similarly remove subwords E where EF € @, L ( F ) < 4, as long as possible to obtain either

Bi+l ... B, C , ... C,

or

B i + l ... B,-.i K-' Ci+l ... Ct.

Combine these cases: B i + , ... BSpiK-'i Ci+l ... C, ( j = 0 * E , = 0: > 0 +. ej = 1 ). Then

j

(7)

A , . . . Ar - I. l j - 6 i B i + l. . . Bs -1 . K-'jCi+l ... C, = 1 in G i + i

If Bi+, ... B,y_i is empty there may be cancellation between H - & i and K - 7 : if so we obtain after cancellation (8)

A , . . . A r - i ( H-.6i)L(K-'i)RC,+1...Ct = 1 in G i + i ________.

Now in case ( 7 ) or (8) the leftmost underlined word "contains at most" ( h -- 5) + 1 = h 4 and the same is true for the rightmost underlined word. The left side of (7) or (8) is either empty or "contains at least" X--3inwhichcasei+l Zs-jand (7')

A r -~ I.H-6i13i+,K-'iCi+lUE 0

or

(8')

A,.-, (H-'i)L(K-~i)RCi+lUE@

and ( 7"

A ...A 1

.

v-r~-l

Ci+2...C, = 1

in ~ i + f + l

in any case. In particular if ( 8 ' ) holds

Summarizing: If B i + , ... B,_i is non-empty (so i = s - j -

1)

J.L. Britton. The existence of infinite Burnside Groups

109

The right sides of (9), (1 0), ( 1 1) have length f 1. If Bi+, ... Bs-iis empty (hence i = s - j ) (9): (lo) hold

The right sides of (9), (lo), (12) have length f 2. Finally in the case when the left side of (7) or (8) is empty we have (9) and (lo) and A , . . - A , - i = 1 , s j + 1 - i - 1 = 0, ~

so that this case can be absorbed into a previous case.

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J.L. Britton, The existence of infinite Burnside Groups

CHAPTER I1 Burnside Groups

1.1. Introduction Let B: denote the Burnside Group with d generators and exponent e (d > 2, e > 0). The argument which we shall give in this chapter proving that B: is infinite for all sufficiently large odd e has the following formal structure. In 8 3 we state some axioms which are readily seen to be true when Y I = 0 (cf. 4.1 ). Assume they are true for some n 2 O.The first axiom implies that B i = I'(Cl, ..., C,, J n ) , where C,, ..., C, and J,* are subsets of the free product

n = (a1,...) a d l a ;

= 1 )...)u ; =

1)

and the notation r ( D ) , for a subset D of n, means the group obtained from II by adding the defining relations Ze = 1 ( Z ED). If J n is empty one easily sees that B: is infinite by using the Morse-Hedlund sequence ( 5 . I , 5.2). If J , is not empty one shows that the axioms are true for n + 1 : this occupies the main body of the chapter. Afterwards it is not difficult t o show that B; = I'(Cl, C,, C,, ...) : the infiniteness follows. (cf. 5.3) I .2. Remark. It was our intention t o derive an explicit numerical value for A such that B: is infinite for all odd e Z A (d 2 2), b u t this aim was abandoned in the course of the proof because of the labour likely t o be involved. However it will become clear t o the reader that an explicit bound is obtainable from the method of

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J.L. Britton, The existence of infinite Burnside Groups

proof, granted that the inequalities (which are almost all linear) are consistent.

1.3. Note: All considerations of consistency are postponed until Section 2 1 , although the reader may prefer t o consult Section 2 1 after reading each section (5,6, ...).

From now on, II denotes the free product

of d 2 2 cyclic groups each of order e . (Our methods would in fact apply to an arbitrary free product provided that no constituent group has an element of order 2.) We adopt the following notation and definitions.

2.1. If X , Y E ll we say X is a C.A.I. of Y if X is a cyclic arrangement of Y or of its inverse Y - ~ . 2.2. For any subset S of II, if A is a subword of B and B say A is a subelement of S.

E S,

we

2.3. Let X be an E.R. word. We say X i s a power if X = YS in I'I for some Y E II and s 2 2. (Then Y is E.R. and L ( X ) = s.L ( Y ) . )

2.4. If W E ABC and A

Z I, C 2 I we say B is properly contained in W. We sometimes write S- = T - t o denote that one of S, T is a left subword of the other: (and dually).

2.5. A point of a word W may be defined as a left subword X of W : it is a left end point (L.E.P.) if X = I, an interior point if I X 2 W a right end point (R.E.P.) if X = W. A point of a subword of W determines a point of W in an onvious way. The point X is

+

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J.1,. Britton, The existence of infinite Burnside Groups

left of the point Y if L ( X ) < L ( Y ) and we occasionally write < to denote “left o f ” for points.

2.6. When we say “distinct subwords S , , S,, ._.,S , of W” it will be assumed, unless otherwise stated, that for i = 1, ..., m - 1 1. L.E.P. Si < L.E.P. S i + l ,or 2. L.E.P. Si = L.E.P. Si+,and R.E.P. Si < R.E.P. S i + l .

2.7. Let A , B be non-empty subwords of W. If neither is a subword of the other then either W = X A YBZ or W F XDEFZ, where A is DE, B is EF and D, E, F are all non-empty: in the first case we say A . B are disjoint and in the second case we say A , B overlap. If A , B are disjoint and Y above is I we say A , B touch. We sayA is left o j ’ B if L.E.P. A < L.E.P. B and R.E.P. A < R.E.P. B . We d o not write A < B however. 2.8. The notation ( A , B ) where A , B are subwords of W means the subword of W generated by A , B , i.e., the smallest subword to contain both. Similarly for three o r more subwords of W.

2.9. I f S,, _ _S,n _ , are subwords of W such that each Si has a nonempty subword J i such that J i is disjoint from all S j , j # i, then the largest such J i is called the kernel of Si with respect t o S , , ..., S,. We also say that kernels exist fo r S , , ..., S , . 2.10. By a parameter we mean an explicitly given function of e (the exponent), not containing variables other than e. For example, W 4+ 6 . All parameters will be positive for all sufficiently large e. 2.1 1 . Let X be E.R, and let g > 0 be any real number. Then S,(X) denotes a subword of some power of X or of X - ’ of length < g e L ( X ) and e g ( X )denotes a similar subword of length > g e L ( X ) .

2. I 2. Cyciic (or circular) words. In II we have (i) if 19 W E II then W is a conjugate of J where J is E.R. or has length 1 : (ii) if A , B are E.R. and conjugate then A is a cyclic arrangement of B . A weak cyclic word is an equivalence class for the relation of conjugacy on n.

J.L. Britton, The existence of infinite Burnside Groups

113

A cyclic word is an equivalence class containing an E.R. element. Thus if X , Y are E.R. then ( X ) = ( Y )if and only if X is a C.A. of Y. A linearization of a cyclic word C is an E.R. word X such that

( X ) = c.

A word, i.e., an element of II may also be called a linear word.

2.1 3. A weak incyclic word is an equivalence class for the relation “ X is a conjugate of Y or of Y-’” on II. An incyclic word is such a class containing an E.R. element. If X , Yare E.R. then (X)‘ = (Y)’ if and only if X is a C.A.I. of Y. 2.14. Let W be E.R. A potential subword of W is either 1. a subword A , B, C of W (cf. 2.1 of Chap. I), or 2. a subword X , Y , 2 of W 2 such that

L ( X )< L(W),

L ( Y )< L ( W )

L ( 2 )< L ( W ) .

The value of the potential subword is B in case 1 and Y in case 2. In case 2 note that W = DEF, Y = FD, D F I, F $ I (also X = DE, = EF).

z

2.1 5. For each cyclic word C assume chosen a linearization Lin(C). By a subword of the cyclic word C we mean a potential subword of Lin(C). In the sequel we shall not distinguish explicitly between a subword of a cyclic word and its value. The concept of kernel may be applied t o cyclic words (cf. 2.9). 2.16. Translates. Let X be E.R. and not a power. Let rn 2 2 and ABC = X m . By the first translate of B t o the right we mean B’ where ABC = A’B’C’,L ( B ) = L(B’) and L ( A ’ )= L ( A ) + L ( X ) : it exists if and only if L ( A B ) + L ( X ) < L ( X m ) :denote it by B’. Dually, B - means the first translate t o the left. A translate of B is any subword obtained from B by applying operations Y + Y’, Y-. Y-.

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J.L. Britton, The existence of infinite Burnside Groups

Let S , , ..., Sp be subwords of X m for which kernels exist: we say the collection S , , ..., Sp is closed t o translation if for each i : if Y is a translate of Si and Y C ( S , , S p )then Y is Si (some j ) . The collection is properly closed to translation if in addition S; C (S,,S p ) :in this case a unique a > 0 exists such that Sf=Si+,

( i = 1 , 2,..., p - a ) ,

l
If X is E.R. and m > 2 the concept of translation can be extended to the cyclic word ( X m ) :if B is a subword then B' and B always exist.

2.16A. Proposition. Let S , , ..., Sp be subwords of ( X m )for which kernels exist and assume that the translate of any Si is an Si.Note that a = p/m is an integer. For any integer k define S , to be S, where k = r ( m o d p ) , 1 < r < p . Then SITis Si+af o r all i. L e t Y C ( X m )and let y be the number of S , , ..., Sp contained in Y. Then

Proof. Let u = y / a , X = L( Y ) / L ( X )and k = [XI so that k < X < k + 1 and X < m. Thus L ( Y ) 2 k L ( X ) . First suppose k >, 3. Then m 2 k 2 3 . Hence each Sihas length < 2 L ( X ) , otherwise St c Siu S:+. Hence y > 1 . Let S b + l , ..., S b + y be the Siin Y . The subword between L.E.P. S b + l and L.E.P. S b + l + ( k - , ) , has length ( k - l ) L ( X ) < ( A - I ) L ( X ) =L ( Y ) - L ( X ) henceSb-l+(k-l)aC Y . Thus .v 2 ( k - I)a - 1 2 ( k - 2)a. Hence u >, k - 2 > h - 3 . The same is true if lc < 3 since then k < 2 and h < 3 < 3 + u. Conversely, let [ ( y - I)/a] = k : then k < ( y - I)/a < k + 1. If y > 0 Y contains Sb+l, ..., S b + ysay. Now y - 1 2 ka and k 2 0. The subword between L.E.P. Sb+l and S b + l + k a has length k L ( X ) s o L ( Y ) > kL(X)andX> k>(-v-l)/u-l = o - l J a - l > u - 2 . If .y = 0 then u = 0 so again X > u - 2. Thus - 2 < h - u < 3 and the result follows.

2.1 7 . Lemma. Let S be a set with an eyiiivalence relation

- . Let

J.L. Britton, The existence of infinite Burnside Groups

115

D = (I, 2 , ..., a + b - d } , where a, b are positive integers and d = (a, b). Let Xi E S for all i>ED. Let X i Xi if i, j E D either i = j (mod a ) or i = j (mod b ) . Then X i -Xi if i, j E D and i = j (mod d ) .

-

Proof. Let q E {1,2,..., d } . Put a’ = a/d, b’ = b/d and A , = Xq+(r-l)d (r = 1 , 2 , ..., a‘ + b’- 1 . D ’ ={ I , 2, ..., a’+ b’- I}. Let i, j E D‘ and i = j (mod a’): then A j Ai.Similarly, if i, j E D’ and i = j (mod b’) then A i -Ai. Without loss of generality a # d, b # d,. a # b: hence a‘ > b’ > 1. For any integer p let Y p = A p where p = p (mod b’) and 1 < p < b’. Let s ED‘: then Y , A , since A , A , . Let k be any integer not divisible by b‘. Then k = xb’ + y , 1 < y < b’. Hence y + a’ E D‘ and Y , = Y y l = A y-Aytu’ Yy+u;- Yk+uc.

-

-

-

-

Let b divide h : then b‘ divides none of

a‘

+ h,

2a’

- - -

+ h, ..., (b’ - l)a‘ + h

-

so Yu’+h YZU’+h “ ’ - Y(bp-l)u,+h Yb’u’+h= Yh. Hence for all integers k, Y , Y k + u FAlso . Y , Y k + b ,hence : Y , Y k + lsince provided k, k + 1 ED’, and (a’, b‘) = 1. Thus A , A1 - A , - ...-Au,+b,-l and the result follows.

-

-

2.1 8. Lemma. I f X , Y E II and X Y = YX (where there are no dots) then there exist J E II,r Z 0 , s 2 0 such that X = Jr, Y = Js. Proof. This is easily proved by induction on L ( X Y ) .

2.18A. Lemma. Let X , Y be E.R. and X e = UA, Y e = UB where e’ Z 1, e“ Z 1. Let L ( U ) Z L ( X ) + L ( Y ) . Then ( I ) a, b exist such that X u = Y b ,a Z I , b 2 1 and (ii) a, b exist such that X = J b , Y = J a for some J E II. Proof. We may suppose L ( X ) Z L ( Y ) . Case 1. L ( Y ) divides L ( X ) . Since L ( U ) Z L ( X ) we have Xis a left subword of U hence of Ye”.Thus X = Y b ,b Z 1. Case 2. Not Case 1. Let V be the left subword of U of length L ( X ) + L ( Y ) . Then L ( X )< L( V )< 2 L ( X ) so V has the form X X L

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J.L. Britton, The existence of infinite Burnside Groups

and L ( X L ) = L ( Y ) . The left subword of V of length L( Y ) is XL and also Y . Thus V = ( Y , Y,)'Y,, a Z 1. Now XL is Y , Y , , Y is Y , Y,, Y , = J r , Y , = J s , r Z 0 , s Z 0. Therefore X = ( Y , Y2)'-' Y , = J ( r + s ) ( a - l ) + r Y, = J r + S , Y , $1, Y , I hence J f I, r 2 1, s 2 1 . Also a 2 2 otherwise L ( X ) = L ( Y 1 )< L( Y ) .

+

2.18B. Lemma. Let C be E. R. and let s be a positive integer. Let Cs 3 XYC. Then X = J', C J' f o r some J E n, t Z 0 , u Z 0. Thus i f C is not a power (cf. 2.3) X = Ct . Proof. X = C" C, where u Z 0 and C = C , C, .,Hence X C C"C, C, C, so C , C, = C, C,. Therefore C , = J r , C, = J s and the result follows. § 3.

Let J , be the set of all E.R. words. For any subset C of ll let r ( C ) be the group

{ a l ,..., adla: = 1, ..., a: = l , Z e = I ( Z E C ) ) Then the Burnside Group B: with d generators and exponent e is

U J ,).

We shall now state some axioms? : immediately afterwards we shall begin t o consider their meaning in an informal manner and discuss their validity when n = 0, 1.

Axiom 1 . There are subsets C,, ..., C,, J,, J , , _..,Jn of II. B: = r(C,, _ _C,, _ , J n ) : C,, ..., C,, J , are pairwise disjoint: C,, _..,Cn are non-empty: J , 3 J , 3 ... 3 J , :J , consists of all E.R. words: Ci c J p l (i = 1, ..., n). Put Gi = r ( C 1 ,..., C i ) , G , = n. If X,Y E II write X 7 Y if X equals Y in G,. Axiom 2. There are relations 7 (i = 0, 1, ..., n ) o n II: ; is =.

t The author hopes that the reader will bear with the unusual numbering of the axioms.

J.L. Britton, The existence of infinite Burnside Groups

117

Axiom 3. There are relations % ( i = 0, 1 , ..., n) on the set of all cyclic words: is the equalit; relation (=, say). Axiom 4. ( X I F ( Y )implies x

7 U-’

. Y . u for some U.

3. I . Definition. Let X , Y E J , . Then X E , Y means (3s, t ) ( X S ) ( Y k t ) , 1 < s < t. Xp”, Y means that some a f ( X ); some eq( Y ) (cf. 2.1 1): here f’ and q are parameters: they are regarded as given, and fixed from now on, even though their explicit definitions are not given at this moment. Similarly for all subsequent parameters. ((L4 1) XB, Y means that some Sf(X) ;some eqI( Y ) ((4’)) X p , Y means X E , Y or X p , Y X o , Y means X E , Y or X u , Y X >, Y means there exist X,, X I , ..., Xiin J , ( j > 1 ) such that (i) X , is X , Xiis Y : (ii) X , a l X 1 a 2 X 2... aJ. XJ. -where each ai is one of E,,,P,, 6,:(iii) ( 3 i ) a i = u, * ( 3 p ) aI.+?. = p , , p > 0. Since q q‘ (this will follow from the explicit definitions of q, q‘ when given), we have X , u, X , u, X , ... u, Xi- p, Xi. ((4 > 4 ) )

3.2. Definition. For k = 1 , 2 , 3 , ... LE consists of all cyclic arrangements of X k e and of X - k e , X € C, . L , means LA. Axiom 5. Let X , , X,, X,, ... be an infinite sequence in J,. Then not X i u , X i + l ( i = 1 , 2 , ...,). Axiom 6. If 0 < x < 1, certain elements of II are called n-bounded by x . Every element of Il is 0-bounded by x . If X is n-bounded by x then so is X - ’ and every subword of X . Informal remark. Usually,

< x < 1.

Axiom 7. X ,yl Y implies X ; Y . Axiom 8. ( X )

=

n-1

x

( Y )implies ( X ) ( Y )

3.3. Definition. A cyclic word is n-bounded by x if every linearization is n-bounded by x .

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J. I>. Britton, The existence of infinite Burnside Groups

By the notation (n)-bounded by x we mean i-bounded by x (i = 1, 2, ..., 1 1 ) : with n - 1 instead of ri we write ( ( t i - 1))-bounded to avoid ambiguity. Axiom 8a. If X e # I in G,,, X is E.R. and ( X ) is (n)-bounded by z, where z > i then for any t 2 1 ( X t ) is (n)-bounded by z + a g . Here u8 is a parameter but z is a variable. ( ( 4 3 1)

Axiom 9. X E J , if and only if X e # I in G , , X is E.R. and (X)is (12)-bounded by str. (Here str is a parameter.) ((S tr)) Put str + = str + u g . (W+N

A x i o ~ n10. I f X is (??)-boundedby b_ and X

;I then X

I.

((b-1)

Axiom 1 1 . z is an equivalence relation o n the set of cyclic words (Ii)-bounde$ by b2

(x) r ( Y )implies (xk) = ( y k )( k z 1).

Kb2N

A x i o m 1 1 a. If W E II there exists W' such that W' ;W and W ' is (ii)-bounded by bmin. ((bmin)) Axiom 1 1 b. Let W e # I in G , . Then there is an E.R. word K such that K is conjugate t o W in G , and ( K ) is (??)-boundedby cb *,I n . ((Cbmin)) Since ~ b< str, ~ any, such ~ K is ~ in J,, . ((cbmi, < str))

A x i o m 1 2. Let 11 > 1 . For any positive integer k there is a nonempty subset of n: its elements are E.R. and it is closed to C.A. and inverses. If 1 < r < 11, 1 < s < I?, L,k n L', f then r = s and 1 = k . L t = L f . L,, = L:z. Also

Li

Axiom 13. If A B E L f ( 1 < i < n, k 2 1) there is defined a real number sAB( A ) such that 0 < ,sAB ( A )< k : if A I then sAB ( A )= 0 : i f B I then s A B ( A ) >k - 2/e:sA-lB-1( A - ' ) = s A B ( A ) .

J. L. Britton, The existence of infinite Brrrnside Groups

119

3.4. Definition. An element of II is a powerelement of 1 if it belongs to some L f: :A is a subpowerelement of L , if it is a subword of a powerelement of L,. If A is a subpowerelement of L , its nsize sn ( A ) is given by

taken over B, k such that A B

E

lf:

Axiom 14. sn ( A ) is finite. Axiom 15. If ABCD E L f: then sBCDA( B )<

(ABC).

Axiom 16. I f A B C E Lf: then

Axiom 16a. If ABC E L i then

Axiom 17. If L ( A ) = 1 then s A B ( A )< e2 = 1/2e. Axiom 18. If A B

E

L i and s A B ( A )< h , then

((€2

((h0 1)

Axiom 20. Let A X E L,, B X E L,, t < n, s < n : say t < s: then (a) and (b) below hold. (a) If t < s then (i) ~ ~ ~ ( ~ ) < r o r ~ - ~ - ~ = ~ ~ ~ ~ ~ - ~ i n C E 1, ((rN

J. L. Britton, The existence of infinite Burnside Groups

120

and (ii) s X R ( X ) < rb ( b ) If r = s then ( i ) sxA (X)< ro and sXB( X ) < ro or (ii) B . K ' = 1 in G t - l .

Axiom 2 1 . Let n > 1. Let 0 < x < 1. Then W is n-bounded by x if and only if any n-subpowerelement X contained in W satisfies St'(X)

< x.

3 . 5 . Definition. A is an n-urelement if it is a subpowerelement of L and sn ( B )> ro for every subpowerelement B of L , containing A . A is n-normal if it is an n-urelement and a subelement of L,. Axiom 22. If W E n, each n-urelement S contained in W is contained in a unique maximum n-urelement S contained in W . I f X is E.R. and no C.A. of X is a subpowerelement of L , then each Iz-urelement contained in ( X ) is contained in a unique maximal n-urelement contained in ( X ) .

o<

Axiom 23. I f A B E L E and o < a < s A B ( A )ando-ff , then for some left subword A L of A , a < sAB (AL) < p.

> E4, ((€4))

Axioiri 24. Let ABC be a subelement of L,, sn(A),sn(C)> ~ 2 4 and A , C ((n - 1))-bounded by b < b2, . Then there is a subelement ALB'CR of L n , ((n 1 ))-bounded by b + q 2 4 such that ABC' n = ALB'CR -1 ~

.\'*(AL ) > srf( A

-~

24,

sn(cR> s"(c) - ~ 2 4 , ((s24 b24 q24, ~ 2 4 9

I.P(ABC) -,P(ALB'CR)I < € 2 4 , s n ( B ' ) < s n ( B ) + d 2 4 .

-

€ 2 4 d24)) 9

Axioin 25. Let A be an n-subelement of n-size > ~ 2 5 A, Y, P p < I I and let A , Y be (p)-bounded by b 2 5 . Then Y contains an nsubelement of n-size > sn(A)- ~ 2 5 . ((s253 b 2 5 , y25))

3.6. Definition. If X , Y are E.R. then ( X )

( Y )means that there

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J.L. Britton. The existence of infinite Burnside Groups

are cyclic arrangements of X , Y that are equal in G, (thus relation on the set of all cyclic words).

is a

Axiom 26'. If (X) ( Y )where both cyclic words are n-bounded by b i 6 , and ((n - 1))-bounded by b 2 6 , , then any maximal n-normal subword U of ( X ) , where s n ( U ) > c 2 6 ' , determines a maximal nnormal subword Im U of ( Y ). ((b;6' b 2 6 ' '26)) ?

r

Axiom 26. Let ( X ) be ((n - 1))-bounded by so and n-bounded by 1 - 2 2 6 . Let M,, ..., M, ( k 2 1 ) be a set of maximal n-normal subwords each of n-size 2 z > 2 2 6 . ((SO, 226 >) If k = 1 assume ( X ) contains a maximal n-normal subword of n-size > 2 2 6 different from M,. Then there are cyclic words X,, X,,X , ((n - 1))-bounded by so + s 2 6 , so, so respectively and n-bounded by 1 - z + t26 such that ((s26 t26 (X),rl x,; X, ; X,. If U is maximal n-normal in (X)of size > w 2 6 then there are maximal n-normal subwords U , , U, , U , of X,, X,,X , respectively such that Im U = U , , Im U , = U,, Im U2 = U , and Im U , = U , Im U, = U , , Im U , = U,. Finally, ((w26)) if Uis anMi then I s n ( U ) + s n ( U 3 ) - 1 I < r26 ifUisnotanMj t h e n I ~ ~ ( U ) - s ~ ( U , ) I < r , ~ . ((726 1) We say that X, arises from ( X ) by simultaneous replacement of MI, ..., Mk. 7

Axiom 26". If V is maximal n-normal in X , of n-size > u26 then V is of the form U , in Axiom 26: thus U exists in (X)of size > w 2 6 and U , , U, exist as in Axiom 26. If I/ is maximal n-normal in X 2 of n-size > u26 then Vis of the form U , in Axiom 26. ((u26 1) Axiom 26-. If X , arises from ( X ) by simultaneous replacement of M , , ..., Mk (using X , , X,) then X;' arises from (X-') by simultaneous replacement of M i ' , ..., M i ' (using X i ' , Xi1). If U' = U-' is maximal n-normal in ( X - ' ) of size > w 2 6 then the corresponding words Ui,U i , U; of X i ' , X i ' , X i ' are Ui' ' U-' 2 3 U-' 3 . 3.7. Definition.

(X); ( Y )means that (X)is (n)-bounded by so

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J . L . Britton. The existence of infhite Burnside Groups

anti either ( X ) = ( Y ) o r ( Y )arises from ( X ) by simultaneous replacement o f some set of maximal n-normal subwords each of size > 1.'. ((r' )) Thus ( X ) + ( Y ) implies ( X - ' ) ; ( Y - l ) . A vioiii 26A. Let ( R A ) ; (SB),where R is max. n-normal o f size > 1.' and S is its (iterated) image as in Axiom 26. Let ST E L n . Tht.11 there is a cyclic word H such that (RA) ; H and any lineari-

iation of H is conjugate in G,

-1

t o T-' . B .

Axiom 26L. (Linear version of A x i o m 26', 26,26", 26A.)

( i ) Axiom 26' holds if X , Y are linear words and X ; Y ( i i ) Axiom 26 becomes: Let X be ((n- 1))-bounded by so and ri-bounded by 1 - z26. Let M,,..., M , ( k 2 1 ) be a set of maximal n-normal subwords each of size > 226. Theiz there exists Y , unique mod G,, I ( ( 1 1 ~-1 ))-bounded by s o and n-bounded by 1 -226 + t26 such that X ;Y . If U is maximal ??-normalin X of n-size > w 2 6 then i f U i s a n hi,,I s N ( U ) + s" (Im U ) - 1 I < r26 if IJis not anM,, I . s ~ ~ ( U ) - S ~U)l< ( I ~Y , ~ . (iii) I f I/ is maximal n-normal in 1' of size > u26 then I/ has the form Im U for some U . ( i v ) I f / l , R / l , ; B,SB,, w h e r e s n ( R ) > r ' a n d S = I m R and S T E L , then ( 3 H ) A , R A 2 ; H , f l B , . T - l . B,. ~

3.8. Note: The definition o f X 1.7.

2

Y for linear words is similar t o

3

Axiom 27. If X is an wsubpowerelement and s n ( X ) > then X contains a subword of the form UUU, where U is E.R. Axiom 28. If ( X ) X ( Y ) where the cyclic words ( X ) ,( Y ) are ( 1 2 ) hounded by h 2 8 , a28 respectively then ( X ); ... 7 ( Y ) .Similarly for linear words: if X ;Y where X,Y are (I?)-bounded by b,, , a28 ... + 1 Y. respectively then X ; ((b28 a2$3)) If W is a cyclic or linear word (n)-bounded by bi5 then there exist cyclic or linear words respectively such that 9

J.L. Britton, The existence of infinite Burnside Groups

123

Axiom 28A. If X Y , where X , Y a r e (n)-bounded by c28 and no linearization of X o r Y is a subelement of L u ... u L , then x: Y. ((C28)) Axiom 28B. Let X be a cyclic word, (n)-bounded by d2* and such that n o linearization is a subelement of L u ... u L,. Then Y exists such that Y is (n)-bounded by str, X Y and if X = 2' for some 2, s then Y = T S for some T. ((d28))

:

Axiom 29. Let X E J,-l, Y E Cn and X > Y. Then either some h f ( X ) contains a subelement of L, of s i i e s 40 or

( 3 s , t , U ) X Sn-= 1 u-l.yt.u,U E r I , l < s < I t l . Axiom 30. Let n 2 2. Let P be a subelement of L , and sn(P) > ~ 3 0 . Let P be ((n - 1 ))-bounded by b30. Then P contains subelements X , , ... , X , of L n P 1 as follows. ( ( ~ 3 0 ,b30)) 1. There exists PQ E L , where spQ(P)> s"(P) - l/e2. 2. There is an integer t > 0 such that e divides t. 3. Either (a) each Xi is maximal ( n - 1)-normal in PQ of size > Y*, or (b) the X i are disjoint and have size > 4 " . If (b) holds then P contains n o (n - 1)-subelement of size > c*. Kr*,q*, c*)) 4. If P' C P and those X i contained in P' are X i , ..., Xi then

where if P = XP' Y then Q' = Y Q X . Also

Axiom 3 1 A. If X ; ;Y then certain pairs E, K E II are called end words f o r X ; Y. Also X ; E . Y . K. Axiom 3 1 B. Let T ; ;Y with end words E, K , where p < n - 1. Let T be an n-subpowerelement, s n ( T )> s31 and let T, Y be (p)-bounded by b < b 3 1 .Then there is a word T L F Y R (p)-bounded by b + q31

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J.L. Britton, The existence of infinite Burnside Groups

and equal in G, t o T . K - l (= E . Y ) . Also Y R , TL are not subelements of L1 u ... U L, and s n ( T L )> s n ( T )- ~ 3 1 . (Dually Y LE'TR = Y . K = E-' . T. ) ((s31 b31 q319 '31 1) 9

9

Axiom 3 1C. Let U ;V with end words E, K , where U, V are (n)bounded by b < f 3 1 . Let U e (S,T ) where S, T are distinct maximal ti-normal subwords of U of size > g31.Similarly, let V = ( S ' ,TI). Then there is a word S L A T r R ,(n)-bounded by b + h3, such that SLA T'R = E . V. Moreover S L ,TIR are n-normal words and s'*(SL)>"slz(S) - j 3 1 , s n ( T r R> ) s n ( T ' ) - j S 1 . SL,TIR are contained in distinct maximal n-normal words S , , Ti and sn(SL)> sn(S1)- ~ 3 s ~ ( T ' R ) >~ n ( T ; ) - ~ 3 1 .

1 ,

( ( f 3 1 r 831 h 3 1 , j31 a31)) 9

Axiom 32'. L , is contained in 1, Axiom 33. There is a subset Rep(L,) of L,. I f A A ' and KS33 1) BB' c Rep(L,), A nrlB and s n ( A )> s33 then A = B and A ' = B'. Axiom 33'. If CAB is a subelement of L, and s n ( A )> t33, ((t33)) then there is a subelement CALEZFARB of L, equal to CAB in G, where Z is a subelement of Rep(L,) and s'*(Z)> s"(A) - T , ~ . Also @33 1) 1. AL and AR have n-size > u33 2 Y; and < w3,. ((a33, w33)) 3.I f A is ( ( 1 1 - I))-bounded by b < b3, then ALEZFARis ( ( 1 1 - l))-bounded by b + ~ 3 3 . ((b33, c33)) 3 . E and F have rr-size < e3,. ((€33 1) 4. A L E , FAR have ri-size < q 3 3 . ((433)) Axiom 33". I f ABC is a subelement o f L, and A , C are subelements of Rep(L,) of size > rn33 then there is a subelement ADC of Rep(L,,) equal t o ABC in G n P l . If further XABCY is a subpowerelement of L, then so is X A D C Y and I s n ( X A B C Y )-sn(XADCY)I

<~

3 3 .

((m33

7

P33 1)

Axiom 33. Let W be a subelement of L , of size > d33. Let W be ((n - 1 ))-bounded by I;, and W = WLDWR,where the right side n-1

J. L. Britton, The existence of infinite Burnside Groups

125

is ((n- 1))-bounded by g3, and W L , W R have size > h,, . Then WLDWR is a subelement of L, . w 3 3 833 J

r;,

t

!

h33))

Axiom 33’. Let ABCD E L, where A and C have n-size >j3,. As in Axiom 33’ let A = ALEZFAR, C = CLE’Z’F’CR.Let ZH, Z‘H’ E Rep(L,). T<&I’ZH is a cyclic f k h g e m e n t of Z’H’. ( ( j 3 , ) ) Axiom 39. Let U = ( Z , , Z , ) be ( ( M - 1))-bounded by b < b,, and n-bounded by big, where Z,, Z 2 are subelements of Rep (L,) of size > s,,.Let Z , , Z2 be contained in distinct maximal n-nw-ma1 subwords Z , , 2, and let s ( Z i )> s ( z i ) - d i 9 (i = 1,2). Let 2,T i 1 E Rep(L,). Then U . Z i l T , n-1 = ZbHT?, where the right side is ((n - 1 ))-bounded by b + c , ~ ,and s(Zk) > s(Z ) - d3, , s(T2R ) > s ( T 2 )- d,,.Alsg-Z), TF are contained in distinct maximal n-normal subwords Zk, TF and s(Zk ) > s ( Z t ) - f39, s(TF) > s(T!) =f39. Finally ZkHT; is n-bounded by Max (bj9 1 - s(Z,)) + 8 3 9 . @39 big s39 4 9 c39, d 3 9 9

9

7

Y

9

>

f39 839)) 9

Axiom 40. N o element of L n has the form J-lAJB where sn(J) > ~ 4 and 0 J is ((n - 1))-bounded by b4,. Ks4, *

640))

§ 4.

From now on we shall suppose that the statement of each axiom has been prefixed by “Let n 2 0” or “Let n 2 1 ” as follows: n 2 0, for Axioms 1 , 2 , 3 , 4 , 5 , 6 , 8 a , 9 , 10, 11, 1 la, 1 lb,31A. n 2 1, for all remaining axioms. 4.1. Lemma. The axioms are true when n = 0.

Proof. Axiom 1 reads “There is a subset J , of n: B: = r(J,) and J , consists of all E.R. words.” This is so by the beginning of Section 3. Axiom 5 will follow if we prove that X CJ, Y implies L ( X ) > L( Y ) . If X E , Y then s L ( X ) = t L ( Y ) > s L ( Y ) :if X O , Y then f e L ( X ) > q‘eL(Y)2 f e L ( Y ) since q‘ >f. ((4‘2 f 1)

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J.L. Britton, The existence of infinite Burnside Groups

Axiom 1 1 reads ''= is an equivalence relation on the set of all cyclic words." This is so. In Axiom 3 1 A if X ; Y , that is, X Y , the only pair of end words shall by definition be I , I.

4.2. hiote, The rernainder of Section 4 consists of an informal discussion of the axioms when I I = 1. This is logically not necessary, but it is hoped that it will help the reader.

4.3. 11 is easy to show that

2 is transitive and not reflexive, hence

it determines a partial ordering on J , (e.g, take n = 0 in 5.5 below). Put X i n C , if X E J , and

Y E J, implies not X

2Y .

Then C, f @.Let k > 1 : put Y in L t if Y is a C.A.I. of X k e for some X in C,. L , means L : . Put G, = r(Cl): thus G = I I / [ L , l . Put L : = L: and L = L , : if A B E L: we say, as in the axioms that A is a subpowerelement of L or a 1-subpowerelement. If X E L: f) L : thzn k = 1 (e.g., taken n = 0 in 5.1 5 below). If A , R E I,, then A . B = I or P(A,B ) < q L ( B ) :to see this take I I = 0 in 5.16 below. Thus G, satisfies the Tartakovskii conditions of ('hapter I, since q < 1 / 5 . ((4 < 115)) I t ' A R E L: put s A n ( A )= k L ( A ) / L ( A B ) .Note that L ( A B )Z 2ke. 'Ihe following "size" axioms hold clearly: 13, 14, 15, 16, 16a, 17. Axiom 1 X is true since h , < 1 : Axiom 20 since q < Y, and Axiom 23 since e4 2 1 / 2 e . Axiom 19 is true if 1 = k = 1 since r, 2 y: the general case can be reduced t o this by considering a suitable A and using Axiom 18. ((h, < 1 ,q < Y o , €4 2 I / 2 e ) ) As in the axioms we have the concepts I-urelement, 1-normal, 1-bounded (ct'. Axiom 21). We prove axiom 22. Consider a word A B C where B is a 1-urelement and AB,BC are 1-subpowerelements, say A B X = Z k e ,Z e E L then B X A = Z;", Zy E L and RCY = T e t L Since s ( B ) > Y, we obtain Z , = T Hence LJXABC'Y = Tck t h ) e . Thus A B C is a 1-urelenient. Axiom 22 now follows. I f '4 is a 1-subelement and s(A ) > Y , + e3 then A is 1-nornial (e.g. take I I = 1 in 5.22 (iii)).

,

,.

J.L. Britton, The existence of infinite Burnside Groups

121

4.4. 1-Replacements. Consider a word X Z Y where Z is a l-subelement, say ZT-' E 1 Then X . T . Y is said to arise from X Z Y by replacing Z : this concept is only useful if s ( 2 ) 2 ro + e3 : then, sZT-1(2) 2 ro so by Axiom 20 if also ZT'-' E then T = T' so

X.T.Y

X.T'.Y.

4.4A. Proposition. Let x 1 = ro + c 3 + e2 + 4/e, x 2 = 1 x l , - ro + x l , x4 = 2e3. Let X Z be I-bounded b y c < x 2 and let Z x3 be maximal I-normal in XZ. Let W' be the result o f replacing 2 in XZ. Let ZT-1 E 1 1. Then W'= X L E T R where L ( E ) < 1, L ( X L )2 L ( X ) - 1 , L ( T R )2 L ( T )- I , T R is I-normul and ~

-

I s ( T R ) + s ( Z ) - 1 I < x4.

W' is 1-bounded by Max(c + x 1 - e 3 , 1 + x4 .- ~ ( 2 )Finally, ). if V is maximal 1-normal in XZ ( o r W ' ) and not Z (or p)and its size is y > x 3 then it determines a maximal 1-normal subword V' of W' (or X Z ) where Is( V )- s( V')I < x l . Proof. We have

hence L ( T )2 2, say T = aS. Next

s s z - i o ( S ) = ~ T z - l ( T )I -/ L ( T Z - ' ) > I - x 2

--

1 / 2 e > ro + e 3

hence S is 1-normal, and so is T. Now W ' = X.aS. No cancellation can occur since Z is maximal normal. If n o amalgamation occurs, W' = XT. Take TR = T, X L = X . TR is T since there is n o cancellation in X - 2 . Now

and

s ( T ) + ~ ( 2- )1 2 0 2 -x4

J. L. Britton, The existence of infinite Burnside Groups

128

If amalgamation occurs X = X'b, W' = X'cS, c = b . a. Take T R = S. X L = X ' . Then ?R; is S and

Note that in any case s(TR)< 1 + x4 - ~(2). Suppose U is normal of size y but is necessarily maximal normal. U n Z is not normal so it has size < ro < x, hence U 4Z. Write U = U , U , U , where U , = U n XL, U , = U n Z, L(U2)< 1. Then

hence U , is normal. Also s ( U 1 ) Zy - e2 - ro - 4/e = y - x5, say. Similarly if U C W ' . Thus W' is 1-bounded by Max(c + x 5 , 1 + x 4 - s ( Z ) )< 1. If U above is now maximal normal then we have y -x5

< s ( U , ) < SJ(u,)+ e3 < sJt(U,)+

E,

(for some J , J ' )

so s(U,)>y-x5

-63

=y-XI.

4.4B. Corollary. Let W = ( Z , , Z,) be I-bounded by c < 1 - 2r, - 3.5, - 2e2 - 8/e where Z , , Z2 are distinct maximal 1-normal of size > 2ro + e3 + e2 + 4/e. Let Zi T;'E 1 (i = 1,2). Put W" T , Z i ' . ( Z l ,Z , ) .Zi'T 2 . Then W" (T;, T i ) where TY, T i are distinct maximal normal, TI'- = T-, -TI' 3 -T, I s(Z i)+ s( Tl!')- 1 I < ro + 3 ~ +, e2 + 4/e

(i = 1 , 2 )

and W" is 1-bounded by Max(s(T;), s(Ti), c + 2(ro + e3 + 4/e)).

Proof. Replace Z , in W t o get W' = ( T i , Z;), say. Then I s ( T ; )+ s(Z ,) - I I < x4 and W' is 1-bounded by Max(c + x1 - E , ,

J. L. Britton, The existence of infinite Burnside Groups

129

1 + x4 -s(Z,)) = c'. Since ~ ( 2 ,>) x 3 , Is(Z2)-s(Z;)l< x,. Since c + x1 - € 3 < x2, 1 + x4 - ~ ( 2 ,<)x2 we may replace Z; to get = (Ti, T i ) where Is(2;) + s(T;') - 1 I < x4 and is 1-bounded by Max(c' + x, - e3, 1 + x4 - ~ ( 2 ; = ) )c". Since s(T;)> 1 -s(Z1)-x4>x3wehave Is(T:)-s(T;')14x1.Thus I s ( 2 i ) + s(TY) - 1 I < X I + ~ 4 .

w

w

4.4C. Simultaneous 1-replacement. Take a cyclic word W = ( Z , A , -.-Z,A,) where s > 2 and the Zi are 1-normal, say Zi T;' E L Let J = {p,, p2, ..., Or ) be a non-empty subset of 1 , 2 , ..., s . Let Qi be Ti, if i E J and Zi, otherwise. We wish t o investigate the weak cyclic word (Q1.A,- ... . & . A s ) arising by simultaneous replacement of the Z . This investigation is easily reduced pi to 4.4A and 4.4B as follows: we give a sketch only. We may write

,.

where Qs+l means Q , and the method is to show ( 1 ) Q j . A i . Q j + l = Q"BiQF+, (2) '(Q,R,,) + L(Q;+i) > L(Qj+l)NOW(2) implies say Q,+, DEF, @+, = EF, Q:+, = DE so 1 . L = E = Q M say. Hence W' = (QyBlQ";'2*..Q~Bs) Q;+1 Qi+lwhich is cyclic. To prove ( 1 ) is trivial if i, i + 1 @ J , while if i, i + 1 E J we may use 4.4B. If i E J , i + 1 c2 J then Q i . A i .Qi+, Ti-Ai.Zi+, and we may use the dual of 4.4A. If r < s assume without loss of generality that Q1 is Z,. Then W, W' have the common subword thus W - W' (cf. 3.6). W' for some Wo.$ut If r = s we show W - W

QY:

1

Oi

then W W,. Now let W, arise from W, by replacing Z y . Then W'. W,- W,. But W, i: W', hence W -1 W,

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= Y in C , . By 8.1 of Chapter 1 , if X and Y are suitably 1-bounded, there are divisions P,,_..,P,, of X and Q1, . _ . ,Q, of Y (we shall simplify this discussion by regarding terms in a division as subwords), and words X,, ..., X , such that X =- X,= - I and for each i either (i) Pi = Q j ,X i - , 2 X i = I and t i = O o r ( i i ) P ; l X ; - l , Q i X i t L l a n d t i = 1. If ti = I , Pi i s 1-normal. Clearly if we simultaneously replace the Pfsuch that ti = 1 i n X we obtain Y . Hence (or directly) a maximal normal subword of X or Y of suitably large size determines a maximal normal subword of Y o r X,respectively.

4.5.Suppose X

,

4.6. We now consider the axioms in turn, for IZ = 1, and comment on those axionis not already covered by the preceding discussion. For Axiom 1 we must first define J , . I f W is E.R. and W e # I i i i G, consider the set of all E.R. words conjugate t o W in G, and choose an element W' from this set such that ( W ' ) has as small a I-bound as possible, sayf'(W). It can be shown that SupJ'(W) is approximately i . Thus if str > c'bmi, 2 Supf(W) then Axiom 1 1 b i s true for I I = 1 and J , may be defined as in Axiom 9. Axiom 1 now follows. In Axiom 2, may for the purposes of this informal 1 discussion be taken to the equality in G , . In Axiom 3 , should be thought of as a transitive relation similar in meaning t o "linearizations are equal in G , " (which is not transitive). Axiom 5 is the only axiom that we shall not attempt to justify here. Next we prove Axiom 8a. Let N C ( X r ) ,t >, 2, where NT- Y e € L , ands(N)> If L(N) Z 2L(X) then by 2.18A, X , 3 J u , Y = J T for some C.A.I. X , of X . Now T = 1 by the definition of C,. Hence X e = 1, 1 1 contradiction. I f L ( X ) < L(N) < 2 L ( X ) then N n N' is not normal, otherwise A' LJN' is normal and has length > 2 L ( X ) so the previous case applirs. Let N = A n where B = N n N'. Then A c ( X ) so d,Y) < l i e + I' + Y, as required. If L(N) < L ( X ) then s(N) < z as required. Axiom 10 simply expresses the fact that in the Tartakovskii

-

5.

7

J. L. Britton, The existence of infinite Burnside Groups

131

Group G I , X = I implies that X contains a subelement of L of 1 large size. Axioms 24,25 are true, trivially. Axiom 26' follows from 4.5. In Axiom 26A, H is obtained from (SB) by replacing S. Axiom 27 is obvious. Axiom 28 follows from 4.5, at least in the linear case. Next we prove Axiom 29. We have X = A ', 01 X , ... aiXi = Y as in 3.1. If ai = E, for all i we obtain ( X u )= ( Y ' ) for some a, b where a < b. Now consider the opposite case. Recall that L ( X i ) > L(Xj+I). We first show that if D is a word of the form 6 f ( X i + 1 )then D also has the form af ( X i ) . If (Xi")= (Xl?ll)then, enlarging s,t if necessary, D'l is a subword of the right side and hence of the left side. If S,(X,) _= E ~ , ( X ~then + ~D'lis ) a subword of the right side since 4' f + 1 /e. ((4' 2 f + 1I.)) Next if C has the form eq ( X i ) and if (Xi") (Xl?:l) then C has the form eq ( X i + l ) . Combining these two results we obtain S , ( X ) = eq( Y ) :but cq ( Y )has size > q. Axiom 31B is trivial. For Axiom 31C, assume U = (S,T ) = ( S ' , T ' ) = V where S, T, S ' , T' are maximal 1-normal. 1 We indicate how to prove that there is a word S L A T I R = U = I/. Let us say that S has property X if S is pi, ti = 1 , that'is, the maximal normal determined by Pi (cf. 4.5). I f T does not have property X then -T = -TI: take (S, T ) for the required word. If S does not have X then S - = S'-: take ( S' , T ' ) .If S, T have X let U" arise from (S,T ) by replacing T : say U" = ( S " , TI'). Then S- = S"- and -T" z -TI: take U" for the required word. In Axiom 33 define Rep(L1) to be L , . The next four axioms are trivial. Axiom 39 follows from 4.4A. For Axiom 40, we obtain from Axiom 20 that B J - A~ J .J - AJB ~

=I

Hence A . A z I . Since e is odd, A J-'AJB, a contradiction.

= I , hence cancellation occurs in

((e odd))

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J. I,. Britton, The existence of infinite Burnside Groups

§ 5.

Take three symbols a, b, c and let s1s2s3-.*be a Morse-Hedlund sequence for a, b, c: i.e., each si is a, b o r c and there d o n o t exist i, p > 0 such that sisi+, ... , s ~ + ~ - equals , S ~ + ~ S ~ +... ~ si+2p-l + , cf. Leech [2].In t h e f r e e p r o d u c t I I = ( a l , . . . , a d : a ; = . . . = u $ = l ) d > 2 , l e t A = u , a 2 ,B = u , u ~ C , = “ : a 2 . Let Smn(u,b, c) denote .s,,,+~s,, +2..-s,when 0 < m < n and put X,, F S , , ( A , B, C ) .

5.1. Proposition. X,, X I I.

does not have the form U X X X V where

Proof. Suppose not. Then X is E.R. Say X = x 1 x 2... x , , n 2 2. Let Y = x 2 ... x n x , . Then X 3 contains Y 2 . Either x or x 2 has the form a : : let 2 be X or Y accordingly. Then 2 = a:’ a:u2‘s, n = 2s. X,, contains 22 = u(A,B, C ) u ( A ,B, C): since u ( A ,B, C ) - u(A,B, C ) we have u(a, b, c) = v(a, b, c), a contradiction.

5.2. Theorem. G,, is injinite. Thus i j J , is empty then B: is infinite. Proof. It is sufficient to show that n o two of X,,, X,,, Xo3, ... are equal in G,. If this were not so, then Xr, ;I for some r, s 1 < r < s. By Axiom 10 X,, is not (!?)-bounded by b , so for some p , 1 < p < n, X,,contains S such that sP(S) 2 b , 2 i (Assume b , > .) cf. Axiom 2 1 . By Axiom 27 S contains UUU where U is E.R., in contradiction t o 5.1.

5.3. Remark. The main effort of the remainder of the paper consists in showing that if the axioms hold for some n and if J , is nonempty then the axioms hold for n + 1. Afterwards it will not be difficult t o deduce that, if J , is non-empty for all n , then B: = r ( C , , C,, C,, ...>.We now prove that the infiniteness of Bf; follows from this. If a finite group G is given by a finite number of generators and defining relations R i = 1 ( i = 1 , 2 , 3 , ...) then there exists h such that R , = 1, ..., R , = 1 are defining relations for G. Thus the finiteness of B i would imply Bf;= r ( C , , ..., C,) = G , for some 1 1 : hence B: is infinite.

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5.4. Proposition. Let X,, X 2 , X,, ... be an infinite sequence in J,. Then not X, >,, X, >, X, 2 *-a.

((Assume q 2 q'.))

Proof. If X p n Y then X un Y. The result now follows from Axiom 5 .

5.5. Corollary. >,, determines a partial ordering on J,. Proof. It is transitive, Suppose X >n X: then X contrary to 5.4.

>n

X

>, X >,, .'.,

5.6. Definition. Put X in Cn+, if X is in Jn and

Y E Jn

=$

notX>. Y .

5.7. Proposition. (i) Cn+lis non-empty. (ii)IfXEJ,\C,+, then ( 3 Y ) Y E C n + l , X > n y .

+,

Proof. Let X E J,. Either X E Cn or X >n X, for some XI E J,. Either X, E Cn+l or XI >, X2 for some X , E J n . And so on. By 5.4 either X E Cn+,or X >n X,>n ... >n X p E Cn+lfor some p 2 1 . 5.8. Proposition. Let Y E C,+,, n 2 1. Then ( 3 2 ) 2 E C,, Y >n-l 2.

Proof. Cn+lc J, c Jn-l. Since Y E J,, Y$! C,. Thus Y E JnPl\Cn. Hence Y >n-l 2 E C,. 5.9. Proposition. (i) Let Y E Cn+,. Then not Y p n 2. (ii) Let Y E Ci, 1 < i < n, 2 E C n + l .Then not Y ui-l 2.

Proof. (i) Suppose Y p n 2. Then Y >n 2, contrary to Y E C n + l . (ii) Suppose Y ui-, 2. Then Y ,2 E Jj-,. Now 2 C J , C Ji so 2 $ Ci. Thus 2 E Ji-,\Cj. Hence 2 >i-l T E Ci. Since Y E Cj we do not have Y >i- T, hence 2 = 2, E j - l 2, ... E j _ 2, = T, say. If 0 < j < k then (2;)iFl ( Z J t r l )for some s, t depending on j such that

,

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.l.L Britton, The existence of infinite Burnside Groups

< s < f , so Z,!

Z,?:l are conjugate in G i P 1 .Hence for some s, t Z s , 7 + rare conjugate in GiP1 hence in Gi. Now T E Ci so T e 7 I . Hence Z e S= I i n G j hence in G,. But Z E J,: this contradicts 5.12 below. 1

5.1 0. Definition. Let h- I . Put Y E L : + ~if Y or y-1 is a C.A. of X k e for some X E C,+].Put L,+, = LA+1 and G,+l = r'(Cl, ..., C,+l). Because Note that if Y E L ; + ] then each C.A. of Y is in of this we shall occasionally consider the elements of Lk,+l as cyclic words .

5.10A. Informal Remark. We are interested primarily in L,+l and k subelements of L,+l: L,+l and subpowerelemeiits enter only incidentally, for technical reasons.

5.1 1 . Proposition. I f ' Y E Lk,+l then Y is (n)-botinded by str+. Proof. Let X E Then X E J, so by Axioms 8a and 9 any power of X is (u)-bounded by sty+.

5.1 2. Corollary. I f X E J,, and s 2 1 then X s # I in G,; hence G, contuiiis elevneiits of infinite order. Y E L : + ~implies Y I in G,

+

Proof. Let X E J,. If X s ;I, s > I , then since X s is (n)-bounded by .sty+ < h , we have by Axiom 10 that X s F I , a contradiction ((str' < b,))

5.1 3 . Corollary. (i) I f W"# I in G, then W has infinite order in G,. (ii) l f ' e is odd then G, has no elements o f order 2. Proof. ( i ) By Axiom 1 1 b some conjugate of W in G, lies in J, so by 5.12 has infinite order. ( i i ) Let X 2 = I . By (i) X e ;I , so if e is odd X ;I .

5.14. Proposition. Let X , Y E J,. (i)+f X p , - i Y then X p , Y .

(ii) I j ' X o,-, Y then Xo,, Y.

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Proof. By Axioms 7,8.

5.1 5. Proposition. Let A E Ci. B E C j and (A").g ( B b )where a, b # 0 1-1 and 1 < i < j < n + 1. Then i = j and la1 = I b I. Proof. Let i < j . Now Be = I in G j hence in Gi- By Axiom 4 Aae = I in GiP1 contrary t o A E JjTl. Let i = j . Then I a I = I b I otherwise A E , B or B Em A where m=j-1.

5.16. Proposition. Let A E L i , B E Li, 1 < i < j (i) I f i < j then p(A,B ) < q ' L ( B ) . (ii) I f i = j then A . B = I or P(A,B ) < q L ( B ) .


Proof. We have B = Y e = UT, A-' = X e = US where L ( U ) = P(A,B ) , and some C.A.I. X ' , Y' of X , Y belong t o Ci, Ci respectively. Let L ( U ) 2 L ( X ) + L ( Y ) . By 2.18A we have Xu= Y h for some a, b > 1. Hence (X"") % ( Y l k b ) .By 5.15 a = h and i = j . Hence X = Y and A . B = I. Let L ( U ) < L ( X ) + L ( Y ) and i < j . If 02 y ' L ( B ) = q'eL( Y ) then L ( X ) > L ( Y ) , so /3 < 2 L ( X ) < f e L ( X ) . ( ( f e 2 2 , q'e 2 2 ) ) Hence X o , Y , so X ' o , Y ' and X ' U ~ Y-' ~contrary to 5.9 (ii). Similarly when i = j , by 5.9 (i).

5.1 7. Proposition. I f AB is a subpowerelement o f 1 then

s n ( A B )< s n ( A )+ sn(B)+ 2/e Proof. Immediate from Axiom 16.

5.1 8. Proposition. Let A 3 E L t and sAB(A)< h , - e 3 . Then s n ( A )= sup sAc(A) over all C such that A C E 1,. Proof. Let A D E 1:. If sAD(A)2 h , then s n ( A )2 h ,

> r,

so

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Hence s A D ( A )< h,. By Axiom 18 there exists A Q in 1 , with s,,(A) = S,,(A). 5.19. Proposition. Let A , B be subpowerelements of L,, s ” ( A )> r, and A C B . Then s n ( A )< s n ( B ) + e3. Hence if A ,B are subpowerelements of L , then A C B implies ,?(A) < Max(ro, s n ( B ) + e3).

Proof. By Axioms 15, 19

5.20. Proposition. Let X

E

1n . Then 1 - 2/e < s n ( X ) < 1 + e3.

Proof. By Axiom 13, 1 - 2/e < sx(X)< 1. Now use Axiom 19. 5.21. Proposition. I f t < s and X is a subelement of 1 and o f 1, then s s ( X ) < r b . Proof. Say X A E L , , X B E L,. By Axiom 20 s,(X) Let X Z E 1,.Then sxz(X)< rb. By 5.18, , s s ( X )< rb.

< rb < h ,

-

e3.

((rb < ho - €3))

5.22. Proposition. (i) I f A is an n-urelement then s n ( A )> r,. (ii) I j A is an n-ure!ement, B is an n-urelement and B contains A thtvi B is an n-iirelement. (iii). I f s n ( A )> r, + e3 then A is an nurelement. Proof. (iii) Let B C A . By 5.19 s n ( A )< s n ( B )+ e3 hence s”(B) > r,. 5.23. Proposition. Let AB be an n-subpowerelement. Then s n ( A )+ s n ( B )- s n ( A B ) is less than 2e3 + ET, r, + 2e3 or 2ro + 2e3 according as both, one or neither of A , B are n-arelements.

i.

Proof. Say ABC E 1 Then ,rn(A)< sABc(A)+ e3 and s n ( B )< sBCA( B ) + e3 in the first case and the result follows by Axiom 16a. In the second case say A is an urelement. Since A C A B and s n ( B ) is less than r, + e3 the result follows by 5.19. The third case is true since -sn(AB) < 0.

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5.24. Proposition. ( i ) If A is an n-urelement and for some B, k A B E 1; and s A B ( A )< h, then A is n-normal. (ii) I f A is n-normal then s n ( A )< 1 + e 3 . Proof. (i) By Axiom 18. (ii) holds since sAB(A) < 1 and by Axiom 19. 5.25. Proposition. Let A be n-normal. If A B and A C are in 1 then B = C i n G n - l . Proof. We first show that there exists A D E L , with s A D ( A )2 r,. For if not, then s A B ( A )< r, < h, - €3 so by 5.18 r, < s n ( A ) = supsAQ(A)overAQ E 1 , . Axiom 2 0 B = D and C = D in G n - l . 5.26. Proposition. Let r, + e3 < b < h, - E ~ Then . the following are equivalent 1. W is n-bounded by b. 2. If U is an n-urelement contained in W then s n ( U )< b. 3. I f U is n-normal and contained in W then s n ( U )< b. Proof. Clearly 1 implies 2 and 2 implies 3 . Assume 3 and suppose 1 is false. Then W contains X where s n ( X )2 b. We show no left subword X L of X satisfies b < s n ( X L )< h,. For otherwise X L is a subelement of L n by Axiom 18; if X L is an urelement then it is normal so by 3 we have a contradiction: if X L is not an urelement then s n ( X L )< ro + e3 < b. Thus in particular s n ( X )2 h,. Take E such that b < b + E < h, - e4 Then Y exists such that sxY(X) > h, - E. Thus 0 < b < h, - - E < s x Y ( X ) and h, - E - b > e4. By Axiom 23 b < s x Y ( X L )< h, - E < h, and we have a contradiction. 5.27. Proposition. Let W be n-bounded by b < h,. Then a subword K of W is an n-urelement if and only if it is n-normal. Each n-norma1 subword o f W is contained iul a unique maximal n-normal subword. Let X be E.R. and let no C.A. o f X be a subpowerelement o f

138

J. I.. Rritton, The existence of injinite Burnside Groups

L,. Let [ X I he n-boianded by b < h,. Then each n-normal subword of' [ X I is contained in u unique maximal n-normal subword.

Proof. By Axioms 18 and 22.

5.28. Proposition. Let X be E. R. and f o r all s > 1 let [ X s ]be nhoirnded hl: h,. Let [X'I contain R where S " ( R ) > 2(ro + e3) + 2le. Thcri L ( R ) < 2 L ( X ) . Proof. .s"(R)< h , hence R is a subelement of L,. By 5.22, R is an urelement hence normal. We may suppose X' begins with R : X t = RT. For any integer N > 3 consider X t N and let the translates of R in it be R , R', R", ... . Assume L ( R ) > 2 L ( X ) . Then R meets or touches R", so R' c R u R". Let R n R' E E : then R' = EF, F C R". At least one of E , F is normal otherwise. s(R +)< s ( E ) + s ( F ) + 2/e < 2(r,

+ e 3 )+ 2/e.

If F is normal then so is R' n R". Since R' n R" = R n R' we see that i n any case R n R' is normal. Hence the unique maximal normal subwords containing R , R' coincide. Thus the subword of X t N generated by all translates of R is normal. Hence so is ( R T ) N - ' R . By repeated use of 5.23.

Since s ( R T )2 s ( R ) e3 we contradict the boundedness of X" taking N sufficiently large. ~

by

5.28A. Nott'. Let A B C E n where A B and BC are subpowerelements and B is an urelement. Then A R C is a subpowerelement (and an ure I e me 11t .)

Proof. A B is an urelement by 5.22. By Axiom 2 2 , = AB. Similarly 13 = BC. Hence B contains A B and BC so contains ABC.

5.29. Proposition. Let A he an n-subpowerelement and let P, Q be

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subwords o f A where sn(P),s n ( Q )> rrzax(sn(A)- a , ro + e3). Let s n ( A )> 2a + 4e3 + E ; . Then P, Q meet and sn(P n Q ) > min(sn(A)- a Hence if ro >, a

-

ro - e3 - 2/e, s " ( A )- 2a

- 3e3

~

€;

- 2/e).

+ 2e3 + ET then

sn(P n Q ) > s n ( A )- a Proof. I f P, Q are disjoint then A

- ro - c3

~

2/e.

CPDQE say. Then

s(CPD) + s(Q.E) < s ( A )+ (2e3 +.eT)

< s(P) < s(CPD) + e3 s ( A ) -a < s(Q) < s(QE) + e3.

s ( A )- a

Adding, 2s(A) - 2a < s ( A ) + 4e3 + e:, contrary t o hypothesis. The result is trivial if one of P, Q contains the other. Now let P, Q overlap say P = RS, Q = ST. If T is not an urelement then s(A)- a < s ( Q ) < s ( S ) + (yo + e 3 ) + 2/e. If T is an urelement then

s ( R S ) + s ( T ) < s ( R S T ) + (2e3+ e ; ) s ( A )- a s ( A )- a

s(RST)

< s(RS) < s ( S T )< s ( S ) + s ( T )+ 2/e < s ( A )+ € 3

Adding, s ( A )- 2a < (2e3+

ET) + e3 + s ( S )+ 2/c.

5.30.Corollary. In the hyp'othesis of 5.29, delete s n ( Q )> sn(A)--a and replace s n ( A )2 2a + 4,; + E ; by s n ( Q )2 a + 4e3 + E : . Then the conclusion of 5.29 holds if s n ( A )is replaced by s n ( Q )+a. 5.31. Proposition. Let HUVWK be a subelement o f L , o f size u Then

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J . L . Britton, The existence of infinite Burnside Groups

s( V )2 s ( U V )- yo

or s( V )

s( V W )- y o

-

e3 - 2 / e

-

e3 - 2 / e

or s( V ) 2 s ( U V ) + s( VW) - a

-

3e3 - E ;

-

2/e.

Proof. If U is not an urelement then

Similarly if W is not an urelement. Now let U, W be urelements. Then

Hence s ( U V ) + s( V W ) < s( V )+ a + 3e3 +

ET + 2 / e as required.

5.32. Remark. Let 1 < s < n. I f W = ABC, where A B is s-bounded by u and BC' is s-bounded by c, and if B is not a subelement of 1 (or more generally if no s-subelement contained in W properly contains B ) then W is s-bounded by Max(a, c). (For if N is any subelement contained in W then N c A B or N c BC or N properly contains B.) §6 Informal Summary In this section we show that Gn is a Generalized Tartakovskii Group and hence derive some properties of it. I n particular if A = I!! in G, and S is a subword of A one can find, under suitable conditions, a corresponding subword of B (by making use of 8.1 of Chapter I).

6.1. Lemma. (i) L , , ..., L n satisj) the Generalizing Tartakovskii

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141

conditions. (ii) I f R = I in G,, R f I then R contains a subword C, L ( R ) > L ( C ) > 2 and ( 3 v , D ) CD E L,, 1 < u < n,

Proof. (i) We prove that the Generalized Tartakovskii Conditions hold if we take e0 = &, = 2 / e , l/p = r + e 5 , l/po = ro + e 5 , 1/pb = rb + e 5 , where e5 is a (new) parameter to be determined. LetAXIE L,,X;'BE L , a n d X 1 ~ X 2 , A X l - = E ~ , ~ ~ l B ~ y F , P = x E . Fy (cf. Chapter I, 6.2). First let t < s and sXIA (X,) > 1/p. ((€5 > 2E2 + 4 / e ) ) Then since 1/p > e2 we have L ( X 1 )Z 2. Hence XI = c1Ud,, X 2 = c 2 U d 2 , x E = d 1 A c l U E L t , F y = U-'c,'RdZ'E Ls.Thus

r < r + es

-

2e2 - 4 / e < s x l A ( X , )- 2e2 - 4 / e < S u d l A c l ( U ) .

in Gt-l where Q E L,. By Axiom 20 P = x E . F y = V . Q . In a similar manner we deal with the case t < s and s X - l B ( X i l )> l/pb and with the case t = s and s X I A ( X , )2 l / p o . ((rb + 2/e + e2 < ro < r)) FiAally we have ((r< 1/5-e5-el)) e l + € , + I / p b < l/po< I/p, 1 > 5 ( l / p + e l ) (ii) By Theorem 4 of Part I, R contains A , A s A ' , A'B E L, for some u , 1 < u < n and S ~ , ~ ( > A 1' ) 3 / p - 4e1 (if u = n replace p by Po)- Now

so in particular L ( A ' ) > 2. Thus A' = c'Cd', A cCd. Let D s d'Bc'. Then CD E L,. Now S ~ ' ~ ( 1 - 3 / p - 4e1 - (26, + 4 / e ) as required.

6.2. Lemma. Let X , Y be equal in Gn and not both I. Let each of X , Y be ((n - l))-bounded b y k , and n-bounded b y ky where k, = 1 - 4 ( r + e 5 ) - I4/e-2e2

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and k: arises from k2 b y replacing r b y yo. Then there are divisions P , , ..., P,, and Q 1 , ..., Q, o f ' X and Y respectively with the properties listed iri 8.1 o j Chapter I. Proof. Let ST E L i where 1 < i < n and let S be *-contained in X or Y . If L ( S ) < 1 then s S T ( S )< € 2 < 1 - 4 / p - 5 ~ 1 <1 - 4/po - 5 E 1 . I f L ( S ) 2 2 then S = cS'd and S' C X or S 'C Y. If i < n then s , ~ ~ (< S 2e2 ) + 4 / e + s'(S') < 2c2 + 4 / e + k 2 = 1 - 4 / p - 5E1. Similarly if i = n .

6.3. Definitions. Let A ;B, where not both of A , B are empty. Let A , B be ( ( n - 1))-bounded by k , and n-bounded by k i . Let P , , ..., P,, and Q 1 ,_ _ _Q,,, be divisions as in 6.2. Consider any Pi of non-zero type. Let Pj! be the largest common subword of Pi and A and similarly define Qi. A point of A is called left nice if it is the left end point of A o r of some Pj or if it is an interior point of some Piof zero type. We define left nice points of B similarly and note that they are in one t o one correspondence with those of A . Dually we have the concept of right nice. Call a non-empty subword S of A nice if its left end point is left nice and its right end point is right nice. Note that there will be a corresponding nice subword of B ; denote it by S ' . The notation S ; S' means that there exist A , B, P , , _ _Pm, _, Q , , ..., Q , as above for which S, S' are corresponding nice words. 6.4. Note. If A ; B where not both of A , B are empty and each is ((11 1 ))-bounded by li, and n-bounded by k: then A ; B. ~

6.5. Note. In 6.3, note that the division P I ,..., P, o f A induces a division of S and similarly a division of S' is induced. We shall define l e f t and right nicc points and nice subwords of S with respect to these induced divisions exactly as in 6.3. For any subword T of S, T is nice in S if and only if T is nice in A. 6.6. Note. IfS ; ;S' and T , T ' are corresponding nice words (of S, S' respectively) then T ; T ' .

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-

6.6A. Definition. If X ; Y with divisions P I , ..., P,, Q 1 ,..., Q, and p < n , we say that X ; Y induces A P B if A , B are corresponding nice subwords (of X , Y respectively) and for j = 1 ..., rn if Pi has type ti # 0 and Pi' C A then ti < p. (Note that if Pi has type ti # 0 then either Pi is disjoint from A or Pi' c A ) 6.7. If Pi has type ti # 0 then Pi f LPjM, Qi = NQ; 0 where L , M , N , 0 have length 0 or 1. Put X!:: = L - ' . X I1-1 1 . N' X J! = O . XJ. . M - l . Then P j - l X ! - ; Q j 4 E lti. With the assumptions of h 3 let S, S' be corresponding nice words. We shall define the end words E, K for S , S ' , with respect t o the given divisions P I , ..., P, and Q 1 ,... Q, , as follows. If the left end point of S is the left end point of Pi put E = XI!:: ; otherwise put E = I . If the right end point of S is the right end point of some Pi' put K = Xi'; otherwise put K = I. Thus in all cases S 7 ES'K. Moreover if S = A B , S' = A'B' and A , A' are corresponding nice words (end words E l , K , say) then B, B' are corresponding nice words (end words E,, K , say) and

E,

= E,

K,

= E i l , K, = K

6.7A. Proposition. With the assumption o j 6 . 3 let Pi have type ti (i = 1, ...)m). Then f o r i = 1, ..., rn either XI!:: = I or

according as ti is >, = or < t i - l ; similarly XI!= I or sD(Xi!)satisfies (*) according as ti is > , = or < ti + ; here

If ti # 0 then Pl!,QI are ti-normal and

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where L:‘= f,!Xl!-lQ:.-lX,!-l,F r Q;XiP,!-lX;I: and X i , Xi-l have ti-size less than r6 if ti < n, r t i f ti = n, (Here k,, k!, r 6 , rz are new parameters with values as in the proof below.) Finally if’j = ti # 0 then

and siinilarly fiw QI A

x.

Proof. I f ti< ti+l then s w ( X i ) < l / p for an appropriate W in the is a subword of X,! and in fact notation of Chapter I, 8.1. Now Xl!I H i i K where H , K have length < 1. Hence s D ( X i )< 2e2 + 1/ p + 4 / e , that is, sD(Xl!)- 2e2 - e5 -- 4 / e < r. Similarly if ti = ti+lor ti > ti+l.Next 1 - 2/e < s E ( f l ! + ) s F ( Q i )+ s c ( X l ~+~sD(Xf) ~) + 6/e, hence sE(P;)+ s,(QI) > 1 - 8 / e - 2(r + 2e2 + e5 + 4/e) = k,. Let lc: arise from the expression for k , by replacing r by ro. Then if ti = 11 we may replace k , by ky . If s ( X i ) > r,, then s ( X I ) < sD(Xl!)+ E, < (r + 2e2 + e5 + 4 / e ) + e3 = r 6 . This remains true if s(Xf) < y o . (Let rz arise from the expression for r6 by replacing r by y o . ) Finally if j = ti < 11 then

s’(Pi)

+ k , > s’(P;) + s’(QI) > k ,

hence ,s’(P;) > k , - k , j-normal. Similarly if j = 1 1 .

> ro + e,.

Thus Pl! is

((k3 - k2 > yo

((k! - k;

>

+ €3)) +€3))

6.7B. If P I , ..., f , , , is a division of S then, if ti # 0, we denote the o r simply maximal ti-normal subword of S determined by Pi by Pi. 6.8. Proposition. Let A C X ; ; Y , where p < n, A is a subelement o f L, und s ( A ) > u1 = 2(rb + E,) + 8/e. Then A 2 A‘AOA‘’and 1= A’ Y M f o r some Y M ( Y M C Y )and

P

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145

Proof. Say X = P,, ..., P,, Y = Q 1 , ..., Q,. Any point of A other than its right end point is either the left end point or an interior point of some Pi. By 5.21, any common subword J of A and a Pi of non-zero type satisfies s n ( J ) < rb. A subword K of A of length 1 satisfies s n ( K ) < e 2 . Using 5.1 7 the resu’lt follows.

6.9. Note. We say A’ arises from A by “deleting partial P’s’’. We shall always assume that A’ is chosen maximally. More generally let X ; Y and let B be any subword of X . If (3C) I C C B, C is a nice subword of X then the union of all such C is nice in X : we denote it by Bo. The corresponding nice subword BO’ of Y is called the nice image of B. If A c B C X and A , , Bo exist then A0 C Bo. 6.10. Proposition. Let A C X ; r Y , where p < n, A is a subelement o f L n and s n ( A )> u3 = 2(rb + e2) + 8/e + ~ 2 5 Let . X,Y be ( p ) bounded by b25. Then A 3 A 0 ;r Y M3 B where B is a subelement o f L n and s n ( B )> sn(A)- u4 where u4 = u1 + ~ 2 5 . A n y such B is called a weak image of A . Proof. By 6.8 A 3 A0 ;r Y M .By Axiom 25 s”(A0)> s n ( A )- u1 > ~ SO

s n ( B )> s n ( A )- ~1

2 5 ,

s n ( B )> s”(A0)- r25

-~25.

6.10A. Note. Let X nyl Y where X , Y are ((n - 1))-bounded by b25. Let Y be n-bounded by x where u3 < x < h , - e2 - 2/e - u4. Then X is n-bounded by x + u4. Proof. If not then A C X , s ( A ) 2 x + u4. Let A’ be the largest left subword of A such that s(A‘) < x + u4. Define c and AL by

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J. L. Britton, The existence of infinite Burnside Groups

= A [ c . Then 2i3 < x + 114 < s ( A L )< s(A') + e2 + 2 / e < + u4 + E, + 2/e < h,. By Axiom 18, A L is a subelement of L n By 6.10, Y contains an n-subelement of size > s ( A L ) 114 > x,

AL .x

-

which is a contradiction.

6.1OB. Note. 111 6.1 1,6.12,6.13 below we shall assume that X , Y are n-bounded by h, (cf. 5.27).

-

6.1 1 . Proposition. Let A C X Y , where p < n, A is a subelement of L, and s'(A) > 215 2 u 3 , wiere u5 is a new parameter. Let X , Y be (p)-bounded b y b,, Thus b y 6.10, A 3 A , ; Y M3 B, where s " ( B ) > sn(A)- u4 > u5 - u4. Let J C Y Mand s n ( J ) > u5 - u 4 . Then J , B are n-normal and determine the same maximal normal sztbword M of Y. In particular M is independent of the choice of B so it may be denoted b y Im(A). Also s n ( I m ( A )> ) s n ( A )- u4 - e3.

Finally, A is normal.

Proof. Since s ( B )> u 3 ,B

3

BO ; ;X M 3 C and

((us 2

2u4 + u3 1)

s ( C ) > s ( B ) - zl4 > s ( A )- 2u4 > us - 2u4. Also

J > J o - K,> K > Ko-J,>.I, s( K ) > s ( J )

-

and

u4 > u5 - 2u4,

.x(J2)> s ( K ) -1114. Note that K and X M are contained in A . Now s ( C ) > Y, so s ( X M ) > s(C) - e3 > s ( A )- 2u4 - e 3 = $ ( A )- a , say. Use 5.30 taking X', K for P, Q respectively. ((u5 - 2u4 > y o ) ) We have

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-

Now K n X M 3 KO n X M J , n Bo 3 N, where s ( N ) > a - u 4 > r 0 + e3. T h u s N C J , C JOcJ , N c B o c B. SinceN is normal and contained in J and B, the result fdlows.

6.1 2. Proposition. Let A C X ; Y , where p < n and s n ( A )> u 6 _ Let X , Y be (p)-bounded b y b25.Then I m ( I m A ) exists and is A , the maximal normal subword o f X determined b y A . Hence if A is maximal normal I s(Im A ) - s(A)I < u4 + e3. Proof.

B is normal and B = Im A . s(B) > S(B>- e3 > u6 - u4 - e3 > u5 (parameter condition number 1 ). Hence B 3 Bo X M 3 C, C = Im 3,S(C)> s(B) - u4 and S ( C )> u6 - 2u4 - e3 > u3 (P.C. 2). By 6.10, C 3 CO- Y m 3 D,S(D)> S ( c ) - U 4 > u6 - 3 U 4 - E 3 > Y o (P.C. 3 ) and therefore s ( Y m )> s(D)- e3. Thus we have s ( Y m )> s(g) - 2u4 - e3. Apply 5.30 taking A , P, Q, a to be B,Y m , B, 2u4 - e3. For this we need u6 - u4 - e3 - 2u4 e3 > ro + e 3 (P.C. 4), u6 > 3 U 4 + 3 9 + €; (P.C. 5), Y o 2 € 3 €; + 2Ll4 (P.C. 6). Hence

-

s( Y

nB )

> s ( B )- ro - e3 - 2 / e

> u6

-

u4 - ro - e3 - 2/e = ul

Now Y m n Y M3 Y m n B so if u7 > ro (P.C. 7 ) then s ( Y m n Y M )2 u7 - e3 > ~ 2 5(P.C. 8). By Axiom 25, Y m n Y M Co n A o IIN where s ( N ) > u7 - ex- ~ 2 > 5 ro + e3 (P.C. 9). Thus N is normal. But N C Co C C C C and N C A0 C A , so = = Im(Im A ) .

-

c

6.1 3. Proposition. Let X F Y where p < n and X , Y a r e ( p ) bounded b y b25. Let A , B be distinct maximal n-normals of X o f size greater than u6 and A be left of B. Then Im A is left of Im B.

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Proof. We have

B

3

BO

- BO'

3

T, T =Im B.

s,

s

By 6.12, # T : but Tare maximal so neither can contain the other. Also neither of S, T can contain the other. Now the left end points of AO, Bo satisfy L.E.P. A0 < L.E.P. go;also R.E.P. A o < R.E.P. BO. Also L.E.P. S < L.E.P. BO' otherwise S is contained in Bo' and S = T. Hence S is left of T. If T were left of F w e would have T C S n T and hence, since T is normal 3 = Thus 3 is left of T. 6.14. Proposition. Let X ; Y with decompositions P , , .. . , Pk and Q1, ..., Q k . Let X , Y be ( ( n - 1))-bozinded b y k , and n-bounded b y k 5 . If1 < j < m < k a n d ti = t , = n t h e n q # p, and Qi # Q,.

ai#

Proof. It is sufficient to prove Q,. We shall do this by assuming Qi = Q,, deducing that Pi = P, and afterwards obtaining a contradiction from this. Thus the subword ( Q i ,QL)5 QSEQL say of Y is a subelement of L,, say Q)EQ;,SE 1,. By 6.7A, 5.25 and since

belong to L,, we get EQLS = X!PI-lXI-l and I I

1-1

S Q ~ E = X I , P ~ l X ~ !in, Gn-,.

Let QI . = NQI.0, I P, P.I = LPIM, I

= L'P,!,,,M', Q , = N'QLO'.

The subword (Pi,P,!,,,)of X is equal in Gn-l to

(Xl!-lEO' -m S X1-1 ! )-lMPi+l. ... . P m - l L ' ( X & l S Q ~ E X ~ - l ) - l

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where the right side is a subpowerelement of 1, by 5.28A. Qj-' Xi-1 Pi is a subelement of L , so by Axiom 24 it is equal in GnWlto ( Q j - l ) L X ' f P i R which , is an n-subelement and is ( ( n- 1))bounded by k4 + q24 : the hypothesis of Axiom 24 is satisfied since k4 < b2, (P.C. I), k! < s(Pi) + s(Qj) < k, + s(Q') implies s(Q') > k! - k, 2 ~ 2 (P.C. 4 2). Moreover, $(PiR)> s(Pi) - ~ 2 > 4 k! - k5 - ~ 2 > 4 ro + e3 (P.C. 3) so PiR is normal and therefore not a subelement of L 1 u ... u Ln-l (cf. 5.21). Similarly for Pl,Xk' Q k l , QLSQ); in particular they are equal in Gn-l t o say Pl,LZQ;lllR, Q Z S f Q i Rrespectively. Thus

where the right side is an n-subpowerelement and each side is ((n - 1))-bounded by k4 + 924 < b25 (P.C. 4) since the two underlined I words are subwords of X , Y respectively. Now let A = Q;-1LXffPIR and B = PALZQ;11lR.By Axiom 24 and 6.7A s(A), s(B) 2 f > u6 (P.C. 5 ) . With respect t o the above equation, Im A , ImB both coincide with its right side, hence by 6.12 A- B. Thus (PIR, P m' L ) is a -I PiR, PhL are normal so Pi " ' = Pl,L and subpowerelement. Now - Pi = P,. Now where (Pi, Pl,) = Pi KPl, we have s(P;) + s( KPI,) < k, + (2e3 + E;) and s(P') < s ( K P ' ) + e3; hence s(Pi) + s(PI,) < k, + (3e3+ E;). Similarly s(Qj) + s ( Q L ) < k, + (3%,+ e r ) . By adding the last two inequalities and using 6.7A we obtain 2k; < 2k, + 2(3€3 + e:). But k! > k, + 3e3 + e; (P.C. 6) so we have the desired contradiction.

6.14A. Proposition. Let S X ; r TY, where p < n, S X and TY are (p)-bounded by b25, S, Tare maximal n-normal and

J.L. Britton, The existence of infinite Burnside Groups

150

Then Im S

=

T and I m T = S.

Proof. Since s(S) > 2 4 , we have Im (Im S ) = S. Now if B is a weak image of S, then s ( B ) > s(S) - u4 hence Im S = has n-size greater than s ( S ) - u4 - e3 2 u6. If Im S # T then T is left of Im S, hence Im T is left of Im(Im S ) , a contradiction. Therefore Im S = T and S = Im(1m S ) = Im T .

-

6. I4B. Proposition. Let A B, where p < n and A , B are (p)bounded by b,, . Let A , B &e subelements of 1n . Let s n ( A )> Max(sZs,ro + r2,). Then Isn(A)-sn(B)I < Y , + ~ e3.

Proof. By Axiom 25, B contains C such that s(C)> s ( A )- ~ 2 > 5 Yo. Hence s ( C ) < s ( B ) + c 3 and s ( A ) - ~ 2 < 5 s ( B ) + e 3 . If s ( B ) 2 s ( A ) then we have symmetry and the result follows. Now let s ( B ) < s ( A ) . Then

0 < s ( A ) - s ( B ) < ~ 2 +5 € 3 and the result follows.

6.1 5. Proposition. Let X ; Y with divisions P,,..., P, and Q , , ..., Q,. Let X , Y be ((n - 1))-bounded b-y k , and n-bounded by k , . Let X ; Y induce A n y ,B and let B be a subelement of 1 with s n ( B ) > u g . For some p let Q, have type n. Then B is not contained in Q D . Proof. Assume B C Q Since Qb is disjoint from B we may assume p. without loss of generality that it is right of B and that X , Y begin with A , B respectively. Now we have say PL-' QbX1; E 1 ,

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151

BUQLS E L , for some S. Note that none of Q 1 ,..., Qppl have type n by 6.14. By 5.25,SBUn=lXLP~-1X~::.Also in Gn-, (InX, PP-,) = XO'(1n Y , Qp-l)Xp-l = X;'BU.N-'X,-,

QF'=

Hence Q ~ S X O ( I n X , P p - , ) L P ~ X ' - ' Qb(SBU)(N-lXp,l.L ) P P' X'-l&il= P I. Now apply Axiom 24 t o PLXL-' Q F 1 ;this is possible since 4 1) k: < s(P;) + s ( Q ~<) k7 + s(Q;), SO s ( Q ~>) k: - k7 2 ~ 2 (P.C. and since PL, are ((n - 1))-bounded by k, < b24 (P.C. 2). Hence there exists PLLX"Q'-lR equal t o P' XI-1 QF1in G n p l and ((n - 1 ))P P bounded by k, + 424 and s(PbL) > s(Pp)- ~ 2 > 4 k: - k7 - ~ 2 > 4 rb (P.C. 3). Let P = (In X,PP-,) .L, Q = (In Y , Q,-,) .N. Then P = A P ' , Q = BQ' say. Now T exists such that X,, T are end words (cf. Axiom 3 1A) for A , B ; hence A = Xi1. B . T. By Axiom 3 1B, since s(B) > U 8 2 ~ 3 (P.C. 1 4) and k, < b31 (P.C. 5 ) there exists BLFAR ((n - 1))-bounded by k6 + 431 and equal t o X , . A in G n - l . A R is not a subelement of L U ... U L n - , and s(BL)> s(B) - ~ 3 1 > U g - Y 3 1 2 S24 (P.C. 6). Hence I = QbSXoPP;LXiQFIR= Q' SBLFARP'P'LX"Q'-'R P 5 . p p . By Axiom 24 there exists QiLS'BLR equal t o Q i S B in G n P l and ((n- 1 ))-bounded by k , + q24; also

Qb

,

s

(

~

>~ s (~ B )~ ) r24 > us -

-

r j l - r24 > Y;

(P.c. 7)

Thus

I = Q~Ls'BLR F A R ~ ' ~ : L X # : Q ' - ~ R -P

____

P

P

P

where the right side is ((n - 1))-bounded by Max(k6 + 924, k , b, (P.C. 8). This is a contradiction by Axiom 10.

+ q31) <

6.16. Definition. Let A c X ; Y with divisions P,, ..., Pk and Q,, ..., Q k . Let X , Y be ((n - 1))-bounded by k , and n-bounded by k,. Let A be maximal n-normal in X,s n ( A )> z i g . We define Im'A t o be a certain maximal n-;lorma1 of Y as follows.

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J. L. Britton, The existence of infinite Burnside Groups

Case (a). (3 i ) Pihas type n, A = pi. Then i is unique by 6.14 since k , < k, (P.C. 1 ) and k , < k s (P.c.3 ) . Put Im'A = Qi. Cuse (b). Not Case (a). Suppose J c A n Pi for some j , where Pi'has type ti. If ti < n, then s n ( J ) < rb < yo + e 3 . If = n then s n ( J ) < ro + e3, otherwise J would be normal and A = Pi.Similarly to 6.8 we find A = A'AoA" Y", s ( A ' ) ,s(AT')< (yo + e3)+ e2 + 2/e. Hence arid I $ A0 s ( A )< s(A0) + (Y where

9

(Y

= 2(ro + € 3

+ e2 + 2/e) + 4/e < u,

-

us -- e3

(P.C. 3 ) .

Thus s(AO)> s ( A ) (Y > u9 - a 2 us + e3 Z u s . Since k , < b,, (P.C. 4) we have by 6.1 1 A0 n ~ Y lM 3 H where H is n-normal. ( I m A o is the maximal normal of Y Mdetermined by H . ) Put Im'A = H , the maximal normal of Y determined by H . F o r later note that A0 is normal. ~

6.17. Note. With the same hypothesis as 6.16 suppose X ; Y induces Z n-l T. Let Case (b) apply and let A o C Z . Then -

Im'A = I m ( Z n A ) .

-

Proof. ( A n Z)O is A 0 . Since A n Z 3 ( A n Z)O = A0 Y M 3 H, we is the maximal normal of T deterhave by 6.1 1 that Im(A n Z )mined by H . Hence Im(A n Z ) = H = Im' A .

6.18. With the same hypothesis Im'Pi = Qi if ti = n . Proof. By 6.7A, k: < s(P,') + s(QI) < k , + s(P,')so s(P: ) > k! ~-k , 2 ro (P.C. 1 ). Hence s(5) > k! - k , (P.C. 2).

-

e3 Z u9

6.1 9. Proposition. Let A C X ; Y where X , Y are ((n- 1 ))-bounded hv h$ m d n-boundcd by k $ , A is maximal n-normal and s n ( A )> l l , O ; theri /m'(Im'A ) cxists atzd equals A .

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Proof. The hypothesis of 6.16 holds ((kb < k,, k$ < k,, ulo_>u,)). If Case (a) applies, then A = pi Im'A = By 6.18 lm' Qi = Pi so the result follows. Now let Case (b) apply. Then A 3 Ao nylY M >H , = Im' A . Now Im'H exists since s ( H ) > s ( H ) - e3 > s(Ao)- u4 - e3 > ul0 - a - u4 - e3 > u, (P.C. 1). We show Case (a) cannot apply t o H. Assume R = Qi and ti = n. Since s ( H ) > ul0 - a - u4 > u1 we have by 6.8 H I H o AoMs ( H o )> s ( H ) - u1 > u g . Now use 6.15 ((kb< k 6 , k$ < k 7 ) ) ;we obtain that H o is not contained in any contradiction. Q p , Q p of type n. But H o c = Thus Case (b) applies and H > ,yLC> D,D = Im' H = Im'Im' A . Now H O and Y M are nice and contain HO, hence Z nylT, where T = H o U Y M , Z - C u A o . S i n~c e~ A O~ C Z_a n_ d H_ OCTwehave Im'A = I m ( Z n A ) , I m ' H = I m ( T n H ) . N o w A o > A - c i > u l 0 - c i > u6 + e 3 . Now A0 is normal and A0 c Z n A , so Z n A has size greater than us and ImIm Z n A = Z n A . Now H O c H n T c H so R n T = R = I m m . Now n T, I m Z n A are maximal normal in T so H n T = Im Z n A . Thus

ei.

-

ei,

no

~

6.20. Corollary. With the same hypothesis if A is left o f A' then Im'A is left of Im'A'. Proof. Similar t o 6.13.

6.20A. Informal Note. Im A is defined only if A is n-normal. Im A is unique. Im'A is defined only if A is maximal n-normal. Im'A is unique. In general if A has a weak image this is not unique. The definition of weak image does not require A t o be n-normal. If X; Y, p < n then X; Y . Thus ImA and Im'A may both exist, but if so they coincide.

6.21. Definition. (Cyclic version of 6.16). Let (X), ( Y ) be cyclic words ( ( n - 1))-bounded by k , and n-bounded by k,. Let no C.A. of X o r of Y be a subelement of L 1 U ... U L,. Let A C ( X ) ( Y ) :

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J.L. Britfon, The existence of infinite Burnside Groups

hence some cyclic arrangements X ' , Y ' of X , Y respectively satisfy X ' ;Y ' with divisions P , , ._.,P, , Q 1, __., Q,. Let A be maximal nnormal with s n ( A )> 219. We define Im'A t o be a certain maximal 17-normal subword of ( Y )as follows: the definition depends o n the choice of X ' , Y ' ,P I , ..., P,, Q , , ..., Q,. X ' 2 (and Y ' 2 )are ( ( n - 1 ))-bounded by k , and n-bounded by k 9 : for if N is a subelement of L u ... u L and N C X ' 2 then L ( N ) < L ( X ' ) since n o C.A. of X is a subelement of L u ... u L,, hence A' is contained in a C.A. of X . Now X'' ;Y ' 2 with divisions

,

,

except that if both P , and PI,,, are of type 0 then the two terms P,, , PI have t o be replaced by the single term P, .P,and similarly for QrTT Q 1. Aiso A is maximal normal in X f 2 :let B be Im'A in the sense of 6.16. Thus B is maximal normal in Y f 2 . As before L ( B ) < L( Y ' ) $0 R C ( Y ) . Define Im'A t o be the maximal normal subword of ( Y )determined by B . 6.21A. Note. We may now delete Axiom 26' provided that, from now on, we interpret Im U of Axiom 26' t o be Im' U in the sense of 6.2 1 . Thus we identify hi6,. b26,, c26 with k,, k,, ii9 respectively. 6.22. Proposition. Let X ; ;Y with divisions P I , , P, arid , Q , Let X , Y be ((11 1 ))-bounded bji k6 and n-bounded by Q1, h , L e t soine P,be of tvpe n : we muy write p, = EPIF Then s r l ( f : '< ) u , , where u , = ( y o + e 3 ) + ?e2 + u8 + b / e ;sirnilurly f o r ~

Sn(F).

Proof. The result is true if L ( E ) < 2, so assume L ( E ) > 2. Note first that if w is a subword of X of unit length and if R.E.P. w is left nice. then either L.E.P. w = L.E.P. X or one of the two end points of LV is right nice. Now K.E.P. E is left nice so E = E , J say, where L ( J ) < 1 and R.F.P. E l is right nice. I f L.E.P. E , is right nice then E , = E 2 E ,

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155

say, where L ( E 2 )< 1 and L.E.P. E3 is left nice and E 3 is nice: this conclusion is clearly also true when L.E.P. E l is left nice. By 6.15 we have s ( E 3 )< u8, so s ( E ) < 2e2 + + 4/e. Finally suppose L.E.P. El is neither left nice nor right nice. Then it is an interior point of some P i and clearly k < i. Let A = Pi n E . Then s n ( A )< ro + e3. Now E = AH say. The result follows if L ( H ) < 1 so assume L ( H ) 2 2. R.E.P. A is right nice so H = H 1 H 2 ,L(H,) < 1 and L.E.P. H , is left nice. H , has the form H3J. Thus E = A H , H3 J and H , is nice, so s ( H 3 )< ug and the result follows.

6.23. Proposition. Let A C X ; Y with divisions P,, ..., Pk and Q1,..., Q k . Let X , Y be ((n - 1))-bounded by kk and n-bounded by kb. Let A be maximal n-normal in X with s n ( A )> ul0. Then Zm'A exists and in Case (a) of 6. I 6 Isn(A) + sn(Zm'A)- 1 I < u2,

where u2 = Max(4ul

+ 4e3 + eT + 8/e, 1

-

k!

+ 2e3):in Case ( b )

Proof. By 6.19 Im'A exists and Im'(Im'A)=A. First consider Case (a). Here A = <.,Im'A = Qi. Write Pi = EPiF. Then s ( P j )< 4 / e + s(Pl!)+ 2u1 since ul0 2 us. Similarly s ( a i )< 4/e + s ( Q ; ) + 2ul. We have P;-lXl!I\ QiXl!E L,. Now s(P;) < s(P;-'X;_Zi) + e3 since by 6.7A s(Pl!)> ro + e3 > yo. Similarly, s(Q1) < s(QIXl!)+ e3. Also

Combining these inequalities we obtain

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J. L. Britton, The existence of infinite Burnside Groups

Conversely s(Pi) < s(pi)+ e3, s(QS) < s ( Q j ) + e3 and by 6.7A, since kh < k , , k$ < k ; , we have k! < s(Pi)+ s(QI). Hence k:

< s(pi)+ s(Qi) + 2e3

and the result follows. Now consider Case (b). In the proof of 6.19 we have H = Im'A and s ( H ) > u l 0 - a - ti4 - e3 = ulo - u 3 . Indeed we may replace u l 0 by s ( A ) to obtain s(1m'A) > s ( A )- u 3 . But since Im'(1m'A) = A we have by symmetry s ( A ) > s(1m'A) - u 3 .

6.24. Proposition. Let A X ; BY, where both A X and BY are ( ( n - 1))-bounded bjs kb and n-bounded b y k $ . Let A and B be maximal n-normal o f s i z e > u l 0 . Then Im'A = B. Proof. Im'(1m'A) = A ) = A and Im'(1m'B) = B. Case 1. A = P iand Im'A = Qi. Here k: < s(P,') + s(QI) < k$ + s(Q& so s(Qi.) > k! - kb 2 u l 0 + e 3 > yo. Now Q;.c Qiso s(Qi) 2 s(QI) - e3 > ulO. Obviously so by 6.20 Im'B is left of, or equal to, B is left of, or equal to, A . Hence Im'B = A and B = Im'(1m'B) = Im' A . Case 2. Case 1 does not hold. Ao' 3 H, H = Im' A . Since A is a left subword Here A 3 A o nrl of A X , A o is a left subword of A X and Ao' is a left subword of BY. If Ao' C B then H C B so H = B as required. Now let B C AO'. B does not have the form Q j (Qi of type n ) otherwise Qj C B C Ao' so P,! C A o C A and = A , a contradiction. Hence

a.

B3BonrylBO'> K,

K=Im'B.

But BO c B c Ao' so BO' c A o . Thus K c A o c A . Hence i.e., Im'B = A . T h u s B = I m ' ( I m ' B ) = I m ' A .

K =A ,

6.25. Proposition. Let R c Y ; Z , where Y, Z are ( ( n - 1 ))-bounded by k i and n-bounded 61) k; and s ( R ) > uio. Then Im'(R) = S exists and, in case (b) of'llejinition 6.16, there exists R M ,SM (subsuch that words o f R , S respectively) each of size > s ( R ) R Mn - 1 S M .

-

J. L. Britton, The existence of infinite Burnside Groups

157

Proof. Put kg = Min(kb, b25, k 6 ) , k; = Min(kb, k7). By 6.19, Im'R = S exists since ui0 2 ul0. Let (b) of 6.14 hold. By 6.19 R 3 Rol:n Z M M , M = S : also s(Ro)> s ( R ) - Q and s ( M ) > s ( R 0 )- u4 > s ( R ) - Q - u4 > ui0 - Q - u4 > u5 By 6.1 1

B C Y Mn y l M o C M where s ( B )3 s ( M ) - u4 > s ( R ) - a! - 2u4 > - Q - 2u4 > ro. Hences(YM)> s ( R ) - a ! - 2 u 4 - e 3 . Now Y MC R and M o C S. The result follows since u'i0 = a! + 2u4 + e 3 .

6.25A. Note. Let R C Y ; Z and let Im'R = S exist. Assume Case (a) of 6.16 applies. Put RM = Pi,S M = Qi; then clearly RM ; ;S M . 6.26. Proposition. Let p < n. Let H C A B where A , B m e ( p ) bounded by b < k52 and H is a subelement o f L P + l of size > u4. Then a weak image J of H exists and there exist U C H , V C J such that U V (end words E, K say) and there exists words U L E V R , P V L F U g , (p)-bounded by b + q 3 1 such that U ; U L E V R .K = E . VLFUR. Also U L , U R , V L , V R are not subelements of 11 u ... u 1,.

-

Proof. Since u4 > u3 + u4 > u 3 and k,, < b25 we have H 3 H o BM 3 J and s ( J ) > s ( H ) - u4 > u4 - u4 > u 3 . Hence L c U - V C J where V is J o . Now s ( V ) > s ( J ) - u l > u4 - 114 - u1 > ~ 3 and 1 k5, < b,, so there are words ULEVR, VLFUR (p)-bounded by b + q 3 1 , and ULEVR = U . K-' (cf. Axiom 31B). V R is not a subelement of L u u L p and neither is U L . Similarly for V L F U R .

-

1..

6.27. Proposition. Let A ; B where A , B are (p)-bounded by b < k5, and p < 11. Let S, T be distinct maximal p-normal subwords o f ' A of size > u 5 . Then 1. S' = Im' S. T' = Im' T exist

I58

J . L. Britton, The ezistence of infinite Burnside Groups

2. A p B induces (SR,T L ) T I L )(end words E, K ) 3 . Words SRLETILR,S'RLFTLRexist, (p)-bounded by b + h31 atid ( S R , T L ) = SRLET'LR. K = E.S'RLI;TLR P 4. S R L , T ' L ~S,! R L , T L R are p-norinal. ( S I R ,

Proof. S' and 7" exist since k,, < k , , k , and u, 2 z i g . If S, S' satisfy ( a ) of 6.16 put S M = P i , S f M= Qi. Now k: < s(Pi) + s(Qi) < .s(P,) + k,,, so s(P,) > k: k s 3 . Similarly s(Qi) > k: - k S 3 . If S , S ' satisfy ( b ) then by 6.25, since us > ui0 and k,, < kb, k;, we have S M SIM where S M , SIM have size greater than s(S) - ~ 1 ; ' ~ . Siinilarly T M T'M. Hence (S', T') (SrM, T f M which ) we may also write (SR , T L ) ( S ' R , TIL);each of S R ,T L ,S I R , TIL has size greater than Min(u5 - .Yo, k: - k 5 3 )- E, > g 3 ] . Axiom 3 1 C is available since k,, < f 3 , and the result follows. ~

-

-

-

6.28. Corollary. In 6.26 or 6.27 let E ' , K' be end words for A B. Then words A i; B. Then words ALEBR , BLFAR exist, (p)-bounded

bv b + q31 b

( f o r 6.26)

+ h31 ( f o r 6.27),

Proof. Consider 6.27. Say A 2 A l S R L A 2 B , = B l T ' L R B 2 .Put A L = A S R L , RR E TtLRB2.Then ALEBR is ( ( p 1 ))-bounded by k,, + 1231 since S R LTILR , are not subelements of L u ... U LPp1. I f it were riot p-bounded by k,, + h31 it would contain a p-subelenient N where N contains SRLor T'LR properly: say N > S R L . Now S = S , SRLS, say. Let S R L Cbe maximal normal in SRL1:'T'LR. Then N c S , SRLC,and ~

Now

J. L. Britton, The existence of infinite Burnside Groups

159

and

so

The required equations follow directly from the definition of end words (6.7). The argument for 6.26 is similar but easier.

6.29. Remark. If A , A , A 3 A 4 A 5 i; B , B , B 3 B 4 B , induces A , i; B , (end words E , , K 2 ) and A , i; B , (end words E,, K 4 ) and if 6.28 applies to A , i; B , , A , i; B, so that in particular A , 7 AkEB!. K,, A , 7 E 4 . B i F ' A f then A ,A,A,A,A,

7A

AkEBFB3BkF'AfA,.

Proof. Trivial (cf. 6.7). §7 Informal Summary We define L:+, (cf. Axiom 12) for any positive integer k . We find that if Y E L;+, then Y is conjugate in G , t o T k for some T E L n + , (cf. 5.10). We show any element T of L n + l contains m subelements of L , where elm, m > 0. These determine km subwords of Y . If Y = A B then, roughly speaking, s A B ( A )is defined to be a/m, where a is the number of these subwords contained in A (cf. Axiom 13).

7.1. Definition. By Axiom 1 1, % is an equivalence relation on the set of cyclic words (n)-bounded by b,. By 5.10 if 2 E L , + then 2 is a C.A.1 of X e for some X in C n + , .By 5.1 1 2 is (n)-bounded by str' G 6 , (P.C. 1 ). Thus there is an induced equivalence relation = on L , + Hence the following is also an equivalence relation on L,,+1:

,.

x xY

or X E Y - ' .

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J.L. Britton, The existence of infinite Burnside Groups

Choose a set S of representatives for this equivalence relation and define Rep(L,+l) by

Thus Rep(L,,+l)C L , , + l .Clearly if X E Rep(L,+,) then

x-'E Rep(L,,+1).

I f Z , Z ' E R e p ( L , , + , ) a n d Z . X Z ' w e h a v e Z - Z"' (in 813 we shall prove Z = Z ' ) : for if 2' and Z r E E ' S where E , E' =: ? 1 then Z' s (Z"')'" hence Z' 3 2'''so Z = 2"".(If 2,Z' are viewed as linear words this conclusion reads: Z is a C.A.I. of Z'.) Put Y in the set L L + l if there exist R , Z,, Z , , ..., 2, such that R E Rep(L,+l), k > I , Z,= ( R k ) ,2, = ( Y )and

z, + 4 4 tz

-+

n-1

...

+

1 20

LA+

Write L , + ~ for Note that Zi E L:,, (i = 0 , 1 , ..., n ) ; this is trivial for i = 0 and if i > 0 we have Z, 2 ... i+TZi or i = n ; now Zi is (i)-bounded by so < M i n ( ~b2*) ~ ~ (P.C.) , and by Axiom 8 (Zi) (Zi); by Axiom 28 (Zi) 7... 7 ( Z i ) .Hence 2, ; ... i+l+ Zi7... 7 Zi, as required.

7

7.2. Proposition. L i + , is contained in

Li+l,hence L n + l c L n + l

Proof. Let Y E Lk,+l so that Y = T k , T E Ln+'. Let R be the representative of T so that ( R ) ( T ) . By Axiom 1 1 , ( R k ) ( T k ) . By Axiom 28 and since str'< Min(a,,, b2,) (P.C. l ) , ( Rf ) ... + ( T k ) , hence Y E

5

Li+,.

-+

7.3. Proposition. Let Z E L n + ]where n > 1. Then either (i) (Z) contains ke distinct subwords, each maximal n-normal yf'size greater than r" (where r" is a new parameter) for some k > 1,or (ii) (2) contains e disjoint n-subelements each of size greater thurl 14 1 1 .

J.L. Britton, The existence of infinite Burnside Groups

16 1

Proof. Suppose (Z) = (Xe) contains R and s n ( R )2 2(r, + e3) + 2/e. Then by 5.28 since str+< h, (P.C. 1 ) we have L ( R ) < 2L(X). Since R is n-normal, and so is any translate of R and since e 2 6 (P.C. 2 ) we can find three disjoint translates: any C.A. of Z contains two so cannot be an n-subpowerelement. By Axiom 22, R is contained in a unique maximal normal. Case 1. ( Z ) contains a maximal n-normal R where s ( R ) > r". Since r" > 2(ro + e3) + 2/e (P.C. 3 ) we have L ( R )< 2L(X). Consider the set of all such R . Note that kernels exist since r" > 2(ro + e 3 ) + 4/e (P.C. 4). The set is closed t o translation, hence the number of elements is a non-zero multiple of e. Case 2. Not Case 1. Some C.A.I. of X , say Y satisfies Y E Cn+,C J , C J,-*. By 5.8 Y >n-lZ' E Cn..By Axiom 29 some a f ( Y )contains a subelement of 1 of size greater than qo: for if Ys n?l U-'ZffU then YSe n:l U-lZ'feU; 1 , contrary to 5.12 since Y E J,. Thus for some C.A. X ' of X , X" = J K where J is a subelement of 1 of size greater than qo and u = { fel (the least integer 2 ye). Thus J = X f U X f where L u < u -1. Hence qo < u s ( X ' ) + s(XrL)+ 2u/e. Therefore s ( X ' ) or s(XrL)is greater than (qo - 2u/e)/(u+ 1) 2 (4, - 2u/e)/u 2 qo/u - 2/e > q o / ( f e+ 1 ) - 2/e 2 u l l (P.C. 5 ) . Since qo > 2 f w e have qo - 2u/e > 0. 7.4. Proposition. Let Y , , ..., Y , be disjoint k-subelements contained in U and let each be of si<egreater than u12. Let k < n and let U k y l V, where U, V are ( ( k - 1))-bounded by k,,. For some i l e t J C (Yi+,, Yi+d) WhereJisasubelementof L k + 1 , s k + ' ( J ) > ~ 1 Then weak images Y i , ..., YL exist and (Yi-,, Yl!+d+2) contains J ' where sk+l(J')> s ( J ) - u4, provided 1 < i - 1 < i + d + 2 < b. I f ( Y i , Y i ) contains K wheresk(K)> ~ 1 then 4 (Yr, Y,) contains K' where s k ( K ' ) > s k ( K )- u 4 .

Proof. Since u 1 2 2 u 3 , k,, < b,, (P.C. 1,2) we have by 6.10 Y u > Y-! Y;'> Y ~ , s ( Y ~ ) > s ( Y u ) - u 4 . N o w

3.

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J.1, Britton, The existetice of infinite Burnside Groups

By 6.10 since 1413 2 2i3 (P.C. 3), L contains J ' , s ( J ' ) > s ( J ) -). But .I' c L c ( Y / - , , Y l ! + d + 2Lastly

using 6.10 K ' exists since 1114 2

143

Z L ~

(P.C. 4).

7.5. Corollary. Let Y , , _..,Y , be disjoint k-subelements contained in I J euch of size greater than x 2 u12. Let k < n and let U z V \t.lirre U, V u w ((lc - 1 ))-bounded b y k , , and Ic-bounded b.y k , , . r o t . i = 0 , 1 , ..., b - d let ( Y i + ,Y,i + d )contain a subelement Ji o j L k + ! , s k + l ( J ; ) > 1%2 "13. Let W f ( Y , , Y,,) not contain a k-subeleiiient oj'size greuter thaii z < u15. Then weak imuges Y i oj' Y , exist ( u =: d + 2, d + 3 , ..., b - - d - l ) , s ( Y L ) > x - 244, and any d + 4 cnri.wutive Y i generate a word containing a subelement of' L k + , of'size > 114. t'iiiully (Y;,,, Y L P d - , )does not contain a k-sirheleinetit oJ'sizc greater than z + u4, providedz 2 u,4 - u4. -

,

Proof. ( Y , , Y,) is not a subelement of L U ... U L k otherwise J o would be, contrary t o i r 1 3 2 rb (P.C. 1). Clearly L ( Y d + , )> 1 since u 1 22 e2 (P.('. 2). Hence W O > (Yd+, . Y h P d p l )WO'i; , VM. We next prove Wok:, V M .I f not, Pi c W o where Pihas type k. Now VM is :) s P i ) + k , , . Thus k-bounded by k , , , so It! < s ( P i ) + s (I< .s(f';) > k: - It,, 2 u152 z contrary t o hypothesis.((k: -- k , , 2 uls)) C W o k--l V M .By 7.4 Yl',exists We now have (Y,,,, Y,-,( u = d + 2, ..., h d - 1 ) and any d + 4 consecutive Y: contain a ( k + I)-subelement of size > y - - u 4 . If ( Y ; + 2 , Yi-d-l) contained a k-subelement of size greater than z + u4 then W would contain a k-subelement of size > z . K U 1 4 - 244 q 5 ) ) -

7.5A. Note. See 1 1 . 5 , 11.6 for an extension of 7.4, 7.5.

7.6. Remark. I f ? ? ( AX, , Y ) ternporarily denotes the nice image of A c X i n Y(c.f.6.C))thenif17(A,X, Y ) = A 1 , n ( A 1 , X 1 ,Y , ) = A , ,..., ti(.4,. X,, Y,) = A , + , , we call A,.+, the iterated nice i i m g e of A in IF,sometimes we omit the word "iterated" if it is clear from the context that iterated is meant.

J.L. Britton, The existence of infinite Burnside Groups

163

7.7. Proposition. Let A C X 2 Y where k < n and A is a subelement o f L k + , , sk+l ( A )> u16. Then ( i ) A contains subelements X , , ..., X u of 1,, and t exists, as in Axiom 30. In particular Ia/t-sk+l(A)I < €30. (ii) In Case ( a ) o f A x i o m 30, X,, ..., XUp3have images in Y (in the sense of Axiom 26), each being maximal k-normal ofsize > 1117 In Case ( b )Xi!+,, ..., Xu-i8 have images in Y f o r some j ' , these being disjoint k-subelements of size > u18. (iii)Let j be 3 in case ( a ) and the least value o f j ' in Case ( b ) . Then j / a < u19and the iterated nice image o f ( X , , X u ) hence o j A in Y contains the images o f Xi+l, ..., Proof. (i) holds since u16 > ~ 3 (P.C. 0 I ) and since X is (k)-bounded by SO < b3, (P.C. 2). Now assume (a) of Axiom 30 holds. Then s k ( X , ) > Y" Z w26 (P.C. 3 ) . If ( Y )= ( X ) there is nothing to d o since r* Z 1417 (P.C. 4): thus we may assume that ( Y )arises from ( X ) by simultaneous replacement of some maximal normals M , , ..., M,, each of size > r ' . Since so < 1 - ~2~ (P.C. 5 ) r' > 226 (P.C. 6 ) , we have by Axiom 26 that there are cyclic words X i , X i such that ( X ) X;-i-Xik( Y ) where X i , X i , ( Y ) are k-bounded by 1 - Y' + t26 and ((k - 1 ))bounded by so + s26, so, so respectively. If X u is an M i , its image Y , in the sense of Axiom 26 satisfies s( Y,) > 1 - s ( X u )- r26 > 1 -so - r26> ~ 1 (P.C. 7 7 ) . If X u is not an M i its image Y , satisfies

cl

s(Y,)

> s(X,) - r26> r*

-

r262

1417

(P.C. 8 ) .

Now assume (b) of Axiom 30 holds. Then s k ( X , ) > q ", the X u are disjoint and A contains n o k-subelement of size > c*. Again we may suppose that Y arises from X by simultaneous replacement of maximal normals M , , ..., M , each of size > Y', since y " > 2418 (P.C. 9). The required j will be 3d + 15, where d is the least integer 2 t . u20 where u20= u I 3+ 2u4 + eiO(P.C. 10). We show first that a - 2j > 0 and d / t - eiO> 2413 + 2u4. Since t Z e and a > ( s ( A )- € 3 0 ) t > (u16 € 3 0 ) t we have i / a < (3(&20+ 1 ) + 1 5 ) / ( u , 6 - € 3 o ) t = ( 1 8 / e + 3"20)/(2'16-f3o)
J. I,. Britton, The existence of infinite Burnside Groups

164

+

(P.C. 1 1 ). But 1iI9 < (P.C. 1 2 ) so a - 2j > 0. The remaining inequality is obvious. We now want to apply 7.5 t o ( X ) = X ; , X'-X; and Xi$ Y ) k. in succession. The boundedness conditions hold since So + s26 < k,,, 1 Y' + t26 < k , , (P.C. 13, 14). Next, dX,)> 4" 2 u 1 2+ 2U4 (P.C. 15) and s k + l ( X i + , , Xi+d) > d / t - eh0 2 "13 + 2214. Finally (XI, X,)does not contain a k-subelement of size > c* < u I 5 - 22i4 (P.C. 16) and also c* 2 uI4 - u4. Hence images Y , of X u in Y exist for u = j + 1, ..., a - j and S ( Y , ) > 4 %- 3214 2 U l 8 (P.C. 17). ~

7.7A. Corollary. u - 2j > uio 2 6 Proof. I f j = 3 then a > e ( u I 6- €30) 2 6

7 . 7 B . Corollary. j / t

<

+ u;,. If j

f 3 then

+ 18/e.

Proof. If j = 3 then j / t = 3 / t < 3 / e < 3 U 2 0 + 18/e. If j # 3 then + 15 < 3 ( t u 2 , + 1 ) + 15. Hence

j = 3d

j/t

< 3u2, + 18/t < 3u2, + 18/e.

7.8. Let W E L ; + , where n 2 1, so that

Z,, - ( R k ) , R E Rep(Ln+l).We shall describe certain subwords of Z,,, _ _ Z,, _ , Z,. First consider Z,. By 7.3 Z , contains m distinct 11-subelements Bi (i = 1.2, ..., m ) where e l m , say m = he and either 1 . the Bi are maximal normal of size > Y", or 2. the Biare disjoint and of size > u l , . Next we describe certain subwords A i of Z n p ,.

J.L. Britton, The existence of infinite Burnside Groups

165

If 2 holds 2, does not contain a maximal normal of size > r". Since r' 2 Y" (P.C. 1 ) we have (2n-l) (2,): put A j = Bi (i = 1,2, ..., in). Now let 1 hold. If (Zn--l)= (2,) again put A i = B j . If not then Zn-l arises from 2, by simultaneous replacement of some set of maximal normals of size > r'. Since r" 2 w 2 6 (P.C. 2) Bi has image A j in Znm1.Moreover s ( A i ) > u18 since 1 -so - Y~~ 2 Y"- r26 Z u18 (P.C. 3,4). Now let y be an r-tuple of integers, r 2 1 , and assume A , is a subelement of Ln-r+l contained in Z n P r and s(A,) > u18. Then if n - r > 0 we have A , C 2n-r.fr Z n - r - l . By 7.7 since u18 2 u16 (P.C. 5) A , contains subelements B . (i = 1 , 2 , ..., a,) of 1n - r , and 7: t, exists, with the following properties. Since q* < r*, s ( B Y j )> q*: BYi (i = 1 + I', , ..., a, - j,) have images in ZnPr-. say A + : since ~ 1 27 u18, > u18;j,/u, < ~ 1 9

Ia,lt,

-s(A,)I

< €30.

((4"

We summarize the above in the following diagram.

Y * . u17 2 ~ 1 8 ) )

In this diagram, superscript i denotes subelement of L i and ( k ) denotes a k-tuple. Since u l l 2 4 * , r" 2 u l l , u l l 2 uI8 (P.C. 6 , 7 , 8 ) all B's have size > q* and all A's have size > uI8.

J.L. Rritton, The existence of infinite Burnside Groups

I66

Call BYi udmissible if A+ exists, i.e., if 1 + j, < i < a, call B , .. . , B,, admissible. We refer t o the above notation as the A-B notation.

-

j,. Also

7.9. Definition. By a proper descendant of any A , where 1-1 is a t-tuple, 1 < t < M - 1, we mean any A , , or any admissible B,,,, where v is an s-tuple for some s such that 1 < s and s + t < n. By ;I proper descendant of any admissible B , we mean A , or any proper descendant of A,. By a descendunt of X we mean X or a proper descendant of X . Lct RD,-,(B,) mean the subword of Z n P rgenerated by all Adescendants o f B i in Z n - r , where M 2 r > 0. BD,_,(B,) is defined similarly if n > r 2 0. The purpose of 7.9A-7.9C below is t o prove: Proposition. I f n 2 r > 0 then kernels exist jiw the collection of ,siibwords AD,,-,(Bi) ( i = 1 , 2 , ..., m ) of Zn-r and similarly kernels exirt l o r BD,, ,(Bi)(i = 1 , 2 , ..., in) where n > r > 0. 7.9A. Let

11

>n

~

r

> 0 and consider the diagram

where y is a fixed r-tuple. Either ( Z n . r - l ) =r ( Z n - r ) or by Axiom 26 there are cyclic words X , , X , such that

By 7.7 the (iterated) nice image of A , in Zn-r - 1 contains all A + and A ” (taken with respect t o Z n - r X , ) contains all adniissibTe B,, . Now let 6 be another r-tuple, let A,, A , be disjoint and let B,, , R,, be admissible. Then it follows that B+ n B , and A,, n A,, are empty. T ~ Uall S descendants of A , are disjoint from all descendants of A

zrx

J. L. Britton, The existence of infinite Burnside Groups

167

7.9B. Proposition. There exist ci, d j such that (Bici,B j d i )is disjoint # i where Bici, .. . , Bidi are admissible and di + 1 - ci 2 7.

from all Ai, j

Proof. This is trivial if the A i are disjoint so we may assume all A are maximal normal of size > r" - r26 (cf. 7.8). By 7.7 and Axiom 30, A i contains B i l , ..., B,,, and t exists such that I a / t - s ( A j ) l < e3,, and if 1 < w < a then s(BiI,B i w )> w / t - ej,,. We show that w can be chosen so that a - 2w 2 7 and w / t - E;,, 2 ro + e3. This will imply that ( B i l ,B i w )is normal and is therefore disjoint from all Ai.j # i: since (Bi w + l , Bi Bi j + l , ..., Bj u - j are admissible and a 2j 2 7 for some j the required result follows. Put w = { a t } , the least integer 2 a t . where a = r,, + e3 + e;,,. Then a 2(w - 1 ) > t ( s ( A i )- €30 - 2a) Z 8 , since ~

~

(r" - Y

, ~ )--

e3,, - 2(ro + e3 + E&,) 2 8 / e

(P.C. 1 ).

T h u s a - 2 ~ 27 7.9C. Kernels exist for the BD,(Bi), i.e., for the Bi, since the B j are either disjoint o r maximal normal of size > 4'2 2(ro + e3) + + 4 / e (P.C. 2). Kernels exist for the A D , - , ( B i )= A i since the A i are disjoint or maximal normal of size > r" - r26 > 2(ro + e 3 )+ 4/e (P.C. 3 ) . This can also be seen from 7.9B. Kernels exist for the BD,-,(Bj) since by 7.9B, ( B j c i B , j d i )is contained in BD,_,(Bj) and disjoint from Ai,j # i, hence from BD, - 1 (Bj). Now consider the AD,-,(Bj). In 7.7 we showed that, in Case (a), the nice image of (XI, X u ) in Y contains the images o f X,, ..., X u - 3 . By the same argument we have in the notation of 7.9B that the nice image of (Bici,B i d i )in Zn-, contains A i , if c j + 3 < 4 < di - 3; there is at least one such 4 . Hence if j # i, Ai, is disjoint from the nice image of Ai in Z,_,, which contains A D , -2(Bj). But Ai, C AD,-,(Bj). Finally kernels exist in all remaining cases since by 7.9A all

168

J. L. Britton, The existence of infinite Burnside Groups

descendants of Ai, are disjoint from all descendants of A j s where i and s is arbitrary.

j #

but first 7.10. We next want t o define sxY(X) where X Y E we make a few preliminary remarks. Let Ri mean AD,(B,) (i = 1 , 2 , ...,m). Thus the Ri are subwords of Z , for which kernels exist. Let Hibe the kernel of Ri. Then Z, F ( L ) ,L = H 1 E , H 2 E 2 . . .H , E m for some words Ei.If L = RST for some R , S, T we can consider the number of Ri contained in S and the number of R i contained in TR. Let ( X Y ) = Z , for some X , Y . Then C,D exist such that L = CD, X Y = DC, and we may consider the number x of Ri contained in X . x is determined once C,D are chosen, but x depends o n this choice. Say Ri+l...., Ri+x are contained in X ; then X contains the kernels of each of these, so L ( X ) > x .

Li+l

7.1 1 . Definition. If X Y E we define s x Y ( X ) and s n + l ( X >as follows. There exist R E Rep(L,+,), Z,, Z , , .._,2, where Z , = ( R k ) , Z , = ( X U ) , Z,, ; ... 7 Z,. There exist A’s and B’s as in 7.8-7.9C. There exist Ri ( i = 1 , 2 , ..., m ) , C, D and x as in 7.10. Put d = m / k . Since ke divides m, d is an integer divisible by e. Put sxy(X) = sup x / d . Put s n + l ( X ) = sup sxy ( X ) , taken over all Y such that

( 3 k ) X Y E L;+l. By 7.10 we have ,yxY(X) < L ( X ) / e ,s n + l ( X )< L ( X ) / e .

J.L. Britton, The existence of infinite Burnside Groups

169

Proof. These results follow from Definition 7.1 1 in a straightforward manner. 7.13. Proposition. (i) Zf Y E L;+, then Y is conjugate in G, to an element of L: (ii) [ L , U ..-UL,+,] = [ L 1 u . . - u L n + , ] where [...I denotes the normal subgroup of II generated by .... +

,

Proof. (i) If X7.Z then X is conjugate to Z in Gi. Hence if X 7Z, where 1 < i < n then any linearizations of X, Z are conjugate in Gi. Hence, using 7.1, if Y E then Y is conjugate in G, to R k for some R E L n + , . (ii) This follows from 7.2 and the special case k = 1 of (i).

Li+l

7.14. Proposition. Let Y E L,k n L t , where 1 < r < n 1 < s < n + 1. T h e n r = s . N.B. It will be proved later that k = h also.

+ 1 and

Proof. The result is true if r < n and s < n by Axiom 12, so assume r = n + 1. If s < n then by 7.13 (i) we may assume that Y is conjugate in G,- to an element Z of Lt. Since Z 7 Z we have Y = Z hence Y ;I. But Y is conjugate in G, to an element T of L$+l, so T ; I , in contradiction t o 5.12. Thus s = n + 1 = r.

7.1 5 . Proposition. Let s n + l ( X )2 $. Then X contains a subword UUU where U is E.R. Proof. By 7.11 there exist Y , k such that X Y E L;+,, and there exists R, Z,, ... such that x/d > $. Hence x # 0, so X contains at least one Ri.Therefore X contains A , for some n-tuple y. Now A , isasubelementofL1=L1 o f s i z e > u 1 8 ; s a y A , U r X e E L , where some C.A.I. of X is in C,, so X is E.R. Now

7.1 6. Proposition. In the notation of 7.8 consider A,, , A where f' y is an ( r - 1)-tuple, 1 < r < n - 1 and t 2 2. Let N be an integer

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170

lrrs thun 1134 Then f o r i = I , 2, ..., N A r l i is l e f t of and disjoin2 f r o m a11 A?[,.

Proof. This is trivial if A , , , A,, are disjoint so we may assume that t = 2 and A ? , , A y r overlap. I f I' = 1 wc have t o prove that A l i is left of and disjoint from all A 2 j ( i = 1 , 2, .... N). By 7.9B, 7.9c', i f c , + 3 < q < d , 3 then A , , is disjoint from the nice image of A , in Zn-, hence is disjoint from :111d left of all A 2 i . Now by 7.9B, ( 3 ~t,,W ) u / t > .s(A1 ) - €30, d I = (1 - w,u -. 2w > 7, t 2 e. Hence -

and the result follows. If r > 1 then as part of the diagram in 7.8 we have

and by 7.7 thc A y i have size > u I 7 , otherwise A , , , A , , would be disjoint. The argument of 7.9B and the relevant part of 7.9C remains valid since +( 1 + e(u17 - €30)) > ~ 1 ~ 14 > N - 1 and (instead of P.C. 1 in 7.9B) ~ 1 --7 €30 - 2(ro + e 3 + eio) > 8 / e . 7.16A. Corollary. L e t X, be the r-tuple ( 1, 1, ..., 1). Let p be an r-tirplc, p # X,. Let 1 < r < I I - 1. Thoti for i = I , 2 , ..., N A h r , is lcft o f u11d disjoint f r o m all A , ] .

Proof. p = X,su wherc u 2 0, s > 2 and u is an ( r - a - 1)-tuple. ConsiderA,,l. A h Q s I. f 1 < i < N then by 7.16 Ah,li is left of and disjoint from all A,,si. If u is not a 0-tuple A,,ll is left of and diswhere k is the first term of u ; say u = k ~ ' For . joint from ativ ( r - u ?)-tuples a , p A , a 1 1 ,is left of and disjoint from A h U s k pTaking . a = A,.-,-,, 13 = u' we have A is left of and disjoint ". from /lo and the result follows (for arbitrary I ) . I f u is a 0-tuple then p = X,s, X r = X, 1 and the result follows. -

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§8 Informal Summary The relations and ; are closely related to ;but they are not equivalence relations. The purpose of this section is to introduce the relation of n-almost equality, denoted by which is an equivalence relation. If S, S' are n-normal and S is a left subword of S' then for any T, T' such that ST E L,, SIT' E L , we have ST = S ' T ' in Gn-, (cf. 5.25); because of this if X S and X S ' are words ( X arbitrary) then they are equivalent in some sense. Thus we consider words of the form X S where X is arbitrary and S is n-normal, calling these right n-objects, and we show how to define an equivalence relation for these, generalizing .; This idea extends to n-objects, which begin and end with an n-normal subword. It turns out that if U ; V and A , B are distinct maximal nnormal subwords of U then the n-objects ( A , B )and (Im'A, Im'B) are n-almost equal.

8.1. Definition. A left n-preobject is a word 0 together with subwords S, X,H , X ' such that 0 = S X = S H X ' and 1. 0 is ( ( n - 1))-bounded by k,, and n-bounded by k13, 2. S is a subelement of Rep(Ln) and s n ( S )> U ~ , , ( ( U ,>~ ro + e 3 ) ) 3. SH is maximal n-normal in 0, 4. s n ( H ) < u,,. Note that S is normal. 8.2. Definition. Let Oi SiHiXf (i = 1,2) be left n-preobjects. Temporarily, write 0 , C ' 0, if X , = X , and S , = SF; it follows that H , =_ H,, X i = X i (cf. 5.27). Call 0, a left n-adjustment of 0, if 0, C ' 0, or 0, C ' 0,. ((k13 < { l o ) )

8.2A. This is an equivalence relation; to prove transitivity, let 0, be a left adjustment of O,, and 0, a left adjustment of 0,. The only non-trivial case is 0, C'0, and 0, C ' 0 3 ;then S , = A S , , S , = BS, say. Now U, V exist such that UAS,, VBS, E Rep(Ln). By Axiom 33, since s(S2)> u , > ~ s3, (P.C. l ) , UA = VB. If say

I12

J. L. Britton, The existence of infinite Burnside Groups

L ( A ) < L ( B ) then B L ( Aj 2 L ( B ) .

B , A so S , 2 S: and 0, C' 0,. Similarly if

8.3. Proposition. Let A S , X , ;B S 2 X 2 , where each side is ( ( n - 1))borriided b y lil, and n-bounded b y k13 and where S , X , and S 2 X 2 are l ~ f n-preobjects. i Assume A , B do not contain an n-subelement o.f'size> 1123. Then either 1 . Si= SfNiSl!',Ni normal ( i = 1, 2) and N , S;'X, ;N 2 S i X , , N, = N,, o r 2. Si SfNiSl!',Ni normal ( i = 1 , 2 ) and J exists suck that N,S;'X, ;J N , S i X 2 , hence S i - l A - l - B S ; ; J, and N;lJN2 is a subelement of'Rep(L,j. Note. A supplementary result is given in 8.5. Proof. Only conditions 1 , 2 of 8.1 will be used in this proof. By 6.2, A S , X , , B S 2 X , have divisions P, , _ _P, _ , and Q, , _ _ , Q, . say. ((k12 G k,, 4 3 G k;)) c'usc 1 . F o r some Pl! of type n , S , and Pi determine the same maximal normal subword of A S , X , : briefly TI = Thus S1= EPl!Fsay and u21 < s(S1) G s(Sl) + e3. If Qf # S , then g2is left of for if Qi were left of T2 then s(g.)< ~ 2 +3 ( r o + e3) + 2 / e < K: - k13 - e3 (P.C. I), but lio < .s(P;) + .s(QI), SO s(Q;) > k: -- k13 > ro (P.C. 2 ) and 3s(Q::) > k: - kl3 - e 3 , a contradiction. Now 6.18 and 6.20 are available since k12 < k 8 , kH and k13 G k 9 , k$ (P.C. 3), u21 - e3 > u I 0 (P.C. 4), k: ~ - -k , , - e3 > u I o (P.C. 5); hence Im' g2is left of Im' E. = Pi = s,.As above I m ' g 2 has size < ~ " 3+ ( y o + e3) + 2/e = d say. But Im' g2has size greater than either 1 - s(S2) - u2 or s(F2) - u 3 , hence in either case is greater than d because both 1 - k I 3- u,, u,, - e3 - u3 > ~23-+ ( y o + e3) + 2 / e (P.C. 6,7). Thus Q . = g2.Now consider S , = EPl!F. Since s ( E ) < u1 (cf. 6.22), s1 n E has size < Max(ul + e3, r o ) ,hence

el!,

s(S1n Pi') > = u,,

~

2Max(ul + e3, y o )

c.

-

4/e.

Similarly s(S2 n QI) > 2 m33(P.C. 8). Put N , = S , n Pi,

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N , = S , n Qi, Pi = P f L N lPiR, QI = QiLN2QiR.N , , N , are normal since $. > ro + e3 (P.C. 9). Thus NilPiL-l Xfr,' QiLN2is a subelement of L,. By Axiom 33" there is a subelement N;lJN, of Rep(Ln) equal to it in Gn-l. Put Si= SfNiSl' (i = 1,2). Now from Pi.... .P, = :'1-1 . Qi. ... - Q , it follows easily that N I S ; ' X , =JN,S;X,. =X Case 2. Not Case 1. As in 6.16 we find Sl = SiSyS;', s(S;)< ( y o + e3) + e2 + 2/e = /3 s(S;') < p, s(S1) < 4/e + 20 + s(Sy) = s(Sy) + (Y and finally SynyIV 3 D,where s n ( D ) > s(Sy) - u4 > u21 - (Y - u4. Similarly S, = SiSiSi, nzlW. Put N , = Sy n W , N 2 = V n Sy . We prove 1 N , nrl N , and either N , or N , has size > s33;the result will follow by Axiom 33. This is clear if Sy C W or W C S ; , since s(S!) > s(S1) - (Y > > s33 (P.C. 10). There remains the case Sy left of W, without loss of generality. Hence V is left of S! and D C V C B S i S ! . Now s ( D n B ) < ~i~~ and s(D n Si) < Max(P + e3, y o ) = P + e3. Thus s(D) < s ( D n S,) + r), where r) = 2623 + (P + e3) + 4/e. Hence s ( D n S y ) > u21 - (Y -- u4 - r) > yo (P.C.11) ands(D n < s( V n + e3. Hence

+

Si

Si)

s(vnsi)>

Si)

U , ~ - ( Y - U ~ - ~ ) - E ~ > S ~ ~

(p.c.12).

If A = B = I note that Sp is a left subword of Si, hence N j is a left subword of Si(i = 1,2). 8.4. Given a left n-preobject SHX' we shall define a certain left adjustment of it. There exists T such that TS E Rep(L,). Now s(TS) 2 1 - 2/e > ro and TSH is an n-subpowerelement (cf. 5.28A) so s(TSH)2 1 -2/e - e3. Use Axiom 23 taking (Y = - e4, /3 = + e4, A = (TSH)-'. This is possible since e4 < 3 - 2/e. Hence for some A L , $ - e4 < s(AL)< + e4. Hence for some ( T S H ) R ,I s ( T S H ) ~-) - 2 I < e4. ( TSH)R does not have the form H R otherwise s ( H ) > $ - e4 - e3 2 u,, Thus (TSH)R is ( T S ) R H and

+

s((TS)R)> 3 - e4 - u2, - 2/e 2 u2*.

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J. I,. Britton, The existence of infinite Burnside Groups

Thus (TS)RIZX' is a left preobject with the size s of its leftmost maximal normal subword satisfying Is -+I < e4. For each 0 = SHX' make a choice of (7'S)RHX' and call it the standard left adjzistinrnt of 0. Denote it by s t L ( 0 ) .Clearly we may assume it is independent of X' in the sense that if 0 = SHX' and 0 , S H X ; are left n-preobjects then for some U s t L ( 0 )= UHX' and st,(O,) = U H X ; . If 0, is an adjustment of 0, then st,(O,) = = SbL(O2).

8.5. Remark. I n 8.3, suppose A 5 B = I and suppose alternative 1. holds. Then Ni is a left subword of Si( i = 1,2) and one of S , , S , is a left subword of the other. Proof. By the remark at the end of Case 2. in the proof of 8.3

Si= NiS!' ( i = 1, 2). Now N , or N , has size > s33and N , n y l N Z .

Say SiTi E Rep L , ( i = 1,2). By Axiom 33, N , = N, and S;l T,= Si T,. Hence S , T , = S , T, so one of S , , S, is a left subword of thc other.

8.5'. Note. In 8.3 suppose A

( B y 6.24, 6.23 since k,, €3

> 1/21 -~€ 3 > I l l ( ) . )

= B = I. Then

< kb, k 1 3 < k$ and ~(3,) 2 s(S1) -

8.5A. Proposition. Let A S , X, ;BS,X,, with divisions P,,..., P, uiid Q , , ..., Q,,, say, where each side is ( ( I ? - I))-bounded b y k,, and n-botinded by k,, and S , X , , S, X, are left n-preobjects ( A ,B being arbitrary). Let Si= SfN.S!' ' I - ( i = 1, 2 ) where either ( 1 ) S , is not contained in any P . of type n ,

kk

S , is not contained in any o f type n , the equation induces N , z N, (hence N,S;'X, ;N,S;'X,), s ( N i )> 1 1 , ~ . o r ( 2 ) f o r sovlze P,! of'typr n, N , = S p P f , N , = S, n Qi, .suj' Pl! Pf,N, Pi),, QI= (L;.,N,Q;,,and s(Ni) > m33(hence J exists m c 8 h that N i l J N , is a sirbelenient of R e p ( L n ) and is equal in Gn-, to Ni'P i i l Xl:\ Qil N , and N , S;'X, ;JN,SiX,).

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Then if i = 1 or 2 is chosen,

where the left side and right side are left adjustments of S , X , , S,X, respectively and STXi is stL(SiXi). Hence S,. ST-,.S;- Si1 = A - 1 . B. n

Note. The reader may find it convenient to note now that if the hypothesis of 8.3 holds then the hypothesis of 8.5A holds (cf. the proof of 8.5B below). Proof of 8.5A. First assume ( 1 ) . lo. Since '-128 2 s33, one of S ; , S; is a right subword of the other (cf. Axiom 33) so we may assume S ; = SiR.(Also one of S;l, Si is a left subword of the other.) Thus S ; N , SLX, ; S ; N , S i X , ; briefly S , X,= S! X,. As usual let Si Hi be maximal normal in S i x i = SiHiXf ( i = 1 , 2). 2O. We next show that, with respect to the equation S , X , = SFX,, Im'(SIH1) = SFH,. Both sides are ( ( n 1))-bounded by k,, < kh and n-bounded by k13 < kb ; also s(S1 H , ) > u 2 , - e3 2 ulo. By definition of Im' (cf. 6.16), S , H , 3 (S,H,)O ,:n (S,H,)O'> H, H - Im'(S1 H,), Now in the present case ( S ,H,)O is a left subword of SIXl containing S ; N , , so ( S ,H,)O' is a left subword of S F X , containing Si N,. Since s ( N 2 )> u28 2 u5 - u 4 , we have by 6.1 1 i.e., S Y H , = Im'(S1 H,). that = 3 O . By 6.23, Is(SIH1)-s(SRH2)I < u 3 . Now one of S;l, S;l is a left subword of the other. We suppose S;l = W ; the argument in the other case is similar. Thus SF = S;N,S, W = S , W. We find an upper bound for s( W). If s ( W ) > ro + e3 then ~

v2

,S;

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J.L. Britton, The existence of infinite Burnside Groups

hencc s( W ) < 3e3 + E ; + u3 + u,, + 2/e = a, say. Thus s( W ) < Max(ro + e3, a) = b, say. 4O. Let KS, E Rep(Ln). Choose ( K S l ) R H , to have size s satisfying Is - + I < e4 (cf. 8.4). Since S , X , = S , W X , we have ( K S , lRX, = ( KS,)R WX,, where the left side is st,(S1 X,).We prove that the right side is a left preobject, hence a left adjustment of S 2 X 2 . s(( KS,)R WH,)

< (t + e4 + e 3 ) + b + u2, + 4/e < k I 3

Since $ - e4 < s < s(( KS:)) + if2,

+ 2 / e we have

as required. 5 O . Alternatively choose ( K S , W ) R H 2t o have size s, Is -51 Now for any W R we have

s ( W R / f 2 ) < s ( W R )+

< e4.

2 / e < Max(ro,e3+ b ) + LL,,+2 / e < $ - e 4 < s .

Hence ( K S , W ) R H 2has the form (KS,)R WH,. Thus (KSI)RX,= ( I Y S , )WX, ~ and the right side is st,(S,X,); it remains to show that the left side is a left preobject. Its size is at most

Also since s < s(( KS,)R)

s ( ( K S ,) R ) 2

+

+ s( W ) + s ( H 2 ) + 4/e we have - €4

-

b - u2, - 4/e > u , ,~

as required. Now assiime (2). 6". Put W P f - l X i l ' , QI. Then W P 12 I - ' N -1I J I N 2 Q' - i2' where J , 3 Pl!,' Xl:: Q;.].Hence J , = J in Gn-,. Note that F ,E exists such that P,!,F- S;'H,, Q12E E S i H , . B y 5.28A, F-' WE is a subpowerelement of L,, since S ( Q ; ) > - s ( ~ f> ) k: / < I 3 2 ro + e3 and similarly for Pi'. Let D arise from F ' W E by replacing J , by J ; -

J.L. Britton, The existence of infinite Burnside Groups

thus D

= H I 1S;-'N,'JN,S;'H,.

177

Then by Axiom 33",

By 5.20, s ( W ) < 1 + e3 and s(WXl!)2 1 - 2/e. Now s(WXl!)< s ( W ) + r: + 2/e (cf. 6.7A) and s(F-' W E ) < s( W )+ s ( F ) + + s ( E ) + 4 / e < s ( W ) + 2ul + 4 / e (cf. 6.22). Thus

and s ( D ) > s(F-' W E ) - p33 > s( W ) - e3 - p33> b , where b = 1 - 4 / e - r: - e3 - p33. Since s ( D ) > b > f + e4, some left subword DL is such that Is(DL)-$I< e4. We have D = DLDR,say. 70. We prove that D L = H i ' M - ' , DR = KH, for some M , K. If DL C H I 1 then we would have - e4 < s(DL)< s(Hl) + + e3 < u2, + e3< f - e4, a contradiction. Thus H i 1C DL as required. Now b < s(D) < s(DL) + s(DR) + 2 / e < f + e4 + s(DR)+ 2/e, so s(DR)> b-$-e4-2/e=c,say.Alsoc>ro+e3 If DR c H,, then s(DR) < s(H,) + e3 < u2, + e3 < c, a contradiction. Thus H , C D R as required. 80. Since N , S ; H , X i = JN,S;H,X; we have X i = O X ; , hence M H , X i = KH,X;. Note that M H , X i is s t L ( S lXl). We show that KH,X; is a left preobject; it will therefore be a left adjustment of S , X, as required. Now s( KH,) = s(DK)< s ( D ) - s(DL)+ 2e3 +

€7

< a - (t- e4) + 2e3 + eT < k13 Also c < s(DR)= s( KH,) < s( K ) + u2, + 2/e, hence s( K ) > c - u2, - 2/e 2 u2, as required. 90. Similarly it can be shown that some left adjustment of S , X, is equal in Gn to st,(S,X,).

8.5B.Proposition. Let Oi = S i x i = S i H i X ; (i = 1 , 2 ) be left n-pre-

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objects a n d let 0, 0,. Then S i x , ;S i x 2 , where Six, is st,(Ol) arid Six, is u left adjustment o f 0,. Hence S;-'.S; ;S;'.S,. Proof. The hypothesis of 8.3 holds so the conditions of 8.3 hold. We deduce th3t the hypothesis of 8.5A holds (hence 8.5B follows). This is clear if Case (1) of the proof of 8.3 holds; we then have (2) of 8.5A. Now let Case ( 2 ) hold. Here N , n-l N2 where N , or N , has size > t i 2 , - a - u4 - 7 - e3 = d , say. By 6.14B, since d > ~ 2 and 5 d > ro + r25 we have for i = 1 , 2 that s(Ni) > d - r25 e3 >, 1428. ~

8.6. We have the dual notions of a right n-preobject 0 = X'HS, a right adjustment and st,(O).

8.6~4.Definition. An n-preobject is a word 0 which is both a left n-preobject SHX' and a right n-preobject X ; H , S , and also X ' f I, f I.

x;

8.6B. A left adjustment of a preobject is a preobject; so is a right adjustment. Let 0 , , 0, be preobjects. Call 0, an adjustment of 0, if it arises from 0, by a left adjustment followed by a right adjustment (equivalently, a right adjustment followed by a left adjustment). This is an equivalence relation. Let the standard adjustment of the preobject 0 mean s t R ( s t L ( 0 ) ) and be denoted by s t ( 0 ) .

8.7. Definition. Let 0,, 0, be n-preobjects. We say O , , 0, are nulmost equal, and write

if there are adjustments 0;, 0; of 0,, 0, respectively such that 0; ;0;.

8 . 7 A . Proposition. ri-almost equality is a11 eyitivalence relation on t h e se t o f n-preo bjec ts.

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179

Proof. Let 0 , % O,, 0, 0,. F o r suitable adjustments 0; of 0 ,; Ol,, 0; of 0,; 0; of 0 , we have 0; ;O i , 0; 0;.Therefore some adjustment of 0; is equal in G, to st(Ol,), which is the same as s t ( O i ) , which is equal in G, to some adjustment of 0;.Thus some adjustment of 0, is equal in G , to some adjustment of O,, as required.

8.8. Definition. An n-object is a word ( S ,T) which is ( ( n - 1))bounded by k;, and n-bounded by k;,, where S, T are maximal n-normal of size > ~ 2 and 4 S is left of T. 8.8A. With each n-object we shall associate certain n-preobjects, as follows. If S, T overlap let S = S,A, T = BT, where A = B E S n T. (Thus S , , T, are the kernels of S, T.) S n T is n o t normal. If S, T a r e disjoint let S = S,A, T = BT, where A , B are not normal (but are otherwise arbitrary). In both cases ( S , T) = S , UT, for some U and s(S1)

>~

2 -(yo 4

+ E,)

-

2/e > t3,.

By Axiom 33' there are words ST, TT of the form S i E Z F S r , are TkE'Z'F'Tr respectively where S , ST, T, ?,=,TT, ST, n-subelements and 2, 2' are subelements of Rep(L,). Temporarily call ST, T: with their given factorizations associates of S, T respectively. Now ( S ,T) is equal in Gn-, to ST UT? which contains ZFSY UTkE'Z'; we show that this last word is a preobject, and we call it an associate of the given n-object ( S ,T ) . First, s ( 2 ) > s(SI)- Y,, > ~ 2 -4 (yo + E , ) 2/e - Y~~ > u2, Next, since k' < b,, ST is ((n - 1))-bounded by k',, + c3, < k12; since neither S12R, nor TF is a subelement of L1 U ... u L n - , the word ( 2 , Z ' ) under consideration is ( ( n - 1))-bounded by k 1 2 ;so is STUTT (cf. 5.32). If (Z, Z') were not n-bounded by k13, there would be an n-subelement A in it of size 2 k13 > k i 3 + Z L > ~ 113 ; by 6.10, since k,, < b,,, ( S , T) would contain a subelement of size > s ( A ) - ~4 2 k13.

,:

~

J.L. Britton, The existence of infinite Burnside Groups

180

The maximal normal subword of ( 2 , Z ' )containing Z has the form ZFSYH' = ZtZ and it is required to prove that s ( N ) < u22. Now

s ( Z )> s(S1) - Y~~

> s(S)

- (yo

+ e3) - 2/e - Y

~

~

.

S ; l f ' is maxitnal normal in ST UTT and has size greater than s(2) e3 > 2124 ~

--

(yo

+ e3)

--

2 / e - Y~~

~

e3 > u6 + u4 + e3

Also s ( S ) > uI4 > ii6 + 144 + e3. By 6.14A, S and STH' are images of each other. Hence

where a = (yo + e 3 ) + 21.9 + Y~~ + u4 + e3. Since ZH E ZFSYH' c STH', s(ZZ1) < s ( Z ) + a + e 3 . I f s ( N ) > yo t e3 then s ( H ) + s ( 2 ) < s ( Z H ) + + 2e3 + ET. Since a + e3 + (2e3 + E ? ) < 2i22 it follows that s ( H ) < 1122. The result follows since yo + e3 < u22.

8.8B. Definition. A k f t n-object is a word SX which is ( ( n - 1))bounded by 1ci2 and n-bounded by k i 3 , where S is maximal n-norma1 of size > 1124. 8.K. Note. The theory of left n-objects can be developed along very similar lines t o the theory of n-objects (which occupies Sections 8 and 9) but it is easier and it may be left t o the reader (in particular the E , K-theory in Section 9 is trivial for left n-objects). Thus with each left n-object one can associate a left n-preobject; two left n-objects 0, 0' are left n-almost equal if there exists associated left n-preobjects P, P' which are left n-almost equal (i.e., some left adjustment o f P is equal in Gn to some left adjustment of P ' j . Similarly for right n-objects.

8.9. Proposition. Let SA T ; ;UB V with divisions P , , ..., P, a n d Q1,...,Q,, so t h a t S A T ; X i ' . UBV.X,. L e t S A T , U B V b e

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181

( ( n - 1))-bounded by k14 and n-bounded by kl5, Let S, T, U, V be n-normal of size > ~ 2 where 5 g# T (i.e., the maximal normals deThen termined by S, Tare different) and #

where each side is ( ( n - 1))-bounded by ki4 and n-bounded by k;, and s(SR)> s ( S ) - u26, s ( T R )> s ( T ) - u26, and similarly for U, V A Is0

Finally each of E, F, E ' , F' is either an n-subelement o f size < 1121 or is a possibly empty subelement o f L 1 U .*. u L,-l.

Proof. We first modify SAT, UBV on the left to obtain an equation ESRAT = E ' U R B V .X m . Case 1. P, has type 0. Then X o = I ; take E = E' = I, S R = S, U R = U. Case 2. P, has type i, 0 < i < n. Then S = P i s R , U = Q;UR. Also s(S) < rb + 2/e + s(SR)hence s(S) < s(SR)+ u26 since rb + 2/e < u26. Now P;-lXA-'Q;X; E L i and X h = X,. Also k , < s(P;) + s(Q;) < s ( P ; ) + + k14 hence s(P;) > k , - k14 > ~ 2 4 By . Axiom 24 since k I 4< b24 pi-1Xi-1Q11 ' . =1 P'R-1 t-' Q;" where the right side is an i-subelement. 1 Let E = EP,d,E' = Q;". Then (1)

E S ~ TA = E' U ~ VB.X, .

The left side is ( ( i - 1))-bounded by k14 + q24 since Pi" is not a subelement of L U ... U Li-l (for s(PiR)> s ( P i ) - ~ 2 > 4 k , - kl4 -r24 > rb). Now consider its i-bound. Since { < s(XA) + d24 and s(&) < r6 (cf. 6.7A) the i-bound is < 2/e + (r6+ d24 + e 3 ) + kl4 = d , say. T h e j-bound ( i = 1 + 1, ..., I I - 1 ) is k14 + 2/e + rb, and the n-bound is k,, + 2/e + rb. Note that

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k14 + 4 2 4 , d and k , 4 + 2/e + rb are < k14 and k15 + 2/e + rb < k i 5 . Case 3. P , has type n. Say SAT P i x , UB V Q; Y . NOWk! < s ( P ; ) + s(Q;) < s ( P ; ) + + k 1 5 hence s ( P ; ) > k: - k15 > ~ 2 4 Hence . Pi-'X;-'Qi = PiR-'E-lQfR 1 and EPiRX = QiR Y . X,. Consider the left side of this last equation. It is ( ( 1 7 - I))-bounded by k14 + q24 < ki4. Now s ( t ) < s ( X i ) + d24 < r: + dZ4hence it is n-bounded by 2/e + (1: + d,, + e3) + k 1 5 4 ro + e3 Next, s(QiR)> s(Q;) - ~ 2 4 * >k: - k15 - ~ 2 > Now either s(QiL)< ro + e3 or s(QiL)+ s(Q1") < s(Q;) + 2e3 + ET. Hcnce s(QiL) < Max(ro + e3, r24 + 2e3 + = a, say. If U C then s ( U ) < s(QiL)+ e3 < a + e3 < u Z 5 ,a contradiction. Thus U = QiLUR say and s ( U ) < s(QiL)+ s ( U R ) + 2/e < a +s(UR) + + 'lie < s ( u ~ + 2126 ) since a + 2/e < u 2 6 . Similarly s = PiLSR. Let E = E, E' = I. Then s ( E ) < r: + d,, < ~ 1 Also ~ (~1 ) holds . and s R E - ~ E 'UR = ~ - 1 x ; l r ~ . There is of course a dual argument where the right sides are modified. Since s(SR)> 2125 - u 2 6 > ro + e3, SR is n-normal; similarly T L is n-normal. Therefore S R A T L is not a subelement of L U ... U L,. This shows that the left modifications and the right modifications are independent and the proposition follows.

ET)

QiL

8.9A. Corollary. I n S.9 assume further that SAT, UBV are ( ( n - 1))bounded hv x < k14 and n-bounded by -v < k 1 5 . Then E S R A T L F , E ' U R R V L F ' are ( ( 1 1 - 1))-hounded by x + uk4 and n-bounded by 1' + dI5where

uiS = Max(2/e + rb, 2/e + r:

+ d24 + e3).

Proof. Immediate from the proof of 8.9.

8.10. Proposition. Let ( S ,T ) = S X , ( S ' ,T ' ) = S ' X ' be n-objects and let S ' 5 S R , X = X ' (hence T = TI). Let ST, T;" be associates o f

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S, T and let P be the associated n-preobject of ( S ,T ) determined by Sr, Let Si* be an associate o f S ' . Then is an associate o f T' and the associated n-preobject P' of ( S ' ,T ' ) thereby determined is n-almost equal to P.

q.

8.1 OA. Corollary. A n y two n-preobjects associated with a given n-object are n-almost equal. Proof of 8.1 0. We have S = S,A, where A is not normal and, if S, T overlap then A is S n T. Similarly S' = S i A ' . Also

ST

S ~ E Z F S ~ ,Si*

s;~E'z'F's;~.

Now S 2 SLSR= SLS',so SkEZFSFA n==l SLS;LE'Z'F'S;RA', each side being an n-subelement. Now s(FSP) < q33 by Axiom 33', so s ( F S r A ) < q33+ ro + e 3 + 21e and hence any subword of FSPA has size < q33+ ro + e3 + 21e + + e3 < ~ 2 3Similarly . for F ' s ; ~ A ' . By the dual of 8.3 and by 8.5, Z = WN, Y , 2' = WIN, Y ' , where N , , N , are normal and S f E W N , S L S ; L E ' W ' N 2 ,N , = N,; note that alternative 2. of 8.3 cannot apply since we are considering equality in Gn-,. Hence

(1)

N , YFSPAX = N , Y'F'SiRA'X.

Now T = BT,, where B is not normal ( B S n T if S, T overlap) and TT is TkE"Z"F"TP, say. Thus X = DT, for some D . Thus the associated preobjects under consideration are

1" z

Z F S A~D T E

'I,

z 'F' S ;

A' D T f E " Z"

respectively; briefly ( Z ,Z"), (Z', Z " ) . Now S f S ( Z , Z " )= S L S ; L E ' ( Z ' , Z ' ' )applying ; 8.5A t o this equation there are left adjustments (Z*,Z " ) of (2,Z " ) , ( Z ' * ,Z " ) of ( Z ' ,Z " ) which are equal in G,. Hence ( Z , Z " ) ( Z ' ,Z " ) as required.

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8.1 OB. For later note that Z . Z*-'. Z ' * . Z ' - l = E-'Sk-'SLS;LE' 8. 1 OC. Proof of 8.1 OA. By the special case of 8.10 in which S' is S . we see that the associated preobject is independent, up to 11almost equality, of the choice of ST. By symmetry it is also independent of the choice of

q.

8.1 1 . Definition. Let 0, , 0, be n-objects. Then 0 ,, 0, are nalrriost eqiiul, 0, f o,, if there exist associated n-preobjects P,,P2 of 0 , .0, respectively such that P, f P,.

8.1 2. Proposition. n-almost equality is an equivalence relation on thc set of n-objects. Proof. This is an easy consequence of 8.1 OA.

8.13. Proposition. Again let ( S ,T ) = SX, ( S ' ,T') = S ' X ' be n-objects with S' SR, X = X',T f T'. Then ( S ,T) 2 ( S ' ,TI). Brief7.v: ( S , T) $ (SR,T). f

Proof. Immediate from 8.10.

8.14. Proposition. Let ( S ,T ) ; ( U , V ) ,where each side is an n-object arid is ( ( 1 1 1))-bounded by k,, and n-bounded by k17. Let S 7' U V have size greater than ~ 1 , ~Then . (S,T ) % ( U , V ) . ~

(

1

,

Proof. Let the kernels of S, T be S , , T,. Then ( S ,T ) = S , A T, and ~ 1 < , s(S) ~ < s(S1) + (yo + e 3 ) + 2/e and similarly for T. In the same way we have ( U , V ) = U , R V , , say. The hypothesis of 8.9 holds since k , , < k14, k17 < k,, aiid dS1) > "28 ro e3 2/e > 2125. Hence by 8.9, 8.9A, ~

~

~

whcrc each side is ( ( P I - 1))-bounded by k,6 + ui4 < k;, and ri-bounded by k , , + zii5 < Li3;s(S1R ) > s(S1) - 11,, (and similarly for T:.

U p , V F ) :E is an n-subelement of size < 1127

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,

185

,

or is a subelement of L u ... U L n - (and similarly for F, E', F ' ) . SPAT: and UFBV: are n-objects since, e.g., -

) €3 s ( s ~ )s ( s ~ -

> 2/28

-

~ 26 €3

> ~24.

_ _ Note that the kernels of Sp, T f are Sp, T f .

We next consider associated preobjects ( Z , , Z,), (Z,, Z,) of these objects (cf. 8.8A): we have SF n=l S p L E , Z l F , Sp" with similar equations for Tf , U p , V ! , hence

We want to apply 8.3, with 2, , Z , for S , , S 2 . By Axiom 3 3 ' , each side of (2) is ( ( n- 1))-bounded by k,, + u;, + c3, < k12.Since and is equal in Grlpl the left side of (1) is n-bounded by k17 + to the left side of (2) we have by 6.10 that the left side of (2) is n-bounded by k17 + ui5 + u4 < kl3. Now s ( S F L E l )< ~ 3 3 , so if E is an n-subelement then s(ESPLEl)< q33+ ~ 2 +7 2 / e , hence any subword of E S p L E , has size Max(ro, q33+ 1127 + 2 / e + + e 3 ) < ~ 2 3 if; E is not an n-subelement, then by 5.21 and 5.19 any n-subelement contained in E S y E , has size < rb + Max(ro, q 3 , + E , ) + 21e < ~ 2 3 Thus 8.3 and 8.5A are available, hence a left adjustment (2; , 2,F, T f R F )of (ZR' 2, F2 T Y F ) is equal in Gn to a left adjustment ( Z ; , Z4F4 VF F ' ) of (2,' Z4F4 V k R F ' ) .By the duals of 8.3, 8.5A, a right adjustment ( Z ; , 2;) of ( Z ; , 2,) is equal in Gn to a right adjustment ( Z ; , 2;)of ( Z ; , 2,). Hence (2,, Z 2 ) (Z,, Z,), i.e., SFAT: UFBVf. By 8.13, S,AT, fi U,BV/,, i.e., ( S ,T ) C ( U , V ) . Q

8.1 5. Corollary. Let U ; V, where U, Vare ( ( n- 1))-bounded by k,, and n-bounded by k17. Let A,B be distinct maximal n-normal subwords in U of size greater than u29. Then Im'A, Im'B exist and (A,B) (Im'A, Im'B). In particular (A, B), (Im' A , Im' B) are nobjects. Proof. C = Im' A and D = Im'B exist since k16 < kk , k17 < k', ,

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1129 2 ul0. We shall show that ( A R , BL) 3; ( C R ,DL) where each side is an n-object and we shall then deduce from 8.14 that ( A R , BL) ( C R ,DL);the result will then follow from 8.13. i n the notation of 6.16, first suppose A (Pi of type n). Then P; c A , Qi c C; say A = A , P , ' A 2 , C - ClQ;C2. Put AR = Pl!A2, CR = Q:.C2.Since k: < s(P;)+ s(Qi) < s(P;) + k,, we have ~ ( A R2 ) s(P,') - e3 > k! - k17 e3 2 2128. Similarly s(cR) > u28. In the opposite case, we have that A has the form A'A'A", where s(Ao)> s ( A )- b, b = 2(ro + e3 + e2 + 4/e) and

=e

~

Since by 6.19 Im' C = A we also have C = C'C°C'',Cony,Co' 3 H , = A . Let p , 4, p', 4' be the left end points of A o , Co, AO', Co' respectively. Then we may assume q is left of p' or coincides with T I ' . NOW s ( G ) > s ( A " ) - ~4 > 1129 - b - ~4 > ~ 3 SO, G ZI G o nrl GO'. Since G c C we have GO C Co n A o ' . Put K ' E C o n Ao', K = CO' n AO. Then A0 = KL, Co F M K'N say. Take AR t o be A0'4" and take CR t o be K'NC". Note that G C CR since G C C n AO'. We show A R , CR have size > u28. s ( A R ) Z $(Ao)- e3 > ~ 2 -9 b €3 2 '128. s(CR)2 s ( G ) €3 > ~ 2 9 b - ~4 - €3 > ~ 2 8 . We define B L , D L similarly. Thus ( A R ,BL) ; (CR, DL) where each side is an n-object. By 8.14 ( A R , BL) $ ( C R ,DL), as required. ~

~

8.16. Axiom. Let P, P' be n-preobjects. Let P be an adjustment of P ' . Let P ; P ' . Then P = P ' . Informal Summary

§9

I n this section we give a deeper discussion of n-almost equality. The main results are ( i ) if ( S , U, T ) is a word where S, U, T a r e distinct maximal ri-normal subwords arid if ( S , U, T ) and its subword (S,U ) are /?-objects then they are not n-almost equal. (One could, e.g., use this t o show that Im'A (cf. 6.16) is independent of the divisions P , , ..., P k Q1,_ _Q_k ,chosen.) (ii) Proposition 9.14 which allows us t o construct a new pair of n-almost equal objects from given pairs, under certain conditions.

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First we consider n-preobjects.

9.1. Definition. Suppose given n-preobjects P I ,P2 such that Pi 2 P, . Choose adjustments P,, P4 of P , , P2 respectively such that P , 2 P4. For i = 1 , 2 , 3 , 4 we have, by 8.6A, Pi = SiHiX,' = X,!'HITi say; put Ui = S i H i , Vi = H I T i so that Pi = ( Ui, V i ) and Ui, Vi are maximal n-normal in Pi. P u t € - U1.U;1.U4.U;1;then~= S,.S;'.S4.S;1 sinceH1- H , and H2 = H 4 . putKE ~ i 1 . v . v - l .= ~ ~ - 1 . .rT - ~ . T 4 3 1 - 2 4 3 1Then P, g E + P ~ . K , We say that P,, P2 produce E and K . Let E(P,, P 2 ) be the subset of II consisting of all such E as P,, P4 vary. Similarly K(P, , P,) denotes the set of all K . 9.2. Proposition. Let E , E' E E ( P l , P,). Then E K , K' E K(P,, P 2 ) then K ; K'.

;E'.

Similarly if

Proof. Let Pi denote an adjustment of P, or of P, according as j is odd or even. Let Pi = (Ui, Vi). Let P, = P, and E' = U , U;' U, U ; l ; it is required to prove that E ; E', i.e.,

Let P,, be st(P1);then also P , , is st(Pi) if j is odd. By 8.5B, since P, = P4, st(P3)equals some adjustment of P4, i.e., ( 3 P , ) P l , = P8, and moreover

Since P, = P6,

Hence P8 = P,,. But P8 is an adjustment of P,, hence by 8.16 we have P8 is P I , ; thus u8 is Ulo; thus (2) and (3) imply ( 1 1. Similarly K ;K ' .

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Proof. Let P i , Pi be adjustments of P,, P,. Let P i , Pi be adjustments of P,. Then we may suppose Pi = Pi, P;' = P i , = U , i 7 - l U i U i l , E , = U2U i - I U; U;' in an obvious notation. By the preceding proof (cf. eq. (2)) we may suppose Pi is st(P2) Pi. Hence and P;' is st(P,). Thus P i = Pg, U i 3 U;' and Pi ; E , . E , = U , U;-'U; U;,E E(Pl,P,) and the result follows by 9.2.

9.4. Proposition. Z ~ EE E(P, , P,) then E-' E E ( P 2 ,P,). I f P , fi P, tlieii P i 1 * Pi' and if E E E ( P i l ,P i 1 ) then E - , E K(P,, P2). Proof. This is trivial.

Now we consider n-objects. 9.5. Definition. Given an n-object (S,T). For a choice of associates SF, TF of S, T there is determined S F E Z F S P U T F E ' Z ' F ' T P = W say. where (2, Z ' ) is an associated preobject and (S,T) n=l W. Write W = W , W , W , where W , = (2,Z'). Although W , depends on the choices of S?, TT note that W , depends on the choice of S? only. and W , depends only on the choice of TT. Temporarily call

W ,W , W, a triplet for ( S ,T ) . I

Now suppose given two n-objects 0,, 0, such that 0, $ 0,. Choose triplets W , W , W,, Vl V 2V , for 0 , , 0,. Now W , * V , ; choose e l , K , such that W,, V , produce E , , K , . Put E =

W,.E1.V~',

K -

V/31.u1.W3.

We say that 0 , . 0, produce E , K . Let E ( O , , 02)denote the subset of II consisting of all E (as the choices vary) and let K(O,, 0,) be the set of all K . Note that 0, ;E.O,.K. 9.5A. Remark. Let (S,7') and ( S R , T ) be objects. Let S:, Tr be associates for (S,T ) and let S ; * , Ti" be associates for (SR,T ) .

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Moreover let TT = Ti". Let A 1 A 2 A 3and B , B 2 B 3 be the triples thereby determined. Then in the notation of 8.10 we have

A,

= ( Z ,Z " ) ,

B,

= (Z',Z " ) ,

A , = SFE, B , = SiLE'.

Then 8.1OB states that a member of E(A ,, B,) is equal in Gn to A ; ' . S L . B 1 (recall that S = SLSR). It is also clear that I € K ( A , , B , ) . Thus SL is equal in Gn to a member of E((S,T ) ,(SR,T ) ) ,proving 9.7 below.

9.6. Proposition. If E , then K ; K ' .

E' E

E ( O , , 0,) then e ;E ' . If K , K'

E

K ( O , , 0,)

Proof. Let W i W ; W ; , V i V;t/'; be another choice of triplets for O,, 0, and let W ; , V i produce E , , K,. Let E' = W i E , Vi-' ; it is required to show that E d . We note that W , fi W i by 8.1OA. Suppose we prove: (a) If e3 E E( W,, W ; ) then I ;W , . e3' W i - ' . Then by symmetry if e4 E E( V;, V,) then I ;V;-e4. V;'. ;e3. € 2 . €4 by 9.3, hence Now

z

Thus for 9.6 it is sufficient to prove (a). Say W, W , W 3 is determined by S:, T: and W i W; W ; is determined by s;,T i . Then s:, Ti determines a triplet W;l W i W ; say. Suppose: (b) if c5 E E ( W 2 , W;') then I = W , E~ W i - ' ; and , (c) if E 6 € E ( W i , W;') then I = W ~ EWi-'. Then (a) follows since e5-€ 6 = e3. Now consider the special case of 9.5A in which SLis I ; this gives (c). Also by 9.5A with SL= I we have I E K ( W i , W i ) ; as another instance of this same fact we have I E K( WZ', Wi-' ). Hence by 9.4 I E E ( W , , W i ) . Therefore e5 ;I ; but W , = W;l, so (b) follows.

9.7. Proposition. Let (S,T ) and (SR,T ) be objects (as in 8.10).

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Then where S = SLSRwe have SLis equal in Gn to an element o f E((S,T I , (sR, 73). Proof. This is immediate from 9.5A.

Proof. Say W, W, W,, V , V , Y, are triplets for 0,, O,, E'E E( W,, V , ) and E , = W, e' V i ' . Say V ; V i V ; , U , U , U , are triplets for O,, 0,, E" E E ( V i , U 2 ) and E , 5 V ;e"U;'. Let e4 E E( V,, U,), e5 = V , c4 U i ' . Then e5 ;E , by 9.6. Let E E E( W,, U,) and € 6 f W ,TUi' ; then € 6 € E ( O , , 03).Thus E I ' E 2 Z E l ' E 5 E WIErEqU;l WIEUil ;€ 6 € 3 .

9.8A. Note. Proposition 9.4 is valid for n-objects. 9.9. Definition. Let 0,, 0, be n-objects with 0, E ' ( 0 , . 0,) denotes the subset of II given by ) X E E ' ( O , , 0,) if ( 3 ~ X;

E E

:0,. Then

E(O,, 0,).

(Thus E ' ( O , , 0,) is an equivalence class for equality in Gn.) Similarly we define K ' ( O , , 02).

9.9A. Proposition. Let 0, 5 ( S ,T ) , 0, = (SR,T ) be n-objects (as in 8.1 0 ) . Then (i) SLE E ' ( 0 , . 0,) and (ii) I E K ' ( O , , 0,). Proof. Let 0 , , 0, produce E , K . Then by 9.7 SL= E , proving (i) n.. Now S L 0 2= 0, ; E . O ~ - Hence K. K ; I , proving (11).

9.10. Proposition. Let 0 , (S,~T ) , 0, = ( U , V ) be n-objects ( ( n- 1))bounded by k , , and n-bounded by k , , such that S, T, U, V have . size > u , ~ Then ( i ) if 0 , ;T 0, with divisions P,,..., P, Q,, ..., Q,, so that 0 , ;X,j'.O,.X,, and bj, 8.14 0 , 2 O,, then X i 1 € E ' ( O , , 0,) and X, E K ' ( O , , 0,).In particular

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191

(ii)if 0 , ;0 , then I E E'(O,, O,), I E K'(O,, 0,).

Proof. Consider the proof of 8.14; using the same notation we have 0, = S , A T , , 0 , = U,BV,. Let 0; = S P A T : , O i = UPBVk. Also eq. (2) holds and

From these two equations we obtain that if K denotes Z , . Z ; - l . Z ; . Z i l then K ; (ESPLE,)-1E'UPLE3.But K E E ( ( Z 1 ,Z,), ( Z 3 ,Z4)), so S,RLE1K(UfLE3)-' E E(O',, 0;). Thus E - ' . E ' E E ' ( O ; , 0;).Let 0; = S , A T f , 0; = U,BVk. Then 0;, 0; are objects and where S , = SFS?, U , = U k UP we have by 9.7 that SF E E'(O;', 0;).Similarly UF E E'(O';, O;), hence E E ' ( O i , 0 ; )(cf. 9.8A). By 9.8, L = SFE-'E'UL-' 1 E ,'(Of',1 0 ; ) UL-' 1 By the dual of 9.9A (ii), I E E ' ( O , , 0;')and I E E'(OY, 0,).Hence n by 9.8 L E E'(O,, 0,). As in 8.9 we have SP-'E-' E ' U1R = ;Si'Xi' U , , i.e., L (t X i ' , hence X i ' E E'(O,, 0,). If 0,, 0, produce E , K then 0 , = EO,K = ~ 6 ' 0 , ~Hence . X , ;K , i.e., X, E K'(O,, 0,).

9.11. Proposition. Let (S, T ) f (S',TI), where each side is an n-object, ( ( n - 1))-bounded by k,, and n-bounded by klg. Let U be maximal n-normal in (S,T ) , U # S, T. Let S, T,S',T' have size > 1130 and let s ( U ) > u i 0 . Then there is a maximal n-normal subword U' of (S', TI), U ' f S ' , T ' , such that (S, U ) (S', U ' )and s ( U ' ) > 1430 Further, (i) Is(U) + s ( U ' ) - 11 < u2 + u4 + e 3 , or (ii) l s ( U ) - s ( U ' ) ~< u3 + u4 + e 3 . Proof. 1 " . Let S , , U , , T , be the kernels of S, U, T. Then ( S ,T ) = S,AU,BT, say. Also

say, where (2,Z') is an associated preobject of (S, T ) . Take any associated preobject ",( Z') for (S',T I ) ;say

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Now (S,U ) is an object since k,, < ki2, k 1 9 < ki3 and S( U ) > ~ j > o 1124 2". Next we show W is an object. By the argument of 8.8A, W is ( ( n - 1))-bounded by k,8 + c33< k i 2 and n-bounded by k 1 9 + u4 < k 1 3 , also s ( Z ) > s ( S 1 )- r33 > 1130 - ( R o + e3) - 2/e - r33 2 '3 + + M a x ( q 4 , u 6 ,~ 2 9 ) Hence . s(Z) > M a x ( ~u~6 ,~u 2, 9 ) 2 u24. Now Similarly W ends with a maximal normal 2' of size > s(S) > 2430 2 2i6 + 114 + e3 so by 6.14A Im S = 2.Similarly Im T = 2'. Hence 2 # 2'. 3". We have ( Z , Z ' ) (Zr',2 ' ) so for suitable adjustments, one of which will be measured

(4) Wow U c A U ,B c W; moreover U is maximal normal in W. Hence U is maximal normal in ( 2 ,, 2 ; ). With respect t o (4) let U" be Im'(U). We may apply 6.23 since k,, + c33< kb, k,, + u4 < k $ , s ( U ) > ii)3,, > u ,o. Hence either Is( U ) + s( U ' ) - 1 I < u2 or I s ( U ) - s ( U " ) I < ~ ~ . I n t h e f i r s t c a s e s ( U " ) >1 - s ( U ) - u 2 > 1 - ( k I 9+ u 4 ) - u2 > 2430 + u4 + e3, while in the second case s( U " ) > s( U ) - u3 > ujo - u3 > ~ 3 +0 u4 + e3. Hence s( U r ' )> ~ 3 > 0 u6. Now U" c (Z",'2 ' ) c W'. Let U' be the image of U" in (S',TI); then U' f S ' , T ' ;(by 6.20 U" # Zy, Z;). By 6.12 I s( U ' ) - s( U r ' I)< ~1~ + e3 since k,, + c33< b,, . Hence s( U ' ) > s( U " ) - 114 - e3 > ~ 3 0 Also . (i) or (ii) holds. We now make use of 8.15, noting first that k 1 8 + c33< k,,, k19+ 114 < k,, and U, U", U' have size > ~ 3 20 ~ 2 9 Then .

( S , U )n (In W ,U )* ( Z ,U )fi ( Z , , U )c (Z;',U " )fi (ZIr,U r ' ) c

( I n W',U " )2 (S',U ' ) .

9.12. Corollary. Let 0 = (S,U, T ) be an n-object, where S, U, Tare (distinct) nzuximul normal-n subwords and s(S), s ( T ) > ~ 3 while 0

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193

s ( U ) > u;,. Let 0 be ( ( n- l))-bounded by k,, and n-bounded by k19.Then 0' = (S, U ) is an n-object and 0' is not n-almost equal to 0. Proof. Assume 0 g 0'. Now Z exists such that 2 is maximal normal in 0, s ( Z ) > ~ 3 0S, is left of 2, Z is left of or coincides with U, (S, U, T ) ^ ( S , Z ) ;namely we could take Z to be U , recalling that u;, > ~ 3 0 Assume . now that Z is chosen leftmost. By 9.1 1, ( S , Z )= ( S , Z ' , Z ) ,s ( Z ' ) > ~ 3 0 (S, , U ) c ( S , Z ' ) ,S is left of Z ' , 2' is left of Z, Z' is maximal normal in (S,Z ) (hence in 0).But (S,U, T ) * (S,U ) (S,Z ' ) , in contradiction to the choice of z.

9.12A. The corresponding result for left n-objects (cf. 8.8B, 8.8C) is as follows. Let 0 = SX be a left n-object containing a maximal n-normal subword U different from the maximal n-normal subword S ; thus 0 = YUZ and Y f I. Let s(S) > ~ 3 0s,( U ) > u;,. Let 0 be ((n- 1))-bounded by k,, and n-bounded by k I 9 .Then UZ is a left n-object not left n-almost equal to SX. 9.13. Proposition. Let 0 = (S, U, T ) , 0'- (9,U ' , T ' ) be n-objects ( ( n - 1))-bounded by k,, and n-bounded by k,, und S, U,..., T' . 0,O' are maximal normal subwords of size greater thun ~ 3 1 Let be n-almost equal and produce E,, K ~ Let . ( S , U ) , ( S ' ,U ' ) be n. E, ; E,. almost equal and produce e 2 ,K ~ Then Proof. 1". 0 = S,AU,BT, say where S , , U,, T , are the kernels of S, U, T . Let subscripts 1,2, ...,6 on the letters E , Z , F refer to S, U, ..., T' respectively. Then (S, U, T )

-

S ~ E , Z , F , S ~ A U , B T ~ E 3 Z 3 F 3 TW ~

say where (2,, Z 3 ) is an associated preobject. Similarly (S',U',T ') n-l S;LE,Z,F4S;RA'U;B'T;LE6Z,F,T~R 3 W' say. Since ( Z , , Z 3 ) 2 ( Z 4 ,Z,) we have for suitable adjustments, one

194

J.L. Britron. The existence of infinite Burnside Groups

of which will be taken standard,

and Z,.z;-l.z$.z,-' E E((Z,, Z,), (Z4,Z6)). Now U C A U I R C (Z:, Z;); moreover U is maximal normal in (Zr, Z;). Similarly U' is maximal normal in ( Z 2 , 2:). As in the proof of 9.1 1 we find that W is ( ( 1 2 - 1))-bounded by t c33 < k ; , , k b , li,, and W is ti-bounded by A,, + i i 4 < k i 3 , k b , k , , . And further

2'. With respect t o (5) let U" be Im' U ; we note that ~ ( l l>) 1i3, > Max(tilo, ~ 1 ~ 9We ) . prove that U" is U'. Now by 8.15

Now (S. U ) fi S , A U , S F E , Z I F I S p A U l fi (Z1, U1) fi (Z:, [ J ) . Similarly ( S ' ,U ' ) fi (22,U ' ) . Hence by (5) and (6) (Z2, U " ) , (22,U ' ) . By 9.12, U" is U ' . 3". From ( 1 ) and the fact that Im' U is U' we obtain, as in the proof of 8.15, that

where each side is an ri-object. Clearly in the usual notation X i ' for ( 3 ) is l. Moreover by the proof of 8.15 UL and U f Lhave size > By 9.10, (ZT, U L ) (22, U L )and I E E ' ( ( Z T , UL), (22, UrL)). Now

J.L. Britton, The existence of infinite Burnside Groups

195

Hence I € E'(ZT,Z,), (Z:, Z,)). Next

z:.zilE E'((Z,*,Z,),(Z4, z5n Hence Z,.Z,*-lZ:-Zil is an element of E'((Z,, Z,), (Z,, Z 5 ) ) ;by the above it is also an element of E'((Z,, Z,), (Z,, Z,)). Now ( Z ,, Z,), (Z,, Z,) are associated preobjects for S , A U , , S i A ' U ; respectively. Hence if S I A U , , S i A ' U ; produce E ~ , Kthen ~ €1 = € 3 .

Since I E E'(S,A U , , (S, U ) ) and I tain E , ; E,, as required.

E

E ' ( S i A ' U ; ,(S', U ' ) )we ob-

9.14. Proposition. ( i ) Let O , , O,, 0 , be n-objects. Let 0 , = XO,, X f I, and let 0 , = 0 , Y, Y I. Thus we have, say

+

O , = ( S ,T, U ) , 0,- ( T , U ) , 0 3 =( T ,U, V ) .

Then X O , Y = (S, T, U, V ) is an n-object. (ii)Suppose, similarly, that 0; = (S', T', U ' ) , 0; = ( T ' , U ' ) , 0; = ( T ' ,U', V ' ) are n-objects. For i = 1 , 2 , 3 let Oi f 0;.and let Oi, 0;be ( ( n- 1 )bounded by k,, and n-bounded by k 2 1 .Let S, T, ..., U ' , V' have . size > ~ 3 1 Then

(S, T, U, V )f (S', T', U',V ' ) . Proof. (i) is trivial since 0, is not a subelement of L1 u ... u Ln (cf. 5.32). (ii). Let Oi, 0;.praduce ei,K~ ( i = 1,2,3). Now by 9.13 E , ;E , and dually K , ;K , . Let S , , T I , U , , V , be the kernels of S, T, U, V ;then (S, T, U, V )= S , A T , B U , CV, say. Let subscripts 1,2, ..., 8 on the letters E,Z, F refer to S, T, ..., V' respectively. Say U = LU, M . Then

196

J.L. Britton, The existence of infinite Burnside Groups

( S , T , U ) n - l S ~ E I Z I F I S ~ A T , B U ~ E , Z , F ,WU ~ M say, where ( Z , , Z,) is an associated preobject. Similarly

Since say, where (2,. 2,) is an associated preobject of (S', T', 17'). (2,.Z 3 )* (Z,, 2,) we have for suitable adjustments (ZT, 2:) = (Z:, 27) and

T', U')K1 = W ' K ~O. n substituting Now W = ( S , T, U ) = el(S', for and using U =U:E,Z,F, U p we obtain (8)

Z r b 7 , S P A ( T IU , l ) M = Z:F5S;RA'(T;, U ; ) M ' . K

~ ,

Similarly by considering 0,, 0; we obtain

where T = PT,Q, T ' = P ' T i Q ' . NOWP(T1, U 1 ) M 0 2 = E2OiK2 = E , O ; K ~= E ~ P ' ( T Ui)M'K] ;, hence by (8) and (9) we have

Z r F , S F A ( T 1 ,Ul)CVFE4ZZ= Z3F,SiRA'(T;, U;)C'V;LEgZg. The left side is an adjustment of an associated preobject of (S, T, U, V ) . The right side is an adjustment of an associated preobject of (9,T', U', V ' ) . Therefore (S, T, U, V ) fi (S',T', U', V ' ) . 9.15. Proposition. Let ( S ,T ) , ( U , V ) be n-objects producing E , K und let them be (n)-bounded buyb where kb4 < b < k24. Let S, T, U, V huve n-size > 1139. Then there is a word S L C V R , (n)-bounded by b + Z L and ~ ~ such that S L C V R ; E . ( U ,V ) ;( S , T ) . K - ~Moreover . S L ,V R ure n-normal and s(SL) > s(S) - ui0,s( V R )> s( V )- uio, s(sL) > s(SL>- i i i 0 and s(P) > s(V)- uio.

J.L. Britton, The existence of infinite Burnside Groups

197

Proof. 1". By 8.8A we may write

S = S'A, T - BT', U - U'C, V = DV' (10)

(S, T ) n:l (SfL E Z F S P RA , B T P LE l 2 , F l TIR)

where (Z,Zl), ( Z 2 ,Z,) are associated preobjects, and adjustments of these are equal in G,: (12)

(Z*FSfRA , BTILE l ZT) ;(Z;F2 U f RC, D V LE 3 Z g )

We may take the left side (Z*,2;) to be the standard adjustment, so that

As in 8.8A, 2,2, ,Z2,Z3 have n-size greater than

and

(13)

~(2 >)s ( S)- (yo+

€3)

-

2/e - Y

~ ~ .

The right sides of (1 0), (1 1 ) are ((n - 1))-bounded by k24 + c33< b,, and n-bounded by k24 + u4 since u 3 < k24 < h, - e2 - 2/e - u4 (cf. 6.1 OA). By 5.1 1 and 5.32 both sides of (1 2) are ( ( n - 1))-bounded by Max(str+,k24 + c 3 3 )= k , + c33< kb . (Z*,2:) is n-bounded by Max($ + e4 + e3, k24 + u 4 ) = k2, + u4 < k $ , since any subelement or in (Z,2, Since (2;' 2;) of L contained in it is in or in _ )._ is a preobject it is n-bounded by k13 < kb, and ZCZT-Fave size > u21- e3 > ul0. By 6.24 Im'(Z") = and Im'ZT = 2; (since - c4 > ulo).Jy 6.23 IS(^) - I < e4 + Max(u2, u 3 ) = p 1 say and similarly for 2;. Hence each of Z*, Z:, Z;, 2; has n-size > -PI - u22- 2/e = a, say (cf. 8.1).

z*

J . L . Britton, The existence of infinite Burnside Groups

198

We wish t o apply Axiom 3 IC t o ( 12). Each side is ( ( n - 1)bounded by k,, + c33< ,;/ and n-bounded by a l where a 1 = Max(k,, + u4, + (3,) < f31. Each of Z*, ZT, Z*, Z* has size 2 - p1 > gsl. By Axiom 3 1C there is a word p'GZ$R (n)bounded by a 2 + h,, , where a , = Max(k,, + c 3 3 , a l ) , s(FL) > s ( P ) - j 3 1 (similarly for and

qR)

,, ( z ; F ~uiRC,D V ' E~ ~ z ; )

z%L<;Z*R = 3

(14)

Let A , f S f L E Z . Z * - l , A 2 =U ' L E 2 Z 2 . Z ~ -Then 1 . E ; A , . A Z 1 , so by premultiplying ( 14) by A s and postmultiplying by Z:-'. Z , F , V R we obtain

*J

z

S ' L E Z . ~ " - 1 z. * L G ~ * R ~. * - 1 . I; v ' R 3 3 3 3

.

2". Let Y = Z.Z*-l.z"L. We shall show that (i) one of Y , P L is a right subword of the other, (ii) one of Y , Z is a left subword of the other, ( i i i ) the intersection B of the subwords Y n Z*L, Y n Z of Y is non-empty and satisfies s n ( B ) > 0 - a5 - 2 / e > yo where a5 = j,l + e3 + (2e3 + (iv) if Z c Y , say Y = ZW then __ Z"L = Z"W, ( v ) if Y C Z , say Z = Y F , then 2"f ZsLF', (vi) Y is a subpowerelement of L , hence is ( ( n - 1))-bounded by s t y + . It will follow that J ' = S ' L E Y G Z ; R is ( ( n- 1))-bounded by Max(a2 + 1 ~li,, ~+ c~, ~ .=) a3 say; to see this we use 5.32 and note that i t N is a subpowerelement of L u ...u Ln-l contained in S f L E Ybut not in Y then N c SrLEZbecause s(Nn Y ) < rb. In the same way if Y , = ZTR .Z;-'. Z 3 we obtain

ET),

J is ( ( I ! later.

-

= S f L E Y G Y 3 F 3V r R

1 ))-bounded by a 3 also. The n-bound of J will be considered

To prove (i)-(vi) we note first that - 2 = -Z* and Z*- = z*L-. The case when one of Z, Z*, PL is a subword of both the other so two is easy; R i s in this case Z,Z* o r PL s ( U ) > Min(P, a o 9 - P , - /31) = 0.

J.L. Britton, The existence of infinite Burnside Groups

199

There remains the case FL C Z* and Z C Z*. Say 2" = T*L R. If s ( R ) > ro + E , then s ( P L )+ s ( R ) < s(Z*) + (2e3 + E : ) and s(Z*) < s(Z*) + E , < s ( F L )+ j 3 1 + E , , hence s ( R ) < a4,where a4 = j 3 1 + E , + (26, + E ; ) . Hence s ( R ) < Max(ro + €,,a4)= a 5 say. Thus R C Z ; otherwise s ( Z ) < s ( R ) + E , < a5 + E , < 0.Thus Z* z PQR, Z = Q R , Z*L = PQ and Y = Q for some P, Q ,R . B above is just Q so /3 < s ( Z ) < s ( Q ) + a5 + 2/e; a4 = a5 since j 3 1 2 y o . Consider the n-bound of J ' . Let FL H , say, be maximal normal in 7?L GT:R. Then S I L E Y H is maximal normal in J ' . I f Y c Z then s ( S I L E Y )< s ( S ' ~ E Z+) e3 < k24 + u4 + e 3 . Now in general s ( A ) > s(AB) - x , A normal, implies s(B) < Max(ro + e 3 ,x + (26, + Hence by Axiom 3 1C, s ( H ) < Max(ro + E , , a31 + (26, + E;)) (= a6 say) and s(SfLEJYH)< k24 + u4 + E , + "6 + 2/e. If Z c Y then Y z ZW and PL = Z* W say. Now

ET).

so s ( W ) < Max(ro + E , , E , + u22+ 2/e + (2e3 + E : ) ) = a7 say. Now s ( S r L E Y H )< s(SrLEZ)+ s ( W ) + s ( H ) + 4/e < k24 + u4 + "6 + a7 + + 4/e = ag say. Thus J' is n-bounded by Max(a2 + h31, a 8 ) = a9 say. Thus J also is n-bounded by 9. 3 " . Y n Z may be denoted by ZL; it contains B so s(ZL)2 /3 - as - 2/e - E , = a l 0 say. Thus J

= StLEZLH Z ; F, V R

for some H , 2;. Also s(Z:) 2 ale. Now S ' L E Z F S ' K A(= S*, say) n:l S and S* 3 ZL. The weak image of ZL, say Z ' , exists since k, + c3, < M i r ~ ( bb31) ~ ~ ,and s(ZL)> a l o > u,. Thus ZL 3 t n y l q 3 2'where t = (ZL)Oand s ( Z ' ) > s(ZL)- u4 > a l 0 - 214 > y o . Hence

By Axiom 3 1 B there is a word qLFtR,( ( n - I))-bounded by k24 + c3, + q 3 1 = a l l and equal in G n P l to E. t n-l q . K-' where

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J. L. Britton, The existence of infinite Burnside Groups

E, K are end words for q L1 u ... u Ln-l and

s

- t . Also q L ,tR are not subelements of zL

Where = elqe2, S* = 8, Ee4, = 8, t86 we have 81q.K-l.66 61qLFER66 and SILEzL= S ' L E 8 s t 8 6 - 83t86 this last word is ( ( n - 1))-bounded by Max(k,, , a l l , s t y + ) and may be written SLFZLR; we note that ZLRis not a subelement of L u ... u Ln-l since it contains E R . Similarly Z:F3 V ' R n z l Z ! L F ' V R Hence ( 1 8)

nzl

J n-1 = SLFZLRHZ!LF'VR= SLCVR,say

Now SLCVRis ( ( n - 1))-bounded by M a ~ ( ka~l l~, s,t y + , a,) < < Min(b25, k24 + "40) and by 6.1 OA, SLCVRis n-bounded by a9 + 214 < k24 + ~ 4 0 because , u 3 < a9 < h, - e2 - 2 / e - u4. 4". Consider ZL = Y n 2. (a) If Y c Z then Y = ZL,say Z = YF, Z* e F L F . Now s ( P L ) > s(z*) - j 3 1 > s(Z*)- e3 -j31 so s ( F ) < Max(ro + E , , e3 + + j 3 1 + (2e3 + E T ) = a 1 2say. Hence

(b) If Y 3 Z then ZL = Z and (19) remains true. Now s(SL)> s(qL)- e3 hence from ( 13), ( 16), ( 17) and ( 19) we obtain

where 4, = ( a l 2+ 2 / e ) + e3 + (u4 + E , ) + ~ 3 1 +(yo + E, + 2/e + y3,). Now s(SL) > '439 - i4b0 > yo + e3 so SLis normal. 5". (a) Each side of the equation S* n ~ Sl is ( ( n- 1))-bounded ) by 6.14B by b,, and s(S) > 1139 > Max(s2,, yo + ~ 2 5 so

(21)

s(S*) < s ( S ) + ~ 2 + 5 €3.

(b) Since Z is normal

J. L. Britton, The existence of infinite Burnside Groups

(22)

201

s ( S ‘ L E Z ) < s(S*) + € 3 .

(c) From (2) we have (23)

s(SrLE Y H ) < s(SfL E Z ) + “6

+ a7 + 4/e

(d) Consider (18). SLCVRbegins with a maximal normal SLand J begins with a maximal normal, namely SrL E Y H . We shall apply 6.14A. SLCVRis ( ( n - 1))-bounded by b25 and Jk-((n - 1))-bounded by a 3 < b25. Now s(SL)> u39- uio hence s(SL)> u39 - uio - e3 > > u6 + u4 + e3. Since Y 3 B,

Hence

6”. The parameter conditions involving form k24

< Xi

(i = 1 , ..., a),

k24

can be put in the

k24 2 / . ~ (i i =

1 , ..., b )

where X i , p i are expressions involving parameters different from k24. Of course it will be shown later (Section 21) that these P.C.’s are consistent. Now define

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J.L. Britton, The existence of infinite Burnside Groups

and temporarily write m for Min(X1, ..., ha).Then the above proof remains valid if k24 is replaced by b where ki4 < b < m ; in particular if b satisfies ki4 < b < k24, (since kb4 < k,, < m).

§ 10 informal Summary I n this section we discuss a problem of which the following is a ;W 2 ;;... ;W , and A is a maximal nspecial case. Suppose W , ; normal subword of W , . Let A , C W 2 be Im' A let A , C W 3 be lm' A,, and so m. Since, for any B , the size of Im'B may be less than the size of B, it is not obvious that all of A , , ._.,A , exist. The problem is to find sufficient conditions, independent of m , for all of A 2 , . _,. A,,, to exist.

10.1. We suppose given a sequence W , , W,, . .. , W , where for each i W iis either a linear word o r a circular word and W i is ( ( n - 1))bounded by k,, and n-bounded by k23. Let 1 < i < m - 1 ; if W i is i.e., Xiis a specified circular let there be given a linearization Xi, C.A. of lin( W j ) ;if W i + l is circular let there be given a linearization Yi+l.

For i = 1, 2 , ..., m 1 assume one of the following conditions is satis fi e d . l a : W;' = W i + l where W i , W j + l are linear or Ib: Xi1 = Y i + l or 2a: W 1. n W j + ]where W i , W i + lare linear, divisions of W i , W j + l being given or 3b: X i ;Y i + l ,divisions being given or 3a: a linear word Ji is given such that W i , W i + l are specified subwords of Ji (here W i 'and Wi+l are linear) or 3 b : W jis linear and is a given potential subword of Yi+l or 3 c : W j + ,is linear and is a given potential subword of Xi. ~

-

10.1A. Assume that each circular W i (i = 1 , 2 , ..., m ) has at least four maximal n-normal subwords of size > 2132; we remark that this condition is stronger than is necessary for the following theory in this section but it is satisfied in all subsequent applications.

J.L. Britton, The existence of infinite Burnside Groups

203

10.2. A maximal n-normal subword A of W i , where 1 < i < m - 1 , determines in a natural way a maximal normal subword f i ( A )of Wi+l (the notation is temporary) under the following conditions. In Cases 1 a, 1b A-' is a subword of Wi+l ; let & ( A ) be this subword. (Thus there is n o restriction on A in Cases 1a, 1 b.) In Case 2a, define & ( A ) t o be Im'A if s(A) > ulo; note that 6.16, 6.19 hold since k2, < k,, kb and k,, < k,, k'g; (ulo > u9). In Case 2b, define&(A) t o be Im'A, in the sense of 6.21, if s(A) > ul0; we note that n o C.A. of W i o r Wi+'is a subelement of L1 u ..-uLn because of 10.1A. Thus in Cases 2a, 2 b , f i ( A ) is defined if and only if s ( A ) > ul,. In Cases 3a, 3b, 3c, if A is properly contained in Wi n Wi+l define J;: ( A ) t o be the subword A of Wi+l. Under similar conditions, a maximal normal subword B of Wi+l determines a maximal normal subword gi+,(B)of W j ;again the notation is only temporary. Note that ifgi+l(Ji(A)) exists it is A , and if ji&i+l(B))exists it is B .

10.3. Now assume further that each linear W j contains two given proper maximal normal subwords hi, pi of size > ~ 3 2 (where , Xi is left of pi);call these extremal for Wi. A subword of Wi is then called central if it is a subword of ( X i , p i ) . Noie that the extremal subwords have size > ul0 since u32 > ul0. Moreover we assume assume In case la,.fi(Xi) = pi+' andJi(pi) = hi+l. In case 2a,fi(hi) = hi+l andfi(pi) = p i + l . In case 3a, at least one of f i ( X i ) , g i + l ( h , + l )exists and is central and at least one of&&), g i + l ( p i + lexists ) and is central; (this statement could be weakened). If W is circular, any subword is called cenzral. The notation A + B , where A is a subword of W i , means that A , B are central and either f i ( A )= B or g , ( A ) = B . Note that if A , A ' are subwords of W i and A + B , B + A' then A' is A . 10.4. Lemma. Let the assumptions of 10.1- 10.3 be made. Let A , be a maximal n-normalsubword of one of W , , W,, ..., W, such that s(A,)> ~ 3 2 L. e t A o - + A l + . . . + A k . T h e n s ( A k ) > u33.

J.L. Britton, The existence of infinite Burnside Groups

204

Proof. 1". Say Ai c Woi; then ai+l is ai f 1. We may assume that the sequence a,, a l , ..., ak has the form p. p + 1 ...)p + k o r the form p , p - 1, ..., p - k ; indeed without loss of generality we may assume it has the form 0, 1 ...) k . Thus AiC Wi. We use induction on k . The lemma is true for k = 0 since ~ 3> 2 2133. For the induction step we can assume k > 0 ; A , , A ..., A k - , have size > u 3 3 ;none of A ..., A , has size > ~ 3 2 ; the pair W,, W , satisfies 2a or 2b (otherwise ~ 3 <2 s(Ao)= s ( A , ) ) . Call a maximal normal subword E l of W , admissible if there exists q , 1 < (1 < k , and a maximal normal subword E , of W, such that E , .+ E,-, ... E1(Ei C Wi) and s(E,) > ~ 3 2 hence ; E,, Eq-,, ..., E l have size > u33. I f W , is linear then A, is left of A and A is left of pl. If W , is circular there exists maximal normal subwords B', C' of size > ~ 3 2 such that (B',A , , C') is a subword of W , . Thus in either case we may consider the rightmost admissible subword B , left of A , and the leftmost admissible subword C, right of A and ( B , , A C,) is a subword of W , . 2". We prove that for i = 1 , 2 , ...) k there exist sequences

,,

,,

-+

-+

,

,

,

,,

where, for all j , Bi and Ciare subwords of Wi. It follows that Bi and Ci have size > " 3 3 . This is true for i = 1. Assume that it is true for some i, where i < k . Now A i lies between Bi,Ci since the operations of 10.2 preserve "betweenness". We suppose (Ci,A i , B i ) C W j(the argument is similar for the case ( B i ,A i , Ci) C Wi). In case l a , l b Bi+, and Ci+, clearly exist. They also exist in cases 2a, 2b since u332 u,,,. In case 3a Ci+l exists iffi(X,) exists and is central. In the opposite case, Di = g i + * ( X i + , )exists and is central. If Di is left of Cio r coincides with it then Ci+l exists. Now let Ci be left of Di. Then, since Di is left of A i we have

so D , is admissible and between A

,,C,, a contradiction.

J.L. Britton, The existence of infinite Burnside Groups

205

Similarly Bi+, exists. In Case 3b B, + , Cj+ clearly exist. In Case 3c let Di =gi+l(hi+l).If D iis left of Ci or coincides with it then Ci+, exists. If Ci is left of Di we obtain a contradiction as in Case 3a. Similarly B j + , exists. 3". Let 1 < i < k . We prove that if Biis left of Cithen ( B , , C,) 5 fi ( B i , Ci) while if Ci is left of Bi then ( B , , Cl)G (Ci,Bi)-'. This is clear if i = 1 ; note that ( B i , Ci) is an object since k2, < k i 2 , k23 < k i 3 , u33 2 u B . Assume the result is true for some i < k . If B j is left of Ci, then

, ,

= (Ci+,,B,+,)-' in Cases la, l b * (Bi+,, Ci+,) in Case 2a, 2b

= (Bi+,, Ci+,)

in Cases 3a, 3b, 3c

by 8.15, since k2, < k,,, k23 < k,, , u33 2 ~ 2 9 Hence . ( B , , C,) fi ( B i + l ,C i + l )or (B,, C , ) c (Ci+,,B i + l ) - l .Similarly if Ci is left of B j . Since the pair W , , W , satisfies 2a or 2b and since B , , C, have size > u332 ul0, B , = g,(B,), C, = gl(C1) exist; indeed by 8.15 ( B o ,C,)" (B,, C,). Similarly (Bo,Ao)*(Bl,Al). We may suppose that the pair W,-,, W, satisfies 2a or 2b, otherwise s ( A , ) = s(A,-,) > 1133 and the Lemma follows. By 6.23 either

In the first case s ( A k ) 2 1 - u2 - k23 2 u33 and the Lemma follows. Thus we may suppose > u33- u3 2 uj, > ~ 3 0 Since . u33- u3 > ~ 1 2 4 ~, 2 we 9 have as before ( B , , A ,)is n-almost equal t o (B,, A , ) or ( A , , B,)-l according as B , is left of or right of A , . We shall suppose that B , is left of A , ; the other case is similar. Thus (Bo,A , ) fi ( B , ,A 1) ( B k , A , ) . Also A0 C (Bopco)* ( B k , C,); we wish to apply 9. I I . As for A , , we find s(B,) > u33 - u3 Z u;, > 2130; similarly for C,. Further, k2, < k,,, k23 < k,,, ck and B , have size > u33> ~ 3 and 0 s(Ao)> ~ 3 Z2 u33 > uj0. By 9.1 1 (B,, C,) contains Ah such that (B,, A , ) fi (B,, Ah). From the proof of 9. I 1 it is easy to deduce that either +

206

J. L. Britton, The existence of infinite Burnside Groups

or

Now ( B k ,A k ) fi (B,, A b ) ;we wish t o apply 9.12 t o yield that A , is A ; . Now 9.12 is available since s ( A k )> ui0, s(Ab) > u33> ui0 and s ( B k ) > 2133 > ~ ( 3 0 Therefore . s ( A k )> u33 and the Lemma is proved.

§I1 Informal Summary The purpose of this section is t o prove 1 1.8. Roughly, this implies that if Z,, 2 ... 7 DE, Un + n ... + 1 DF, Zn E Rep L,+, and D is large enough in DE, say sDE(D)> c , then 2, = D ' E ' , Un = D " F ' , D' ; D", sntEr(D')> c - E and El-' .F' is conjugate in G, t o E-'. F . Hence a cancellation hypothesis between two elements of L n + l is reduced t o a relation ; ;between subwords of two corresponding elements of Rep L,,

1 1.1 Proposition. Let (1)

X

Y (end words E, K )

wlzcre li < n and X , Y a r e ( ( n - 1))-bounded by k I 6 and n-bounded by k 1 7 . Lc>tX = X O X , X , X3 X 4 , Y = Y o Y , Y , Y 3 Y4. L e t (1) induce

(Thris 0 < t < k, 0 < s < k ) . Let t 2 1, s 2 1 , and let T , , T , ht? maxirnal t-normal in X a n d contained in X , , T i , T i be maxirnal t-normal in Y and contained in Y , , S , . S , be rnaxinzal s-normal in X arid contained iFi X 3 , S ; . Si be maximal s-rzormal in Y and contained in Y 3 .

J.L. Britton, The existence of infinite Burnside Groups

201

In ( 2 ) let Im' Ti = Ti ( i = 1 , 2 ) ; in ( 3 ) let Im'Si = Sf ( i = 1 , 2 ) . Let T , , T 2 have t-size > ~ 2 9 let ; S,,S2 have size-s > ~ 2 9 Thus . by 8.15, since k,, < k,,, ( T l , T 2 ) ( T i , T i ) ;let this pair o f objects produce E , K . Similarly let ( S , ,S 2 ) , ( S i ,Si) produce E ' , K ' . Then (i) ( T , , S , ) ; E.(Ti,S;).K' (cf. 2.8) (ii) if X,Y a r e E. R. and K = E-l then where

7

we have A

;K ' - ' . B . E - ~ .

Proof. (i) By the proof of 8.15

(4)

( T F , T i ) 7 (Ti". T i L )

where each side is a t-object and TF , TiR have size > u28. Similarly T;, T iL have size > u28. By 9.10 (i), if the pair of objects (4) produce E , , K , then E , 7 E,, K , 7 F4,where E,, F4 are the end words for (4). Similarly

and E , 7 E,, K , 7 F , . By 9.7,9.10 (ii) and 9.8, E 7 T ; . e 4 . T ' L - l ;hence E 7 T:.E,.TiL-'. Similarly K' 7 Si"-'.F,.S;. Now ( T F ,S i ) ;E 4 - ( T i RS, i L ) . F , by definition of end words. Hence (i) follows. (ii) Say X = U(TF,S i ) V , Y = U'(TiR,S t L V'. ) Then we have V; F;' V ' K ,U r; EU'Ei1, SFAT: = VU, $RTiL = V'U';hence (ii) follows.

11.1A. Note. In the same way we have that if X,Y are E.R. and ( ( n - 1))-bounded by k,,, n-bounded by k,, and X; Y (end words E, E - l ) and if T , , T2 are maximal n-normal in X of size > ~ 2 and 9 Im'Ti = Tl! ( i = 1 , 2 ) t h e n A ;K - ~ - B . Ewhere - ~ , (T,, T2),(TiT , i) produce E , K and ( X ) = ( ( T , , T 2 ) A ) ,( Y ) ( ( S ,, S 2 ) B ) .

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J. L. Britton, The existence of infinite Burnside Groups

1 1.2. Definition. Let X , , _..,X q - , and Y,, ..., Y , be given where for i = 1 , 2 , ..., q - 1 either X i 2; Yi +,or Xi Yi+,. A k-path where 0 < Ic < n is a sequence

where X i 2; Y i + , (or X i

Y i + l )inducesAi z B i + , (i = 1, ..., q - 1).

11.2A. Let S be a k-subelement contained in A , . We say that S has inzages throughout the k-path (1) if 1. S is maximal k-normal in X , : 2. I m ‘ S = T exists with respect to A B,; 3 . T is maximal k-normal in Y,; 4. either q = 2 or T C A , and T has images throughout the k-path A , x B 3 > A 3 . . . r B ,.

z

11.2B. Let S be a (k + 1)-subelement contained in A , , k + 1 < n. We say that S has weak images throughout (1) if 1 . the weak image T of S in B2 exists; 2. either y = 2 or T c A , and T has weak images throughout the k-path A2 B3 A , ... B,. Sometimes we shall omit the word “weak” in using 1 1.2B. I

1 1.3. Note. Let S, T, U be disjoint ( k + 1)-subelements having weak images throughout the li-path (6). Then the iterated nice image of T exists throughout the k-path; for the iterated nice image of T is disjoint from and between the weak images of S, U (cf. 2.6).

11.3A. The conclusion of 7.5 can now be expressed as follows. There is a (k - 1)-path, with respect to U V , beginning with a , Y b - d - l have weak images subword o f ( Y 1 , Y,) such that Y d + 2..., throughout. Recalling that in the proof of 7.7, in case (b), we used 7.5 three times in succession, the conclusion of 7.7 may be expressed as follows. In Case (a) of Axiom 30 there is a k-path, with respect to

209

J.L. Britton, The existence of infinite Burnside Groups

beginning with a subword of A in which each of Xi+*, ..., Xapi has images throughout. In Case (b) there is a ( k - 1)-path with respect t o ( * ) beginning with a subword of A in which each of Xi+l, ..., X a P i has weak images throughout.

11.4. Remark. Suppose that W , I W , ... W,. Put X i = W i (i = 1 , ..., 4 - l ) , Yi = W i( i = 2 , ..., 4 ) ; thus Xi Yi+, ( i = 1 , ..., 4 - 1). Let (6) be a corresponding k-path. Let C , be a subword of X , containing A Then the iterated nice image of C, exists throughout, i.e.,

Moreover the terms of (7) contain the terms of (6) respectively.

,

Proof. A is nice and contained in C , so A i c Cy , say C: Hence B , C C, . Thus A C C, , A C C! and so on.

,

; C,.

11.5. We consider two conditions: Condition A. If 1 < s < n and X is a subpowerelement of L,+, and of L, then sn+' ( X ) < rb. (cf. 5.21, noting that in 5.21 t and s are < n . ) Condition B. If A ; ;Y , where A is an ( n + 1)-subelement, p < n - 1, s n + l ( A )> ~ 2 and 5 A , Y are (p)-bounded by b25 then Y contains a subelement of Ln+, of size greater than s ( A )- ~ 2 5 . (Condition B is implied by, but does not imply, the statement that Axiom 25 holds for n + 1.) Thus in Conditions A, B if we replace n by n - 1 we obtain true statements. 11.6. Remark. Assume Conditions A, B. Then 7.4,7.5 remain valid if k = n. Proof. Using Condition A it may be verified that 6.8 holds if A is a subelement of L n + l and p < n - 1. Hence using Condition B it may be seen that 6.10 holds for A a subelement of L,+, and

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J.L. Britton, The existence of infinite Burnside Groups

p < n - 1. The proofs of 7.4,7.5 are now seen t o remain valid for k = 11; (in the first sentence of the proof of 7.5 Condition A is needed again).

1 1.7. Proposition. Let

where 1. Z n E Rep(Li,,) 2.O 2i36. 5 . I f r = 0 asslime Conditions A , B . Then

Zn-, = ( J E ' )

un-rn E r

(JF')

where: 1. (,JF') is ( ( n- r))-bounded b y u35. 2. E - ' . F n = T - ' . E ' - ' . F'. T f o r some T. -- Y 3. I f r > 0. J contains all A-descendants of B , , ..., B, in Z n P rexcept possibly jbr the five k f t m o s t of B , and the five rightmost of B,. I f ' r = 0 , .I 3 ( B i + s ,BgPs), where k ( g - 2s)lm > k g / m - ~ 3 7 . 4. In U n p r U n p r - , - ( D F ) let the iteruted nice image of D in U,,+,. be D* say and in Un-r - ( J F ' ) let the nice image o f J in Un-, be .I" .my; then J" is contained in D*. In particular J " , D* exist. +

Note. I f r = 0 we interpret 4 of the hypothesis t o mean that the A-descendants of B 6 , ..., B,-5 in Zn-r-l are contained in D (where g ' i s such that 1 < g < m and Icglm > u 3 6 ) .

J.L. Britton, The existence of infinite Burnside Groups

211

We take a similar interpretation when r = 1 in 3 of the conclusion; also in 1 1.8, conclusion 3, when p = n - 1. 1 1.7A. Proposition. Proposition 1 1.7 remains valid when modified as follows. Let UnPr,Un-r-l be linear words and let

where GDH is ((n - r - 1 ))-bounded by u35 ( 1 , 2 , 4 , 5 of the hypothesis being unchanged). Then Un-r ,,fr G'JH' is ((n -r))-bounded G'.E'-1.H',3remainsvalidandJ*C D* (where J * , D * are defined by analogy with 4). Proof of 11.7. Let m' = m/k. 1". Write Z, U, U', k for ZnP,, Un-r, U n P r Pnl-,r. Then 1 < k G n and

Z

z( D E )

uz U' k Z 1 ( D F ) Let r > 0, i.e., k < n. Let the A-descendants of B , , . _ . B, , in Zn-r be Vi (i E I ) . (Thus I is in one-one correspondence with the set of r-tuples y such that appears in the diagram of 7.8 and begins with one of 1 , 2 , ..., g. If i # j the subwords V i , Vi of Z n P r may coincide.) Each Vi is a (k + 1)-subelement contained in Z so is (k-bounded) by so. By Axiom 30 since so G b30 and s( V j )> u18 2 s30 we have, that V j contains k-subelements Vii of size > q*. Since q" 2 ~ 3 we 0 have, if k 2 2, that each Vij contains subelements Viik of L k - l o f size > 4 " ; and so on. Summarizing:

Let r = 0 (k = n). Put Vl E (Bl, B g ) and put Vli = Bi (i = 1 , ..., g ) . Then V , C Z , E L n + l , V l j is an n-subelement and s( Vlj) > 4". Thus (8) also holds for r = 0 but i has the fixed value 1.

J. L. Britton, The existence of infinite Burnside Groups

212

2". By Axiom 26 cyclic words Y , , Y,, Y,, Y , exist such that

(Possibly Z = Y, = Y , = ( D E ) or U 2 Y, = Y , = U'.). We say a subword A o f 2 c k w s a t-path if there is a t-path, with respect t o (9) followed by ( lo), beginning with a subword of A . 3 " . Consider a fixed Vi and assume the Vii are maximal normal; more precisely i f r > 0 assume the Vii satisfy (a) of Axiom 30, while if r = 0 assume the Vli (= Ri)are maximal normal of size > Y" (cf. the beginning of 7.8). If Y > 0, Vi is an A-descendant of some Bi so has size > 1418; recall that i i I 8 > uI6. By 7.7 Vii ( j = 4, ..., a - 3 ) have images V &in ( D E ) of size > ii17 > 116 and they are contained in the iterated nice image of ( V i l , Viu)in ( D E ) . Now (cf. 7.8) these V i are the A-de= ( D E ) so by 4 of the hypothesis V i sccndants of Vi in Z,, ( j = 10, ..., a - 9) are properl-v contained i n D ,so are maximal normal in ( D F ) . I f r . = 0 then by 7.8 the Ri (= Vli) have imagesAi (= Vii say) in Z,, of size > r" r26 > u I 7 . Writing a instead of g, A 4 , _ _nu-., _ , are contained in the iterated nice image of V , = ( I / , , , V l u )in ( D E ) and again Ai = Vii ( j = 10, ..., a - 9) are properly contained in D. Since S ( V / : > ~ )1 i 6 , Vi, ( j = 1 1 , ..., u - 10) have images Vl; in U' contained in the nice image of ( Vl! Vl!u~ 9 ) in U' and by 6.1 1 they have size > s( V i ) )- u4 - E , > zi17 - 214 -- E , > u26. Now U + U ' , so by Axiom 26" the Vl; have images Vly in U. Further, V'.' ( j = 14, ..., (I - 13) are contained in the iterated nice image of /I ( VI' , Vi-lo) in U. Summarizing, Viclears a k-path and Vii ( j = 14, ..., a - 13) have images throughout. a - 26 > 4 since

,.-,

~

,,

u

> e( V i ) - E , ~ )> e(u18 - €30) 2

29

( r > 0)

J.L. Britton, The existence of infinite Burnside Groups

213

4". Assume Y > 0 and suppose that Vi,, ..., Viasatisfy (b) of Axiom 30. In the present case, by 7.7, and 1 1.3A there is a ( k - 1 )-path, with respect to (2) beginning with a subword of ( V j ,, V i a )in which V.. ( j = p + 1, ..., a - p ) have images throughout ( p here is the j of IJ 7.7 (iii)); let V;j be the image in ( D E ) of Vii. The Vii are the Adescendants of Vi in ( D E ) , so Vl) ( j = p + 6 , ..., a - p - 5) are contained in D . We shall apply 7.5 four times in succession; it will follow that there is a ( k - 1)-path with respect to ( 3 ) beginning with a subword of ( VI!p + 6 * Vl!,-p -5 ) in which some of the Vl>have images throughout. Thus Vi will clear a ( k - 1)-path in which some of the Vii have images throughout. The words Z, Y , , Y 2 ,( D E ) , U ' , Y 4 ,Y,, U are ((k - 1))-bounded by so + s26 and k-bounded by 1 - Y' + t 2 6 . ( D F ) is ((k - 1 ))-bounded by u35.Since U' krl ( D F ) and since Max(so + s26,u ~< b25 ~ ) we have by 6.1 1 that ( D F ) is k-bounded by 1 - Y' + t 2 6 + u4. Since Max(so + s 2 6 , u35)< k,, and 1 - Y' + t26 + u4 < k,, , all the words are ( ( k - 1))-bounded by k,, and k-bounded by kl,. Let a = ~ ) +3~ ~1 +3 6u4. Let d be the integer defined by d - 1 < ta < d (where t arises from Axiom 30). Let u38 be defined by u38 = u18 - ~ 3 0 14a - 14/e. We have a / t > s( V i )- €30 > u18 - ~ 3 0 . We show a - 14d > e . u38.Indeed a -- 14(d - 1 ) > t ( u 1 8 - €30 - 14a) = t( 14/e + u38) 2 14 + eu38. Write Yi for Vji. Any d consecutive Yis generate a subelement of L k + , of size > d / t - E;, 2 a - E;, = ~ 1 +3 6 ~ 4We . have s ( Y j )> q* > u12+ 6 ~ 4Next, . since ( Y l , Y,) contains no k-subelement of size > c*, it does not contain a k-subelement of size greater than u I 5- 6 ~ 4since , c* < u15 - 6 ~ 4 In general, if 7.5 is applied n times in succession the number of Y's having images throughout is b, = b - 2n(2n + d - 1); (we used b , = b - 2(d + 1 ) in 7.5 and b3 = b -2(3d + 15) in 7.7). In the present case the number of V j jhaving images throughout is a - 14( 13 + d ) - 10 = a - 14d - 192 (since ten are 'lost' in passing from ( D E ) to ( D F ) )> e . u38 - 192 > 3. 5". Let Y = 0 and suppose that V l l , ..., Vl, are small in the sense that they are disjoint and of size > u l , (cf. the beginning of 7.8).

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J. L. Britton, The existence of infinite Burnside Groups

Since Conditions A, B hold, 1 1.6 is available. Recall that Vli is Bi. Any s consecutive Bis generate a subword of Z (= 2,) hence a subelement of L,,,, of size > s/m' (cf. 7.1 l ) , so of size > s/m'- E;,. ( B , . B g ) does not contain an n-subelement of size > Y" (cf. 7.3). Thus the argument of 4" remains valid for Y = 0, where u l l , Y" take the place of q*, c* since ! i l l > u12 + 6u4 and r"< u , -~6 ~ 4 . 6 " . Let V;+l where y is a ( k + 1 - p)-tuple occur in (1) and let p 2 1. We say V , satisfies ( R ) if all VYisatisfy (b) of Axiom 30, V, clears a ( p - 1)-path and V;i ( i E I,) have images throughout; here I , is a set of integers of the form {s,

+ 1 , s, + 2, ..., s7 + t,} , t,

2 4.

We say V , satis,fies (S) if all V . satisfy (a) of Axiom 30, V, clears r' a p-path and VTi(i E I,) have images throughout. We have already proved that each V;+l satisfies (S) or (R), if we interpret 'satisfies (a) or (b) of Axiom 30' when r = 0 t o mean 'satisfies 1 or 2 at the beginning of 7.8'. Assume V:', satisfies (R) and let i, E Zk = s, + 2, ..., s, + t, - 1 we shall prove that VP. satisfies ( R ) or (S) (if p 2 2). 72 0 V , clears a ( p - 1)-path and all VTi (i E I,) have images throughout, so the iterated nice image of VTi0exists throughout the ( p - 1)-path by 11.3. Let J = Vri0,Ji = First assume the Ji ( j = 1, ..., a ) satisfy (a) of Axiom 30; we shall prove that J satisfies (S). Since p < k, all the words of the ( p - 1)path are ( ( p - 1))-bounded by Max(so + s26, u 3 5 )< k,,, k,,, k19, kb, k $ ) .The path begins say ZM Min(kl% Y , 3 Y r pzlY r , w h e r e J c Z M and J o pzlJo'> 50'' J0'O'

-

-

P-1

Clearly Jo'C Yiw but by considering 1 1.3 we obtain Jo' C y r . p p l Since s ( J i ) > Y * > Min(u2,, z i l o ) , the Ji have images JI in Y p and (I;,Jh-l) c f l ' ;moreover by 8.15 we have ( J , , J s ) pyl ( J i . J i ) for Y , S E { I , ? ,..., a } , r < s . Now s ( J l ! )> Y * u3 > Y * - u3 - u4 - E > M i n ( z ~ulo), ~ ~ , so J i , _ _JL-, _ , have images J ; , ..., JiPlin Y$ and (J;', J i - 2 ) C J0'Of. Also (I:, J i ) n (J:, J i ' ) for Y , s E {2,3, ..., a - 1) , Y < s. Hence for this range of Y, s (Jr , J , ) fi (J:, J:). ~

215

J. L. Britton, The existence of infinite Burnside Groups

Let 3 < x (Jx-lj Jx+l)

< a - 2; we prove s(J:) > r* - u3 - u4 - E , . We have (Jl-1,J:+G. Since s ( J x )> r* > M a x ( ~ui0) ~~,

then by 9.1 1 there exists Jx such that (J,-

J,)

e

(J:-

1, TX)and

s(Tx) > r* - u3 - u4 - E, > M a x ( ~ t , ~ui0) , By 9.12, Tx is J l and the result follows. By repeating the above argument we find that images of J 8 , ..., Ju-7 exist throughout the ( p - 1)-path. Note that a > e ( s ( J )- E ~ >~ e(q* ) - €30) > 17. Finally assume the Ji ( j = 1, ..., a ) satisfy (b) of Axiom 30; we show that J satisfies (R). Consider the part of 4" from 'The words Z, Y , , Y2, ...' t o the end; it applies in the present case since s ( J ) > q* 2 u18, and we obtain that the number of J i s having images throughout is (at least) a - 14( 13 + d ) > e . u38 - 182 > 3. 7". For later, note that the part of the argument of 6" from "First assume the Ji ..." t o "... J 8 , ..., JUp7 exist throughout the ( p - ])-path" can be applied t o the situation of 3"; with V i , Vji, k replacing J, J i , p - 1 respectively. T o see this, note that all words of (2), (3) are ( ( k - 1))-bounded by Max(so + s26, u35)< < Min(k16, k18, kh) and all words except ( D F ) are k-bounded by 1 - r' + t26; but since U' kIl ( D F ) and Max(so + s26,u35) < b25 we see from 6.1 OA that ( D F ) is k-bounded by 1 - r' + t26 + u4 < < Min(k17,k,,, kb). The result is that (in the notation of 3")

(i E I ) be denoted by I/, , Vf 8". Let the leftmost, rightmost sasay; these coincide if and only if r = 0. Choose i, E I ; then tisfies (S) or (R). If it satisfies (R) and k 2 2, choose i, E Z;' .'Then satisfies (S) or (R). If it satisfies (R) and k 2 3, choose Vil i, E j z . Then K l i z i 3 satisfies (S) or (R). Eventually either V i , *,., i , satisfies (S) for some w , 1 < w < k , or V j ,i , j k satisfies (R). We consider, in particular, two such sequences:

I;,

,,,

216

J. L. Britton, The existence of infinite Burnside Groups

(i) i, = 1 and i2. i,, ... chosen minimally. Denote this sequence X = { ~ l~ , 2 ...} , ( 0 1 = 1). (ii) i, =f'and i2, i,, ... chosen maximally; call this sequence p = { b l . b,, -.} ( b , =f). We shall assume that both V , and V p satisfy (S); if one or both satisfy (R) the subsequent argument becomes easier since 0-paths are easily dealt with. Let V,, Vp be subelements say of L t + , , L,+, respectively. Then 1 < t < k , 1 < s < k , h is a ( k 't 1 ~- t)-tuple and p is a ( k + 1 -s)tuple. V, clears a t-path and V A j(i E I,) has images throughout. The VAjsatisfy (a) of Axiom 30; the case when r = 0 and V , satisfies (S) is easy but exceptional and will be considered later; thus V,, Vp are disjoint. Let the two leftmost V k i (i E I h ) be say T,, T 2 . Let the two rightmost of the V p i(i E I p ) be S , , S 2 . Let their images in ( D E ) be T i , T i , 5'; ,S; and let their images in U be T i , T i ,S;',S i . Then ( T , , T 2 )7( T i ,T i ) 7( T i , T i ) . We have, say, 2 = ( ( T , , S2)ar) D E E ( T i , S ; ) D R E D L and U r ( T ; ' , S i ) y . We now use 1 1.1 three times; this is possible since

by

Min(r", P ) - u,

1 - r'

+ t26 < 1

- u4 - E ,

~

r'

>Z

C ~ ~ ,

+ t26 + u4 < k17.

If ( T , , T 2 ) ,( T i , T i ) produce E , K and (S, , S 2 ) ,( S ; ,Si) produce then, making use of 9.8 we obtain (Tl,S2) E.(T;,S~).K' and (Y K ' - ~ D ~ E D Let ~ E (-T~i ,.T i ) , ( T i ,T i ) produce E , , K , and let ( S ;, Si),(S;',Si) produce e 2 ,K ~ Using . 1 1.1 four times we obtain similarly ( T i ,Si) 5 € , ( T i , Si) K~ and DRFDL = ~z'y~;'. Let ( T , , T 2 ) ,(T:, T i ) produce e4, K~ and let ( S , , S,), (S;',S ; ) produce e 5 ,K ~ Then . E'E, ; e4, K ~ K'. = K ~ Hence . E ' , K'

x

J, L. Britton, The existence of infinite Burnside Groups

217

9". We shall show that words Oi, Wi, W;", 4i, E,, F, exist such that

(14)

~ i ~ - 0z ~~ ~W W~ ; ' ~ E ~ W ~

(1 5 )

W283. K

; ~

WkF, W i R @ ,

where W y , Wk,W Y L , W i R are non-empty. We shall then have the following. ( T i ,S;')y e i l . ( T 1 , S 2 ) .~ s l . =7 V ' , where

z

V ' = $1 W ; L E o W y 0 2 W ) F o W i R 4 3 y ,

-

V , where V = ( V ' ) . For the required J in the conclusion of Proposition 11.7 take W F 0 , W ) and take E l - W;0,cu0, W k , F' F W"R @ 3 y $ lW i L E , , where W i W k W r (i = 1,2). Part 2 of the conclusion will follow since by ( 14), ( 1 5)

U

,

so E'-'. F' is conjugate in G, to E-'. F (by 8"). Obviously the nice image J* in U of the subword J of '6 is contained in ( T ; , s;). 10". If a subword of Z clears a u-path (cf. 2") whose last term is T , say, then it induces a u-path with respect to (3) whose first term of D and whose last term is T. By 1 1.4 the iterated is a subword nice image D* of D with respect t o ( 3 ) contains T. Hence D* 3 T i , D* 3 S;; hence D* 3 (TY, T;'). 11". Let r > 0. Each Vi (i E I) is, in the notation of 7.8 of the form A;-'+, where y is an r-tuple. I f r = 1 , t h e V i ( i E I ) a r e A l , ..., A , c f . 7 . 8 s o w e m a y t a k e I = (1,2, ..., g}, 6 = Ai, f = g. Thus kernels exist for Vl , ... , Vf.

218

J.L. Brifton, The existence of infiniteBurnside Groups

If Y 2 2 consider the A-descendants of B , , ..., B, in Z n - r + l ; temporarily denote the leftmost, rightmost by X,Y respectively and the remaining ones by Wi( j E J ) . By 7.16A, if N is the largest integer < u , the ~ first NA-descendants of X in Znpr are left of and disjoint from all A-descendants in Zn-r of W j ( j E J ) and Y . (Dually for Y . ) Thus there is n o loss of generality in denoting these N A-descendants by V,, V2, ..., VN or in assuming (i) f i s an integer ( i i ) f ' 2 2N. The rightmost NA-descendants of Y in Zn-r may then be denoted by V:-N+l, _..,Vf.Kernels exist for V,,._.,VN and for VfPN+] , . _ . ,l / f S l n c e N ~ ~ 1 3 4 - 1 > 5 w e h a v e N ~ 6 , 1'212. Also g > eu36 2 12 so g 2 13. 12". Let Y > 0. Now T, c Vk c V , , S , C VpC Vf. Put (TI ,S2)= T1AS2, (Ti', S;') = T;'BS;. Note that A 3 ( V 3 , Vf-,). Recalling 8.8A, 8.8B we have

TI

t

~

TFEZFT:, l

= T~~LE'z'F~T~~R 1 1

1 t-1

where Z, Z' are subelements of Rep(Lt). As in 9.5

for a suitable Z * of Rep(L,) such that one o f Z , Z * is a right subword of the other, and a suitable Z ' * with similar properties. Hence

~, SimilarlyS2 s = l S k E 2 Z 2 F 2 SSg

S i L E 3 Z , F 3 S g R .B y ( l l ) ,

Next we show that each side of ( 1 7) is ((k))-bounded by b Z 5 . T F R S L is not a subelement of L u ... u L k since it contains V 3 and s k i l ( V , ) > u I 8 > Y; (cf. 5.21); hence we need only show that H = Z*FTp A S ; is ((/{))-bounded by b25. Now Tp AS," C 2, so is (k)-bounded by so. Z * F T P is a subword of a t-preobject so by 8.1 is ( ( t I))-bounded by kI2 and t-bounded by kI3. Now T: is n o t a subelement of L 1 u ... U L f P 1 so H is ( ( t - 1))-bounded by Max(k12,so) G 625.

J.L. Britton, The existence of infinite Burnside Groups

Next consider the t-bound of H. Where T2 t:l TY r f l T;'LE"Z"F"T;'R say, we have

219

T k E 1Z , F , TF ,

where each side of (*) is a t-preobject and one side is standard. We claim that z"has size < + e4 + Max(u2, u 3 ) = a say. This is true if the left side of (*) is standard. In the opposite case I z'" - I < e4 ; but by 8.5' I Z* - zQI < u3 or I Zy: +"Z ' - 1 I < u2 so the result is true in this case also. Let N be a subelement of L , contained in H . Note that Z*FT: is maximal normal in H . Let N F N , N 2 where N 2 = N n AS:. If N 2 is non-empty then N , is not normal and s ( N 2 )< so, hence s ( N ) < so + ( y o + e 3 )+ 2/e. If N 2 is empty then s ( N ) < a + e 3 . Hence H is t-bounded by b25 since Max(so + ro + e3 + 2/e, a + e 3 )< b25. Let t < i < k. If N is an i-subelement contained in H then s(N n Z * F ) < rb, so H is i-bounded by so + rb + 2/e < b25. The left side of (10) contains A hence V4 so applying 6.10 V4 3 V t V i 3 V i since s k + l ( V 4 )> u I 8 > u3. Here V i has size > s( V 4 )- u4 > rb. Similarly for VfP3. (1 7) has the form (1 8)

A , J $ 4 2q 3 A 3

;B , ViB2 v;-3t33

whereA, =Z*FTPAL,A3=ARS,LE2Z2*. We prove ( V i , V;-3)C B ; we may then write B , = Z'*F'TYRBL, B 3 - B R S ; L E 3 Z ; . Let W = V3\V4. Then W 3 Wo W ' > X where X is a ( k + 1)-subelement of size > rb. Now Z'*F'TYR is a subelement of L t so does not contain W' by 5.21. Thus V i C B. (Since W is disjoint from V4, W ' is disjoint from Vi.) Dually VjP3C B. From (1 1 ) we have say

-

V;-3=X3. V;-3.X4,

A 3 = X 4- l . B 3

J.L. Britton, The existence of infinite Burnside Groups

220

-

Axiom 3 1 B is applicable to V t V i since V:, V i are ( k ) bounded by so < b31 and by 6.8 s( V t )2 s( V4) - u1 (since u18 > 111) > 2iI8 - 1 4 , > Hence there is a word ViLCVtR( k ) bounded by so + q31and equal in G, t o V i ,X , where ViL is n o t a subelement of L u ... u L and s( > s( V4) tl -~ 3 > 1 u18 - u1 -

,

Vp)

r3, > rb. Similarly V:L3DViF3 = X , V j P 3 .Now

= T l A L V p 1 2 V;-3ARS,

(12')

(T,,S,)

(13')

( T y , S i )= T;BL V i B , Vip,BRSi.

We show

(15')

Vf-3 o A R S2 .

K5 - ~=

VoL f - 3 DV'R f-3 BRS,.

The right side of (7') = T;BL V i .X , = T;BLX;' V s =

€4,T , ( Z * F T ~ ) - ' ( Z ' * F ' T ; R ) B L XV:~ l (by 16)

= €qlT1(AIAL-l)-l(BIBL-')BLX-lVO 1 4 = €ql T , A L

q.

Similarly ( 1 5 ' ) is proved. Thus (1 2'), ( 1 3 ' ) , (14'), (1 5 ' ) have the same form as ( 12), ( 1 3), (14), (1 5 ) , taking W,,W,,W;l, W ; t o be V!, V i , Vi-3. Thus 2 of the conclusion of 11.7 is proved. In the present case J is V F A , V?F3 and

q3,

JF'

3

V t R A , V;t3 D ViR3BRSiyT;BL Vi" C

which is (k)-bounded by so + y 3 1 < u j 5; thus 1 follows. By 9" the nice image J* in I/ of the subword J of V is contained in ( T r ,S;'). By 10" D* 3 (Ty,S:) 3 J*. Thus 4 follows. Finally J 3 A , 3 ( V 6 , Vf.51) so 3 follows. 13". Let Y = 0 and let C', satisfy (R) (cf. 6"). Thus the V l i (= B i ) are disjoint, V , clears an ( M - 1)-path, V l j (i E 11)have images

J. L. Britton, The existence of infinite Burnside Groups

221

throughout, I , = {sl + 1, ..., s1 + t , } , 1; = {sl + 2 , ..., s1 + tl - 1) (for some sl, t l ) , c V kc V1 $ , + 2 * S 2 C V,, C V1 s l + t - 1 . Since V , clears an ( n - 1)-path, (4) may be improve& by replacing k ( = n )by n - 1. Now T 1 A S 23 A 3 ( V , v, $ + t , - 2 ) . Put a = s1 + 4, b = s1 + tl - 3. Then if a < p < b we have &om the C B . Hence analogue of ( 1 7), ( 18) V l p3 Vyp

-

( T 1 , S 2 )= T I A LV f a A 2VybARS2 (Ti,,!$)

=

T;'BLViaB2 V i h B R S i .

As before

hence U nzl (H-y)where ( H y ) is ((n - 1))-bounded by so + q 3 1 < ~ ~Now ) . U is n-bounded by so hence by 6.10 ( H y ) is proving 1. We have U (H-y). n-bounded by so + u4 < 2 and 4 follow as before. Vlb-l)=(Ba+l,Bb-l)= J i s here V ~ ~ A so 2 contains(V1a+l, V ~ ~ =(BO+lr Bg-Q ) , ~ i n c e g = 2 s ~ + t ~ s ob a= +g + 1.Wemustprove (g - 2a)/rn' > g/m'- u37,i.e., (2sl + 8)/m' < u37. In 4" we have ( 1 4 d + 192)/t < 7 a + 103/e, so analogously from 5" we deduce 2 sl/m' < 7 a + 103/e, where a = ekO+ ~ 1 +3 6 ~ 4Hence . (2sl + 8)/m' < 14a + 214/e < u ~thus ~ 3;holds. 14". Let Y = 0 and let V , satisfy (S). V , clears an n-path and Vli (=Bi) ( j = 14, ..., a - 13) have images throughout (cf. 3"). Also a = g > m'u36.Put T , = B14, T2 = B15,Sl = Bg.. 14,S2 = B,- 1 3 , and let their images in U be T ; , T;, Sy,Sg. Say

< Min(b25,u

Again we have (1 1) and the two equations immediately following (1 1). We wish to apply 9.15 t o ( T , , T 2 ) ,( T i , Tg). Each is (n)-bounded by so; also ki4 < so < k24. Now s ( T l ) > Y" > 2139 and by 7"

222

J.L. Britton, The existence of infinite Burnside Groups

s(T;') > r" - u3 - u4 - e3 > 2139. Hence by 9.15 there is a word T;'LCTF, (n)-bounded by so + ~ 4 and 0 equal in Gn to .(T,, T2). Similarly there is a word S:DSiR = (S,, S 2 ) . ~ J ' .Here TItL, 1 TF , Sp, are n-normal. Consider T;'L CTF JS: D S i R y = V' Then U ( V ' ) ,and ( V ' )is (n)-bounded by so + ~ 4 < 0 u35 (we note that i f N is a subpowerelement of L, contained in ( V ' )then N cannot properly contain TF since T2 is maximal normal in Z . Similarly N cannot properly contain T;'L,SFSiR). Thus 1 , 2 , 4 follow as before. J contains B17, ..., Bg-16 so it remains to prove that 2s/m'< u37 where s = 16. But 2sIm' = 32/m' < 32/e < u37.

€4'

SiL

1 1.8. Lemma. Let

where the jollowing conditions hold. 1 . Z, E Rep(LE+,). 2. p = Min(n, k ) Z 1 . 3. I f ' p = n theri Conditions A , B hold. 4. I11 the notation o f 1 . 8 , all A-descendants of B , ,,, B, in Zo are contuined in D, where g is such that 1 < g < m , kglm > u36. Th('11

zp

53

(DPEP)

where: 1 . ( D , F p ) is (p)-bounded b y u35. 2 . E ; I . F p ;T - ' . E - ' . F . T f o r s o m e T. 3. I f p < 11 then D, contains all A-descendants of B , ,,.., B, in Z p except possibly jor the five k f t m o s t o.f B , and the five rightmost of B,. I f p = n, D, 3 ( B , + s , B g p s where ) k ( g - 2s)lm > kg/m - u37. 4. In Up ; ... 7 Uo = ( D F ) let the iterated nice image of D in Up be denoted by I,; in (*) let the nice image o f D p in Up be Dp*; then Dp* c Ip./ti purticiilur D i , I , exist.

J.L. Britton, The existence of infinite Burnside Groups

223

Proof. Call the conclusion P ( p ) . Then P ( 0 ) is true, taking D o , E o , Fo t o be D, E, F respectively. Assume P ( q ) is true for some q , 0 < q < p . Then D: c Zq. We wish t o apply 11.7 to Zq+l

--f

Zq = D , E,

This is possible since if r = 0 then n - 1 = q < p < n , so p = n ; hence Conditions A, B hold. If X C Uq let f ( X ) denote the iterated nice image of X in Uq+ 1 , with respect to U q + l -,Uq. By 11.7 we obtain Zq+l=

Dq + 1 E q + l

and J* c D*, where J* is the nice image of D, + 1 in Us + under (20) and D* is the iterated nice image of D, in Uq+l under ( 19). Now f(D,*) = D * , I q + l = f ( l q ) and D,*+l =J*. Since D: C Is C Uq we havef(D;) c f ( Z q ) i.e., D* C l q + lBut . Dq*+l= J * C D * , hence q + 1 = Iq+P

11.9. Lemma. (Linear version of 1 1.8 (cfi 1 1.7A)). Let

where U,, ..., Uo are linear words. Let 1 , 2 , 3 , 4 o f the hypothesis of 11.8 hold. Then

where 1. G p D p H p is (p)-bounded by u35

224

J.L. Britton, The existence of infinite Burnside Groups

2. c . E - ~ . H =G . E - ~ . H p P P P' 3. Part 3 o f the conclusion of 1 1.8 holds 4. I I I Up+. ... .+ U, f GDH let the iterated nice image of D in U p be l p ;in (*) let the nice image of D, in Up be D;; then D: C I p .

§I2 Informal Summary Axiom 2 0 may be regarde.d as the main axiom. In this section we prove (in 12.2, 12.18) that part (a) of Axiom 20 holds for n + 1. (The rather deeper result that part (b) holds is given in Section 13.) We also prove that Axiom 25 holds for n + 1. 12.1. Consider the following condition. Condition C. If A nrlC, AB E L n + , and CD E L , , then L ( A ) / L ( A B )< 2141.

12.2. Proposition. Assume Condition C. I f D E E L n + , , D F E 0 < k < n then sDB.(D) < rb.

Lk+l

Proof. Assume sDE( D ) k rb. Put E = 1 / e 2 . Then by 7.1 1 Z,, 2 ... 7Z , = ( D E ) ,2,E Rep(L,+,). There exist B i , denoted without loss of generality by B , , ..., B, such that all A-descendants of them in 2, are contained in D , 1 < g < m, g / m > rb - E . Also zi 2 ... f Zb ( D F ) ,zi E Rep(Lk+1). Case 1. k = 0. ( l a ) Let n = 1. If Z , is different from 2, then by 3.7 A , , ..., A , are maximal 1-normal and contained in D. Now

g > m(rb

~

E)

> e(rb - l / e * )> 1

,

so we have the contradiction that a subelement D of L contains two distinct maximal 1 -normal subwords. Hence 2, = Z, . Thus D L E L,, DF E L , . By 5.16 L ( D ) < q'L(DE). But I L ( D ) / L ( D E )-g/m I < 3 / e (cf. 2.16A) hence g/m < 3 / e + q' < rb - l / e 2 , a contradiction. ( 1 b) Let n > 1. Let A , be an A-descendant of one of B , , ..., B,

J.L. Britton, The existence of infinite Burnside Groups

225

in Z , . By 7 . 8 , Z , 3 A , 3 BYi and C Z , (i = 1 + j , ..., a - j ) . All A + C D . If the B+ are maximal I-normal we have the contradiction that two distinct maximal I-normal subwords are contained in D . Now recall 7.5,7.7. Any x consecutive BTi's generate a subword of A , of size > x / t - E;,. Hence any x + 12 consecutive A+'s generate a word containing a 2-subelement of size > x / t - ej0 - 3u4. In particular, taking x + 12 = a - 2j, D contains a 2-subelement of L 2 of size

where P temporarily denotes u18 - 6 ~ 2 0 €30 - 4 8 / e - E;, Now 2 rb + u4 > rb in contradiction to 5.2 1. Case 2. k > 0 . By Lemma I 1.8 since g / m > rb - 1/e2 > u36

- 3u4.

(2a) Let n > k + 1 . Now if A , is any A-descendant in Zk+l of B,, A , 3 BTi,A . C D k . If the A + are maximal ( k + ])-normal then Y' by (T) Z i contains two maximal ( k + 1)-normal subwords (cf. 3" of 1 1.7A), a contradiction. Hence we may assume as in Case 1 that D k contains a ( k + 2)subelement of size > 0.Now k + 2 G n so by (t)Z i contains a ( k + 2)-subelement of size > 0- u4 2 rb, a contradiction. (2b) Let n = k + 1. First suppose Z n P l is the same as Z,. Then Dk is a subelement of L n + , hence of L,+l. From (2a) if N is a subelement of L 1 U ... U L n - , contained in Dk then s " + l ( N )< rb. By considering 6.8 and (t)we have D k 3 D! DL C Z i where S"+'(D;) > Sn+'(Dk)- U 1 . But S(Dk) 2 ( g - 2)/m > Yb - l/e2 - 2/e. NOW I L(D,)/L(DkEk) - ( g - 2)/m I < 3 / e SO

J . L . Britton, The existence of infinite Burnside Groups

226

contradicting Condition C. Finally let Z n - , be different from 2,. Then B , , ..., Bg are maximal ri-normal, D, contains A,, A , hence by (?) 2; contains t w o maximal n-normal subwords, a contradiction.

12.3. Corollary. Condition C implies Condition A. 12.4. Note. I f A is a subelement o f Rep(L,) of size > ,s, then there is a unique I? such that A B E Rep(L,); this is by Axiom 33 since A r i - 1 A . Temporarily denote this B by A ' .

-

12.5. Definition. Subelements A A , of Rep(L,) of size > s, satisf.ii.(0)i f A , A ; i s a C . A . ofA,A;;satisfy ( I ) i f ( A , A i ) - ' i s a C . A . of A 2 A ; . 12.5A. Let each of i, j , k be 0 or 1. I f A l , A 2 satisfy (i) and A 2 , A , satisfy ( j ) then A A , satisfy ( k ) where i + j = k (mod 2).

,,

12.5B. A

A cannot satisfy both (0), ( 1 ).

Proof. Assume not. Then Z = A , A ; is a C.A. of Z - ' . Say Z = CD, Z-' = DC. Then CD = C-'D-' so C2 = I, D2 = I . Since e is odd, fl = Go has n o elements of order 2, hence C = I , D = I , Z = I which is a contradiction. ((e odd)) 12.6. Proposition. Let EZF ,yl AZ"B where each side is ( ( n- 1 ))bounded by k25 and 2, 2" are subelements oJ'Rep(L,) o f size > ~ 4 2 . Assume that iJ'X is a subelement o f 1 contained in E or F then s ( X ) < 4,. Then 2, 2" satisfy ( 0 ) . Proof. Since k25< b25 and ~ 4 22 u , we have Z" 3 Z"O nyl W 3 A where s ( A ) > s(Z") - 1 1 4 . Now A c E Z F ; consider A f' Z = J . Either J is A o r J is Z o r without loss of generality A c EZ and A n E has Thus in any case size < u;, so s(J) > s ( A ) - 2/e - 4,.

J. L. Britton, The existence of infinite Burnside Groups

221

-

s ( J ) > ~ 4 -2 u, - 2/e - ub3 = or say. Now A 3 .I> JO X , where X C Z"0 C Z " , s ( J 0 ) > or - u1 > s,,. Hence by Axiom 3 3 , X = J o . Thus Z " , Z satisfy (0).

12.7. Proposition. Let X , Y be (n)-bounded by k26 and X ; Y with divisions P,, ..., Pk and Q,, ._.,Qk. Let T, U be maximal n-normal in X , Y respectively of size > u4 and images o f each other in the sense of 6.16. A s in Axiom 33' let T ,.,yl T L E , Z , F , T R ,

U n = l ULE 2 Z 2 F 2UR

Then (i) if T = &, U =- Qi (where P i , Qi have type n ) then Z , , 2, satisfv (1) of 12.5; (ii) in the opposite case, Z , , Z2 satisfy ( 0 )o f 12.5.

Proof. (i) In the notation introduced in 6.7 we have T 3 Pl!, U 3 QI, Pl!-lX,'l\ QiX,' E Ln. Now s(P,') + s(Q:.)> k! hence s(Pl!)> k! - k26 > t3, and Pi = PiL E,Z, F , Pi' R , s(Z,) > k; - k26 - r33 > s3, so Z,Z; E Rep(L,). Similarly Q ; = Q:.LE4Z4F , QIR, Z 4 Z iE Rep(Ln). By Axiom 33' (using k: - k26 > j,,) Z;'Z;-' is a C.A. of Z4Zi, i.e., Z,, Z4 satisfy (1). Now U L E 2 Z 2 F 2 U Rnyl U = ULQ:URn-1 = ULQiLE,Z,F4QiRUR

so by 12.6 Z,,Z2 satisfy ( 0 ) (u4, - Y , ~ > ~ 4 2 k! ; - k26 - r3, > ~ 4 2 ; q3, + E, < 4,;ro < ub3;k,, < b,,; k26 + c , ~< k,, .) Similarly Z , , Z , satisfy (0). Hence by 12.5AZ,,Z2 satisfy (1). (ii) 6.25 is available since k,, < k i , k26 < k i and u4 2 ui0. Hence there are subwords T , , U, of T, U each of size > s ( T ) > (= or, say) and such that T , l : n U,. As in the proof of > u44 8.9 we deduce that

,

,

for some H , , H,, K , , K , which are subelements of L u '.- U L - ; we require k26 < k14, or, > u2,. Moreover s( Tr;') > s( T , ) - 22126,

228

J.L. Britton, The existence of infinite Burnside Groups

s ( U F )> s ( U 1 ) ~

2 ~ 1 and , ~

By Axiom 33', since a ,

each side of (*) is ( ( n - 1))-bounded by

-

2u26 > t3, and a ,

H , T M LE 3 Z , F , T Y R K ,

nfl

< b,,,

H , UYLE, Z , F, UY" K ,

where each side is ( ( n- ]))-bounded by a , + c3, < k25; also s n ( T y L E 3< ) q 3 , and s(Z,) > s ( T f l )- r3, > a1 - 2u26 - r3, > ~ 4 2 . If X is a subelement of L , contained in H , T f l L E , then s ( X ) < rb + 2 / e + Max(ro,q3, + k,) < 4,.By 12.6 Z,, Z, satisfy (0). Let T = T ' T , T " , U - U ' U U " , T,= T k T y T p , U , - U ~ U ~ U ~ . Now T L E , Z , F , T R n=l T ' T I!, T ~ L E 3 Z 3 F 3 T Y R T ~12.6 T " . is available since k,, < b,,, k26 + c,, < k25, ~ ( 2 > , )s ( T ) - r3, > u4 - Y,, > 1142 (by above s(Z,) > ~ 4 2 ) If. X is an n-subelement in T LE , then s ( X ) < Max(q3, + E , , y o ) < 4,. Hence Z , ,Z , satisfy (0). Similarly Z,, Z , satisfy (0). Hence by 12.5A Z , , 2, satisfy (0).

12.8. Proposition. Let

where euch side is ( n t b o u n d e d b y k,,, induce Y , ; X , (end words E,, K , s a y ) and Y , ;r X , (end words E,, K 5 ) . Let R , S be maximal n-normul of size > ZL,, and images oj'each other in (1). L e t

and hence

J.L. Britton, The existence of infinite Burnside Groups

where the three right sides are (n)-bounded by k,, let the image o f S in Y be S'. Then S' is R .

-

229

+ u46.For ( 2 )

Proof. By 6.25 we have R M SM(end words E', K' say) since k27 < kit k; and u45 2 uio. (If R = pi, S = let R M = Pf,SM= QI ; then s ( R M )> k! - k27.) Say R = R L R M R RS, = SLSMSR. Thus

a

-

Y , Y 2 Y 3 R L R M X , X 2 X 3 S L S M , Y , Y 2 Y 3 R L R M= Y , Y t E X F X , S L S M .K'. We claim that

Y , Y 2 Y 3 R fi Y , Y 2 Y 3 R L R M Y , Y,LEXFX3SLSM +

fi

Y , Y,LEXFX3Sfi (In Y , S ' )

where the terms are right objects. The first and second terms are right objects since k24 < u46 + k27 < ki2,ki3 ; u452 ~ 2 ;4s(RM)> M i n ( ~ .y~o, ~ k! - k27) > ~ 2 +4 E ~ They . are almost equal by 8.13. The third term is a right object similarly. Apply 8.15 to (2); this is possible since k27 + u46 < k16,k,, and u45 > u29.Hence the fourth and fifth terms are right objects and are almost equal. The third and fourth terms are almost equal by 8.13. Now 6.23 is applicable to (2) since k27 + u46 < kb, kb and u45 > ul0. Hence s(S') > s(S) - u3 > u45 - u3 > uj, or s(S') > 1 - u2 - k27 > uj0. By 9.12A, R is S' (k27 < k,, k,g ; u45 > u;o>. 9

12.9. Proposition. Let ( i ) Y ; Y L A Y Rand (ii)A ; A L B A R ,so that (iii) Y ;Y A LBAR Y R, where all these words are (n)-bounded by k2, . Let S C A and let S be maximal n-normal in Y L A Y R . Let T C B and let T be maximal n-normal in Y L A LBAR Y R . Let s(S), s( T ) > u47.Let S' be maximal n-normal in Y . Let A L , A R be nonempty. Let S ' , S be images of each other in ( i )and let S, T be images of each other in (ii). Then S ' , Tare images of each other in (iii).

230

J.L. Bntton, The existence of infinite Burnside Groups

Proof. With respect to (iii) let T' = Im' T ; by 6.23 this exists since k2, < lib, k$ and ii47 2 21 ; also either I s( T ) + s( T' ) - 1 I G u2 or I s( T ) - s( T ' )I < u 3 . There are two similar inequalities for S, S'. Using right n-objects, (In Y , T ' ) Y L A L (In B, T ) (cf. 8.15, noting that k2, < k,,, k17 and u47 > u2,). Similarly, ( I n Y . S') 5 Y L ( l n A , 5') and (In A , S ) * A L ( I n B , T ) . BY premultiplying both sides of the last expression by YL and using the transitivity we obtain (In Y , S ' ) * (In Y , T ' ) . From this follows by the dual of 9.12A that S' is T' (we note that k28< k,,, k19 and each of S', T' has size at least Min( 1 - u2 - k28, ~ 4 -7 u 3 ) > ui0 > u ~ ~ . )

12.10. We introduce the following temporary notation. In the notation of 7.8, i f X is a family of A-descendants in Z n p r ,where 1 < r < n - 1 , let D ( X ) be the following family ofrl-descendants in Z,z 7- 1 . let Y arise from X by deleting the leftmost 1 1 and the rightmost 1 1 members, for each A , E Y put all A+ in D ( X ) . D 2 ( X ) means D(D(X)). '

12.1 1. Proposition. Assume the following: Z,, ; ... 7 Z, where Z, E Rep(L,+,); Z p = ( C D )f o r some p , 1 < p < 11; C contains: U , , ... B, i f ' p = n all members of' D n p p - l ( A 9 , .... A,-B) if p < n, where g/m > ~ 1 Y ~ C ~where ; Y is (p)-bounded b y k29. Then there exist 8;- C ' , D' such that 1. zp3 z;-, ( C ' D ' ) . 2. YM z,C' j b r some subword YM o f Y . 3. I n 2..' r,;l Z p ; Zip,, C' contains all members c1.f D'l "(A,, A,_,).

-

,,

$

. . . I

Proof. I " . Say P 1 , ..., P, and Q1, ..., Q, are divisions for Y ; C. If Q ihas type p denote by Ri the maximal p-normal subword of Z p containing Qi. Since k! < s(Pi)+ s(Qi) we have s(Qi) > k! - k,, > r' + e3; hence s ( R i )> r'. Let Z;I-l arise from Z p by simultaneous replacement of all R i ; if there are n o Ri let ZA-, = Z p . Then Zp ; Z;-, so by Axiom 26 ZppI1 Y , Y , Zb-,, say. t a s e I . p < 11.

23 1

J.L. Britton, The existence of infinite Burnside Groups

Let the members of D n - p - l ( A 9 ,..., Ag-8) be ai (i E I ) ; they are ( p + 1)-subelements. As in 1 1" in the proof of 1 1.7 denote the leftmost members by a , , a,, ... and the rightmost members by ..., of-,,af.Consider the A-descendants of any ai in Zi-,. By 7.8 and 4" of 1 1.7 ai contains p-subelements ais (s = 1 , ..., a say) and either (a) ai clears a p-path, or (b) ai clears a ( p - 1)-path, in which ai i + l , ..., ai a-i have images throughout; denote the image of ais in Y , , Y,, ZAP, by ais, a;.:, pis respectively, s = j I- 1, ..., a - j. If (a) holds these maximal p-normal. The &, s = j + 1 , ..., a - j , are the A-descendants of ai in ZAP,. If (b) holds we see from 7.4 and Axiom 30 part 4 that i + 1 , aj' a-i ) in Y , and the three similarly defined subwords of Z p , Y,, ZLp1each contain a ( p + 1)-subelement of size > ((a - 2j) - 4h)/t- hu4 - where h = 0, 1 , 2 , 3 for Z,, Y , , Y,, Z i - , respectively. Hence since s ( a i ) > u18 each contains a ( p + 1)-srtbelement of size greater than u where

u = (u18 - 6 ~ 2 0 € 3 0 ) - 36/e - 12/e - 3u4 - E

; ~

(cf. 7.7A). Since u > rb each is not a p-subelement. In this case, contains a p-subelement J then a j conmoreover, if (pii+,, tains a p-subelement of size > s ( J ) - 3u4; hence by Axiom 30 s ( J ) < c* + 3u4 (note that ~ 1 +4 2u4 < c * + 3u4). 2". We prove that Z;-l is p-bounded by /3 = $. + e3 where

t = 1 -k!

+k29+€3+Y26.

Suppose M is maximal p-normal in ZL-, of size 2 t . By Axiom 26", since t > u26, there exists N , maximal normal in Z p such that I s ( N ) + s ( M ) - 11 < r26; for if I s ( N ) - s ( M ) I < r26 then t < s ( M ) < s ( N ) + r26 < so + r26 < k29 + r26 < t , a contradiction. Thus N is replaced in Z p -+ ZL-,, i.e., N is of the form R i . Hence s ( M ) < 1 + r26 - (k! - k29 - e3) = t , a contradiction. Similarly Y , is p-bounded by 0. Since Z p Y , we have by + s26 < b,, , 6.1 OA that Y , is p-bounded by so + 214 < p; u3 < so < h, - E , - 2/e - u 4 ) .

-,

6;

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J.L. Britton, The existence of infinite Burnside Groups

Since so + s,~< and k29 < 6,each of Y , Zp, Y , , Y 2 ,ZL-l is (p)-bounded by 0. 3". We use the following notation for the p - or ( p - 1)-path clearedbyai: Hi- J i > Ki- L i 3 M i - N i . T h e n a i 3 H i , J i C Y , , Li c Y , and Ni c ZLp1.Now Hi c ai C C ; Y so Hi 3 C. Di ' P where Ci is Hi" (with respect to C ; Y ); that Hi" exists will appear below. First suppose the ais satisfy (a) of Axiom 30. Then they are maximal p-normal of size > r* > r* - 3u3 > u9 so by 6.16 their images in Y (with respect to C- Y ) exist since 0< k,, k,. 'In particular a:' exists. Now Hi 3 ais so Ci = Hi" 3 a k ; hence Ci contains properly two of the air. Using 6.28,6.27, which are available since r* > Y * - 3u3 > u5 and /3< k,,, and putting 0,= p + q 3 1 ) we have the following. There are words CF Fi0: , DF EiCF (p)-bounded by PI such that Di ;DFEiCF. Ki ;El. C&F.DR where Ef, KI are end words for Di Ci. None of D,: CF, 0: is a subelement of L 1 u ... u L,. The last part of (*) holds because of the four words properly contains a maximal p-normal subword. Now suppose the ais satisfy (b). Then from 1" Hi contains a ( p + 1)-subelement Wi of size greater than u. Now u > u , so Ci = HP 3 Wo,s ( W 3 ) > s(Wi) - u,.6.28, 6.26 are available since 0 < k52, u > u4 so we have (*) again. Hence (cf. 6.7)

-

-

(3)

brr

Y ; (In Y , Dk)E(CF,CF)F(D& Fin Y )

where E = E l , F E Ff; the right side is (p)-bounded by 0,. 4". If the ais satisfy (a) of Axiom 30 then all of ais,aIS,a;,, have size > r* - 3u3; this is by 6.23 since 0< k;3, kb and r*-2u3>u10and 1 - p - u 2 > ~ * - u 3 . Note that (Cp, C t ) contains (H4,Hf-3) properly. Just as we obtained (1 ) we can find an equation (4)

BiS

J. L. Britton, The existence oiinfinite Burnside Groups

233

and similarly

(6)

(M,,M~-~ M)~;E ''(N;, ' N;-~)F'' ' M ~ R _ ~

Combining (1) t o (4) we have for some H, K

and the right side of (7) is (p)-bounded by PI. 5". Let 8 be a temporary parameter. Let V be maximal p-normal in ZLP1of size > 8 2 u26 and contained in ( N l o ,N f - 9 ) . By Axiom 26'' it has images S, T, Usay in Z p , Y , , Y 2 respectively. Also either I S(S) - S ( V )1 < Y26 or S(S) > Y'. We have u c ( M 8 , M f - 7 ) C (L8,F f - 7 ) , T c ( K 6 , Kf-5) C (J6, and S is contained in ( H 4 ,H f - 3 ) which is properly contained in (C:, C;) C C Y . Let Im'S = R C Y ;this exists by 6.15, and Im'R = S , since /3 < Min(kb, kb), r' > ul0 and s( V )- r26 > 8 - r26 > ulO. Hence by 6.23 Is(R) + s(S) - 1 I < u2 or Is(R) - s(S)I < u3. By 12.8 R , S are images of each other in (1 ); for 0< k27, Min(r', 8 - t-26) > u45 + u 3 , 1 - u2 - so > u45,0,< u46 + P. Now S C ( H 4 ,H f P 3 ) (J4,J f - 3 ) 3 T ; in fact S, T are images of each other (Im'S = T but since s(S) > ul0 also Im' T = S ) . Now

-

-

Min(r'-2u3, 1 - u 2 - s 0 - u 3 ,

1-

P - U ~ ) > ~ - Y ~ ~ - ~ U ~ulO, =~>

hence V,S, R, T, U have size 2 y. By 12.8 since y > u45, S, T are images of each other in (2) T, U are images of each other in (3) U, V are images of each other in (4). By 12.9 since 0,< k28 and y > u47 R, V are images of each other in (5). 6 " . We first claim that (N,,, N f P 9 )= A(NFo,NF-9)C= ABC for some A , B, C where A , C are not subelements of L1 u ... u L, and B is nice with respect to ( 5 ) , hence Y M ; ;B for some Y M .

234

J. L. Britton, The existence of infinite Burnside Groups

To see this, first let a,, satisfy (b) of 1". Then N,, contains a subelement X of L P + , . Let N , , = A A , where A is a subelement of L u ... u L, and A is chosen maximally. Then A , $ I by 1" ; say A , = c A 3 . Let A , = A , A , where A , is a maximal subelement of L I LJ ... LJ L,. Then A , I since otherwise X C A l c A 4 and s ( X ) = s ( AI ~ A n4 A') G 4/e + 2 ~ ;+ E, < u < s ( X ) . T h ~ i sthe required A will have the form A c A . ~ If a,, satisfies (a) then it is easy t o select A . Next we prove that in

,

+

(8)

Y"

,

-B P

p may be replace by I, - 1. Assume not. Then some Pi, Qi for ( 8 ) have type p ; let R' = p7 C Y Mand V ' = Qi C B . Now V ' C ZA-l. Let V 3 V' be maximal normal in Zlj-, and let R 3 R' be maximal normal in Y . Then V C (N,,, N f p 9 )by the choice of A , C. Now k: < s(Pi)+ s(QI) < p + (s( V ) + e3) so s( V ) > k: - p - e3 > 8. Similarly s ( R ) > k: - p - e3 > 8. Using the notation R , S, T, U, V of 5" (which is consistent with the present use of R , V ) choose for each of R , S, T, U, V a fixed decomposition as in Axiom 33' (e.g., V P = ,V L E Z F V Rsay); this is possible since y > t33 and p G b33.We say that a pair of them satisfy (0) or ( 1 ) according as the corresponding pair of Z's satisfy (0) or ( 1 ) of 12.5; note that each Z has size y - Y~~ > s33. By 12.7, R , V satisfy ( 1 ) (since PI < k,,, /3< k,,, y > u,,). By Remark 12.12 below, one row of the following partial table holds, where an entry 0 or 1 is read as the statement that the indicated pair satisfies (0) or ( 1 ) : R,S

S, T

T, U

U, V

0

0

0

0

1

0

0

0

1.

R, V

By the definition of ZL-l and by 12.7,

R. S satisfy ( 1 * S is replaced (i.e., S is an Mi in Axiom 26)

J.L. Britton, The existence of infinite Burnside Groups

235

Thus the column R , S has entries 0, 1, 1 in rows 1 , 2 , 3 respectively. By 12.3A the column R , V has entries 0, 0,O respectively, i.e., in all cases R , V satisfy (0). By 12.5B we have a contradiction. (6) now reads YMP,: B and since B C Zi-l 1, 2 of 12.1 1 hold. B contains (N12,Nf-ll) so contains all A-descenda-its in Zi-l of all ai except a l ,...,al1 and afPl0,..., af.Hence 3 holds. 7". Case 2. p = n Recalling the definition of the B , (cf. 7.8), first suppose the Bi are maximal normal of size > r " . C contains Bi (i = 1, ..., 1 2 ) ; let their images in Y be Yi and let their images in Z;-l be Ai.As in 2", Zi-lis p-bounded by and as in 3" Y; (InY, Y f - ) E ( B ~ , B ~ - I ) ~ ( Y ~Y, F) ai n d ( B 4 , B 3 ) = ;B,LE'(A!?,Ak-4)F'B . ( r " > r * ) . Hence Y 7 H(As %- , A gc- 4 ) K og- 3 ' and YM ( A 7 , , where the right side contains A , , ..., Ag-8 as required. Finally suppose the Bi are not as above. Then by 7.3 (Z,) does not contain a maximal n-normal subword of size > r " . It follows that Yl:n C, otherwise by the beginning of 1" 2, would contain a maximal n-normal subword of size > r' > r " . Moreover there are n o replacements in Z , -+ Zn-,. Thus Z;-l = ZA-l is just Z n - l . Also Zn --= Zn-l. F o r the required Y M , C' take Y, C.

-

12.12. Remark. When we prove, later, that Axiom 26 holds for n + 1 it will appear that, in the notation of Axiom 26, (i) if U is an Mi then exactly one of Im U,,Im U2 arises as in Case (a) of 6.16; (ii) if U is n o t an Mithen both Im U,,Im U , arise as in Case (b) of 6.16. Further, (iii) Im U always arises as in Case (b). We shall therefore add these properties to Axiom 26. 12.13. Corollary. With the hypothesis of 12.1 I , Y contains an ( n + 1)-subelement of size Z glm - 40/e.

pzl

Proof. We have by 12.1 1 Z , 2 ... Zp -+ Z i P l = C ' D ' . Now Zb-l is ( ( p - 1))-bounded by so (by Axiom 2 6 and so < b2*,a28 so by Axiom 28

J.L. Britton, The existence of infinite Burnside Groups

236

and 12.1 1 is again available. Hence by induction

where H, = C"D" and Y m ; C " , i.e., Y m z C" and C"contains all members of Dn-'(A,, A g - 8 ) . Hence C" contains an A-descendant in H , of A , , and one of (cf. 7.9). Hence C" contains all A descendants in H, of A , , , ..., A g - 2 0 . Hence s(C") 2 ( g - 40)/rn 2 g/m - 40/e. 12.14. Proposition. Condition B holds.

Proof. Assume p > 1 since the result is trivial if p = 0. We have Z,, ... + Z , F A B , where Z , E Rep Ln+l and all A-descendants in 2, of B , , _..,B, are contained in A and g/m > s ( A ) - E > ~ 2 5 for , . since some preassigned E satisfying 0 < E < $ ( A )- ~ 2 5 Now b 2 5 < b& we have by Axiom 28 say A , ; ... 7 A , 7 A . Hence A = A (cf. Axiom 26L part (ii)). By 1 1.9 since p < n and -+

P P s25 2136

zP r A'B'

A

;A , ;GA'H.

where A' contains all A-descendants of B , , ._.,Bg in Z p except for the five leftmost of B , and the five rightmost of Bg. They are ( p + I)-subelements; denote them by ai (i E I ) and denote the leftmost four and the rightmost four by a l , ..., a4 and af-3,..., af respectively. Then ai3 ay ; ;a: C A . Also (a:, a y ) ;;(a;, a>) C A where each side is (p)-bounded by b 2 5 (since so < b 2 5 ) . Let Pi be a weak image of aiin ( a ; , a>) and let y i be a weak image of pi in Y , withrespect t o A - Y , ( i # l , f ) . T h e n Y M = ( y 2 , y f f - l ) = = 7 5 . W 3 2 , P ~ - l ) ~ ~(cf. ~ 6.29) p , and (P4,Pf-3) = P t E ' ( a f , a f - 3 ) F1PRf - 3 P

hence

Y M ;H(aF, a F p 3 ) K = H A t M K say. Hence Z p 3 (AtM)' ;r Y M M and (AtM)' contains (a6,afP5).

J.L. Britton, The existence of infinite Burnside Groups

231

We wish to apply 12.13. We have b25 < k29. If p + 1 < n let the A-descendants of B , , ..., B, in Z p + l be S,(J E J ) of which the leftmost, rightmost are 6,, 6,. Then A' contains all A-descendants of these except for the first five of 6, and the last five of 6 , hence (AfM)' contains all except for the first ten of 6, and the last ten of 6 , . Thus (AfM)'contains all descendants o f B 2 ,..., Bg-,. The hypothesis of 12.13 holds if ( A f M ) 'contains all A-descendants of g' - 38 consecutive Bi's for some g' > m . u48, that is if (*) g - 2 > m . u48 - 38. Put g' = g + 36. I f p = n - 1 , A' contains A,, ..., A , - ~ so ( A ' ~ ) 'contains A , , , ..., A,-,'. The hypothesis of 12.13 holds ifg' exists such that g - 20 2 g' - 16, g' > m . U 4 8 , that is (**) g - 20 > m.2148 - 16. Put g' = g - 4 in this case. NOW ( g - 4 ) / m > ~ 2 5 4/e > U 4 8 SO (*) and (**) hold. By 12.13 Y contains an ( n + 1)-subelement of size greater than g'/m - 40/e 2 g/m - 44/e > s ( A )- E - 44/e 2 s ( A )- ~ 2 5 since r25

> 44/e.

12.15. Proposition. I f 2, ; 7 CD, where 2, E Rep(L,+,) and C is an n-subelement o f s i z e > 1149 then Z n p 1 contains an n-subelement of size > s ( C )- ~ 3 7 u4. 1..

Proof. Say ZA-, nfl ... 7 CE, where CE E L,, ZA-, E Rep(L,). Using the A-B notation for this sequence let all A-descendants of B,, ..., B, be contained in C ; we may assume g / m > u49 > u36. By 11.8, 11.5

where C' 3 ( B , + s , Bg-s), ( g - 2s)/m > g/m - u37 > 2.449 - u37 > 1 1 3 . Hence s"(C') 2 ( g - 2s)/m > u49 - u37.Finally C' has weak image in Zn-, of size > s ( C ' )- u4 so the result follows ( 2 <~ b25). ~ ~

J. I,. Britton, The existence of infinite Burnside Groups

238

12.16. Proposition. Assume Condition C. Then Axiom 25 holds f o r + 1. Thut is, i j A ;r Y where A is an ( n + l)-subelement of size > ~ 2 5 , < 11 + 1 a n d A , Y are (p)-bounded by b,, , then Y contains an ( 1 1 + I)-subelement ofsize > s ( A )- ~ 2 5 . 11

Proof. This is true if p < n by 12.14 so we assume p = n and A Y. Now Zn 2 ... 1 Z , = A B for some B, Zn E Rep(L,+l), where all A-descendants in Z , ofB,, _..,B, are contained in A and ~ A~ n 2 . ... 7 A , by we may assume g/m > s(A) > s25 > ~ 1 Also Axiom 28. Now 1 1.9 is applicable by 12.3, 12.4. Hence

A

;A,,

GA'H

and A' contains ( B l + sB, g p s )where ( g - 2 s ) / m > g / m - u37. Cuse 1. The Bi are maximal normal of size > r". Let their images in A be Ciand let the images of the Ci in Y be D i(i = a, ..., b) where a = 1 + s, b = g s. Then Y M = (Do, Dh)= = DbE(Cf+,, CkPl)FD; and ~

Hence Y M ;HArMK , where A ' M 2 (Bu+6,Bh-6) (Bs+7,Bg-sp6). Also Y M M (R:+7, ( B s + 8B, g P s p 7=) A" say. Let g' = g 2(s + 7). Then ~

g'/rn = ( g

~

2 s ) / m - 14/m > g/m - 1 ~ 3 7 14/m ~

By 1 2.13 Y contains an ( n + 1 )-subelement of size > g ' / m - 40/e - 1 / 3 7 -~ 54/e 2 s ( A )- r25,since ~ 2 2 5 ~ 3 + 7 54/e 0725 k29). C'asc. 2. Not Case 1 . Y ,A contains an n-subelement of Here Z,, Z,2-l. Since A

> s(A)

J. L. Britton, The existence of infinite Burnside Groups

239

size > k i - b25 - e3 > u49 (cf. 6.7A). By 1 2 . l 5 , Z n = ZnPl contains an n-subelement of size > ( k i - b,, - e3) - u37 - u4 > y " , a contradiction.

12.17. Proposition. Let k < n a n d let A , T A , x B , SB, (end words E", K" say) where each side is (k)-bounded by k30, induce T x S (end words E, K say). L e t TT' E Rep(L,+,), SS' E L k + l a n d let T, S be of size > use. Then

z

Proof. We shall prove (*) TI-' E . St-'. K. Now (cf. 6.7) A , = E " . B , . E - ' , T = E . S . K , A = K-1.B2.K"sofrom(*)follows A , . T ' - ' . A 2 --El'-B 1 . E - 1 E . S ' - ' 2 . K . K - 1 . B 2 K " a s required. We apply Axiom 3 1B to T S . This is possible since uso 2 sgl and k30 < b31. We obtain

where S L ,SR have size > s ( S )-r31 > uso- ~ 3 > 1 ro + e3. If we had S = SLfSRthen by 5.23 s(SL)+ s(PSR)< s ( S ) + 2e3 + e r , s(SR)< s(fSR)+ e3. Hence s(S) < 2r3, + 3e3 + E:, a contradiction since uso > 2r3, + 3e3 + e r . Hence S = PQR, SL= f Q , SR= QR for some f , Q, R . Hence by 5.3 1 s(Q) > Min(uso - ~ 3 1 - ro - e3 - 2/e, uso - 2r3, - 2(e3 + e r ) - 4/e) = a say. By (9) and (10) we have

TLE'QF'TR 7 T, T L E ' z E . f , FITR

R . K.

Now 22 ... T S S ' , Z E Rep(Lk+,) and by Axiom 2 8 since str+ < b,, T x 7 T L E ' Q F ' T R .By 1 1.9 since k30 + q31< a > u36 + l/e2 1..

240

J.L. Britton, The existence of infinite Burnside Groups

and (1 1)

u. w - l .

v; T ~ E ' ( R S ' P ) - ' F . T R .

Now u, T a r e subelements of Rep(L,+,) and for some T M (1 2 )

z

TM u0 (end words E l , K , say)

Let T = T LT MT R, u F u l uo u2. Then T L Uu,. ET1, TR = K;'. u2 V . Now s(uo) > $(a) - u1 > s33so b y (4) and Axiom 33 we have T M= uO, TR TITL = u2 Wu,.Further E l = K , = I. Thus by ( 1 1 )

and (*) follows.

12.18.Proposition. L e t D E E L k + l , D F E k < n . Let sDI:( D ) 2 r . Then E - ' . Pis conjugate in G, to an element of l n + l .

By 1 I .8 since Y > ~ 1 3 6

> r - u~~ - l / e 2 and E - ' . F is conjugate in G, to E ' - l . F'. F r o m ( l 3 ) , D ' > D f 0 z Z F > S a n d S 3 S o T > T'where T C D'O and ,sk+l(s) > s ( D ' ) - u 4 . Since soc s we have SOJE L k + l for some J. Also TT' E Rep Lk+l for some TI'. Now

s(D')

J.L. Britton, The existmce of infinite Burnside Groups

24 1

We have s ( T ' ) > s(S) - u4 > r - u37 - l/e2 - 2u4 > y o , hence s(T') < s ( T ) + e3 and s ( T ) > r - u37 - l/e2 - 2u4 - e3 > use. Let Z , = (So&), D' = 0; TO', and let Y , 2 be end words for S o T. By (1 3) 2-1D' F'D; Y - ' , SO S0 a = Y . TD', F'D; Y - ' . By 12.17 J - ' . a = YT"-'D;F'D;Y-' ; but we may take T " to be D', E ' D ; , so J - l a = Y.D;-'(E'-'F')D\ Y - l ; ( T and S o are ( k ) bounded by so < k30). 2". Let s be the maximal ( k + 1)-normal subword of 2, determined by So;say s- ASOB, ( T M ) = 2,. Then a = BMA. Say S U E L k + ' . Now Z k + l ,z1Zk so Sdetermines a maximal ( k + 1)normal subword of Zk+l,say R', by Axiom 26" (since s(s)Z $ ( S o )- e3 > u26). Either Is@') + s(9) - 11 < r26 and s ( R ' )> r' or Is(R')-s(S)I < r26.Hence s(R')> Min(r', s(So) - e 3 - r26)= r', since

-

z

-e3

-

r26 - u1 - u4 + r - u37 - l/e2 > r'.

By Axiom 26A, there exists H such that Zk+ll:k H and any linearization of H is conjugate in G, to U - l . M . Thus H E L,+l. But A S O B U E L k + l andSOJE L k + l , so by 5.29 J z B U A . In G,, E - l F is conjugate to E'-lF' which is conjugate to J-1 a - A - l U-' B-' BMA . Thus E - l F is conjugate t o U-'M hence to

H.

3". Case 2. k = 0. The argument of 2" remains valid if we take S, S o to be D, a to be F and J to be E ( E ' = E, F' = F ) . § 13

Informal Summary In 13.1 1 it is proved that the remaining part (b) of Axiom 20 holds for n + 1 . Afterwards some other axioms are proved for n + 1 including Axioms 19,40,33.

24 2

J.L. Britton, The existence of infinite Burnside Groups

13.1 Definition. If X and Y are cyclic words then X Y means either X = Y or for some p , 1 S p S n, X,Y properly contain maximal p-normal subwords S, T respectively, say X E ( S A ) , Y ( T B ) ( A I Z, B 9 Z), such that SAS, TBT are p-objects which are almost equal; let them produce E , K say; and further SA ;E.TB.C'. 13.2. Note. Let X be E.R. and not a power and let f > 2 be an integer. For some ~ i u, let X f be n-bounded by u and ( ( n - 1))bounded by u. Let S, T be maximal n-normal in Xf and let S' be T. Then X g is n-bounded by ZL and ( ( n - 1))-bounded b y u (g = 1 , 2 , 3 , ...). Proof. Let N be a p-subelement in Xg ( p < n ) . Let the translates of S in W = X X g X be S j (i = a, ..., b ) ;these are maximal normal in W . I f N is in some S j then N C S c Xf.If N contains some Si then p = n since rb < ro + E~ so N = S j = S c Xf.Consider the remaining case. Now N c (So,S,) so N C ( S i ,Si+h)where i is maximal, h is minimal. If p 2 2 then N > (S,+,,Si+h-l) 3 S i + , , a contradiction. Hence iVC ( S l ,S f + , )= ( S ,T ) C Xf.

13.3. Notatio!. If A , B are subwords of W then ( A ,Bh) means ( A ,H ) .B-' : ( A ,B ) means A-'. ( A ,B). 13.4. Proposition. Assume 1. X f is ( ( n - 1 ))-bounded b y k31 and n-bounded by k,, where X is E.R., not a power, and f a 2 . S , , ..., S p are maximal n-normal sitbwords o f Xfo f s i z e > u s , and properly closed to translation; thus ( 3 a ) Si = Si + a (i = 1, ..., p ~

1< p a and )

-a.

3. Similavljt f b r Yg uiid T , , ..., T p ;say

Ti+=Tj+,,( j = 1 ,..., / ? - h ) , 1 G p - b . 3 . For some k 2 u

+ b - d + 2 , where d = (u, b ) we have

J.L. Britton, The existence of infinite Burnside Groups

(Si, Si)2 (Ti, Ti) whenever 1 < i < j

243

< k.

Then (i) (Si,Si)g (Ti, Ti)whenever 1 < i < j < p . (ii) Zf ( S , , S,), ( T , , T,) produce E,, K, and if w, where 3 < w < p , is divisible by a and by b then A

A

(S,,S w + l ) ;

E,.

( T , , TW+,).ei1and ( X W l u ) (Yw/'>

13.4A. Remark. Under the above hypothesis we may take p as large as we please, without loss of generality; for by 13.2 we may increasef and then form the set of all translates of S,, ..., Sp in the new word; similarly for Yg. Proof of 13.4. We have k,, < Min(k;,, k,,), k32 < Min(kk3, k,,), ~ 52 1 M a x ( ~~~ ~ ~, ~ 1 . (i) It is sufficient to prove (i) when p = k + 1. Let D, z (S,, S,+,), C, = ( T u ,T,+,) ( u = 1 , ..., k - 2). Then D,, ..., D,-, have period a in the sense that D, fi D,,, ( t = 1, ..., k 2 -a): indeed D,= Dt+,. They also have period b since -

. . . , p - 1); then Say(S,.,S,,,),(T,., T r + l ) p r o d u c e ~ r , ~ r + l (1,r = el, K,+,. Since a diby 9.13 and its dual the pair in (1) produce h vides w, ( S , , S w + l )has the forrn X;"'" where X , is a C.A. of X . Sim-

J. L. Britton, The existence of infinite Burnside Groups

244

(T,,,,,, T w + 2 = ) ( T , , T 2 )so clearly E, ;E,+,. Hence xwla = ywlbt-1. I n 1 1 1 Since w 2 3 , ( T , , T 2 ) C Y r I b ,( S l , S 2 ) C X r l a so (X;"la)= (Y;"lb) and hence ( X W ~ Q s )( ~ " 1 ' ) . 13.5. Proposition. Let Xfand S , , .._,Sp satisfy hypothesis I o f 13.4. Let (sC+], s,+,) g ( s d + l , s d + , )-' and assume that whene i w i, j satisfy c + 1 < i < j < c + rn we have

where g ( i ) = c + m + d + 1 - i. Then m


Proof. Assume m 2 a + 2. Without loss of generality we may suppose c < d, 1 < c + 1 < a , 1 < d + 1 < a and c = 0. Thus (3 By 13.4, since m 2 a + a - a + 2 we have

(4) and (2) holds if 1 < i < j < m + d . Call the left side, right side of (4) say A , R. Let A , B produce E , K ; let B , A produce E " , K " ; let A-1 , B-1 produce E', K ' . Then by 9.8A I-

';IK

1

K 1

= n

= &-1 7

11

;K'f-1

But A-' is B, B-l is A so we may take E" ;E ' , K" ;K' and we get e K . Now A ;E . B . Kso I ; B . E . B . E .By 5.13 we deduce B . E ;I that is, E ;A . Now (3) implies

245

J.L. Britton, The existence of infinite Burnside Groups

and in the same way that (4) follows from (3) we obtain (4')

As above if A ' , B' produce e0, I C then ~ e0 ; A ' . Let ( S , , S,), ) produce e l , q . Then by 9.13 ;e0. But (si:m + a si m + a- 1 7

'+

so comparingAwith (4) we have by 9.13 e l ;E . Thus e0 ;E and A ; A', i.e., ( S m + d , S m + d + , ) ;I. This contradicts 6.1 since k3, < k1, k32 ky13.6. We recall that - A = -B means that one of A , B is a right subword of the other (cf. 2.4). Although this relation is not transitive in general, it is transitive if we restrict A , B, _..to be subwords of elements of Rep L,? of size > ,s, (cf. 8.2 or Axiom 33).

13.7. Proposition. Let X e and S , , ..., Sp satisfy hypothesis 1 of (Sc+l, S , + , ) where 1 < c + 1 < a and assume 13.4. Let (Sl,S,) that whenever i, j satisfy 1 < i < j < m then ( S j ,Si) (S,+i, S j + i ) . Let m 2 a + 2. Then either c = 0 or T , Q exist such thut

where a' k,, + c,

= a/d,d = (a,c). Also (Q"")

is ( ( n - I))-bounded b y

+ c , ~and n-bounded b y k,,.

Proof. Assume c f 0. Since m Z a + a - a + 2 we may take m as large as we please by 13.4. Let 1 < i < j . 1". We prove ( S i ,S,) 2 ( S j + d , S , + d ) . Put D, = (S,, S,+i- i) (u = I , 2 , ...). Then D , fi D,+,because D, D,,,,. and D, D, +,. SO by 2.17 D, fi D,+d. 2". We have k31< b,,, k31 + c3, < Min(ki2, k,,, b 3 9 ,k I 2 )and k32 + u4 < Min(ki3, k , , , b i g , k I 3 ) . Let the kernels of S , , ..., Sp be A K,, K , , __., K p - , , A , . Then s ( K i ) > u S 1- 2(ro + € 3 ) - 4/e > Max(r3,, Z L +~ E,). We may write f-

,,

J. L . Britton, The existence of infinite Burnside Groups

246

A , 5 A ; K , , A , = K P A'P where Ki = K j + a (i = 1 , 2 , ...,p - u ) . We may also write ( S , , S p )= A ; K , C , ... KR-, Cp-l KpAb and Ci = Ci+, (i = 1 , 2 , ..., p - u - 1). ( K , , Ka+,) is a C.A. of X , say X , , and X , = K , C , K 2 C, ... KaCa. Now as in Axiom 33'

,.=,

Y = K ; C, ... K:Ca. Thus ( X ' ) nI-l ( Y e )and ( Y e )is 1))-bounded by k31 + c33 and n-bounded by k,, + u4 by 6.1 OA ( ~ <3 k 3 2 ) . Now ( S i ,Si)= S L ( K i , Ki)S: ( K j , Ki) (Kz!,K i ) ( Z j ,Zi). Hence ( Z i ,2;)fi ( Z i + d Z, j + d ) Note . that if ZiB and DZi denote temporarily the maximal n-normal subwords of ( Z i , Zi)containing Zi, 2;then since s(Fi K F ) < q3, and B = Fi K,? H where H is n o t normal we have

so X ,

((M

-

Let ( Z i ,Zi), ( Z i + dZ, j + d )produce E ~ ; , K ~By , . 9.13, eii is independent o f j (mod G,) so we may write ei instead of ei,; similarly

K.. =K . 'I I'

*

Since ( Z i ,Z;) = ( Z i ,Zi+,).(Zi, Zi+l)-l we have A

h

(5)

( Z i ,Zi)= 'i.(zj+d, Zj+d).E:?

Take i = sd, j = (s -1

I

+ 1) d, s 2 1. Put Hs = (Z,,,

'sd . H s . E ( s + l ) d = H s + l .

Next

h

Z(s+l)d);then by (5),

J.L. Britton, m e existence of infinite Burnside Groups

24 7

But (a' + l ) d = d and (a' + 2 ) d = 2d (mod a) so ede2, -.curd ; I. h

Hence ( Z d , ;((Z,, Z 2 d ) . ~ d 1 ) u ' . 3". Consider the word A = (Z2,-,, Z2,)-e;l -2,. We shall prove that A = Z ) d _ l H Z f for some H where S ( Z ~ , - ~>)~ ( 2- d,,~ and s ( Z f ) > ~ 5 2 . Write Z, 2' for Z,, Z,, respectively. For any j > d , the pair of . are equal preobjects ( Z , Z j ) ,( Z ' , Zj+,) produce Ed, K ~ Adjustments in G,, say

fu+d)

(6)

A

~

(Z*,Zj?)= (Z'*,Zj?+,) (left side standard).

In particular, - Z * - -2 and - Z ' r -Z'*. Also E, =Z.Z*-l.Z'*.Z'-' by 9.1. HenceA= ( Z 2 d - l , Z ' ) . Z f - 1 . ~ d 1 . Z(Z2,-,, _ z'>.(z'*)-1z*. By Note 8.5A and the proof of ( 2 ) of 8.5A we see that if ( 2 ) of 8.5A holds then (ii) Z*-'Z'* is a subpowerelement of Rep L,; if ( 1 ) of 8.5A holds then by 8.5 we have (i) Z*F, = Z'*F2 where F , or F2 is I. 4". Assume (i). Here A = (Z2,-,, Z ' ) . F 2 .F ; l . In ( 2 ) let Z*B and Z'*B' be maximal normal. If F 2 = I then by 8.5' Is(Z*B)-s Z"F,B')I < u 3 . As in the proof of 8.5A we obtain

s(F,) < Max(ro + e 3 , u 3 + 2 / e + (2e3 + E T ) + e3 + a 2 ) = a 3 say Similarly if F , = I then s(F2)< a 3 . Suppose F , is I . Then A = (Z2,-,, Z ' ) . F 2 . Now -2'" f - Z ' hence there is no cancellation or amalgamation in the product Z'. F2 and -Z'*F2 = - Z ' F 2 . Now - Z -Z* = -Z'*F2. Thus A ends with Z ' F , and -Z'F, = -Z. For any i, since s ( S i ) > ~ 5 we 1 have by Axiom 33'

-

Hence s(Z'F2)> s ( Z ' )- e3 > - e3 > ~ 5 2 Also . s(Zi) < k32 + u 4 . Suppose F2 is I. Then - Z ' = -Z'* -Z*F,. Now

~

~

24 8

J. I,. Britton, The existence of infinite Burnside Groups

s(F, ) < a 3 < 0- e3 < s ( 2 ' ) - e3. Hence 2' = Z " F , say. Thus - Z " F , = -Z*F, and - Z " = -Z*= -2. A ends with 2". Next ~ ( 2<' )~(2") + s(F,) + 2/e < ~(2") + a 3 + 2/e. Hence s ( 2 " ) > ~ ( 2 -' )a 3 - 2/e > P - a 3 - 2/e 2 ~ 5 2 Thus . if (i) holds then A begins with Z 2 d - 1 . 5 " . Assume (ii). Then K , Z*-'.Z'* = K 2 R ' where R ' E L , and K , or K 2 is I. Also A E (22d-1, Z ' ) . R ' - ' . K;'. K,. Now T-l 2' E Ln for some T and -R' -2'" -2'; hence T-12' = R ' . By Axiom 39, (Z,,-,, Z ' ) . Z ' - l T n-1 = ZL 2d-1 E T R say, so A Z k d - l E T R . K ; l . K,. We prove - TRK;' = - 2 K ; l . If K , = I we have - T R = - T = -R'-'

. s ( T ~ K ~ ' ) > - ~ ~ ~ - E 1~- 2- / ~e 2~ ~~ 5-2 u. ~ + Suppose K 2 is I . Then - TR F -2KT' and K,Z*-'Z'" = R' . From ( 2 ) and 8.5', 1 - u2 < s(Z*B) + s(Z'*B') < s(Z*) + s ( B ) + + s ( Z ' * )+ s ( R ' ) + 4/e < s(Z*) + s(Z'*) + 2a2 + 4/e. Since 2" and 2'" are normal we have by 5.23 s ( K , ) + s(Z*) + ~ ( 2 ' < " )s(R' ) + + ( 3 e 3+ E : ) + (yo + 2 ~ ~Also ) . s ( R ' ) < 1. Hence s(K1) < u2 + 2a2 + + 4/e + yo + e: + 4e3 = d ' say. Now s(TR)> s ( T ) - d,, > 1 - 2/e - k,, - u4 - d,, = d" say. Since d" > d' + e3 we have T R = UK-' 1 for some U. Also s ( U ) 2 s(TR)- s ( K , ) - 2/e > d" - d' - 2 / e > ziS2 > yo + e3. In particular U is normal. Since - UKT' = -2K;' we have -- U -2. Also A 3 zkd- EU. Finally s(Zid-,) > s(22d-1) d39.which is also true when K , is I. Z i d p l UZR E A ' , say (for some U ) 6". Thus in all cases A where ,(Zkd_l)> s ( Z ~ ~-d3,, - ~ )~ ( 2 :> ) ~52. If ( I ) holds the ( ( 1 2 1))-bound of A' is the same as that of (Z2,-, , Z2,) and the n-bound exceeds that of ( z 2 d - 1 , z 2 d ) by 2/t. + a3 + € 3 . If ( i i ) holds the ((12 1 ))-bound of A' is the same as that of ~

-

249

J.L. Britton, The existence of infinite Burnside Groups

A" = ZkdPlE T R while the n-bound of A' exceeds that of A" by 2/e + s ( K 2 )+ E , . Now s(Z*) + s(Z'*) < s(Z*B) + s(Z'*B') + 26,

s ( K 2 )+ s ( R ' )

< s(K2R')+ (yo + 2e3)

1 - 2/e

< s(R')

< 1 + v2 + 2e3

hence S ( K 2 ) < 2/e + v2 + 2e3 + ro + 2e3 + 2 / e = 7, say. Hence A' is ( ( n- 1))-bounded by k31+ c33+ c3, and rz-bounded by Max(B,g), where 9 = k,, + u4 + 2/e + a , + E, and cp = Max(kn + u4, 1 - p + e3) +g3, + 2/e + y + E,. 7". ( Z , , Z 2 d - 1 ) = Z d V Z 2 d - 1 s a y . P u t B - (Zd,Z2d).Ed1'Zd.I f d = l thend=2d-lsoB-A-Z;UZt.Ifd>l 0 .

B - ( Z , , Z ~ , - ~ ) . Z Z ~ - ; ( Z , , - ~ , ~ ~ , ) E , ' . Z ,=Z,VZ2,-,

UZ:

Hence in any case B = 2; WZf where ~ ( 2 :> ) s(Z,) - d,, and s ( Z ~>) ~ 5 2 NOW . s(Z;) + s(Z;) - s ( Z d ) > s(Z,R)- d39 > ~ 5 -2 d39 > 3E3 + E T so Z,".Z(;'.Zi Thus ( Z ~ , ~ u ((Zd, + ~ Z2d)-~(;l)"' ~ ; = (B.Z;')a'=Z;. (WZy)"'ZL-l d Hence Z;-'.(Z,, ZU+,).Z; = (WZ;)"'; but the left side is a linearization of Y , so ( y e ) ( ( W Z ! ) " ~ ) . By 5.3 1, where Z , = PQR, Z: = PQ, Z; = Q, Z t = Q R , either s ( Q ) 2 Min(P - d,,, ~ 5 2 ) yo - E , - 2/e > ro + e, or s(Q) 2 u52- d,, - 3e3 - - 2/e > ro + e,. Also B = PQWQR. We discuss the bounds of (WQ)" = K, say; for this purpose we may regard a' as being as large as we please. Since Q is not a subelement of L u ..-uLnP1 the ((n - 1))-bound of K is at most that of QWQ hence of B , i.e., k31+ c3, + c , ~ . For the n-bound first note that A is not an n-subelement, hence neither is B since B 3 A . In general if a word H begins with a subelement Z of Rep L , and Z is normal and if ZB is maximal normal in H call B the left excess of H (in 2).

=,zf.

ET

250

J.L. Britton, The existence of infinite Bumside Groups

In Case (i) (cf. 4") the left and right excess f o r A is the same for (Z,,-, , 2 ' )so each has size < ay2(cf. 2"). In Case (ii) (cf. 5 " ) the left and right excesses for A are the same as for Ztd-l E T R , so by Axiom 39 each has size < a ; = Max(ro +E,, f 3 9 + (2e3 + E : ) ) . Put C I = ~ Max(&,, a;). If B S A the left excess of B is the left excess of (Z, Z2d-1) so has size < a i ; the right excess of B is the same as that of A . Let Y , , Y , be the excesses for B . Then PQWQR = B = PQY, X , = X , Y,QR. The Q in QR is not contained in PQ Y , , so XI = X i R , X i $ I. Similarly X , = P X ; , X i I I . Hence QWQ = Q Y , X ; = X i Y,Q. Now K > QWQWQ = X i Y,QY,X; so il: = Y , Q Y , is maximal normal in K , and s ( N ) < s ( Y 2 )+ 2/e + kB < a; + 2/e + k B , where k , denotes the n-bound for B given below. Any n-urelement in K is either in N or in QWQ so its size < Max(kB, s ( N ) + e 3 ) . Hence K is n-bounded by k, + a ; + 2 / e +

+ € 3 < k3,.

VZkd-l is the same as that of Since the right excess for Z, liZ2,-, its size is < a,; hence kB = Max(o,cp) + d , + 2/e + E , . 13.8. Proposition. Let X be E.R. and not a power and let (Xf), whew f > 2 , be ( ( n 1))-bounded b y k,, and n-bounded b y k,, and be such that no linearization is a subpowerelement o f 1,; similarly j b r Y and Yg. Let (Xf) 3 ,$ ; ;77 C ( Y g ) where L ( , $ ) / L ( X > ) u5, and not ( ,yl q. Then there exist maximal n-normal subwords S , , _ _S,, _ , o f ( X f ) ,which are contained in ,$and size > u33,and maximal n-normal subwords T , , ..., T, o f ( Y g ) ,contained in q and o f size > ~ 3 3 sutisfying , the following conditions. 1. L((S1, s, ) ) l L ( X )> u53 - 4. 2. Im'iSi)= Ti and I m '( Tj )= S j (i = 1 , ..., m ) . 3. SlT = Si+a(i = 1 , ..., m - a ) for some a where m - a > 1 (cf. 2.16). 4. I f T i c (TI , T , ) then Tl?= Ti+ ( i = 1, ..., m - b ) for some b, 111 - b 2 1 . 5. I f ( X ) = ( Y )then denoting the translates of the Si in (Xf) b y S , , . . _ ,S,, Sm+,, ..., S p , we huve for some c , 1 < c + 1 < a, that (Ti, Ti)= (Sc+j , Sc+i) whenever 1 < i < j < m ; in particular T1. - S C + l .. 6. I j ' ( X )= ( Y - ' ) then for some c, 1 < c + 1 < a, ( T i , T i ) = iSc+i, Sc+i)-l whenever 1 < i < j < m. ~

J. L. Britton, The existence of infinite Burnside Groups

25 1

Note: ( Y )f ( Y - l )since II has no elements of order 2 (since e is odd).

Proof. 1". Let t and q have divisions P,,P2, ... and Q , , Q,, ... . Since t contains at least one Pfof type n we may consider the nnormal subword A of of maximal size. Since k! < s(Pj)+ s(Qi) < s(Pi)+ k,, we have s ( A ) > k! - k,,. By an argument similar to 5.28, L ( A )< 2 L ( X ) . Let the translates of A that are contained in I: b e A l , A , , ...,A,. Kernels exist since k! - k,, > 2(ro + E , ) + 2/e. By 2.16A, I L(t)/L(X) - u 1 < 3 , SO ~ 5 <3 u + 3. Also L ( t ) < L ( ( A 2 ,A,-l)) + 4L(X). Let Ai be the maximal normal subword of determined by Ai ( i = 1, ..., u ) . These are distinct and of size > k! - k,, - E , . Let the image of in q be Bi (i = 1, ..., u). For any i, either Bi has the form Qj of type n , or by 6.23 Is(Bi) -s(&,)l< u3. ( k N < k ; , k,, < k $ , k; - k,, - E, > ulo). In either case s(Bi) > k! - k,, - E , - u,. &,..., A',-, are maximal normal in ( ~ f )B,, ; ... , B,are maximal normal in ( Y g ) .Call X 2 ,Xu-,, B,, BU-, extremnl. As in Section 10 call a subword of t central if it is contained in (A2, similarly for q. Note that L((X2,Xu-,)) > L ( l )- 4L(X) > (us, - 4) L ( X ) . 2". We consider the following operations. (1) U + V where U is central and maximal normal in (or q ) of size > ul0 and V is Im'(U); thus V is central. Call this operation u. (2a) U + U' where U and U' are central in t; or in q. (2b) U -+ U- similarly. Call these translation operations t , , t 2 . (3) The following operations gt, g, are defined only if ( X )= ( Y ) . Let the distinct maximal normal subwords of (Xf) of size > u54 be U , , ..., Ur. Define Ui for any integer j to be U, where j = k (mod v ) and 1 < k < r. Then for all i, U: = Ui+h where h = r / f . Since uS4< k! - k,, - e3 - u,, (A2, is (Uil, UI1. ) and ( B 2 ,BU-,) is (Ui2, U j 2 )say. Since u - 1 > u5, - 4 2 2 we have A , = Uil+h so i , + h < j , . We say Uiz corresponds to Uil+, where i, = i, + s (mod h ) and 0 < s < h ; further say Uj2+ corresponds to U j ,+s+ for u = 0, 1, ..., z where z = Min(j2 - i,, jl - i, - s). Put gq7(uj2+u) = u j l + s + u , gt(uil+s+")= Uj,+u. Thengq(B2) exists. If z = j 2 - ii then g,(BU-,) exists: if z = jl - i, - s then exists.

q.,

xi

,

xu-,);

xu-,)

gt(zu-,)

25 2

J . L . Brittoii, The existence of infinite Burnside Groups

( 4 ) If (X)= ( Y-' ) we define fi,fq as follows. (1, A. u - l )is ( U i , , Ui,) and ( B 2 ,B i l - , ) is (U;:: U;:) say. For the same range of u as before putfo(U;\,) = %+,+, .f&Ui,+,+,) = u~~+...fv(~,-~) exists and one offq(B2), f C ( A u - , )exists. 3 " . Call a central subword of t (or q ) initial special if it is either extremal or the maximal normal subword of (or q ) containing a Pi! ( o r Ql-) of type n : its size is > k! - k,, - e3 - u3 > ~ 3 2 . Call a central subword of t o r q speciul if it arises from an initial special word by a sequence of the above operations. Clearly any special word is maximal normal. We claim that each special word has size > u3,. This is by Section 10; for we can associate with any sequence of operations a sequence W , , W 2 , ... where each Wi is one of [, q if the operations . t i ,j , do not occur, or one of t , t - ' ,q ,q-' if these operations do occur; thus operation ( 1 ) above corresponds t o the operation fi of 10.2 Case 2a and W i = Wi+' = t (or Q);(2a), (2b), (3) above correspond t o the operation f ; of 10.2 Case 3a. For example consider ( ? a ) with U. I / + central in t . Then t = AUB = CU'D say where L ( C ) = L ( A )+ L ( X ) . [ is a subword of some power X N of X: X N = l 2433 2 ul0, Im'(Si) = Tiexists; hence u(Si)= Ti, so Tiis special in q. Thus the special words in 7 are precisely T , , _.., T,. If S; c ( S , , S,) then t l ( S i ) = S; so S; is special. Thus for some a S,! = Si+* (i = I , 2 , ..., m - a ) and 1 < m - a . If Ti c ( T , , T , ) define b by T i = Tb; then Ti' = Ti+ ( i = 1 , ..., m - b ) and 1 < m - h. Finally we note that 5 and 6 hold. 13.9. Corollary. With the hypothesis of 13.8 either ; q induces t ' ;;Q'where L( t ; ' ) / L ( X )> 1153 -- 6 and L ( q ' ) / L (Y ) < 5 or (iil of 13.4 holds.

J.L. Britton, The existence of infinite Burnside Groups

253

Proof. Since Im' S j = Ti, there are words t' = (SF, Sk ) and q'= ( T P , T k ) such that ( S 2 ,S,-,)C t ' ; q' C ( T , , T,). Now ( S , , S,) = U(S2,Sm-,) V for some U, V ;also L ( U ) < L ( X ) , L ( V ) < L ( X ) . Hence

>~

53 6.

If T i ( T , , T,) t h e n L ( ( T , , T,))< L ( Y ) + L ( T ; ) < 3 L ( Y ) . If T i c ( T , , T,) and if m < a + b - d + 2, w h e r e d = (a, b ) then L((T,, T, ) ) / L ( Y< ) 3 + mlb and mla > - 3 + L((Sl, S,, ) ) / L ( X ) > us3 - 7. Since d 2 1 we have m
< b/(uS3 - 8)

a(uS3- 7 ) < a + b m/b

< 1 + l/(uS3 - 8 ) < 2

since us3 2 9. HenceL(q')/L(Y) < L(( T , , T,))/L(Y) < 5 . There remains the case T i C ( T , , T , ) and m 2 a + b - d + 2. By 13.4, 8.15 the desired result follows, since k34 < Min(kI6, k31), k35 Min(kl7, k32).

13.10. Proposition. Condition C holds. Proof. Assume n o t ; then A nrlC, AB E L,,,, CD E Ln and L ( A ) / L ( A B )2 ~ 4 1 Let . p be minimal such that A ; C ; then p 0. Since sty+< k,,, k,, (cf. 5.1 1 ) and u4, > uS3/ethe hypothesis of 13.8 holds. Say A B = X e , CD = Y e . By 13.9 (also cf. 3.1) either (i) A ' ; C' (hence Cr),L ( A ' ) / L ( A B ) > (us, - 6)/e 2 q', L(C')/L(CD)< 5/e < f ; or (ii) for some integers a, b # 0 , ( X e b ) (Ye'). If (ii) holds then X e b ;T - ' . Yea.T for some T. Now Y e = CD ;I hence ( A @ 7 I , contrary to 5.12.

7

254

J.L. Britton, The existence of infinite Burnside Groups

Thus (i) holds. Now X is a C.A.I. of X,, where X o E Cn+,C J n C J p and Y is a C.A.I. of YoE Cn C J n - , C J p . By 2.1 I , A' nyl C' has the form e q . ( X O )nyl h f ( Y 0 ) ,i.e., Y O ~ n - l X contrary o to 5.9 (ii).

13.10A. Note. We have now proved that conditions A, B, C hold (cf. 12.3, 12.4). 13.1 I . Proposition. Let DE, DF E L n+ and let sDE( D ) 2 ro. Then

D E ; DF. Hence by 13.10, 12.2 and 12.18, Axiom 20 is true f o r n + 1. Proof. We have

...T DE ( 2 ) Un 2 ... j+D F where Z,, Un E Rep(L,+]); say Z , = ( X " ) , U,, = ( Y " ) .Using the A - B notation for ( I ) let all A-descendants of B , , ..., Bg in D E be contained in D.If E > 0, in particular if E = l / e 2 , we may assume that g/m > s D E ( D )- E . By 11.8, since ro - l / e 2 > u36 (1) 2,;

un

-

D'F'

( g - 2s)/m > g / m - u37 > ro - I / e 2 -- u37> u1

Now D ' 3 D'O ; U y ; we may write 2," for D'O, so Z," ; U," . By 12.2 if A is any subelement of L i and of L,+], where i < n , then s n + ' ( A )< rb; thus 6.8 holds for n + 1. Now D'= C U ~ D say;' as ~ in~ 6.8 ~ a, and p1 have ( n + ])-size < rb + e2 + 2 / e . Let w be the least integer > m(rb + e2 + 2 / e ) . If cwl 3 ( B l + s B,,,) , then s ( a l ) 2 w / m > rb + e2 + 2 / e , a contradiction; hence ( B w + s + 2B, g - s - w - l ) ~D'O- Z,". Now by 2.16A

255

J.L. Britton, The existence of infinite Burnside Groups

hence

>

-

3/e + ro - 1/e2 - u37 - 2(rb + e2 + 41e = a sa

Thus L(Z,")/L(X) > oe >, uS3. Case 1. f u," . Let 2," U," where p is minimal; thus 0 < p < n. Since str+< MinQh-,,, k 3 5 )we may apply 13.8 and 13.9. By either (i) Z,"" ; U,"" (hence ;), L(U,"")/L( Y )< 5 and L ( Z / " ) / L ( X ) > [are] - 6, or (ii) the conclusion of 13.4 holds; in particular a, b ) E). ,exist such that if w 2 3 , a l w , blw then ( X W l u ) ( Y w / b (hence Assume (i) holds. Now X is a C.A.I. of X o and Y is a C.A.I. of Y o , for some X,, Yo in Cn+,. Since [ a e ]- 6 2 qe and 5 < f e we have Z,"" = e q ( X O ) ,U,"" f 6 f ( Y o )and Yo p n X o contrary to 5.9(i). Hence (ii) holds, and a = b , otherwise X,p,Yo or Y o p n X o . Take w = a e . From 13.8, (S,,S,)CZ," and (Tl, T,)C U,". Also (cf. (ii) of 13.4) ( S , , S,) g ( T , , T 2 )and

z,"

-

(7)

(S,,S2)

€,.(Ti, T 2 ) ' K 2

Z' = E .U'

(8)

n p

1

n

.€-I 1

r\

where Zl, = ( S , , S u e + , )and Ul, = ( T , , T U e + , ) . Zl, is a C.A. of X e , i.e., of Zn and UE, is a C.A. of U,. We have say Zl, 3 (S,, S,)Z', Ul, = ( T , , T 2 )U'. By (7), (8) we have (9)

K2

Since Z,"

C

+

P

A

;u'-€i'

-2'

D' we may write D' = Dl(S,, S 2 ) D 2 .Now Z ,

Since Un D'F' induces U," ( S , , S 2 ) , then by 1 l . l A ,

= D'E'

so

; 2," , which induces ( T , , T 2 );

J.L. Brittorz, The existence of infinite Burnside Groups

256

(1 1) since zi33 2 1429 and M a x ( ~str+) ~~< , Min(k16, k 1 7 ) . By (9), ( 10). ( 1 1 ) I?' ; E ' ; therefore E ; F and DE ; DF. cusc 2. M z = UM . Some C . i . of has the form 2 : Z' and some C.A. of U, has the form Uf: U ' . Now L(Z:)/L(Xe) > a > q , so by 5.16 (ii) we have Z ' - U ' . Now 2 : C D ' ; say D ' - D l Z M D 2 but ; 2, = DIE', so Z ' = D, E ' D , . Since u,, D ' F induces u$ = , u' ; D , F'D,. Herice E' ;F ' , E ;F, DE ;DF.

zl

zt

13.1 2. Proposition. Let the hypothesis o j 13.7 hold and let

X'E Rep(Lr,+l).Theti c = 0.

where a' 2 2. We Proof. Assume c # 0. Then ( X ' ) ,lz-l( T ' ) may suppose that X E C,l+l,hence X E J , , . Now ( X " ) , ( T e ) ,(Q"'") are (n)-bounded by M a ~ ( +k c33 ~ ~+ c 3 9 ,k32+ u 4 , k 3 3 ) < Min(c,*, b,, d,,), hence by Axioms 28A, 8, 1 1 we have (X') (Q""). By Axiom 28B (Q"'") (P"") where the right side is (n)-bounded by str < b, ; hence

( X " ) (P"") I f Pa ;I , then X e (t I contrary t o 5.1 2. Hence P E Jn by Axiom 9. By ( * ) XE,,P hence X p , P ; this contradicts 5.9 (i) since

XE

(;I+,.

13.13. Proposition. Let A A ' , B B ' E R e p ( L n + l ) Let . A ; B, with di~~isioris P I , ..., P, Q 1 , ..., Qr suy aiid let L ( A ) / L ( A A ' > ) u 5 5 . Then A = B , A' = R', r = 1 , arid P I has type zero. Proof. It is sufficient t o prove that n o Pihas type > 0; for then r = 1 and P , has type 0, so A = B. By 5.16 (ii) since uS52 4 we have AA' = BR'. Lct p be the maximal type occurring and assume p # 0. Thus A; ;B . Now 1155 2 1 { 5 3 / ~ ) .As in Case 1 of the proof of 13.1 1, since [ 1155 cl 6 2 ye, we have (X') ( Y e ) ,i.e., ( A A ' ) ( B B ' ) (hence X). ~

P

P

J. L. Britton, The existence of infinite Burnside Groups

251

Since AA' and BB' E Rep(Ln+l), AA' is a C.A.I. of BB' (cf. 7.1). By 13.8 parts 5 and 6 we have

( T i , Ti>= <sc+i, Sc+j)fl,l < i < j < m

(12)

and 1 < c + 1 < a. Now m > 2a since m/u > L(Sl, S , , ) / L ( X ) - 3 > uS3- 7 2 2 so m > a + 1 and m > a + 2 (cf. 2.16A). By 13.5 we may replace the f 1 in ( 1 2) by + 1, since 2133 2 u 5 1 , str+< k31, k 3 2 . By 13.7 either c = 0 or T, Q exist such that

(X")

( T e ) (Q''e) where u'

> 1.

By 13.12 we have c = 0. Some Pihas type p . For some w we have PI!C S , (cf. the beginning of 3" in the proof of 13.8); hence QIC T , = S,+, = S,. We shall prove ( * ) P I !n QIhas size > ~ 4 0 . Assuming (*), let J = P,! n QI; since Pl!-lXf:,' QJXfE L,, some element of L, has the form J - l A J B , in contradiction to Axiom 40, since str < b40. It remains to prove (*). Now k! < s(Pf)+ s(Qi) and by 5.1 1 s(S,) < str+. If Qi. C Pf then J 3 QI and s(J) > k! - str+2 ~ 4 0 .Simiiarly if PI!C QJ. If Pf,Qi were disjoint (say PI!is left of QI) then abbreviating L ( X ) t o X and using 5.23 +

Pf+K


Adding,

which is a contradiction since k! If PI!,QI overlap, say PI!= DE,

> str+ + 5e3 + E : + y o . = EF; then

Ql!

J.L. Bntton, The existence of infinite Burnside Groups

258

DEF

< str+ + e3

DE+F

< D E F + yo+

EF

< E -t F + 2 / e

k!


S(E)

> k!

2e3

Hence -

(str++ e3 + ro + 2e3 + 2 / e ) 2 ~ 4 0 .

13.14. Proposition. No element of L n + ' has the form J - ' A J B , wherc. s w ( J ) 2 y o , W f J B J - ' A . Proof. We have ( i ) 2, + ... + J B J - l A where 2,E Rep(L,+'). Taking the inverse of every term we have (ii) z,' -+ ... + JB-' J - ' A - ' , Denote (ii) also by U,, + ... + JV where V = B - ' J - ' A - ' . As in the proof of 13.1 1 , Zf;' ; ;U y , L(Zf;')/L(Xe) > a 2 u55 where a is as in proof of 13.1 1 . By 13.13 Zf;' = Uf;' and 2, is a C.A. of U,. Thus Z,, is a C.A. of Z,;'. This is impossible since !Ihas n o elements of order 2, as e is odd.

13.1 5. Proposition. Let DE, D F E 1 ,+ and sDE(D) > y o . Then

Proof. As in the proof of 13.1 1,Z, = DIE' and Un D'F'. Further U ~ ; ; D ' O ( = Z ~ ) . A s i n13.14Uf;' r D f 0 a n d Z , i s a C . A . o f U , . Now

Let 1' be the iterated nice image of D in U, with respect to U,, ... + D F . Let D * be the nice image of D' in U, with respect t o U, - D'F'. By 11.8, D * c 1'. But D * is U:, so 1' 3 D * 3 ( B w + s + 2B,g - s - w - l ) . Hence in U, + ... + D F , all A-descen-+

259

J.L. Britton, The existence of infinite Burnside Groups

dants of B w + s + 2 ..., , Bg-s-w-l are contained in D. Therefore

sDF(D) 2 ( g - 2 ( + ~ w + l))/m

since e3 = l/e2 + u37 + 2(rb + e2 + 4/e). 13.16. Corollary. Axiom 19 is true for n

+ 1.

Proof. The above proof remains valid if DE E L',+,

, DF E

Lt+l.

13.17. Proposition. Let X Y E L n + l . As in Definition 7.1 1 let a sequence Zn -+ ... -+ Z = ( X Y ) and A's, B's, C, D be chosen. Let u denote this set of choices, and in the notation of 1.1 1 denote x / d (here k = 1 so d = m ) by s x y , , ( X ) . Let r be another set o f choices. Then (i) Isxy,.(X)-s,y,.(X)I< e3+4/e (ii) s x y ( X ) > s x y ,,( X ) 2 s x y ( X ) - e3 - 4/e. Proof. Consider o, which may be called a set of choices for the pair X , Y . u determines in an obvious way a set u' of choices for Y , X . Let the number of R icontained in Y be y . Then x + y < m < x + y + 4 and y / m = s Y x , , ' ( Y ) .Hence if we put

then 0 < t , < 4/e (since e I m). Case 1. sxy,,( X ) > yo + e3. Since s x y , , ( X ) > y o , the proof of 13.15 without change implies that sxy, ( X ) > sxy,,( X ) - e 3 . Hence sxy, ( X ) > ro so again S x y , , ( X > > S x y , . ( X ) - € 3 - Hence

so (i) holds in this case. Case 2. Not Case 1.

260

J. L. Brittori, The e.xisterice of infinite Burnside Croups

By ( l ) , s ~ so by Case 1

~Y ),= 1~ - ~t , -( sxY,,(

X ) 2 1 - 4 / e - ro - e3 > yo + €3

By ( 15), ( 1 3) and the equation obtained from ( 1 3) by replacing

u , u' by r , r' respectively

Now I t, - t, I < 4/e hence (i) holds in case 2 also. To prove (ii) we may, given E > 0, choose T such that

The result (ii) now follows from (i).

13.18. Corollary. Let A R C € L,,+l. Then

Proof. Given

E

> 0 choose

u such that

For u let a, b, .Y be the number of R i (in the notation of 7.10) contained in A , B, A B respectively; then a + b < x so for choices u', u" determined by u in the obvious way

By adding these four inequalities the result follows since E : = e3 + 4/e.

J.L. Britton, The existence of infinite Burnside Groups

26 1

13.19. Note. Recalling 7.12 we now have that the following "size axioms" are true for n + 1 : Axioms 13, 14, 15, 16, 16a, 17, 19. 13.20. Proposition. Axiom 40 holds for y1 + 1.

This contradicts 13.14.

13.21. Proposition. If CD E Rep(L,+,) then ~sCD ( C )- L(C)/L(CD)I< E ?

+ 3/e.

Proof. Take a choice u based on the trivial sequence CD 2 CD nzl... 7 CD. Let c be the number of Bi contained in C. Then sCD,.(C)= c/m. Since by 7.3 the translate of any Bi is a Bi we have by 2.16A I c/m - L(C)/L(CD)I< 3/e. By 13.17, I S C ~ , . ( C ) - S ~ ~ ( C e3 ) I <+ 4 / e = ~ ? . T h e r e s u l follows. t

13.22. Proposition. Axiom 33 holds for n + 1 Proof. Let AA', BB'E Rep(Ln+l),A ; ;B , s n ' ' ( A ) s33 > y o , s A A , ( A > ) s""(A) - e3 > s33 - e3. Now

> s33.Since

L(A)/L(AA'> ) sAAt(A)-e,*-3/e > s33 - e3 - e r By 13.13, A

=B

and A'

-

3/e > 2

~

F B'.

13.23. Note. In the proof of 13.22 we proved slightly more than was necessary, namely, if P,, ..., Pr Q , , ..., Q, denote divisions for A; B then Y = 1 and P, has type 0 (cf. the beginning of 13.13). Thus from now on we may interpret Axiom 33 in this strengthened form.

~

~

.

262

J.L. c-'tton, The existence of infinite Burnside Groups

5 14 liiformal Summary Another key axiom, Axiom 30, is shown t o hold for n

+ 1.

14.1. Proposition. Axiom 23 holds,Cor n + 1, i.e., i f A B E L:+l and 0 < a < /3 < sAB ( A )and p - a > e4 then for some left subword A L o f A , a < s A B ( A L< ) p.

Proof. For some u, we have (3 < s A B , ~ ( A ) Now . s A B , ~ ( A= ) x/d where x,d are integers and e divides d . For any A L

where y is an integer, 0 < y < x. Also if y is an integer, 0 < y then for some A L we have (1). Now by 13.17 (ii)

<x

so if (i) a < y / d < - e3 - 4/e then a < sAB ( A L )< p as required. Thus it is sufficient t o show there is an integer y such that da

< y < Min(d(0

-

e3 - 4 / e ) ,x).

This is so if (ii) d(P - a - e3 - 4 / e ) 2 1 and (iii) x - da > 1. Now (ii) holds since - a > e4 = 5/e + e3 2 l / d + 4 / e + e3, and (iii) holds sincex/d-a>p-a> l / e > l/d.

14.2. Proposition. Let Y E Lh,+l n L : + l . Then h = k. Hence by 7.14, Axiom 12 holds for n + 1. Proof. Suppose not; then h

< k say. We have

Z f k + ... -+ Y = Y B where A

= B = 1. By

11.8,

(2,Z ' E Rep(Ln+l))

J.L. Britton, The existence of infinite Burnside Groups

26 3

Z h E Y'A'

;B'. Now s( Y ) > h - 2/e > 1 - 2/e so s( Y ' )> 1 - 2/e > u1 + s33 so from (*) (also cf. 6.8) Y ' 3 Y'O = C ; D say and Z r k= DD',Y ' = Y , CY, and s ( C ) > s33.We claim that CL,DL exist such that CL DL and one of them has size > s33 and L(CL) < L ( Z ) ,L(D1) < L ( Z ' ) . CY2A' Y , = 2: where 2, is a C.A. of 2.Let C' be the left subword of C such that C' = C n 2,. Now ~(2,) > 1 - 2/e > s33 + u1 and A' - u37

-

-

so C'O D, where D, is a left subword of D.If D, C 2' the result follows. If Z ' C D, then s(2') > s33 + u1 so 2" CL. By Axiom 33 for n + 1 CL = DL and 2'= 2,. Now (*) induces CL- DL SO (cf. 13.23)DD'=CY2B'Y,= C Y 2 A ' Y l .But

-

Thus Se(k-h);I for some S E Cn+,.Now S E Cn+,C J,, contradicting 5.12. 14.3. Proposition. Axiom 30 holds for n We first need a lemma.

+ 1.

14.4. Lemma. Let 1 < w < n . Let S,, ..., sk be subelements o f L , and contained in a word X . Let T,, ..., Tk be subelements o f L , contained in Y . Let X , Y be (w)-bounded by where q < k52, kS3. Also assume either ( A ) X ,ylY , the Siare disjoint of size > u4, Ti is a weak image of Si, there is a positive integer N such that k > 2N + 5 and neither (Si+,,S i + N )nor ( T i + , ,Ti+,,,) is a subelem e n t o f L l u . . . u L, ( i = O , l ,..., k - N ) o r ( B ) X ; Y , S i i s m a x i ma1 normal of size > us and Im'Si = Ti ( i = 1, ..., k ) , N = 5 and k>2N+5. Then ( i ) ( T I ,T k ) 3 M ; M ' (end words E, K ) ;M . K-' ; MLJ;MtR where the right side is (w)-bounded by q + = q + Max(u4,q 3 1 , h31); M r R3 ( S N + 3S, k - 3 ) and M L , M r R are not subelements o f L l U ... U L , and (ii) ( T , , T k ) 3 M ; M ' ; M ; M L E M i MFMR

264

J.L. Britton, The existence of

infiniteBurnside Groups

wlierc the right side is (iv)-bounded by q + and M L ,M R are not siibclernents O j ' L 1 U '.. U L W ; M f M 3 ( s N + 3 , S,-N-~). Proof of 14.4. (A) For any Si we have by 6 . 2 6 , Si 3 Uiw:l Vi C Ti (end words E i , K i ) and there exist words UF D j VF , V k Fi UF ( ( w 1))-bounded by q + q 3 1 where U k , V F , V F , UF are not subelements of L LJ u L , v _ l . I f 1 < a < b < k for some a, b then

,

t..

and the right side is ( ( i t )- 1))-bounded by q + q31 < q + .Since the left side is \v-bounded by q the right side is w-bounded by q + 114 < q+. I f 1 < a < c < h < li then

( V , , V C ) = ( V l , V:)I.JUf,

Uc).K;'-J.K;'say

and since .I c ,/', J is (w)-bounded by q+.Moreover ( V l , V i - )3 ( T 2 ,T u P l ) which is not ;I subelement of L u ... u L if a 2 > N and ([Jf, U ( , )3 (So+ , S,-- ) is not such a subelement if c - a - 1 2 N. Now ( T I ,T k )3 (I/, , V,) so (i) follows if we take a = N + 2 , b = ~- 1. c = 2. I n ( + ) i f a - - -2 = N a n d l i - h - 1 = N then ( V , , V,") and ( V F , V,) are not s11belenlents of L 1 U ... U L w . Also (U,", U k ) 3 (SN.3, Sk-N-2). (B) By 6.27 if 1 < i < k then (SF, SF+,) ( T F , TF+,)(end words Ei,K j + l ) and there exist words SFL Di T k Z , TFL FiS): (w)-bounded by q + lz31.Also by the proof o f 6.27 Si 3 Ui V jC Ti(i = 1 , ..., k ) a n d ( U i , U i + , ) - ( S F , S F + l ) , ( V i , V j + l ) ~ ( T TF), + l ) . T hU~i~- V i has end words Ei. Ki. By 6.28, ( V , , V,) = ( V , , TFL)F,(SkR,U,), K - l E .I. K i ' where J is (w)-bounded by q + h,, < q+.Now k (I/, , T F L )properly contains T , which is maximal normal so (I/, , TFL) is not a subelement of L U ... U L,. Similarly for (SkR,U,). Now ( T , , T,) 3 ( V , . V,) and (SkR,U k )3 (S8, S k - 3 ) . Also

, ,

-

-

-

-

J.L. Britton, The existence of infinite Burnside Groups

265

Proof of 14.3. 1". We have 2,; ... 7 PQ where Z , E Rep L,+, and we may assume spQ,u ( P ) > s(P) - 1/e2 > ~ 3 0 1 /e2. P is (n)-bounded by b30. Using the A-B notation, Z , contains subelements Bi of L, (i = 1, ..., m ) ;let R i = A D o ( B i ) .Then R , , ..., R , say are contained = a/m. We shall prove that each R i i:i P contains in P and s,,.(P> a subelement of L n . This is trivial if yz = 1 so let n 2 2. Consider the diagram Zn-r

-+

n-r Zn-r-1

An-r+1 Y

where y is an r-tuple; here n > Y > 0. Say the range of i is 1 , 2 , ...,f for B Y i to exist; 1 + j , ..., f - j for A+ t o exist. Recalling 7.7 we have say z n - r n x l X I ,yrX 2 n~ Z n - r - 1 . 2". If the BTi are disjoint of size > q* then by 1 1.3A we have say

,

where N , C X , , N4 C X,, N6 C Z n P r - a n d f o r i = j + l , . . . , f - j there exist Cri and DYi such that BTi C N , , CYiC N 3 , DYi C N , , A+ C N6 and the weak images of B T i , CYi,DTi are C y i ,D T i , respectively. Also each of BYi, C r i , DYi, ATi has size > q" 3u4 2 U 8 5 . Also j = 3d + 15, d 2 t . u~~> d - 1. By 7.8, s(A,) > U 1 8 so f / t > s(A,) - €30 > u18 - c30 by Axiom 30. Let N = d + 12. Since S W + l ( B y " + ,B, , u + d ) > d / t - €)30 2 U 2 0 - €)3o = U 1 3 + 2u4 we have, X, u+N) where X ranges through the symbols A , B, C, D,s ( X , > ~ 1 3 - ~ 4 >r b ( u = j,...,f-j- N);hence(X,u+l,X,"+N)isnot a subelement of L u ... u L w . The Ni are ((w - 1))-bounded by so + s , ~< ks,. Since N , is w-bounded by so we have N 2 , N 4 , N , are w-bounded respectively by so + u 4 , so + 2u4, so + 3u4 < k56 ; thus all the Ni are (w)-bounded by ks6. 3". If the BTi are maximal normal of size > r" then for i = 4, ...,j- 3 there exist CYi,DYi in X , , X , respectively such that -

,

"+,,

J.L. Britton, The existence of infinite Burnside Groups

266

Im' RTi = C T i ,

Hm'CTi = D T j ,

Im' DTi = ATi

Z,-, is w-bounded by so;X , is w-bounded by so + u,. We now

make the assumption that (AT 1,ATf) is w-bounded by ks4. We prove that (DT1,DTf)is w-bounded by k,,. Suppose not. Then it contains a w-subelement S of size Z k,, > ro + e3. Let be maximal normal in X,. Then (DTl,DTL)3 and s(9) Z s(S) - e3 Z k,, - e3 > u26. By Axiom 26", S has the form U2 in Axiom 26. By 6.21A and 6.23, using 1 - r' + t26 < k$ and so + s26 < k', and k,, - e3 > i d l o , we have either l a :

s

s

and either 2a:Is(U,)-s(U3)I<

u3 or 2b: Is(U2)+s(U3)- I I G u2.

Hence s(Ul) > u I o since k,, - u2 > ulo. Hence

-

e3 - u g > ulo and 1 - (1

- r'

+ t26)'-

3 : Is(U)-s(U1)I< u3 By 12.12, 1 b and 2b d o not both hold. If 1a holds then k,, - e3 < s ( U ) + 2u3 < so + 2u3 < k,, - e3, a contradiction. If 1 a does not hold then 2a holds and s ( U 2 )< k,, + u3 < k,, - e3, a contradiction. Thus (XT1,XTf) is (w)-bounded by

(x= A , B, c,D).Also each xTihas Size > Y"

-

3U3 2 U s 6

4". We show that the number of B T i ,f - 2j, satisfies

(2)

(f- 2 j ) - 4 - 4(N + 2 ) > 2N + 5

where N is defined t o be 5 if the BTi are maximal normal as in 3" ; ( N = d + 12 in 2"). Thus in the situation of 2" we must prove that

J.L. Britton. The existence of infinite Burnside Groups

f - 2(3d

+

15) - 4

> 6(d +

12) + 13, i.e., f

> u18 - e3,

-

12u2,

>

-

267

12d + 119. Now

131/e.

Denote the last expression by u87. Note that the inequalities remain true if u18 is replaced by s ( A , ) . (2) follows since us7 > 0. If N = 5 we must show that f - 2j > 47. But f - 2j > uio > 47 (cf. 7.7A). 5". Since kS6 < Min(ks2, k,,), u85 > u4 and us6 z u5 we may apply 14.4 three times to obtain

say where the right side is (w)-bounded by k,, = k56 +

+ Max(u4, q31, h31). Also X, Y are not subelements of L and

U ... u

L

The left side of the equation is the subword of Z,-, generated by all A-descendants of A , except for the first two and the last two. If A , C ZnPr and A,, A, are disjoint then similarly (AYj+3'A,f'-j'-2);

X(AyM,A!)Y'

say where the right side is (w)-bounded by k,, and X , Y ' are not subelements of L u ... u L,. The left side is the subword of Zw-l generated by all the A-descendants of A,, A , except the first two of A, and the last two of A , . 6". Take any fixedAi (1 < i < a). We claim that if O < s < n - 2 and A , , A , are the leftmost, rightmost A-descendants ofA, in 2, then

26 8

J. I.. Britton, The existence of infinite Burnside Groups

for sonie P, 0 and some A: c A L , A: C A R , where the right side is (s)-bounded by k,,. This is true for s = 0 since b30 < k58. Assume it is true for some s, 0 < s < 12 2. Let A , , A , be the leftmost, rightniost A-descendants of A , in ZS+,; then A , l + i = A , and A, = A,. The left side of ( 2 ) is (n)-bounded by b30 so the right side is (s + 1 )-bounded by b30 + Z L < ~ k5, and has the form Pp'(A 3+[, A , f ' I ' - 2 ) Q ' Q . BY 5" ~

I

,,

where the left side is ((s + 1 ))-bounded by k S 7 . Hence

R , , y ~ lP P ' X ( A Y ,A,") YQ'Q whcrc the right side is (s + I)-bounded by M a ~ ( +b zc4, ~ ~k 5 7 )= k 5 8 and (s)-bounded by Max(kS8,k S 7 )= k,,, thus is ((s + 1))-bounded by k S 8 . The A-descendants o f A i in Z n - , are say A i ..., A , f-i hence I-?, P"(AM1+i,AM. .)Q". Now X ' A M Y 'n!l ( A i 3 + j , f-l and s"(AM) > i l g 7 ~E ; ~> 2.13, hence R i n:l HiAM Ki say where the I))-bounded by k,, < b2,. R , is (n)-bounded by right side is ( ( 1 1 b3()< h,, hence by 6.10 Ri contains a subelement Siof L , of size > its7 -. ~- 114 > qq:namely a weak image of AM, 7 " . I f Case 2 of 7.8 holds then the Ai are disjoint; hence the Ri are disjoint (cf. 7.7 (iii)). For the X , , ..., X , of Axiom 30 take S , , _ _So. _ , N o w Z,, does not contain a maximal normal subword of size > Y" and ( Z t l p 1=) (Z,*).Suppose that P contains an n-subelenient of' size > c". Since c * 2 i149, Z n - , contains an n-subelement T of sire > cq: -~ 114 > Y" + e3 (cf. 12.15) so s(T) S ( T )e3 > i f a, contradiction. 8". Now assume that Case 1 of 7.8 holds. Then the A i are maximal normal and kernels exist but the A i are not necessarily disjoint. Let h = u or u 1 , whichever is odd. Then R , , R,, R , , ..., R , are disjoint and ~

where the right side is

((12

~

1 ))-bounded by k 5 8 . Let i range over

J.L. Britton, The existence of infinite Burnside Groups

269

1 , 2 , 3 , ..., a and let j range over 1 , 3 , 5 , ..., b. A weak image of A? in (3) is still Si.Now by 4", 5" s ( A r )> (us7 - u18 + s ( A i ) )- ej0. But by 7.8 s(A,) > Y" - Y , ~ ;hence s(AM) > 2487 - 2418 + Y"- Y26 -E ; ~ = uS8> u5. Therefore Im(AM) Next I

=Isi.

for some 2 where the right side is ( ( n - 1))-bounded by k58. Write (5) briefly asA = B. In (5) a weak image ofAM C B is AM C A CA since (InA, A:) n-l (In B, A:). Hence Im5(i:(C B ) ) where the subscript 4 refers to eq. (5). By 6.1 1 if Im X exists then Im(X) = Im X ; so Im (Ax(. B ) ) = AT (C A ) ) . We wish to apply 6.13. Now s(Ai ) > U 8 8 - E 3 > u6 Im5(Ay)>s(AM)-U4-E3> 2 4 8 8 - 2 4 4 - E 3

3d

> u6. With respect to (4) let Im(Im5A?)

= Ci. Then by 6.13 Ci is left of C i + l (i = 1,2, ..., a - 1); the Ci are maximal normal in - 2u4 - 2e3 > Y * . For the required X , , ... , X , ( R , R,) of size > of Axiom 30. take C , C3=.., Ca-l; they are maximal n-normal in PQ. Also Ci = Im(A . ) - Si3 SiC Ri. Let P' C P and let the number of C,, ..., Ca-, in P I be y and the number of R , , ..., R , in P' be p. We prove that I p - yI < 6. Say R i + l ,..., R i + pc P'; then R 2 u + l R , 2 u + 2 ..., , R 2 t + lC P' where 2t + 1 - 2u 2 p - 2 and S 2 u + l S, 2 t + l are in P'; hence so are C 2 u + 2C, 2 t .Thus y 2- 2t - 2u - 1 2 p - 4. Now say C i + , ,..., Ci+.YC P'; then C2u+l.C 2 t + lare in P' where 2t + 1 - 2u > - - - ~ S O R , ~ + ~ ,aRr e, i~n P- '~a n d p 2 2 t - 2 u - 3 2 y - 6 . In Case 2 of 7.8 if y is the number of S , , ..., S, in P' then y > p 2 y - 2 so again Ip - yl < 6. NOWsPQ,(P') = p/m S O by 13.17

,

d-

hence I y/m - sPQ(P')I < e3 + 1 O/e < E since I spQ0(P) - s(P) I < 1/e2 we have

; ~ .If

P' = P then p

~y/m - sn+l(p)l< 6 / e + l/e2 < ~ 3 0 .

=a

and

270

J. L. Britton, The existence of infinite Burnside Groups

14.5. Proposition. Axiom 10 holds f o r n + 1.

,,rl

Proof. Let X be ( ( n + 1))-bounded by b m . Let X I. Since Axiom 20 holds for n + 1 (cf. 13.1 I ) , we have that 6.1 holds for n + 1; hence if X f I then X contains a subelement C of L, (1 < e, < n + 1) of size greater than k , or Icy. But s(C ) < bm < Min(ky, kl). Hence X = I.

§15 Informal Summary Propositions 15.3, 15.3A below correspond to the informal results 4.4A and 4.4B. Using these propositions we can discuss simultaneous ( n + 1)-replacements and prove that Axiom 26 holds for n + 1 (cf. 4.4C). 15.1. The statement that Axioms 18,22 and 33' hold for n be called Condition D, + or simply Condition D.

+ 1 will

I5.lA. It is convenient to proceed as follows. Assuming Condition DI2+,we shall develop in this section a theory of ( n + 1)-replacements, say T n + ,. Since D, is true, so is T,. Then, using T,,, we prove Dn+l. 15.2. Let W = X A Y where A is ( n + 1)-normal. Call W' a weak reX . B-'. Y and A B € for some B. placement of A in W if W'; By 5.25 for n + 1 any two weak replacements are equal in G,.

15.3. Proposition. Let the (linear) word HZ = BDZ be ( ( n- 1))bounded by k,, and n-bounded by k,7. Let ZT-' E Rep(L,+,) and let DZ be maximal ( n + I)-normal in BDZ where s ( Z ) > s(DZ)- 1 . 4 ~ ~ and "58 > s ( 2 ) > u59. Then B D Z . Z - l T C ' T R where C ' T R is (n)-bounded by k,,, s(TR)> s ( T ) - u77 and

arid as usual T R denotes the maximal normal word containing TR.

J. L. Britton, The existence of infinite Burnside Groups

211

Proof. 1". By Axiom 1 la, BD.T C-', where C is (n)-bounded by bmin. Put H = BD.Then H . T . C ; I. To apply 8.2 of Chapter I we require that if SS' E L i , i < n , and S is *-contained in H , T or C then sssl(S)< 1 - 5/p - 5e1 - eo = 1 - 5(r + e , ) - 12/e = 0 say (if i = n , replace r by r,, and 0 by Po.). As in 6.2, this is so if H, T, C are ( ( n- 1)-bounded by 0'= 0 - 2e2 - 4/e and n-bounded by 0; = Po - 2e2 - 4/e. Now Z is (n)-bounded by str+< 0'< 0;;C is @)-bounded by bmin< O f < 0;;H is ( ( n - 1 ))-bounded by k,, < 0' and n-bounded by k,, < 0;. Hence there are divisions H , , H 2 , H , of H ; T , , T 2 , T3 of T ; C,, C2, C, of C and

H, ; TI'

(end words U, I )

T3 ; C;'

(end words V ,I )

; Hi'

(end words W, I )

C,

for some U, V, W. Let H i be the largest common subword of H, H , (possibly H i is I ) . Then L ( H ; ) 2 L(H,) - 1. Similarly define H i . Now define H i by H = H i H i H i . Similarly T = Ti T i T i and C = C; Ci Ci. By 5.19 if H i has size > ro then s ( H ; ) - e3 < s(H,) < e2 + s ( H ; ) + 2/e. Choose H i 2 of maximal length such that H i F H i , H i 2 and H i 2 Z is a subelement of L . Then H i 2 Z C HZ 5 BDZ, so H i 2 Z C DZ. Since s ( Z ) > u59 2 ro + E , > ro we have +

,

s ( H j 2 Z )< s ( D Z ) + € 3 < s ( Z ) + u57 + € 3

Now H , Z . 2-' T , ;U and H,Z ; T;'Z. Let Zdenote the maximal normal subword of H,Z containing 2. We shall prove that

XL. Britton, The existence of infinite Burnside Groups

212

By 12.16 and 13.10, Axiom 25 holds for IZ + 1, hence so does 6.10. Thus (Z)' ( 2 ) O ' 3 M where M is a subelement of Ln+' and s ( M ) > s ( Z ) u4. Also s ( M ) < s ( T I I Z )+ e3 hence s ( Z ) < s(Tf'Z) + + e3 + u4. Again, taking a weak image of T;' Z , say N , H 3 Z > N, s ( N ) > s(T1' Z ) - u4. By 6.1 1, since s ( T ; ' Z ) > s(Z)- e, > 1159 - e3 > z i 5 and Max(str+, k,,, k37)< b2, we have Im(TI'2) = @. Now Z C H 3 Z and s ( Z )> uS9> us- u4 hence 2 = N (again b y 6.1 1). Hence N C Z a n d s ( N ) < s(z) + E , and s ( T i 1 Z )< s ( z ) + e3 + u4, proving (2). Now H 3 Z is H ; Z or c H i Z ( L ( c )= 1 ) so either (i) C H;Z and 2.- H j 2 Z , or (ii) Z-cH;Z and Hj f Hi2. In either case,

-

~

z

Adding ( l ) , (2), (3), (4) we find - a < s(l';-lz)-s(z)< u57+(Y

where a = 3e3 + u4 + e2 + 2/e. Next we show that .s(Ti)< 2e2 + 4/e + rb = PI say. If L ( T 2 )> 2 then by 8.2 of Chapter I T2 cAd where A is a subelement of L u ... u L,!. Now T i = c'Ad', c - c', d - d' hence s n + I ( T ; )< 2e2 + 4/e + s n + ' ( A )< 2e2 + 4/e + rb. If L ( T 2 )= 1 or 2 then L ( T i ) = L ( T 2 )< 2e2 + 2/e. If T2 = I then L ( T ; ) < 1 so

s(Ti)< €2. Thus

1 - 2/e < s(Z-' Ti T i T i ) < 4/e + s(Z-' Ti ) + p1 + s ( T ; )

< 4-je + s(Z) + u57+ a + p1 + s(T;)

Hence T i is normal ( 1

- u58 -

( 6 / e + u57+ (Y + P1) > ro + E 3 1.

J.L. Britton, The existence of infinite Burnside Groups

213

s(2-l T i ) + s(Ti T i ) < s(Z-' Ti T i T i ) + (2e3 + e r )

< 1 +(2e3 + E:). Also

-

e3 + s ( T i ) < s(Ti T i ) and s ( Z ) < s(Z-' T i ) + Q! so by adding,

T , with end words V',I. 2". We have "1' ; Case 1. Cf' nzlT,. T3 contains say Pi where k: < s(Pi) + Qi < s ( P i ) + bmin.Hence T i contains a normal subelement of L , of size > k: - bmi, - e2 - 2/e - e3 > u l l . By 7.3, Ti T i T i 2 - l contains rn distinct maximal normal subwords of size > Y" which as in 7.8 are denoted by B , , ..., B,. Let u of them be contained in T i . By 13.17 (ii) and 13.16 (111)

u/rn Z (s(T;)- e 3 )- E,

- 4/e.

Now T , 3 T i 3 (Ba+',..., Ba+u).6.27 is available since Max(str+,bmin)< k and r" > u s . Let Bl!be the image of Bi in ,9' C;'. For i = a + 2, (Bi ,B : k l ) ; ( B F , Bb+ ) which we abbreviates to B' ; B (end words E ' , K' say); thus Cii= (CilYBf(Cf')rand T3 = T i B T ; . Also there is a word BiRLABkz ;B'.K'-'. Let

D 0 -= c-l 3 C 2- ~ ( C ~ ~ ) ' B ; ~ L A B ~ LT;. ,R, Then Do = Cjl C;'(CI')'B'K'-'Tj ;C-'. Now Bi.RLABLFlis (n)bounded by Max(str+, bmin)+ k,, = h say. BIRL and B$' are not subelements of L U ... u Ln-' hence Do is ( ( n - 1))-bounded by h (cf. 5.32). Since Do ; C-l , Do is n-bounded by bmin+ u4 (cf. 6.1 OA). Hence Do is (n)-bounded by h + u4 < ks0. Do has the form C;' C;'(Ci')LATR. T j contains Bi+3,..., B,,,; hence

s ( T R )Z ( u -4)/m z s ( T i )- 2e3 - 4/e - 4/e

> (1 - s ( Z ) )

-

(6/e + us7+ Q! + pl)

= 1 - s ( Z ) - p , say

-

2e3 - 8 / e

J. I-. Britton, The existence of infinite Burnside Groups

214

Now

Cuse 2. Cil nylT,. If (i) of 7.3 holds we may proceed as in Case 1. Thus we may assume that T ; Tk TiZ-' contains B,, ..., B,, disjoint n-subelements each of size > u,,. Again (111) holds. 6.26 is available since Max(str+,bmin)< k52 and u l l > u4. Hence for i = a + 2 (in particular)

a weak image Bi. of Bi exists and Bi." -BY (end words E ' , K' say) and there is a word BIMLAByR= B:.". K'-' , ( ( n- 1 ))-bounded by Max(str+,bmin)+ q 3 1 = X' say. As before there is a word D = C;lC;l(Cil)LATR, (n)-bounded by X' + u4 < k50 and again (* *) holds. > -u77. Thus in both casess(TR)-s(T)> - p - ( 2 e 3 + R . T IS normal since I - u58 - p > ro + e3. We now have s(?)>s(TR)-e3>

>

1

~

1 -s(Z)-p-e3

(s(DZ) + e3) - p

- e3

hence

(1V)

s ( D z ) + S(F)- 1 > - p

-

2E3

2 -u78

3". Write M for T R ; then Do E B'M = B'M'TR say for some B'. By Axiom 33' for n + 1, B'M'TR ; B'MLE,Z,F,MR (since s(M) > 1 - 2158 - p - e3 > Max(tg3,r33 + u , ) and Max(X, X') + u4 < b3,); also the right side is (n)-bounded by X" = Max(X, X') + U4 + + c,,. Now s(Z,) > s(M) r33 > u1 and s(TR)> 1 - u58 - p > u1 so Z:, (TR)' exist, and Z, 3 2: 2 ; ' . Define T,, T2 by (TR)O= T, T2, T I = Zy' n (TRIO. If T, has size > a' (defined below) then since a' > u 3 and h" < b,, we have Y F , M R contains a weak image J of T2 where Z, XZYY;also s ( J ) > a ' - u4. Let J - J , J , where J , Yn J . ~

-

-

J.L. Britton, The existence of infinite Burnside Groups

215

Either s(J1)< ro or s(J1)< s ( Y ) + e3 < u2 + e3 (cf. 6.8). We have

s(J)

< s(J1) + s(J2)+ 2 / e

a'- u4 < Max(ro, u2 + e3) + 2 / e + s ( J 2 ) s(J2)

> a'-

u4 - Max(ro, u2 + e3) - 2 / e > yo

s(J2)

< s ( F I M R )+ e3 < q33+ e3

so defining a' by q33+ e3 = a'- u4 - Max(ro, u, a contradiction. It follows that

+ e3) - 2 / e we have

s((TR)O)< s(T,) + s(T2)+ 2/e < s(T,) + a'+ 2 / e . By 6.8, since U(TR)' = T R for some U, s ( U ) < u,; hence s ( T R )< u2 + s((TR)') + 2/e < s(T,) + a' + 4 / e + u2. Hence

s ( T 1 )> 1 - u58 - p - a'- 4/e - u,

> Max(sg3,ro + e3).

By definition of T , , TR 3 T , ; S , = 2: n (TR)O'C 2,. Hence T , = S , since Axiom 33 is true for y1 + 1. Where 2, = Z,,S, Z,,, T R = T R LT 1 T R R then (cf. 13.23) T , T R R ;S , Z 1 2 F , M R . Thus Do (t B'MLE,Z,, T , T R R which is (n)-bounded by A'' (cf. 5.32). Let D'=M L E l , and Z ' = Z,, T * T R R .NOWM L E , Z , , T , is a subelement of L,+, since it is contained in MLEIZ1,T , T R R is a subelement of L n + , and T , is normal. Hence D'Z' = M L E , Z , , T I T R R is a subelement of L n + , (cf. 5.28A). Note that - - Z ' = - T. Now D'Z' is ( n + 1)-normal, D'Z' (t B'-' .Do = M = T R . Hence (i) ~ ( D ' z ' > ) s(TR) - u4 - e3 (ii) s(D'Z') < 433+ s ( Z ' ) + 2/e. Case 1'. Either Z' C T or 2' = VT where s n + l (V ) < Y;. Then 2' < s ( T ) + e3 or s ( Z ' ) < r i + s ( T ) + 2 / e so in either case ~ ( 2< ' )r; + s ( T ) + 2 / e + e3, hence by (i), (ii) and (*)

276

J.L. Britton, The existence of

infiniteBurnside Groups

s(F)+ s ( ~ ) 1 < u4 + e3 + q33+ 4/e + rh + e3 + --

+ (26, + E : )

=A

say. Since s ( D Z ) - uS7< s ( Z ) we deduce (V)

S(F)+ ,'$(LIZ) 1 < U s 7 + A < u 7 8 . -

Cast. 2'. Not Case 1'. We shall obtain a contradiction. Since -2' = - T we have Z'=: VT, sn"(V)> r i . In particular V + I . Now

Now-Z-'T=-T-Z'=-VThence-Z-'--Vand Z- = V-'-. Therefore BDZ- = BDV-'-. Hence the d o t can be removed from ( i ) Next . we show DBV-' is (n)-bounded by k37. If V-' = ZH for some H then BDV-' = BDZH. Now Z is not a subelement of L u ... u L n , BDZ is (n)-bounded by k,, and ZH is nbounded by str+< k,7. If V-' $ ZH then Z = V-' K for some K and BDV-' C RDZ which is (n)-bounded by k37. V-'D'-'B'-' = = (B'D'V)-' C (B'D'Z')-' which by above is (n)-bounded by A"; V-l is not a subelement of L u ... U L,, . Hence the left side if (t), with d o t removed, is (n)-bounded by Max(A", k 3 7 )< b _ in contradiction t o Axiom 10.

15.3A. Corollary. Assume further that RDZ is ( n + 1)-bounded by b whcre u79 ~ 8 <3 h < 1 - u g 3 . Then (i) C ' T R is ( n + 1)-bounded by Max(b + uk3, 1 - s(DZ) + Z L ~ ) (ii) I f Q is maximal ( n + 1)-normal in C'TR where Q is n o t TR and s(Q) > ";9 then there is a maximal (n + 1)-normal subword Q* c?f'RDZ, not DZ, and -

J. L. Britton, The existence of infinite Burnside Groups

217

(iii) If P is maximal ( n + 1)-normal in BDZ, not DZ, s(P)> u ; ~ then there is a maximal ( n + 1)-normal subword P* of C'TR, not F , where I s(P)- s(P*)I < ub. (iv) If Q** exists it is Q ; if P** exists it is P. (v) If B = Z'Cwhere 2' is a subelement of Rep L n + l of size > u80 we may take C' = Z r L K(for some K ) in (i)-(iv) where s(zIL) > s(z') - u81.

Proof. 1". First assume only that Q is ( n + 1)-normal in C'TR, Q @ FR and s ( Q ) > q9. Now C'TR = Do = "5' C;l(CI1)LATR 7 C-' = Ci' c;1 c;' so

(5) Also

(6)

c-1= c - ~ L A T R 1 n 1

D 1 say

T3 ; CI' (end words V , I )

Write Q1 = Q n C;', Q2 = Q n C;', Q3 = Q n CIILATR. 2". We show that D, is ( n + 1)-bounded by a 2 where

Suppose not. Then it contains an ( n + 1)-subelement S of size

2 a 2 > u 3 + u4. Let S' be a weak image of S in C;' (under ( 5 ) ) (the

right side of ( 5 ) is (n)-bounded by A"). Then s(S') > s(S) - u4 > u3 Let S" be a weak image of S' under (6). Then s(S") > a 2 - 2u4 > ro + e3 and s ( T 3 )2 s(S") - e3 > ro. Thus we have uS9+ s ( T ) - 1 < s ( Z ) + s ( T ) - 1 < (2e3 + E;)

s ( T 3 )< s ( T ) + e3 and s(S) - 2u4 < s(S")

< s ( T 3 )+ e3. By addition

we obtain s(S) < a 2 , a contradiction. 3". We show that s(Q3)< Max(ro + c 3 , u6 + u4 + 2e3) = P2 say. Assume not. Then Q3 is normal and of size > us + u4 + 2 ~ ~ . Let G3, T-it be maximal n o r m a u n D, and containing Q 3 , TR . Now

D,C Do s o ?

C T R . If

g3= TR then Q3 C TR n Q; hence the

subword TR u Q of Do is an ( n + 1)-subelement containing

so

J.L. Britton, The existence of infinite Burnside Groups

278 -

is T R;+us Q C

TR which contradicts the hypothesis. Thus

Q3 # T R . NOW r3

s ( T R ) > s ( T R ) - € 3 >1 - S ( z ) - P - € 3 >

>U6+

1-U58-P-€3

u 4 4 €3

and s(&) > s ( Q 3 ):e3 > u6 + K + ~ e3. By applying 6.13 to (S), (6) we see that Im Im T R , Im Im Q 3 are different maximal normal subwords of T 3; this is impossible. 4". Thus s ( Q ) - s ( Q , ) < s(Q2)+ s ( Q 3 )+ 4/e G r-6 + P2 + 4/e = p3 say and s(Q1) > u79 - p3 = p4 say > u 3 . Now Q1 C Cjl H l . Let Q' be a weak image of Q1 in H I . Then

-

> Max(ro + e 3 ,e3 + us7 + 2e3 + €,*I. In H , H 2 H 3 Z , clearly Q f and Z are disjoint. Let Q* be the maximal normal subword of HZ containing Q'. Then where = p3 + u4 + e3 we have s(Q*) 2 s ( Q ' ) - e3 > s(Ql) - u4 - e3 > s(Q) - i l M . We prove that Q* # DZ. If Q* = DZ then Q' C DZ so C D. Also Qf

hence s(Q') < e 3 + us7 + 2e3 + E ; , which is a contradiction. We now have s(Q1) < s(Q') + u4 < b + u4 SO s(Q) < s(Ql) + p3 < b + u4 + p3 = b + ug3< b

+ ulg3. -

Now suppose Q , is some ( n + 1 )-normal subword of T R .Then S'(Q0)

< S(F)+ € 3 < u78 + 1 - S ( D Z ) + € 3 .

J.L. Britton, The existence of infinite Burnside Groups

Since u78 + E , bounded by

< 2.48

279

we have proved (i); indeed C' TR is ( n + 1)-

5". If Q is now maximal normal then since s(Q1) > p4 > u6 we have by 6.12 that Q n C;' and Q* n H , are images of each other, with respect to C;' H,, in the sense of 6.1 1. 6". Now assume only that P is ( n + 1)-normal in BDZ, P P DZ and s(P) > u79. We argue similarly, as follows. Let P, = P n H , , P2 =_pnH 2 , P, = P n H,Z. First, s(P3) < pz: (Assume not; then let 2, P, be maximal normal in H3Z. Thus Z C DZ. Also p3 # 2. Now H , T i 1 (end words U, I ) so H,Z Ti'Z. Since uS9- E , > u6 we may use 6.13 and obtain a contradiction.) Next s ( P ) - s(P1)< 0, and s(P1)> p4 > u6 2 u 3 . Now P, C H I C;'; let P' be a weak image of P, in C j l . Then

-

-

-

-

s(P') > s(P,)

- u4

> p4

-

u4 > Max(ro + E , , A

+ p + 3 ~ +, E : ) .

In C;'C~'Dl f D o , P' and TR are disjoint; let P* be maximal normal in Do containing P'. Now s(P*) 2 s(P') - E , =

> s(P,) - u4 - € 3 2 s(P) - p3 - u4 - € 3 =

s(P)- us4

If P* = TR then P' C T K so where

Tx = U T R we have P' C U and

By Case 1 ' of 3" of 15.3 and ( * * ) s ( Z ' )< s(TR)+

YX + 2/e +

E,

+

+p + 2e3 + E? and s ( p ) < u4 + E, + q33+ 2/e + s(Z ) so s( TR)< s( T R ) + A + p . Thus s(P') G A + p + 3e3 + ET, a contradiction. Also if P is maximal normal then P n H , and Thus P* # TR. P* n C;' are images of each other with respect t o HI C;'. 7". If Q is maximal normal and of size > u79 + u84 then s(Q*) > u79;write P for Q*. Then by 5" Im(P n H , ) = Im(Q* n H , )

-

280

J. L. Britton, The existence of infinite Burnside Groups

so P* n Cil = Q n Cil. Since a normal subword is contained in a unique maximal normal subword, P* = Q. Thus Q** = Q and I s(Q) - s(Q'I) I < ug4.In particular since u ; 2 ~ u79 + us4 and u84< ufMwe have (ii). A similar argument with the P of 6" yields ( 5 ) . 8". Finally consider (v). Here BDZ = Z'CDZ. Since ~ ( 2> ' )us0 > 1179 we may write P for Z' and then P , = Z' n H , has size > s(P)- p 3 > p4 > u,. With respect to H , Cil (which has end words I, W ) we have V = Py .;; N say and s( V )> s(P1)- u,. V is a left subword of Z ' ; N is a left subword of Ci'. Thus V - N is analogous to the situation of 2" in 15.3, where Ci' ; ;T,. Thus N ;VLENR where the right side is (n)-bounded by Max(A, A') + u4 and s( V L ) > s( V )- 2e3 - 8 / e ;also N R is not a subelement of = p3 + u1 + 2e3 + L u ... u L,_, . Thus s( VL) > s(P) - usl where + 8/e. Where Ci' = NM, we have BD. T ; C-* ;Do = C'TR and D, -Ni14C;'(Ci')LATR ;V L E N R M C ; l ( C i l ) L A T R= Db say. Thus Db is ( ( n - 1))-bounded by Max& A') + u 4 ; by 6.1OA it is nbounded by Max(A. A') + 2u4 < b,, and ( n + 1)-bounded b y 7 + u4. Thus (i) still holds since L L +~ u4~ < uk3 and u78+ e3 + u4 < ub8. If Q is maximal ( n + 1)-normal in Db of size > u ; ~ then since 21b9 > u6 we have, with respect t o Do = Db, that Im Im Q = Q and Is(Q) -s(Im Q)I < 1i4 + e3. Hence s (ImQ) > ub9 - u4 - e3 > u79+ U g 4 , so Is((Im Q)*) -s(Im Q)I < u84. Since us4 + u4 + e3 < u b , we see - 294 > u6. (ii) still holds. Also (iii) still holds since

-

15.3B. Corollary. With the hypothesis o f 15.3 and the further hypothesis of 15.3A, but with n replaced by n - 1 throughout (so that, e.g., ZT-' E Rep L,) let BDZ = J X for some X , where J is a subelement o.f Rep Ln,l of size > 2196 2 ~ 3 0 thus ; J contains subelenzents C , , ..., C, o j L , as in Axiom 30. Then we may take C ' T R = J L K F R f o r some K, where JLcontains C,, ..., CU+,, ..., CaPu where (a - v)/t > (a - 2v)/t > s ( J ) - u97 and t is as in Axiom 30. Proof. BDZ ;B D . T n z lC ' T R . Let P,, P2,... Q , , Q,, ... be divisions for J X = BDZ - - n = - C' TR . There is a Pi of type n , otherwise ZT-' n f l I, and pi = Z, Qi = T R . Also P,, ..., Pi-, have type at most n - 1. Let .I = J , J 2 and let C,-,, ,.., C, be the C's in J,. Then

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s(J2) 2 (u + l)/t - ebOand I s(J) - a / t I < €30. We shall have s(J2) > u9g = Max(u3, u4 + rb, u1 + r b ) if (u + l ) / t > ejO+ 2498, which holds if we define the integer u by u + 1 > t ( & + u98)> u ; we note that €bO + u < u96 - e30< a / t so 1 < a - u < a. Hence J, 3 J; J,' 9% 3 F where s ( F ) > u98 - u4 > rb and

-

Ji

s(J;) > u98- u , > rb hence neither J;' nor is a subelement of L u ... u L , . Hence J;, J;' $? 2, TR respectively. Let J J, J;J4 and C'TR = LJ;'M. Then J, n:l L. Now J, 3 J, 3 C,, ..., Ca-up2. Let u = u + 4. Then (a - u - 2 ) / t > (a - 2u)/t > s(J) - €30 - 2 ( ~+; u98 ~ + 4 / e ) = s(J) - ug7. Now Ca-u-2 C J, n:l L ; let C" be a weak image in L. Then by 6.26 with H = Ca-u-2 we have C ' T R = (In J, C ~ - u - 2 ) E ( C " R Fin , T R ) .The right side has the form JLEP where P 3 J i ' M 3 TR and J L 3 C, , ..., Ca-". 15.3C. Note. Let JAJ' be ( ( n- 1))-bounded by Min(b2,, b31) where J, J' are subelements of Rep L , , , of size > ug6.Then JAJ' nzlJLY J t R where the right side is ( ( n- 1))-bounded by Max(str+, bmin)+ q 3 1 ,and J Lcontains C,, ..., Ca-" as in 15.3B, and similarly for J I R .

Proof. Consider JAJ' n - l M, where M is ( ( n- 1))-bounded by bmin and consider J,, J,, J, as in the proof of 15.3B. 15.4. We now discuss simultaneous replacements in cyclic words; we leave the easy modification of the subsequent argument necessary to discuss linear words to the reader.

15.5. Definition. Let W be a cyclic word (n)-bounded by so and (n + 1)-bounded by 1 -z,~. Let the max. ( n + 1)-normal subwords of size > 226 be N , , ..., Nh (h Z 0). If h = 1 assume L(NI) < L ( W ) . Let u be a sequence ul,..., uh where each ui is 0 or 1 and ui = 1 implies s ( N i )2 z (z is as in Axiom 2 6 ; the set of Ni such that ui = 1 corresponds to the set M,,..., M, of Axiom 2 6 , but we do not assume here that k 2 1.). If h = 0 let u be the empty sequence. Any cyclic word W , arising from the following construction is said to arise from W by simultaneous replacement of those N i such that ui = 1. ( W , is not unique.)

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1 O . Consider Ni; it may be written ABC where A , C are of maximal length such that s ( A ) ,s ( C ) < yo + E , ; B is called the formal kerrzel of N , and we have s(B) 2 s(Ni)- a, > 226 - a, = = a, 2 Max(t3,, yo + e 3 , s33+ r,,), where a, = 2(ro + E , ) + 4/e. 2". Let ii # 0. Let K , , ..., Kh be the formal kernels of N,,,, N h ; then K , , _ _K,? _ , are disjoint. (If C , , ..., c h are the kernels of A', , _..,Nh then Kic C, (i = 1, ..., h ) . Choose a word Kf = KkEiZiFi K r as i n Axiom 33'. Then K , ;Kl! and s(Zi) > s ( K i ) - r,, > s33,so there is a unique Ti such that Zi T;' E Rep L,+,. Let Wfarise from W by replacing each Ki by KI. Then W ; is (n)-bounded by so + c33< so + s26 and W W ; .Let W, arise from W by replacing each K, by Kl! and each A i such that L ( A i )Z 2 by a word A ; of the form .44Di4F(A: $ I, A ; $ I ) which equalsAi in Gn in such a way that W , is (n)-bounded by so + s26 (for example, each A ; could be A i ) . W , has the form (Z,B,...Z,B,) and W z W , . Define Q, t o be Zi if ui = 0, Ti if u l = 1 . Consider the linear word Q 1 . R , . . . . . Qh.Bh. First let Iz 2 2, and consider Q1. B,. Q2. We prove this is equal in G I , to a word U = Q ; where U is (n)-bounded by k,, and .s(Q:) > s(Q1) uU2,s(Q2 ) > s(Q2) - us,. There are three cases (i) Z , R , . T2 (and its dual), (ii) T , . B , . T,, (iii) Z,B,Z,. Cuse (i). Write Z , B,Zz_- BDZ where 2 = Z,, DZ = Then R = Z ; C where Z ; = Z,\Z,. We wish t o use 15.3 and 15.3A (v). Since W is ( 1 1 + 1)-bounded by so + u4 the hypothesis except (t) s ( Z ) > s ( D 2 ) 1157 are satisfied (so + s26 f k,,, k,,; a, - r3, Z u S 9 ; so + i r 4 < Po where Po = Max(so + u 4 , u79- us, + l/e2 ) < 1 - us,; a, - r33 2 / e - yo - e3 > use.) To verify (f) note so + ii4 f that W - W , induces K , = K k E,Z2F, K f ; considering Im 2, : Z , 3 Z f - Z:' 3 B , B = Im Z , . Now 2;' c K2 so B c K , and Im Z , = Im Z, = K2 = N,. Hence s ( N l )> s ( Z 2 )- u4 - E , and s ( 2 ) > s ( N ) - a , - r33 > s ( Z ) - uS7 since uS7Z u4 + e3 + a, + r3,. Thus Z , B,. T , ;Z i L KT; and the result follows since uU22 u g l , ZA~2 , i/77 and k50 f k,, . Case (ii). Write Z i L KTF in the form ZDB where ZiL 2, ZD = and let Z" T;\Z. We check the same hypotheses as in Case (i). k50 < k,, , k 3 , .

Qk fl,

~

z2.

~

~

z

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J.L. Bntton, The existence of infiniteBurnside Groups

s(Z2)> s ( z 2 )- e3 > z - al - r,,

so + u4 + E,

- E,

= P,.

< u58. s ( Z i L )> (a2 - z26+ s(N1) t ;but

from Z , B , Z 2 n + = lZ i L K T ; ~ e h a v e s ( Z ; ~ ) < v+, s ( Z 1 ) < v 3 + u 4 + e3 + s ( N , ) ; hence s(ZiL)> s ( Z i L1 - TI where TI = u 3 + 214 + e3 + z26 + t - a 2 . Now 77 < us7.s(T) 2 1 - 4/e - (so + c , ~ ) . -

1 -so-

1.477 - 6/e - c,, -Yo-

e3 2 ugO

Where ZiL7-l E Rep L n + , we have ZiL7-l = Z , TI' hence T,.B,- T2 = T~ K Z f f Rfor some K , where s ( Z " ~>) ~(2") - us1 and ~ ( 7>~s(7) ) - u77.If T , contains T ~ then , T~ has the form T f ; now since s ( Z i L )< s ( Z l ) + E , we have s(7) - u77> s(T1)- us2 If T~ contains T I the correspondsince u82 2 4/e + 3e3 + el* + ing result is true trivially. Finally Z f f Rhas the form TF and has size

> s(T2)- us1

-

u77- 2/e - yo

-

e3 2 s(T2)- u82

Case (iii) is easy and is left to the reader. means Thus Qi.Bi.Qi+, ;Q: YjQi"+,(i = 1 , ..., h ) where Q1. Also s(Q:), s(QF) > s ( Q i ) - ~ 8 and 2 s ( Q i ) > ~ 6 BY . 5.29, with a = u82, since v6 - u82 > ro + E, and v6 2 2uS2+ 4e3 + we have that QF, intersect; say Q i zFiGiHi, Q: FiGi, QiR =- GiHi. Write for Gi. Also s(QM) > s(Q,) - v7 where v7 = Max(uS2 + yo + e3 + 2/e, 2 2 +~3e3~ + ~ + 2/e). Let W, = Y;.. Y,); this is the required W, in the case Z y C W,, so under consideration. If ui = 0 for some i then W,; define W 2 to be W, in this case. If ui = 1 for all i let W,

ET

QF QM

(QF

Qr

ET

QM

J.L. Britton, The existence of infinite Burnside Groups

284

W , arise from W , by using the sequence 0, 1, 1, ..., 1. Then W , ,- W 2 and W 2 nTl W,. Thus in both cases (7)

Now let h = 1. Here W = ( N , C ) = ( K , A ,)for some C and W , = ( Z , B 1 ) .Also Q1.B, = Q,.B,.Q,. Q;'= QiY,Qy.Q,' and QY QF = QY so as before take (QY Y , ) for the required W,. (7) follows if u1 = 0, but without further conditions on C (7) does not follow if u1 = 1. As an example of a condition on C sufficient for (7) see 15.6C. Finally consider h = 0. For the required W , take any cyclic word (n)-bounded by so such that there exists a cyclic word W , such that W W , W , and W , is (n)-bounded by so + s26 (e.g., W , = W, = W).

QI1.

15.5A. Note. If W is a kth power in n, i.e., W = P k for some P, and if ui = 1 if and only if s ( N i ) > f then clearly at least one W, exists which is a k t h power in n. 15.6. We introduce the concept of a simultaneous t-replacement modulo a set of ( t + 1)-subelements; t < n. Let W be a cyclic word (t)-bounded by so. Let S,, ..., Su be disjoint subwords of W , where each Si is a ( t + 1)-subelement of size > 1i99 ( u 2 0). Let the maximal t-normal subwords of size > 226 in W be N , , ..., fi-h (12 Z 0 ) . Let Max(u, h ) Z 2. If h # 0 let u be the sequence u , , _ _uh _ ,where ui= 0 or 1 and if ui = 1 then s(Ni) > Y'. If h = 0 let u be the empty sequence. We shall define a cyclic word W a called the result of simultaneously replacing the N j with ui = 1 modulo S , , ._., S,. Call Sipreadmissible if it does not properly contain an Ni. ( I f h = 0 all Sj are preadmissible.) In this case we can write Sj = ASiB where A is the left subword of maximal length such that s ( A ) < rb and dually for B . Call Sj admissible; thus s(Si) 2 s(Sj) -- 2rb - 4/e. I f h Z 1 let the formal kernel of Ni be Kiand call K , , ..., Kh admissible. Let there be k admissible words; they are disjoint and k Z Max(u, h ) >, 2. I f A is a subelement of L, where s ( A ) > t,, make a choice, arbi-

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285

trary but henceforth fixed, f, ( A ) of a word of the form ALEZFA as in Axiom 33'. Let W , arise from W by replacing each admissible Si by f,,,(Si.>, each admissible Ki byff(Ki). Now Si r f t + , ( S ; )and Si is (t)-bounded by so < b,, so the right side is (t)-bounded by so + c,,. Any Pi of type t would have size > k! - so - c,, > 226 + + E , > ro + E , , hence pi has the form Ng and Si would meet K g . W , . Write Thus Si f y l f f + , ( S i )and hence W f,+,(Si) = SiLEjJiFiSiR, f f ( K i )E KkEiZiFiKp. As before let Qi be Zior Tiaccording as uiis 0 or 1. Corresponding to Q1. B l - Q2 in 15.5 we now have two cases 1. J F S f R A K : E 2 . Q 2 (and its dual) 2. J , F , S !,R ASiLE i J , (the case 3. Q1. F , KPAKkE,. Q2 has already been discussed in 15.5). Since s(Ji) > s(Si)- r3, > ug9- 2rb - 4/e - r,, > u96 we have by 15.3B, 15.3C that the word 1 or 2 is equal in G, t o (1') JkY,Q! or ( 2 ' )JFY1$. We recall that 3 equals (3') QkY,QF. Although Y , is not unique we now make any choice of Y , to be fixed henceforth. of Qi intersect, the In 15.5 we had that the subwords intersection being . We now observe that J k , J F intersect, in J Y say, where s(JM) > s(Ji) - ug7- E ; ~ . Put W , = Y;.. Y k )where M t is or J Y . Then W , W , unless k = h and all ui are 1 when, as before, W , W , W,. Thus in any case W fTl W , W , W , . W , is the required W a .

QY

([y

,$F

Qk, QR QM

15.6A. The word (1') is a function of S,AK, and 0,;the word (2') is a function of S,AS, only; the word ( 3 ' ) is a function of K , A K,, u1 and 0,. Indeed, writing ( l ' ) , (2') or (3') as ,$kYEF,we may say that 5:. is a function of S , or a function of K , and a,;$,; is a function of S , or a function of K,, a,. Hence is a function of Si or of Kiand ui.

[y

15.6B. Suppose u Z 3 ana consider thesubword V = (Si-,, Si, S i + l ) of W . Let u and u' be such that W a , W"' exist. Assume that if Nu C V properly then uu = u; ( u = 1 , ..., h ) . We show that W u , W u ' have a common subword. Each Si contains at least one admissible word (Si or K i ) ; let A,, A,,, , A n be admissible words contained in Si-,, Si,Si+l respec-

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tively; thus 1 < I < rn < n < lc without loss of generality. Then the subword cf) is the same in W" and W"'. Since Si-,, Si, Si+,are disjoint, their iterated nice images with respect t o W - W , - W , - W,, say SLl, S:, S:,, are disjoint, so S: C ($l, $:, ); hence S: for u is the same as ST for u'.

(c,", [E,

EE,

15.6C. Note. I f h = 1 , W = (N, C) and C contains a ( t + 1)-subelement S of size > ugg we can consider a replacement of N , modulo S. (cf. the case h = 1 of 15.5.).

15.6D. Clearly a simultaneous t-replacement modulo a set of ( t + 1)subelements is a simultaneous replacement (cf. 15.5).

15.6E. Informal remark. In 15.5, a simultaneous replacement in a cyclic word W is discussed by subdividing the word into linear pieces. Unfortunately for some applications this subdivision is not fine enough because there may be too few maximal normal subwords in W . This is the reason for the construction in 15.6. 15.7. We consider 7.8 and specify certain A-descendants to be called prejerred A -descendants. Take any A-descendants A , in 2, where n > Y > 0. This contains B Y j (i = 1 , ..., a ) as in Axiom 30. We define the centre C(A,) of A , t o be the formal kernel of A , if s(A,) > Min( 1 - so - Y ~ r *~- t-26) , = Y * - Y~~ ; otherwise C(A,) = A , . ..., B, a-i where The admissible B , (cf. 7.8) are B , j / t < 32120 + 18/e (7.7B). If C(A,) $ A , we can choose w such that s(B, B , w ) > Yo + 2e3, namely (cf. 7.9B) w is the least integer 2 t(ro + 2e3 + ejo). Then ( B , w + 2 , Bya-w-l ) c C(A.) and (a - 2w - 2 ) / t > r* - r26 - ~ 3 0 4 / e - 2(r0 + 2c3 + E&,) = b say. Since 3 ~ + 118/e ~ <~(yo + 2c3 + cjo) + l / e we have j < w + 1 hence B , + 2 ,..., B , a-w-l are admissible. Let p = w + 1 in this case; if C(A,) = A , put p = j . Then B, l + p ,..., B, a - p are admissible and contained in C(A,); also (a - 2 p ) / t > Min(b, u i O / e ) . Recalling 7.7, the iterated nice image of ( B , l + p ,B r a - p ) con. A , l + p + j , . . . , A y , - P - j the preferred tains ( A , l + p + j , A , u ~ p - i ) Call

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.ATio r the preferred first A-descendants of A , . Note that (a - 2 p - 2j)/t > Min(b, u i o / e )- 6 ~ 2 0 36/e. Thus the iterated nice image of C ( A , ) contains all preferred A , i. A preferred A-descendant of A , is defined by 1. a preferred first A-descendant of A , is a preferred A-descendan t ; 2. if X is a preferred A-descendant of A , then any preferred first A-descendant of X is a preferred A-descendant of A , . A prej'erred A-descendant of Bi is Ai or any preferred A-descendant of A i . A preferred A -descendant is a preferred A-descendant of some Bi. Two A-descendants of the form A , i , A , i+l are called consecutive. 15.7A. Proposition. The centres of all preferred A-descendants in 2, are disjoint ( r = 0 , 1 , ..., n - 1). Proof. This is true for r = n - 1. Assume it true for some r, n - 1 > 0. Let P,Q be preferred A-descendants in .Z-, and different subwords of Zr-,. If P is A , and Q is A , for some y (then i # j) then clearly C(P) is not C(Q). Now let P be A , and Q be A , where y # 6. By the induction hypothesis C(A,), C ( A , ) , and hence their iterated nice images, are disjoint. Hence P, Q are disjoint and so are their centres.

2r

15.8. We specialize the definition of L n + l as follows, by fixing certain choices which so far have been arbitrary. Let R E Rep Ln+l be fixed. Put R , =_ R . Take a set B , , ..., B , as in 7.3; these are subelements of L n (> r" or > u l l ) ; if a choice is involved make an arbitrary but henceforth fixed choice, subject to (*) below. If R does not contain a maximal n-normal subword > 226 put Rn-, = R and A i= Bi (i = 1 , ..., m). Now let N , , _..,Nh be the maximal n-normal subwords of R of size > 226 and let h > 1. Choose a sequence u such that ui = 1 implies s(Ni)> r'. Modulo the empty set of ( n + 1)-subelements define Rn-l to be R u . We next describe a set S of disjoint n-subelements in R n - l . Consider A-descendants A i (i = 1, ..., m).I f the Bi

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QY. -

have size > Y" then since Y" > z26 A , has the form F o r S take the formal kernels of A 1, ..., A , . = 0 for all i since Y' > Y", so If the B , d o not have size > Y" then (T, K nr-l R" for some K . Since the formal kernels Ki have size R,, > yo > uI1 we may in this case assume (*) each Biis a Ki.Let (ZM , Z z 1 ) = ZM 1 - 1 EZE 1. Then the iterated riice image of Bi in 1-1 R u is contained in E hence so isAi. Since the Biare disjoint so are the A i . For S take the A,'s, making a fixed choice. Now let R n P 2arise by simultaneous ( n - 1)-replacement in R n T 1 modulo S. Also write S, -1 for S. If R r has been defined for some Y, n - 2 2 Y > 0, let Sr be the set of centres of all preferred A-descendants in R , . (The members of Sr are disjoint.) Let R r - l arise from R , by a simultaneous Yreplacement modulo S,. Fina!ly L,+l is the set of all R , obtained by this construction.

§ 16 Informal Summary The main difficulty in this section is the proof of Axiom 2 2 for ( n + 1 ). We also deal with Axioms 33', 3 1B, 3 1C and several minor axioms. N.B. In some sections of Section 16 the discussion is informal and details are left t o the reader.

16.1. The reader may now verify that Axioms 26,26", 26A, 26L, 26- and 39 hold for ( n + 1). 16.2. Denote by (X, Y, p ) that part of Definition 13.1 after " 1 < p < n". Thus if X, Y are cyclic words then X ,yl Y means X = Y or for some p , 1 < p < n + 1, ( X , Y, p ) holds. 16.3. Note. Axioms 4 , 8 and 28A holds for ( n + I ) . Y choose p to be minimal such (If X _ that _ X Y. Then X = Y or, by considering the normal subwords Pi,Qi where Pihas type p , ( X , Y, p ) holds.)

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16.3A. Proposition. I f X nTl Y then a maximal ( n + 1)-normal subword S of one o f X , Y , where s(S) > uFo, determines a maximal ( n + 1)-normal subword I m * S = S' in the other such that either

16.3B. Proposition. Let (S, T ) ( S ' , T ' ) where each side is a p object, p < n, and is (p)-boundd by k59. Let S, T, S ' , T' have size > ~ 3 0 Let . R be an n-subelement of size > ~ 9 1 contained , in ( S , T ) . Then ( S ' , T ' ) contains an n-subelement of size > s ( R ) ~ 9 2 . +

~

Proof. We have k,, < k,,, k19. Recall the proof of 9.1 1. Let S , , T , be the kernels of S, T ; then ( S ,T ) = S , A T , say. An associated preobject ( Z , Z ' ) of (S, T ) contains A . Similarly ( S ' ,T ' ) = S ; CT; and some associated preobject (Z",Z') contains C. Suitable adjustments of these preobjects are equal:

Some subword R M of R is disjoint from S, T and is such that s(R') > ~ 9 -1 2rb - 4/e > u3. Thus R M c A C ( Z , , Z ; ) . Since k,, + c33d b,, and k,, + u4 d b,,, the right side of ( 1 ) contains a weak image V of RM and s( V ) > s(RM)- u4. Some subword R' of V is contained in C and is such that s ( R ') > s( V )- 2rb - 4/e. Thus R ' C (S', T ' ) and s(R')> s(R)- ug2 since ~ 9 =2 4rb + 8/e + u4. 16.3C. Corollary. Assume further that s ( R ) > ug32 2.191. Then ( S ' , T ' ) contains a maximal normal subword R" of size 2 s ( R )- u94. I f R R , are distinct maximal normal subwords each of size > ug3 then R;' f R;.

,,

Proof. s(RM)> ug3- 2rb - 4/e > u 6 , hence Im R M is maximal normal in the right side of (1) and of size > ug3- 2rb - 4/ e - u4 - c 3 = t, say. Also s(C n (Im RM)) 2 t - 2rb - 4/e > ro + e3 and the maximal normal subword R" of ( S ' , T ' ) thereby determined has size

J. I,. Britron, The existence of infinite Burnside Groups

290

2rb - 4/e -- E ~ Thus . s ( R " ) > y - ug4 since 4rb + 8 / e + u4 + 2e3 and the first part follows. The second part follows from 6.13.

>, t

--

((94 =

16.3D. Note. Let X X Y where X,Y are (n)-bounded by k,, and X is ( n + 1 )-bounded by x where 1 < x < ho e2 2/e - i d 4 . Then Y is ( n + 1 )-bounded by .Y + ~ 4 ~ . ~

~

Proof. Recall 6.1 OA and use 16.3B. 16.3E. Proposition. Axiom 1 1 holds for ( n + 1 ). Proof. Let X nyl Y and YiTl Z . We may assume ( X , Y , p ) and ( Y ,2,y) otherwise it is trivial t o deduce that X nylZ . Say q Z p . Then Y contains a maximal q-normal subword and by 16.3C it determines a maximal q-normal subword of X . Hence ( X , 2, q ) , so

x ",Y1 2.

16.3F. Proposition. Let B arise from A by a simultaneous ( n + 1)rcplucrment, wliere A , B ure ((11 + 1 ))-bounded by k60. Then A ,,ylB . Proof. We have A we have A nyl13.

W,

n+l

W,

nTl

R . By Axioms 28A, 1 1 for n

+1

16.3G. Proposition. Axiom 28B holds f b r n + 1 Proof. We are given a cyclic word X , ( ( n + 1))-bounded by d28. X is 1-bounded by d,, < 1 -~z , ~so by simultaneous replacement of all maximal 1 -normal subwords of size > we obtain say X I , 1-bounded by + r26 < str (cf. Axiom 26). Now d28 < k60 and + r26 < k6, so X X I . Consider the following hypothesis p ( t ) . There is a sequence X,X I , . _ . ,X, where X i arises from Xi-1 by simultaneous replacement of all maximal i-normals of size > i. (Write Xip1(i) Y ) . (i = 1 , ..., t ) ;X X , ; X , is t-bounded by 12 + r16 and ( ( t 1 ))-bounded by s o ; if X is a k t h power in II then so is X , .

7

7

~

J.L. Britton, The existence of infinite Burnside Groups

29 1

Then P( 1 ) is true. Assume P(t) holds for some t where 1 < t < n. Since i + r26 < so < k59 and d28 < k,,, X, is ( t + 1)-bounded by d28 + u4 < 1 - 226. Hence Xt(:)Xt+, say, where Xt+, is ( t + 1)bounded by i + r26 and (t)-bounded by so. By 16.3F since d28,s0< k6,,Xt t ~ l X t + l . H e n c e X X , t+l X r + , . H e n c e P ( t + l ) t+l holds (cf. 15.5A). Hence by P(n + 1) X nylXn + where X n + is ( n + 1 )-bounded by 2 1 + r26 and (n)-bounded by so < d28; if X is a kth power then so is

,

,

Xn+l*

By Axiom 28B (for n ) X n + l % X;+, where &+, is (n)-bounded by f + r26 + u4 < ks9. Now so < ks9 so X;+l is ( n + 1 )-bounded by + r26 + u4 < str. Also X nFlX n + , nylX;+,.

16.3H. Proposition. Let X X Y where X , Yare (n)-bounded by k61. Let X contain at least four maximal n-normal subwords o f size > u s . Then W exists such that X, W Y and W is (n)-bounded by Max(k61, ki4) + ~ 4 0 . Proof. SAS has the form (S, U , , U,, U,, S) where the Ui are maximal normal with images Viin TBT,so that TBT = (T,V , , V,, V,, T). Now by 9.15 SAS = SLE(V r , V$)FSR where the right side is (n)bounded by MaX(k61, k i 4 ) + ~ 4 0 The . subwords SL,SR of S overlap, say S = LMN, SL= LM, SR = MN. Hence ( S A ) = (MNAL) and MNAL

= L - l . S A S . S - l . L = L-'.LME(VP,

V k ) F = M E ( V F ,V k ) F .

,

Thus ( S A ) - (ME( V p , V $ ) F ) . We now show that (ME( V P , V 1 ) F ) (TB).Recalling 13.1 we have S = E . T . K . Now S L E V f = e . ( T , V , ) and V$FSR = (V,, T). u. Hence

which is a linearization of TB.

16.35. Proposition. Axiom 28 holds for n Proof. 1 Let X bounded by b,, , O .

+ 1.

nrlY , where the linear words X , Y are ( ( n + 1 ))-

respectively. Now b28 = so. Consider the set H,

J.L. Britton, The existence of infinite Burnside Groups

292

possibly empty, of Pi, where Pi has type n + 1. Then by simultaneous replacement of all such we have X n;l Z say. Clearly 2 ;Y and since Z is (n)-bounded by so we have by Axiom 28 (for n ) 2+ ... + Y . Hence X n ; f l 2 ; ... 7 Y . n 1 2". Now let X nyl Y , where the cyclic words X and Y are ( ( 1 2 + 1 ))-bounded by b 2 8 = so and a28 respectively. Case 1. Not ( X , Y , IZ + 1 ) Then X Y . Let X n ; f l X ' by the zero simultaneous ( n + 1)-replacement (i.e., either ui = 0 for all i or u is empty). Then X n W , W 2 iI X ' where X ' is (n)-bounded b y so. Hence by Axiom 28A X X ' , so X ' Y . By Axiom 28 (for n )

8

X

X

so

+ 1 ) holds. Here SAS n$l TBT where X = ( S A ) , Y = ( T B ) . Adjustments of corresponding preobjects are equal: S'AS" n+l T'BT". Again consider the set H defined as in I", but for the new equation. By Axiom 40. S'E N if and only if S" E H . Now any maximal normal subword N except S of X determines a maximal normal N ' in S ' A S " ; let X X ' by simultaneous replacement of all N such that N' E H together with S if S' E H . Then X ' is (n)-bounded by so and X' X , hence X' Y ;indeed X ' Y . Hence (1 ) holds and since X *tl X ' also ( 3 ) holds. 3". Given a cyclic word W ( ( n + 1))-bounded by b& < d,, there exists by 16.3G a word Y , ( ( 1 1 + 1))-bounded by str, such that W nyl Y . By 2" Y n71 ... 7 W . Cusr 2. ( X , Y , n

X

16.4. Definition. In Axiom 30 let b be the smallest integer < t(rb + ejo); then if h < a , ( X , , X,,,) and ( X a p b ,X,) each have size > ( h + 1 ) / t eiO> rb. Call X i central for P if 3 + b < i
N o t e : If s ( P ) > q* then central X i exist since (a - 2b

--

4)/t

J.L. Britton, The existence of infinite Burnside Groups

293

-

- € 3 0 - 2(rb + ~ 5 - ~4/e)> 0. I f P C A B then Po (cf. 6.8) exists and contains all central X i . Recall that the word “central” has been used in a different sense in Section 10.

> q*

16.4A. Definition. Let V be a ( t + 1)-subelement of size > q”, where 1 < t < n. V contains t-subelements Vi as in Axiom 30; if they satisfy (b) denote by pt+l V any central V j .If the Vi satisfy (a) there are no words pt+l V ;in this case call ( V i , V i + l )where V j and Vi+l are central a protected t-object of V . If some p t + l V exists consider any J = p , p t p t + l V where p u - l J does not exist. Then u 2 2 and J is a ( u - 1)-subelement. If u 2 3 then the u - 2-subelements Ji contained in J by Axiom 30 satisfy (a); call ( J i , J j + l ) where both terms are central a protected (u - 2)-object of V. If u = 2 J is a 1-subelement; we say that V contains a 0-object (even though we shall not define “0-object”.) 1..

16.4B. Note. If V is as in 16.4A and V C B 7 C, it is easy to deduce that there is induced VM ;T CMwhere either u # 0 and VM contains a u-object 0, where 0 is a protected u-object of V , and 0 O‘C CMfor some u-object 0’,or u = 0 (hence VM = C M )and V contains a 0-object. 16.4C. Proposition. Let A be a word containing two q-objects (S,T ) ,( S ’ , T ’ ) ,where q < n, such that they produce E , K . Let

(4)

(In A , (S,T ) );(InA , ( S ‘ ,T ‘ ) )- K

Then the subwords (S,T ) , ( S ‘ ,T ’ ) of A coincide. Proof. It is sufficient to show T, T‘ coincide so assume say T is left of T’. By cancelling (In A , from ( 4 ) we obtain

f)

(5)

T i (T, T ’ ) - K .

Adjustments of associated preobjects are equal:

294

J.L. Britton, The existence of infinite Burnside Groups

where

(8)

T L E Z * ;( T , T ' L E ' Z ' * )

The left side of (8) is a q-subelement. Taking divisions for (8), any Piof non-zero type has size > rb so by 5.21 n o Pihas type > q . Hence in (8), ;may be replaced by =. (8) now has the form 9 M = ( M , , A!,) where M , , M , are distinct maximal q-normal sub4 words of size > u,, and M is q-normal. By 6.19, Im'M, = M = Im'M, hence M , = M,, a contradiction. 16.5. Proposition. I f A D , D and DCare ( n + l)-urelements then so is ADC. Hence Axiom 2 2 holds f o r n + 1.

Proof. 1" . We may assume that D is normal. For let D 5 D'X where D'is ( n + 1 )-normal and put A' = A , C' = X C . Then A D ' X 3 AD' 3 D' so AD' is an urelement. Then A D 5 A ' D ' X 3 AID' 3 D' and D'C = DC so A'D' and D'C' are urelements. Thus if A'D'C' is an urelement then so is A D C . 2". We have

and

Un 2 ... 7 Uo 5 D Y A = D K for some X , Y , H, K, where 2, = ( B k ) ,Un = (R") and R, 13' E Rep L,+l. Since s ( D ) > y o , the situation is similar t o that of 13.1 1 . As in 13.1 1 we use the A-B notation for the 2-sequence; say B , , ._.,B , C Z , and all descendants of say R , , ..., B, in 2, are contained in D. Then we may assume kg/m > r,. As in 13.1 1 we have by 11.8

J. L. Britton, The existence of infinite Burnside Groups

Z,

= D,H,,

295

Un 5 D, K ,

and there is induced U," ; Z,". Moreover L(Z,")/L(B) > a , where Q is as defined in 13.1 1. Since Q > uS5 (P.C.) we have by 13.13 Z," = U," and B = B'. Let the B's for Un be B', , ..., BL,. Then for some a 2 1 BI+, = Bi for all i and rn' = lea. Similarly Bi+a = B j for all i and m = kea. But we may take Bj = Bi. for all j . Write R , for ( B w + s + 2 ,Bg-s-w-l)C Zn and RL for the corresponding subword of U F ; then RI, = (BL+s+l,B&w-l). Un - D , K, induces U," = Z," and hence induces RL = R,. We have

Z,

= R,S,,

U,

RLT,,

for some S,, T,.

Z,

= D,H,,

U,

0,K,

( r = 0, I , ..., n )

Also

where D o , H,, KO= D, H, K respectively. 3". Assume inductively the following. 1 . Z, = R,S,, U, = RI T,, R , = RI, R , C D,. If U is any subword of R , let f r ( U ) be the corresponding subword of RI. 2. R , (Bi+,, Bg-,), h = W + S + 1 ; R n - i ( A , + , , Ag-h-i); if r < n - 2 , R , has the form (PI, P2, Q1. Q2) where P,, P2 are two consecutive preferred A-descendants of B3+hand Q1, Q2 are two consecutive preferred A-descendants of Bg-hp2. 3. U, D, K,?nduces RI = R,. 4. If r < n the images under f, of the preferred A-descendants (for the Z-sequence) contained in R , are the preferred A-descendants (for the U-sequence) contained in R:. 5 . (RnSnRI,Tn) ... (RrSrR: Tr) Call this induction hypothesis P ( r ) . Note that P(n) is true. 4". If P(0)is true then the proposition follows. To see this note that R , C D o = D , say D = X R , Y ; then So = Y H X . By 3 Rb To DK induces Rb- R,, hence To= YKX. Now by 5 RoS,RbTo E ;:1 and (RoSoRbT,)= ( R , YHXRbYKX)= (HDKD) = ( D K D H ) = = (DYADCX). Hence ADCis a subpowerelement of L , + l ; since it contains D it is an urelement. Thus it remains to prove that if P ( r ) holds for some r where n 2 r > 0 then P(r - 1 ) holds. ~

296

J. I,. Britton, The existence of infinite Burnside Groups

5". This subsection, 5", is mainly devoted to recalling some arguments from the proof of 1 1.7. Consider

I n 1 1.7 we obtained from this an equation

Having obtained ( 1 0) we defined D, , Hr , K, by D, = 2," , H, Z ' , K, = F, U y R U ' U y LE,., where Z, = (2," Z ' ) , U, = (U," U ' ) . Since R , C D, and U, D , K,. induces RI = R , we see that (1 0) induces RI = R,. We recall how ( 1 0) was obtained. 2, contains a t-object ( T , , T 2 ) and an s-object ( S , , S,); U, contains objects ( T i , T i ) , ( S ;, S ; ) where ( T , , T,) ( T i , T i ) (producing E , , K~ say) and ( S , , S,) ( S ;, S ; ) (producing E , , K , say). (i) I f r < n then ( T , , T,)c V , , (S,,S,)C Vfwhere V , , Vf are the leftmost, rightmost A-descendants of B,, ..., Bg in 2,. (ii) If r = n and B , , B,, ... are disjoint and of size > u l l (cf. 7.8) then for some u,u ( T ,, T,) C B , ,( S , , S,) C B,. (iii)If Y = n and B , , B,, ... are maximal n-normal of size > Y" then for some 11, u ( T , , T,) = (B,, BU+,), ( S , , S,) = (B", BU+,). By a D-word we mean in these three cases respectively an Adescendant of B , , ,.., B, in 2,; a B,; a subword (Bi, B i + , ) . Write D, for V , , B, or (B,, B L l + , and ) define D, similarly. Note that ( T , , T,) is a protected object of DT in cases (i), (ii), while in case (iii) (Ti,T 2 ) DT. Let D,, D,, D, be D-words such that D,, D,, D,, D,, D, are disjoint (each term being left of the next) and D, C R,. D, contains an object, say a u-object ( U , , U 2 )and in the language of 11.7 D,, which we also denote by D,, clears a u-path with respect t o (9) in which ( U , , U,) has images throughout; in particular there is an object ( U ; , U ; ) in U, such that ( U l , U,) $ ( U ; , U ; ) (producing € 2 , K , say).

7

J. L. Britton, The existence of infinite Burnside Groups

291

Say ( T l ,5'2) T IAS,, ( T i , 5';) E T i BS;; we have ( T , , S 2 ) = e l *( T i , S ~ ) - and K ~

(where C, D are defined by (1 1)). In (1 l), TT, Ti* are such that

T I ' . q . T' = T *

(12)

1 t (

1 )

-1.

T'* 1

ThusA=(D,)RA1D2A2D3A3D4A,(D,)L for someAi ( i = 1, ..., 4). Still recalling 1 1.7 we have Di 3 D: D;. (i = 2,3,4), these being induced by ( 1 1); let ( q D F ) O V so that V is a left subword of D. Then V is disjoint from D i so Ti* C 0;.Now D = D; DiB' D; B"Di D; say and

-

-

Since D (13) (1 4)

E

T;*BSF, T;"

C

Di and dually SF C D; we have from

B = R L E ( D i R ,D,OL)FBR

for some B L , B R . ( 14) is the required equation ( 10) ( BC ( T i ,S;) C Ur , ( O F , D ~ ) ACC Z , ) . 6". Now (U,, U 2 )C D , 5 D, C (Di", D:L) so by ( 13) since either (U,, U 2 )= D , or ( U , , U 2 )is a protected object of D , there is a u-object ( U , , 02)C D and using 9.10 (i) if (G,, U 2 ) ,( U , , U 2 ) produce e5,K~ then

Let ( T l , U 2 )= T,AoU,, ( T i , U ; ) = T i B o U2. Just as we obtained (1 1 ), so we may obtain

where (1 2) holds and hence U;- K

~ U.jl - =

U p -UT-'. Hence

298

J. 1,. Britton, The existence of infinite Burnside Groups

( U , , U 2 )has image (0, , U 2 )in D with respect to ( 1 1 ). Moreover ( U , , U,) c Di,.Let ( U , , U2>,( U , , 0,) produce e4, u4. Then (In C, U 2 )= (In D , U 2 ) . ~ 4

(18) and

(In C, U i ) - (In D,U;)

(19)

(end words 1,K say)

is induced by ( 1 1 ) where K = U F . u4. UF-' and U , = U k U F , U , = Uk U F . Since ( U , , U 2 )C D, and ( U , , U 2 )C D;, D: is disjoint from ( U , , U,) and D i is disjoint from ( U , , U 2 ) ;thus in (1 9), D! is contained in the left side and 0; is contained in the right side. Hence (In D ,

ii2L)

(In D,

U 2 ) . ~ 4=

= (In

D, D i L ) E ( D Y , U ; ) . K - l

and

(20)

(In D,D k L ) E ( U F , U,)

From (17),(18) (In D,U;).u2 = (In D,U,).u4; but ( U ; , U i ) , produce e ; l . e 4 , u4.u;1. By 16.4C, U,is U i . It follows that U 2 )= ( U ; , U ; ) so u2 = K ~ Thus . the left side of (20) can be replaced by (In D, U;).u2 and hence by ( I 5 ) we have (InD, O2).uJ1 = ( I n D,U ; ) ' u 2 . Since 02),( U i , produce e 5 ' e 2 , u 2 . u 5 we have by 16.4C that U2 is U i . Since ( 14), i.e., ( l o ) , induces R: = R , and hence ( 13) induces RI = R , and since D, = D, C R , we have f,((U,, U,)) is U2) hence is ( U ;, U;); in particular ( U , , U 2 ) ( U i , U i ) . 7".Consider (1 ) and, recalling 1 1.7, let (U;', U i ) say be the image of ( U l , U 2 )in D r - l ; let ( U i " , U;") be the image of (U;', U ; ) i n U,. Then ( U ; , U ; ) is the image of ( U i " , U i " ) in U,. Thus in the Z-sequence, U2 has image U; in Z,-, and in the U-sequence U; = f r ( U 2 )has image U;" in U,-,. 8". We prove that if J is maximal r-normal in R,, and properly contained in it, and if s ( J ) > Y' then J is replaced in Z, + Z,-' if and only if f,(J) is replaced in U, + U,-

( U , , i?,)

(o,,

(ol,

u;)

(o,,

,.

,.

J.L. Britton, The existence of infinite Burnside Groups

299

R, c D,= Z," = (04".Dp).Assume r < n. Where ( V l , Vf)= = I/, C, Vf we have C, 3 R, 3 J . Now I/, , Vf are (r + 1)-subelements

and by 1 1.7 they clear paths with respect to (9). Thus J has image J" has image say J' " in U,-, and 9'" has image say say J" in D,-,, J' in U,. Also J' c ( T i ,Si).Since J C ( D p , D,"'),J has images J in D by (13) and j in D by (1 1). Making use of right r-objects we can follow the argument in 6" with J in the r6le of U2. We conclude that f,(J) is J ' . If r = n then since s(J) > r', all Bi are maximal normal of size > r" and we obtain the same conclusion by a simpler direct argument. Recall 12.5 and if J is replaced in 2, + Zr-l put 8 = 1 , otherwise put 3 = 0. If J' is replaced in U, -+ U,-, put cp = 1 , otherwise put cp = 0. Since Im'(J") = J'" with respect t o Ur-l rrl D r - l K r - l so by 12.7 J " , J'" satisfy (0). Since J = J ' we have 8 + cp = 0 (mod 2) and the result follows. 9". Let the preferred A-descendants in R , be Pi (i E I ) . Their cetres Ci (i E I ) which we denote also by C,, Ca+l,..., C,, are disjoint. The set 8, for Z, contains c,, ..., C, hence the iterated nice image of C, (a < u < b ) in Z,-, is equal in II t o the iterated nice image of f,CC,) in U,.-,. Define R,-, as the word generated by the preferred first A-descendants of P, ( u = a + 1 , ..., b - 1) and let RI-, be generated by the preferred first A-descendants of f r ( P , ) (u = a + 1, ..., b - 1). Then R,-, = RI-,. Next, since the iterated nice image of R , in Z r - l , and hence that of C, (a< u < b ) , is contained in D,-,C we have R,-, C Recall that ( U , , U 2 )C D, C R , C (DyR, In Case (i) D, is an A-descendant; we now restrict D, to be a preferred A-descendant, not the two leftmost or the two rightmost, in R,. Making use of the available choice of ( U l , U 2 )within D, we may say that f r p l ( U i )is U;"; indeed fr-l((U;', U;')) is (Ui", U;") SO (U;', U i ) (Ui", U;"). Since ( U i " , U;"), (U;', U;') correspond in U r - l - D r - , K,-, the following lemma implies that Ur-l -D,-,K,-, induces ( U i " , U i " ) = (U;', U;'). It follows that RI-, = Rr-l is induced since if X is any non-empty subword of R,-, such that f,_l(X) = X is induced, then RI-, = R r - l is induced.

"8").

300

J. I,. Britton, The existence of infinite Burnside Groups

16.5A. Lemma. Let

where ( U , , U2j, ( V , , V,) are u-objects and with respect to (2 1) lrn' U, = Vi (i = 1,2). Also let ( U l , U 2 )= ( V , , V,). Then (1 ) induces ( U , U,) = ( V 1 , V2). 9

Proof. If X c ( U , , U z )let f ( X ) be the corresponding subword of ( V ,, V,); thus X = f ( X ) . Now the two u-objects produce 1,1so we have

Since Im' U, = V 2 and U2 = V , we see that Case (b) of 6.16 applies hence by 6.25 U p urlV p (cf. Axiom 40). Let j be minimal such that U y 7 V Y . If j = 0 then ( A U F ) ; ( B 1 ,V y ); also UY = V Y , so where U2 = U k U y UF and V 2 = V k V y V: we have by (22) U: ;V:. Since U2 5 V 2 ,this implies that UF = V:, otherwise a subword of U would equal I in C,, contrary t o Axiom 10. Thus f ( U y ) is V y and the required result follows since (20) induces = VM 2 ' Now let j # 0; we shall obtain a contradiction. Take a Pi of type j ; then Pl!-l Xl!Ii Q i X t E L, (cf. 6.7). Let pi, Qi be the maximal normals of U,, V 2 respectively determined by Pi, Qi. Now f(P,) C V 2 and # Q iby Axiom 40; say f ( p i ) is left of Q i . Thus V , = V;(J'(P,~), Qi.) V i and U2 = U i Pi U i say. Now ( B 1 ,QI) V i= = ( B , , V 2 )= ( A P i ) U; = ( B , , Qi)X,'U; hence V i = X;. U i . But ( f ( P i ) .QI) V i = Pi U; so ( f ( P , ' ) ,Qi).Xl! P,!. By Axiom 24 Q)XE!Pf-'= QI" W ( P i L ) - l = H say. Write H = K L where K is of least length such that s ( K ) > s ( H ) ;hence K n Q:." = N is normal and ( f ( P ; ) ,K ) = L-l. This has the form ( M I ,M 2 ) = M , which leads t o a contradiction as in 16.4C; for f ( P l ! )and K are j-normal, but if ( f ( P l ! jK , ) were a j-subpowerelement, then ( f ( P l ! )N, ) and hence ' f ( P i ) ,0;) would be subpowerelements, contrary t o f(pi.)# Qi.

qf

f(c.)

J.L. Britton, The existence of infinite Burnside Groups

301

16.6. Let Z E Rep(L,+ ,) and let Z contain maximal n-normal subwords Bi (i = 1, ..., m ) of size > r f . Let 2 ; C', say Bi replaced or not according as pi = 1 or 0 and let 2 ; D' using pi = 0 or 1. Let Z 2 C' nzl... 7 C and Z 2 D'nzl... 7 D.Now let "j

= pi (i = 1, ..., q )

CY. 1 =

0; (i = q + 1, ..., m )

Then Z ; B' say by replacing Bi if ai = 1. Making use of the standard choices (1 5.5A) one can show

16.6A. In particular if D is 2 (cf. I6.3A) we have constructed an element of L n + , of the form CMEZM F. Let C = C M Xsay. Then X ;E Z MF. Let CM= C, C, C,; then C3EZMFC1is a subelement of L,+, equal in G, to the subword C3XC, of C. In this way Axiom 33' may be shown to hold for n + 1. 16.7. Axiom 33" is easily proved to hold for n

+ 1.

16.8. Consider Axiom 33'. Say Z , 2 ... 7 A B C D . Now if the equation A ; A L E Z F A Rarises from Z" ; ... 7 A Q say (cf. 16.6A) then Z" = Z T for some T. Since A is normal Z , = 2" i.e., Z T is a C.A. of Z,. Similarly Z'T' is a C.A. of 2,. 16.9. Consider Axiom 33. We have Z , -+ ... Zo = W V E L n + , and W ; WLDWRwith divisions say P,, P,, ... and Q1, Q z , ... . Let the of type n be H , , H,, ..., H,; Any Qi of non-zero type is contained in D.By 12.15 Zn-, contains maximal n-normal subwords K , , ..., K,. The image Li of Ki in 2, in the present case is > r'. Let the maximal normal subwords of size > rf in Zn be Bi (i = 1, ..., m ) and in Z , -+ Zn let pi = 1 if B j is replaced, 0 otherwise. Now Li = Bfci,(i = 1, ..., s). Let Zn + Z' by replacing Bi if (i) pi = 1 and j does not have the form f ( i ) ,or (ii) Pi = 0 and j = f ( i ) . One then shows -+

-,

302

J. L. Britton, The existence of infinite Burnside Groups

WLDWRVn -=1 WLLPWRRV Hence WLLPWRR = W ' is a subelement of L n + l equal in Gn-l to WL D WR and W LD W R may be written W t LD ' W R . By induction W LDWR is a subelement of L n + l .

16.10. Axioms 3 1 B and 3 1C for n + 1 read briefly: 3 1 B. If T ; Y where T is an (n + 1)-subelement then there is a word TLFY R . 3 1C. If (S, T ) nyl(S', TI) where S, T, S ' , T' are maximal ( n + 1)normal there is a word S L ATIR. To prove 3 1B, T contains subelements U j of L, by Axiom 30 for n + 1. If these are maximal normal let two of them be U, V . Let U' = Im' U V' = Im' V;then ( U R ,V L ); ( U R V, I L )and by Axiom 31C we obtain the desired result. If the Ui are not maximal normal then UIC TnzlY , hence Uy nyl YM and we may use Axiom 31B. Now consider 31C. Let the divisions be P I ,P 2 , ... and Q1, Q 2 , ... If S does not have the form Pi of type n + 1 then SL; StLand we may use 31B. Similarly for T. There remains the case when S = P i T = P1. ;Pi and Pi of type n + 1. Now ( P i ,Pi) n ~ (Qi, l Qi) and it is sufficient to show that a word Pi" GQR exists which X;-l1, ( Q i , Qi). BY Axiom 33' for n + I

c.,

nrl

P i = P ; E I Z , F I P F -Piy,

Qi=QkE3Z3F3QF=Qiy

Pi= PFE2Z2F2PF =Pi*, Qi = Q F E , Z 4 F 4 Q F =

QT.

Now P T - l X ; ! I Q T X i t L n + l and contains (2;'. Z , ) hence J, K exist such that K Z i 1 J Z 2E Rep(Ln+l) and J = E i l P:-lX;!l Q4 E,. Similarly Q,?XiP;-' 3 (Z4;2;') and K'Z4J'Z;' E Rep(Ln+l). Now

( Z , F I P F ,P:E,Z2)J'-'

n ~ J l( Z , F , Q ; ,

QFE4Z4)

J.L. Britton, The existence of infinite Burnside Groups

In the left side replace Z,J'-' in G , t o Zk WZ:. Hence

Pi" E l ZF WZF F4 Q f

303

by K ' Z , . By 15.3 the result is equal

n ~ X;-l1(Q: l

E 3 Z 3F , Q p ,

QF E 4 Z 4F4 Qf )

The left side is PTL WQ*'. Now Pi ;P: so P?' = PL EP*LR J and pfEp:LR'WQ*RiF@ is'the reSimilarly QTR = QTdLFQR quired word.

16.11 . Axiom 2 4 is easily shown to hold for n + 1. 16.12. Consider Axiom 18. First note that if Z E Rep(Ln+l) then one can find a sequence Z , + ... Zo frDm Z t o Z , i.e., Z, = Z , = Z such that (i) every Zi has the form Xi" and (ii) for k = 1 , 2 , 3 , ... -+

is a sequence from zk to z k . Now suppose AB E L;+,, s A B ( A )< h,. Then Z k 2... 7 AB, 2 E Rep(Ln+l). By Axiom 33' for n + 1,

B

x B ~ E Z , F B ~ ,z,z,= z k .

Now Z ,

=Zk-lZL,Z -

Hence Z

-+

... -+

Z L Z RZ , R = Z,, and

ZLRFBRA B LE Z L L; thus A is a subelement of

Ln+1.

16.13. The reader will easily verify that the following Axioms hold for n + 1: 2 , 3 , 6 , 7 , 2 1 , 3 1 A . 16.14. Note that Condition D now holds.

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5 17 Informal Summary This section is mainly concerned with the proof of Axioms 1 1 a, 1 1 b for n + 1. In the case of 1 1 a, one would like simply to take W' to be a word of smallest length such that W' = W.The author does not know whether or not such a W' is ((n + 1))-bounded by any parameter or in particular by 1, so in the proof given an alternative to "smallest length" has been used. 17.1. Proposition. Let A c X; Y where A is a subelement of L n + , and s ( A ) > u60. Let A 3 A o ; Ao' 3 B where B is a weak image o f A . Let A , B be (n)-bounded by b < Min(b2,, b31). Then there is a word A L E B R ,(n)-bounded by b + q31 where neither A L nor B R is a subelement of L U ... U 1,. Proof. See the proof of 6.26 and take u60 = u4.

17.1A. Informal note. We shall be interested in applying 17.1 , 17.2 when the (n)-bound of X,Y is greater than the (n)-bound of A , B.

-

17.2. Proposition. Let A C X ; Y where A is a subelement of and s ( A ) > u61. Let A 3 A' A'' 3 B where B is a weak image of A . Let A , B be (n)-bounded by b < b25,b31. Then there is a word A LEBR, (n)-bounded by b + q31 where s(AL) > s ( A ) - 2462 and B R is not a subelement of L LJ u 1,. Also s(A)> s(B)- ~4

(*I

-~ 3 .

s(B) > s ( A ) > u61- > u3. Hence by 6.10 C Bo' -Proof. Bo B, s(C) > s(B) Also s(C) > 2u4 > Now C c BO'c Ao c A , so s(C) < s ( A ) + e3 and follows. C

- u4 - u4.

u4

C

-

(*)

yo.

Now s(C) < s(B0')+ e3 hence s(Bo')2 u61 - 2u4 - e3 > ~ 3 1 , ro + r31 and there is a word BorLEBOR,(n)-bounded by b + q31 and equal in G , to BO'. K-' = E . Bo for end words E, K. BoR is not a subelement of L u -.. u L , and s ( B O ' ~> ) s(Bo')- r31 > yo. Now A = ZBO'T, B =X'BOY'say. The required word A LEBR is ZBOrLEBOR'Y'.

J.L. Britton, The existence of infinite Burnside Groups

3 05

We have

since u62 = 2u4 + 2e3 + ~ 3 1 . For later we require u 6 1 >

u62

+ ro + e3.

17.3. Notation. We use the notation ai (i = 1 , 2 , 3 , ...) for certain expressions which will be defined explicitly in the present section (Section 17). 17.4. Definition. Let U be a linear word (n)-bounded by k39. Call a sequence of subwords A,, ..., A , (m 2 1) of U linearly admissible if 1. Each A iis a subpowerelement of Rep(Ln+l) of size > 2463. 2. A , , ..., A , are pairwise disjoint. 3. ( A i ,A i + l ) is not a subpowerelement of L n + l (i = 1, ..., m - 1). 17.4A. Definition. Let U be (n)-bounded by k39. If U has no linearly admissible subwords put Dk+,(U) = 0. Otherwise, put

(taken over all linearly admissible sequences).

17.4B. Note. Dk+,(U) is finite.

,,

Proof. Take any linearly admissible sequence A ...,A , s(Ai) < L(A,)/e, hence 2 s ( A i )< ZL(Ai)/e < L ( U ) / e .

17.4C. Note. If

o;+l(U)

> 0 then D ; + l ( u ) >

. By 7.1 1,

2463.

17.5. Definition. Let W E II. By Axiom 1 la, since k39 2 bmi,, ( 3 W ' ) W g W', W'is (n)-bounded by k39.

3 06

J. I-. Britton, The existence of infinite Burnside Groups

Hence D L + , ( W ' ) exists; put

17.5A. D n + , ( W ) is 0 or > u63 17.5B. I f Wl

;W 2 then D,+,(W,)

,

= D,+,(w,).

1 7.5C. Proposition. D , + (W) is finite. Proof. It is sufficient to show that there are only finitely many W' as in 17.5. This follows from the next proposition since bmin< so. 17.5D. Proposition. If W, E n and W, is (n)-bounded b y so, then there ure only finitely many W,such that W , is (n)-bounded b y k3, und W, ;W,. Proof. Consider any W,. Then by Axiom 28 (so 6 b2*;k,,

<

Now if W, 2 X and W , 2 Y by simultaneous replacement of the same set of maximal n-normal subwords of W, then X .fl Y ; hence there is a finite set X , , ..., X , such that W, 2X i (i = 1, ..., N ) and if W , 2 Y then ( 3 j ) Y nl: Xi,1 < j < N . Say U,-, Xi.Now Un-, n=;l W,, so Xin-l W,. Now Xiis ( ( n- 1))-bounded by so, W, is ( ( n - I))-bounded by k,, so by an induction hypothesis there are only finitely many choices for W,.

,:

17.6. Proposition. Let R', C' E II. Let U be E. R. and let the cyclic word ( U ) be (n)-bounded b y k3,. Let R'. U . R'-' ;C'. Let A , , ..., A s be disjoint subwords of ( U ) where each A i is a subelement of Rep(L,+,) of size > ~~3 and ( A i ,A i + , )is not a subelement of L n + , ( i = 1 , ..., s ) (where A , + 1 means A , ) . Let C s ( A i ) > a,. Then C' ;X V Y , X V Y is (n)-bounded b y k,, , V is a subword of ( U ) , C s ' ( A i n V ) > C s ( A i )- a,, whcre s ' ( X ) means s ( X ) i f s ( X ) > u63 and 0 otherwise.

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17.6A. Note. s ' ( X ) < s ( X ) < s' ( X ) + u63. Also if A B is a subelement of L n + , then s'(AB) < s'(A) + s'(B) + 2uS3+ 2/e. Before proving 17.6 we need a lemma. 17.6B. Lemma. Let A , , ..., A,, B,, ..., B, be disjoint subwords of X where A , is left of B , (and 2.6 applies). Let X be (n)-bounded by k,,. Let A , , ..., A , and also B , , ..., B , be linearly admissible. Let

1

1

'

Then there is a word X ' , (n)-bounded by k3, and equal to X in G,, and there is a linearly admissible sequence C,, ..., C, for X ' such that

Hence D , + , ( X ) > a

+ b - us,.

Proof. Assume ( A , , B , ) = A,EB, is an ( n + 1)-subpowerelement, since otherwise the result is trivial. If s(A,) < us9 then A , ... ,A r - , , B , , ... ,B, is admissible and c = a + b - s ( A , ) 2 a + b - u89. Similarly if s ( B , ) < u89. Now let s(A,), s(B,) > 2489. Now us9 > rb. By Axiom 33" for n + 1 there is a subelement A , D B , = J of Rep Ln+, equal in G, to A , E B , . Where X - X , A , E B , X 2 put X ' - X , A , D B , X , . By 5.1 1, 5.32 X' is (n)-bounded by k3,. Also X ;X ' . Now s ( A , ) + s(DB1) < s(A,DB,) + (2e3 + e,*) and s(B,) < s ( D B 1 )+ e3. Hence S ( A r D B 1 ) ~ S ( A , ) + S ( B 1 ) - U 8 9>S(A,)>U63. ThusAl ,..., A , _ , , J , B,, ..., B, is admissible and

,

c = a-s(A,)

+ s(J) + b - s ( B 1 ) 2 a + b - u8,.

Proof of 17.6. First note that u63 > u,, u60; bmin,k,, Max(b,,,i,, str+)+ q31 < b25, b31. Put N = Z s ( A i ) ;then N > a,.

< b,,

and

308

J. L. Britton, The existence of infinite Burnside Groups

1O . Take C such that C' ;C and C is (n)-bounded by bmin (Axiom 1 la). There is at least one R such that (*)

R is (n)-bounded by bmin and U' exists such that

U' is a C.A. of U and R . U ' .R-' C. Namely, take R to be equal to R' in G, and (n)-bounded by bmin. Of the R satisfying ( * ) choose one such that D,+,(R) < Inf D,+,(R) R + 11e3. Let R . U' ; D where D is (a)-bounded by bmin. We can apply 8.2 of Chapter I since bmin,k,, < 1 - 5(r + E,) - 12/e (cf. the beginning of the proof of 15.3). Hence there are divisions R,R,R, of R , U , U , U3 of U ' , D , D , D , of D where R,; Uil (end words 8, I ) U3 ; D , (end words cp, I ) R, ; D , (end words I, $). As in 15.3 write R = R; R;R', and similarly for U' and D . 2". We have U ; C ( U ) . We shall obtain a contradiction from

+

Zs'(Ai n U ; ) > a,. Let Pi = Ai n U ; and let the Pi of size > u6, be P p , ..., P4 (since

a 2 2 0 there is at least one). Then Zf3 s(Pi) = Z s f ( P i )> a,. Now Pi C U ; C U , so we may consider a weak image Pi of Pi in R i l (i = p , ..., 9). Say RJ1 = X ' P i Y ' , U , = XP4Y. By 17.1 there is a

word P i E P i R , (n)-bounded by Max(bmin,str+)+ 9 3 1 . Also, if s ( P 4 ) > u61 then by 17.2 s ( P t ) > s(Pq)- u62. Put R 3 = X P t E P i R Y ' ; thenR, ;R i l . Let R * - R,.R,. Y f - l P i R - l E - ' P 4L - l X - l Then

R " - R 1 . R , . Y ' - l P i R - l E - l and

< D,+,(R) -~l/e3. We h a v e R3 > XP,">P, ,..., P 4 - , , P ~ . N o w R ;R * a n d R * i s (n)-bounded by Max(Max(bmin,s t y + ) + 9 3 1 , bmin,k,,) = P1 say.

J.L. Britton, The existence of infinite Burnside Groups

309

Now 0,= k39.R" is (n)-bounded by Max(bmi,, s t y + ) + q 3 1 . Hence (cf. 5.28A, 5.22) q-1

C s(P,) + s ( ~ f ; ) , i f s ( P i ) > yo + e3

(yo

+ e3 > u 6 3 )

Dn+,(R)2 s(Pi), otherwise. P

If s(P,) < then C 4 - l s(Pi) > a 2 and Dn+,(R)> a2 - u61 P > 11~3. If s(Pq) > u61 then s ( P i ) > s(Pq) - u62 > yo + e3; if also s(P:) > ro + e3 then s(Pq) - us2 + s(P:) < s ( P k ) + s ( P t ) < s(Pq) + + 2e3 + E:. Thus s(P:) < Max(ro + e 3 , us2 + 2e3 + e;') = a 3 . Now 2 / e + s ( P i ) + s(P:) + C;-l s(Pi) > a2 hence

D,+,(R)> a 2 - 2 / e - s ( P : ) > a 2 - 2 / e - a 3

> l/e3;

thus the result is true if D,+,(R") = 0. Now let D,+,(R") > 0. Then S' exists, (n)-bounded by p1 and R" ;S ' , and there is an admissible sequence B , , ..., B, for S' such that

Let Ci be a weak image of Biin R". There is a word B i ECF (n)bounded by str+ + 2 q 3 1 ,since bmin< s t y + . Also if s ( B k ) > u61 then s ( B i ) > s(B,) - u62. Hence the word R ** where (In S', R i ) E ( C F , Fin R " )

R**

is (nkbounded by P1 since str+ + 2q31 < k39.Also R** R* = ~ r l p L - 1 ~ -so1 4

;R". Now

~ * * p L - 1 ~ - R* 1 == R (7

n

n

where there is no dot on the left side and it is (n)-bounded by pl. Consider the sequence B , , ..., B k - l B p , PF-', Pi!', ..., Pi' where

310

J.L. Britton, The existence of infinite Burnside Groups

X o means X is omitted if s ( X ) < ro + E,. By 17.6B, k- 1

0

q-1

0

cs(Pi)-a4

Dn+l(RP cs(B,)+S(B~)+s(P;)+ 1

P

where a4 = ~ 4 ~ ~ . From the first part of 3" follows

) E;s(Bi) = s(BiL)-s(BR);if Now (C:-'s(B,) + s ( B ; ) ~ s ( B k )< u61 this expression is 2 -u61, while if s(Bk)> then as before s(BF) < a, and the expression is s ( ~ k -) s ( B ~ ) -s(@)

- 2/e

> -a3

- 2/e.

Hence D + l ( R ) 2 -ay4+ a 2 - 2 M a x ( ~2/e ~ ~+, a 3 )+ E t s ( B i ) = - a5 + E ffl s ( B i ) say. Now D,+,(R) > a5 + D n + l ( R f f-) E 2 D n + l ( R " )+ + 1/e3 since we shall define a 2 by the equation a5 = E + 1 /e3. This proves that D,+,(R") < D n + l ( R )- l/e3. 4". Let R" ;T where T is (n)-bounded by bmin.Then C ;T . U". T-' and D n + l ( R )> D n + l ( R " )= D n + l ( T ) a, contradiction. 5". Thus we have C s'(Ai n U ; ) < a 2 . Now N < E s(Ai)= E s'(Ai) and C s f ( A i )< 3(2u63 + 2/e) + s'(A, n Ui>+ c s'(Ai n U ; ) + + C s'(Ai n U;). But s'(Ai n U ; ) = 0 , since 2e2 + 4/e + rb < u63 (cf. 1" of 15.3)

Let Qi = A i n U; and say Qr, ..., Qs have size > u63 (there is at least one since Z: s(Qi)2 N , > a1 - 3(2U,, + 2/e) - a2 > 0. Now Qic U 3 - D, so we may consider a weak image Q: of Qi in D, and there exists a word EQ,", (n)-bounded by str+ + q31 and

elL

D, ;(In D,,Q;")E(Q,", Fin U,)

z D",say

J. L. Britton, The existence of infinite Burnside Groups

311

andD; Dl.D2.D"=-D*,say.HenceC;R.U',R-1 ;D.R-'= = D * . R - l . Put H E R - ' ; then there are divisions D:DgD; of;*, H2 H 3 of H and C , C2C, of C and D3 Hi' (end words 8, I ) (end words cp, I ) H3 C3 (end words I , $) Cl Dr Now -D* -D* = - -(@,Fin U 3 ) ;let V = 0; n Fin U 3 ) . Then J , D:* exists such that Fin U 3 )= JV, 0; = D;* V. 6". Assume that 2: s'( V n Q i ) > a 2 ;we shall obtain a contradiction. ~ ~ ; Put Si = V n Qi and say S q , _..,S, have size > ~ 1 then E i s ( S i ) > a2. Now Si C V C D:; let S; be a weak image in HI' of Si.There is a word SiLES:, (n)-bounded by str+ + q 3 1 and

h,

-

(QF,

(QF,

HI'

;(In H i 1 , SrL) E(S:, 4.

Fin 0:)

where the right side is (n)-bounded by pl. Now U' = J'JV say, and V = V , Vo for some V l , where Vo = (SR,Fin 0 ; ) . Thus H-'= 1 n TEVo = Ro,say, where T = (In&', S f L )Let RE = H i l H i l TE and R t = R: V o ;thus R t 3 Ro. Also R ;H-f =.Hi1H i 1H-'1 = R:. N o w C = R.Ul.R-1 = R " V .J'JV I / . V ~ l R ~ - ' = R ; I . V o J ' J V l . R ~ - l , n 0 0 1 0 where U l = VoJ'JVl is a C.A. of U' hence of U. We shall show Dn+'(R;)< Dn+,(R)- l/e3. The method is the same as before but it may help the reader t o give a sketch. R z 3 Ro 3 Vo 3 S t , S q + l ,..., Ss. Next it can be shown that R: is (n)-bounded by k,, and hence as before

so the result is true if Dn+l(R:)= 0. Now let Dn+,(R(d)> 0. Then Sh exists, (n)-bounded by k3, and R: ;Sb and there is an admissible sequence B , , ..., B, for Sb such that E s(B,) > Dn+'(l?:)- l/e2. Let Ci be a weak image of Bi in R:. Then there is a word

R ~ * = . ( I n S A , B ~ ) E ( C ~ , F i n R : ) a n d R ~R*Z; s o R $ * V o ; R : ; R . R t * V o contains subwordsB1, ..., B,_,, B k , S R ,S q + l ,..., S, and proceeding as before we again obtain the result. As in 4" the result just obtained leads to a contradiction.

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J. L. Britton, The existence of infinite Burnside Groups

Hence we have Z; s'( V n Q i ) G a2. 7". Now

Therefore Z s'(J n Q i ) 2 N,- s'(Qr) + sf(@) - a 2 - 2uS3- 2 / e > 0 . Fin U 3 )= JDT. Now D*D Hence J I so D* E 1 2*D* 3 F J"( QF, Fin U 3 )E J"JDT so DTD; = J"J. Write J 3 J , J2where J , F J no:. Then

+

(QF,

C ~ (n JQ ~<) C ~ ' (nJQ, ~+) C ~ ' (nJQ~~+)2uS3+ 2/e. But s'(J2 n Q i ) = 0 so

C ~ ' (nJQ, ~2)N,- S ' ( Q , ) + ~ ' ( Q F ) a 2 - 4u63 - 4 / e = N2 (say) > 0. -

-

Hence J , $ I and J , has the form (DT)R C DT C,. Let Ti = J , n Qi = J , n A i n U; and let the Ti of size > u63 be Ta, ..., Tb (there is at least one). Then Z,bs(Ti)> N 2 . Let Ti be a weak image of Ti in C,. Then

where the right side is (n)-bounded by k,, and contains (Ta, ..., Tb-,, T k ) = V say; it is the required X V Y . Also V C U; C ( U ) . Now

J.L. Britton, The existence of infinite Burnside Groups

313

NOW if s ( T b ) > U then S(Tk)> S ( T b ) - U s 2 > 2461 - U62 > 2463, hence S ( T ; ) = s ' (61L T ~) and s ( T b ) = s ' ( T b ) . Therefore s ' ( T ~>) S ' ( T b ) -

- u 6 2 . If s ( T b ) < then S ' ( T b ) < 2461. Hence s ' ( T ~- s)' ( T b ) is greater than - u 6 2 or 0 - U61 so in either case it is greater than -2461. Similarly, s'(Qp) -s'(Qr) > - u 6 1 . Therefore

C ~ ' (n A V )~ 2 -

2

+ N~ , -~a2 -~42463 - 4/e

if we define a1 to be 2a2 + 5 ( 2 U 6 3 + 2 / e ) + 2~~~ 17.7. Corollary. D n + l ( C f 2 ) C. s ( A i ) - a l

- 2(ro + e 3 ) .

Proof. This is trivial if the right side is negative so let E s(A,) > a 1 + 2(ro + e 3 ) . Let Pi = A i n V . Those Pi of size > 2463 have the form (i) P a , ..., P b . Also b

cS(Pi) cS'(Pi) > C s ( A , ) =

a

- "1

> 2(ro + E 3 )

From the sequence (i) delete Pa if it is not normal to obtain (ii) P a + l , ..., P b , where k

b

c S(Pi) z =

a+l

a

S(Pi) - s(Pa)2

b

c S(P,) a

- (Yo

+ € 3 ) > Yo + € 3 .

From the sequence (i) or (ii) delete P b if it is not normal. For the new sequence 2 s(pi)2 2,"s(pi)- 2(ro + e 3 )> o so it is non-empty; it is also admissible, so

17.8. Definition. Put W E J n ( x ) if W is E.R. and ( W ) is (n)-bounded by x , and W e # I in G,.

J. L. Britton, The existence of

314

infiniteBurnside Groups

17.8A. Note. By Axiom 1 1b, if W E II, We # I in G, then some conjugate U of W in G, is in J n ( k 3 9 ) ,because cbmi, < k39. 17.9. Definition. If U EJn(k39)we define B k + , ( U ) as follows. Call A , ... , A, admissible for U if 1. A 1 , ..., A , are disjoint subwords of ( U ) . 2. Each A i is a subelement of Rep(L,+l) of size > u63. 3. (Ai, A i + l ) is not a subelement of L,+, ( i = 1, ..., q ) A , + , = A , . Put Bi, + ( U ) = Sup C:=, s ( Ai) (or 0 if no admissible sequence ex is ts). 17.9A. BL+, ( U ) is finite. 17.10. Definition. Let W E II, We # I in Gn+,. Define Bn+,(W) to be Sup BL+,(U) where U is conjugate to W in G, and U E J, ( k 3 9 ) . 1 7.1 OA. Proposition. B,

+

,( W) is finite.

Proof. Choose U, where U, is conjugate t o W in G, and U, E J,* (k39). Consider any U E Jn(k39)conjugate t o W in G,. Then R'. U .R'-' ;U, for some R'. Let A , , ..., A s be admissible for U. Then by 17.6, 17.7 either C s ( A i )< a l or

B,

+

( U, ) 2 &(A

-

a

,

-

2(ro + € 3 ).

Thus C s(A,) is bounded above, as required.

17.1OB. Note. I f W; # I in Gn+ and W,, W2 are conjugate in G, the11Bn+1(W1)=Bn+l(W2). 17.1 1 . Proposition. I f We # I in Gn+l then there exists K such that (i) K is conjugate to W in G,+l. (ii)I f X is conjugate to W in G,, then B n + l( K ) < Bn+l( X ) + l / e 3 . (iii) K E Moreover, f o r any such K the cyclic word ( K ) is ( n + 1 )-bounded by

k40.

Proof. Let L be conjugate t o W in G,+,. Then Le # I in G n + l and

J. L. Britton, The existence of infinite Burnside Groups

315

B,+l ( L ) exists. Of all such L choose L' such that B n + l ( L ' ) < InfB,+l(L) + l/e3. Thus L ' i s conjugate t o Win G,+l. Now L f e # I in G, so some conjugate K of L' in G, is in J , (k3,) and

B,+l(K)=B,+,(L').

It remains to show that ( K ) is ( n + 1)-bounded by k40. Assume not; then ( K ) contains S where S is an ( n + 1)-subelement of size 2- k40, so n o linearization of ( K ) is a subelement of L u ... u L,. By Axiom 28B, since k3, < d,,, ( K ) (K') where (K') is (n)bounded by str = k41. Also ( K ' ) contains an image S' of S where s(S') > s(S) - u ~ Say . ( K ' ) = (SIT'). Since k,, - us4 > t33 and str < b,, we have S' ;SfLEZFSfR = S " , where S" is (n)-bounded by k41+ c33< k,, and by 5.32, so is ( S " T ' ) .Also s(Z) > s(S') - Y 3 3 > - Y 3 3 - U a + k40 = CYy6 and (116 > MaX(U63, "1 + 1 /e2 + lie3). We have (S"T') = ( A Z ) for some A , and ZT-l E Rep(L,+l) for some T. Thus 2 n ~ T.l Let A . T ;C where C is (n)-bounded by bmin(cf. Axiom 1 la). We may write C = Q-lC'Q where Q is chosen maximally. Now write X conj, Y if X,Y are conjugate in G,; then

W conj,+l K conj, K'

= S'T' ;S"T' conj,AZ

,fl

A . T; C conj, C'.

Thus W conj,+,C'. H e n c e B , + l ( K ) < B,+l(C')+ l/e3. Now K conj, A Z E J , (k3,) and 2 is admissible so B,+,(K) 2 ~ ( 2 Therefore ). Bn+l(C')> 0. Hence B n + l ( C ' )= Sup B h + l ( U ) over U such that U conj, C' and U E J , ( k , , ) . Thus U exists with an admissible sequence for U , say { A i l , such that

Since C' conj, U we have by 17.6 that C' ;X V Y , (n)-bounded by k3, , where V C ( U ) and C s'(Aj n V )> I:$(Ai) - al. Put Pi = A i n V ; then pi, ..., p k , say, have size > 4 3 and c,!s(P,) > c s(A,)- a 1 . C' is (n)-bounded by bmin;let Pl!be a weak image in C' of P i . We have say C' C i ( P i , P i ) Ci. If j < k

J.L. Britton, The existence of infinite Burnside Groups

316

C'

;c;P;"E(P;,

P;) F P ~ =;c".

This is true also when j = k if we interpret (Pi", Pk) to be Pi".C" is (n)-bounded by k39.Now A . T; C- Q-l C'Q ;Q-' C"Q (no dots and D = Q-l C"Q is (n)-bounded by k3,. There are divisions A1A2A3 o f A , TlT2T3of T a n d D 1 D 2 D 3 o f D a n d A 3 - T i 1 , T3 D3, A , D , . Let ni, ...,nk mean PR, Pi+l, ..., Pk-,,Pk and let J Qi = ri n D,. Assume Z: s ' ( Q i ) > s(T) + a7 > 0; we shall obtain a contradiction. Let the Qi of size > uS3be say Q,, ..., Q,. Then Ey s(Qi)> s ( T ) + + a 7 . Let Q; be a weak image of Qi in T3. Then

-

(1)

-

T, ;(In T3. QiL)E(Qp,Qk)F(Q:, Fin T3)- T;', say

(if I = m interpret (Q,",

Qk)as Q,"). T;' is (n)-bounded by k3,. Now

and by 5.32 the last word is (n)-bounded by k39.By Axiom 28 since str+ < b28 and k,, < a28 we have 2-' T 2 7 Z-' T, T2T;' hence Hence (Q,", 2-' T, T, T;' E is a subelement of L n + l . If ( A , , , , A , + , ) would be a subelement I + 2 < m then (Q,,,, Q1+2) of L , + l ; hence I + 2 2 m. Let

Qk)

If rn - 1 = 2 then either Qp nor Qf;, is normal. If m - I = 1 then not both of Qf , are normal. Hence one of the summands H of (2) satisfies

QL

x < 2(ro + E 3 ) + s ' ( H ) . Now

317

J. L. Britton, The existence of infinite Burnside Groups

so

s ' ( H ) - s ( T ) > a7 - 2u61- 2(ro + € 3 )

> (Yo + € 3 ) + u4. Now s( T ) 2 0, so in particular s' ( H )= s ( H ) . From (1 ), T , contains a weak image H' of H and s ( H ' ) > s ( H ) - u4 > yo + e3. Hence

s ( T )2 s ( H ' ) - €3

>s(H)

- u4

- €3 = s ' ( H ) - u4

-

€3.

+ 3(ro + e3) + u4 + e3 we obtain a Hence, defining a7 to be contradiction. Hence C. s'(Qi) < s ( T ) + a7. We have

Cs'(ni)< C sf(nin D,)+ E sr(ni n 0 2 +) C sf(n n 03)+ 2(2u6,+ 2/( Now

so

J. L. Britton, The existence of infinite Burnside Groups

318

C ~ ( A , ) - ~ ( l/e’-T ) >I -

~ + ~ ( z ) - ~ ( T )

Thus p > 0. Let Ri = xi n D,.Let the R i of size > u63 be, say, R , , ..., R , ; there is at least one: CL s(Ri)> p. R j has weak image Ri in A , ; say A = X,(RL, R k ) * X X Then ,.

;X , R: E ( R f , R k )FRLRX2Z = G, say.

AZ

G is E.R; and ( C ) is (n)-bounded by k,,, so G E Jn(k3,). Now K conj, A 2 ;G. Consider the sequence RUR R u + l ..., R , _ , , R k ,

(3)

9

9

z

of subwords of ( G ) and put E - s’(RF)+

+ s’(Z). Then E

s ’ ( R j )+ s‘(Rk)+

> s ‘ ( R ; ) + s‘(Rk)+ s ‘ ( Z )+ p -s‘(Ru) -s’(R,)

Now s ( 2 ) - s( T ) > 1 /e2 + a9 + 1 /e3 so the last expression is greater than C s ( A i ) + 1 /e2 - 2u61+ a9 - “8 + 1/e3. We can obtain an admissible sequence B , , B,, ... from (3) as follows. Delete RF if not normal; delete R; if not normal; then apply 17.6B to obtain a linearly admissible sequence S , , ..., Sp say, where C s(Si) 2 E - 2(ro + e3) - u89. If p = 1 the sequence is already admissible. If p > 2 , apply 17.6B t o s,, ..., S p , S l t o obtain the required B , , ... , B, where C s ( B i )> E s ( S i )- u89. Hence Bn+l ( K ) = B n + l ( A Z )2 S(Bi) > E - U66 where u66 = 2(ro + e3 + ug9).Hence

J.L. Britton, The existence of infinite Burnside Groups

319

since a9 > 2u61 + "8 + u66, i.e., 2a6 - I - (ro + 2 ~- ~ 1 /e 2) - I /e3 > 2 U 6 , + a8 4- u66. But We proved above that c S ( A i ) > Bn+l( K ) - l/e2 - l/e3 so we have a contradiction.

17.1 2. Proposition. Axiom 1 1b holds for n

+ 1.

Proof. Let We # I in Gn+ Of the L such that L conj, + W choose L' such that Bn+,(.L') < Inf B,+,(L) + l/e3. By Axiom 1 lb, since Lre# I , there exists KOsuch that KO is E.R., KO conj, L' and (KO) is (n)-bounded by cbmi, < k39. NOWBn+l(Ko)= Bn+l(L') and KOE Jn(k39). By 17.1 1 (KO)is ( n 1)-bounded by k40. Now k40 = str. Thus (KO)is (n)-bounded by cbmi, and ( n + 1)-bounded by str. Since cbmi, < str < str++ q31 G so, (KO)is ( ( n + 1 ))-bounded by so. Make a simultaneous ( n + 1)-replacement in (KO)by replacing those maximal normal subwords, if any, of size > f . Say (KO)+(K,). Then by Axioms 26,26" no maximal normal subword of (K,) has size 2 + r26,since 5I + r26> u26 ; thus (K, ) is ( n + 1 )-bounded by f + Y , ~+ e3 = k42. It is (n)-bounded by so. Similarly (K2) exists such that (K,) + (K2) and (K,) is n-bounded by k42 and ( ( n - 1))bounded by so ; and so on to obtain finally ( K n + , ), I-bounded by k42-

Now let S be a ( p + 1)-subelement ( I < p < n ) in (K,+,) of size 12.15 it determines a subelement S' of L p + l in (Kn+l-p) and s(S') > s(S) - u65 where us5 = u37 + u4. Hence s(S) < s(S') + + u65 < k42 + u65 = Cb,i,. Thus ( K n + l )is ( ( n + 1))-bounded by cbmi,. Finally Kn+, conj,,, KO conj, L' conj,,, W.

> u49. By

17.13. Corollary. If We $; Z in G , +1 then K exists such that K conjn+lW;X conjn+lW implies Bn+, (K) < Bn+,(X) + l/e3, and K E J n + lwhere , Jn+lis defined as follows. 17.13A. Definition. Put Y E Jn+lif Ye # Zin Gn+,, Y is E.R. and ( Y ) is ( ( n + I))-bounded by str (cf. Axiom 9). Proof of 17.13. For the required K, take the KO of 17.12.

3 20

J.L. Britton, The existence of infinite Burnside Groups

17.14. Note. Axiom 1 1a is just a linear version of Axiom 1 1 b. The proof that Axiom 1 l a holds for ( n + 1 ) is left to the reader. 17.15. Proposition. Let T be E.R. and let ( T ) be (n)-bounded b y k45. Let T e # I in G n + l .Let B n + , ( T )Z ~ 7 1 Then . B n ( T )> 2B,+,(T). Proof. As a preliminary, note that k,, < b 2 5 , b30, b 3 3 2.463 ; - u4 > "30; Min(q*, r* - 2(ro + e 3 )- 4/e) > t33and :u63 - €30 - 1/e2 > rb + ekO+ 2/e. U exists such that U conj, T, U E J , ( k 3 9 ) ,and

there is an admissible sequence A 1, ..., A , for U such that

C S ( A>~B ,), ~( T )

-

l/e2 2

u71-

l / e 2 > al;

say, K t U . R - l ;T . By 17.6 T ; X V Y where X V Y is (n)-bounded by k,,, V C ( U ) and 2 s'(Ai n V ) > 2 s ( A i ) - a 1 . Let Pi = A j n V , and let those Pi of size > u63 be P f , ..., Pg, say. Then EFs(Pi) > C s ( A i ) - a l > B , + , ( T ) - 1 / e 2 - a l > B,+l(T)since ~ 7 > 1 2(1/e2 + "1). For i = J ; ..., g let Pl!be a weak image in T of Pi; then s(p1!)> s(pi) - u4 > s(Pi)since 2163 > 2u4. By Axiom 30 for n + 1, Pl! contains subelements Ui; ( j = 1 ...)ai) of L,. Let Cii be the kernels of the Uii (i fixed); thus if the Uii are disjoint then Cii = Uii. Now (Axiom 3 3 ' )

say where Cb , C: are not subelements of L u ... U L n - l and Cly is ( ( n - I))-bounded by k,, + c33.Also s(Zii) > q* - r33 > u63. Let Pi!' arise from Pi!by replacing Cii by C i for all j , and let T" arise from T by replacing each Pi' by Pi''. Then T n ~ T" l and T" is ( ( n - 1))-bounded by k,, + c33< k3, (cf. 5.32). For fixed i suppose the subword (Zi a - l , Zi b + l ) of PI!'is a subelement of 1 , and a < b. Then Ci", Ei,(Z,, Zib)Fib C: = U is a subelel cib)c Pi! so ment of L , say U V E L,. Now U n ~ (Cia, UV ,,zl(Ci,, Cih)V . Since (Ci,, Cib) is ( ( n - I))-bounded by k,, ( c i a , Cjb) VE L n' Since a < h. this implies that the Uii are not maximal normal

J. L. Britton, The existence of infinite Burnside Groups

321

(cf. 5.28A), i.e., Cij = U i j . Thus (Ui,, Uib) is a subelement of

1,. By Axiom 30, s(Ui,, Uib)> (b + 1 -a)/ti aj/ti > s(P~) - €30 > s(Pj)- €30.

- E;,

and

Next we show there are positive integers ci, k , such that (i) (ci - l)/ti - €5, > rb (ii) kici + 1 < ai. Let c, be the least integer > (rb + ejo) ti + 1 = 6 say, then (i) holds. Now kici + 1 < ki(6 + 1) + 1 < ai if ki < (a, - l)/(6 + 1). Next,

= Ei say

and Ei > 1. Thus for ki we may take [ E i l . Hence k j > E j - 1. Put Ais = Zii where j = 1 + sc, (s = 0 , ,.., ki). Then we have shown that (A,,, A , s + l ) is not a subelement of 1, (cf. 5.21). Thus the sequence { Ais}, i = J ..., g, s = 1, 2, ..., ki, is admissible for T" (since, e.g., (Ai-l A , , ) > (Aio,Ai,)). Hence

i = f s=l

i=f

We show that us3(Ei- 1) > &(Pi) and it will follow. that B n ( T )> 4 2 s ( P i ) > 2B,+,(T) as required. Let a = ~ 3 +0 l/e, p = rb + €j, + 2/e. Then Ei - 1 = (3 s(Pi) - a - p)/p > 4s(Pj)/u63 provided that

17.16. Proposition. If ( U ) is (n)-bounded by k,, and Bn +1( U ) < U 7 1 then ( U ) is ( ( n + 1))-bounded by k46. Proof. If not then ( U ) =_ ( U ' S ) where S is a subelement of L,+l and s(S) 2 k4, 2 51 , t33.Now S ; sLEZFS' E S' where s' is (n)-

322

J. L. Britton, The existence of infinite Burnside Groups

bounded by Thus

k46

+ c3, < k,,,

which is a contradiction

and so is U'S'. Also s ( Z ) >

(k46

2'

-

r33 > d .

< b3,).

17.1 7. Proposition. B: = QC,, ..., Cn+'.J n + l ) . Also C , , ..., J,* ure pairwise disjoint. Hence Axiom 1 holds for n + 1. +

,

Proof. Call the right side H say. Let W E II. If We = I in G,+' then W e = I i n H . I f W e # I i n G n + l t h e n b y 17.13 t h e r e e x i s t s K E J , + l conjugate to W in G,+l. Hence in H we have I = K e = T-' . We.T for some T . So again We = I in H. are disjoint since Cn+lC J,. If X E J n + , then C , , ..., en+' X " # Z in Gn+lwhile if X E C, then X e = I in G, hence in G,+,. 17.1 8. Proposition. Zj TCTD = Pe E Ln+l and L ( T ) 2 L(P) then TC = Pk for some k hence (TC)en ~ I.l Proof. This is trivial (cf. 2.18B). 17.19. Proposition. Let Y A Y B E L , + l , s ( Y ) > ~ 7 2 Then . if Y i s (n)-hoiinded hy k,, we have ( Y A ) e = I in G n + l .

,.

Proof. We have Z + ... + Y A Y B , Z = P e E Rep L, + The first Y ; Y L E Z I F Y Rand the second Y ; Y ' E ' Z 2 F ' Y r( ~ 7 2 t3,; 2 k47 < b3,). Also (2, C Z 2 D )= Z and 2,CZ2 ;Z , FY A Y' E'Z2 by Section 16. Consider Y L E Z I F Y R;Y ' E ' Z 2 F ' Y r . Let J be a weak image of 2,in the right side (k47+ c , ~< b25;~ 7 -2 r3, > u 3 ) . Then s(J n Z,) > ( ~ 7 2- r33 - u 4 ) - 4/e - 2Max(ro, q3, + E , ) = 6 say. Now 6 > u1 and, where 22" E ( J n Z2)0,s ( Z Y ) > 6 - u1 > s , ~ . Say Q; then Q has the form Z r . By Axiom 33 for n + 1 Z y 3 Z Y . We may write Zi = Z ~ Z (i ~= 1,2). Z By ~ 13.23 YL EZ: ;Y'E'Z;. By 13.2 1 and Axiom 19, since s3, 2 ro we - ( E : + 3/9) - E , > l / e . Now have L ( Z M ) / L ( Z>)

Zr -

J. L. Britton, The existence of infinite Burnside Groups

323

Z - ( Z F Z F Z p C Z ~ Z F Z F D ) and ZF = Z F hence (Z? Zp CZi)e n Now

Hence ( YA)e n

~ I ,l by

17.18.

~ I.l

17.20. Proposition. Axiom 8a holds f o r n + 1 ; i.e., if Xe f I in Gn+l and (X) is ( ( n + 1))-bounded b y z 2 then (X') is ( ( n + I))-bounded b y z + ag. Proof. If not we may assume that ( X ' ) 3 S where S is a subelement of L',+~ of size 2 z + a s = y say. Now S = Y k Y L where Y is a C.A. of X is a C.A. of X and k is a non-negative integer. Since Y LC Y C ( X ) we have k 2 1. Clearly Ye I in G n + l . If k = 1 , S = Y L Y Y L .Since s ( S ) 2 y and s ( Y ) < z we have s ( Y L ) > y - z - 2 / e = a s - 2 / e > ~ 7 2 . B y 17.19(YLYR)enT1', a contradiction ( z < k47). If k = 2, S = Y Y Y L .One of Y , Y , Y L has size 2 1 ( y - 4/e) > r,, hences(Y)> 5(y-4/e)-e3 (5.19). Now :(; -4/e)-e3 > 1172. Now apply 17.19 to the subelement Y Y of we obtain Ye n=+l I. If k = 3 we proceed similarly using ( y - 6/e) > Max(r,, e3 + u72>. I f k > 3 t h e n k = 2 s + O whereO=Oor 1 . 2 s + 1 > k>-3hence s 2 2. One of Y s , Y s , Ye YL has size Z f (v - 4/e) hence s(Ys) > ~ 7 2 as above. Thus (i) Yse I. One of Y S p 1Y, has size > i ( i ( 4 - 4/e) - e3 - 2/e) = a (say) > r,, sos(Ys-') > a - e3 hence (ii) Y(s-l)e I. By (i) and (ii) Y e nyl I , contradiction:

+

nrl

nrl

3 24

J. L. Britton, The existence of infinite Burnside Groups

§ 18 Summary In this section we prove Axiom 8.16.

18.1. Notation. We shall write u A instead of Im'A (cf. 6.16). We shall also use the notation PA as follows. Suppose B is a subword of X and also a subword of another word Y ;then any subword A of X such that A C B is also a subword, which will be denoted b y P A , of Y . Of course A = PA.Thus p maps certain subwords of X to subwords of Y . 18.2. Proposition. Let A X C 7 B X D , where each side is (j)-bounded by k,,, j < n and A or B is Iand C o r D is I. Let M be maximal i-normal in A X C and properly contained in X . Let s ( M ) > ~ 7 3 As. sume vM exists and let uM, p M be different subwords of BXD. Then there exists k > 1 such that for s = 1, ..., k ( i )(up)'M exists and has size > u74;(vp),M meets C, or (ii) (vp),M' exists and has size > 2'74; (vp),M' meets D, where M' is pM. Proof. 1 O . If J is Max j-normal in A X C or B X D and s(J) > ul0 then by 6.23 since k,, < kk, kb, uJ exists and either s ( u J ) > s ( J ) - u3 o r s ( v J ) > 1 -s(./)-u2> 1- ~ ~ - k ~ ~ = a s a y . 2". pM is maximal normal in B X D . Both uM, upM exist since 1 i 7 3 > i d l o . Now upM is different from M otherwise VM = uupM = p M by 6.19. Now s ( v M ) > Min(u,, - u 3 , a) > ul0 so if upM is left of M then by 6.20 uupM is left of uM, i.e., p M is left of v p p M , i.e., M' is left of vpM'. Thus without loss of generality M is left of u p M ; we shall prove (i). We use the notation K , = p M , M , = u K , , p M i = Ki+,, M i = vKi. upM = M , has size Z Min(a, u73- u 3 ) > ~ 7 +4 u3 > I t is sufficient t o prove for s = 1 , 2 , 3 , ... . (*) M , exists * s(M,) > u7, and M,-, is left of M , (where M , means M ) . For if k is the largest integer s such that M , exists then pMk cannot exist, otherwise M k + , = upMk would exist because u74> ul0; hence M , d X , i.e., M , meets C. We may assume M 2 exists otherwise (*) holds. Also

J.L. Britton, The existence of infinite Burnside Groups

325

Now consider the following s(lCZ2)> Min(a, s ( M l ) - u 3 ) > statement P(t), t 2 2 : M , exists and has size > 2174, ( M , M 2 ) P ( 2 ) is true since fi (M,-2, M , ) and ( M , M , ) + i 1 u74 > U24 and k48 < k i 2 , k13 (cf. 8.8). Assume P(2), ..., P(t ) true. Then we may assume M,+, exists; otherwise ( * ) holds. Hence M i is left of Mi+l ( i = 0, 1 , ..., t ) . By 8.15 since k4, < k,,, lc17;

7

2174

> u29;

and similarly ( M t - l , M , ) fi ( M , M , ) . Apply 9.1 1 taking S, U, T,S ' , T' to be M,M,,M,,M,-,, M,,,; this is possible since S ( M , + ~ ) > Min(a, u74 - u 3 ) > ~ 3 0 Min(a, , u73- u 3 ) > u j , and li,, < k18, k19. We conclude that U' exists where ( M , - ] , U ' ) fi ( M , M , ) and s ( U ' ) > Min(a - u g , u73- 2u3) - u4 - e3 = > uj,. Thus 0 have U' ( M t - , , U ' ) (M,-,, M,). By 9.12 since u74 > uj, > ~ 3 we is M,. Now s(M,+,) > Min(a, s(M,) - u 3 ) 2 Min(a,P - u 3 ) > Thus P(t + 1 ) is true and (*) is proved.

18.2A. Corollary. The proposition remains true if placed by u73+ ~ 9 0 , + ~ 9 respectively. 0

1174

are re-

Proof. Examine the P.C.'s in the proof of 18.2 and use - u90 > 2u3 + u4 + e3 + u74 and u74 = u73- ~ 9 0 .

QI

18.3. Proposition. Axiom 8.16 holds for n + 1, i.e., if 0, O'are ( n + 1)-preobjects and 0 is an adjustment of 0' and also 0 n ~ 0' l then 0 = 0'. Proof. 0 is (S, T ) where S, T are subelements of Rep(Ln+,) of size

> u21, hence ( n + 1)-normal, and S, T determine distinct maximal ( n + 1)-normal subwords S, 7 and where s=SH then s ( H ) < u22, Similarly for T. 0 is @)-bounded by k,, and ( n + 1)-bounded by '13.

Similarly 0' = ( S ' , T ' ) . Now S F AS,, S' = BS, where A or B is I . Similarly T = ToC, T' f ToD , C or D being I .

.1.L. Britton, The existence of infinite Burnside Groups

3 26

1". Let ( S i , T o )f S O X ' ;then 0 = A S o X ' B , 0' = C S o X ' D . Let H, K be the kernels of S o , To. Then ( S o ,To)= HA K for some A . D exists such that D is (n)-bounded by bmin and H A K ; D.Now s ( H ) > s(So) - ro - e3 - 2/e > u21 - ro - e3 - 2/e > u l , so H > Ho; ;Ho' and s ( H o )> s ( H ) - u l . H o has the form HL and Ho' the form DL,so H L ; ;D L . Similarly KO = K R ; D R . Also D FZ DLD M D R for some D M . Since s(HL)> u21- ro - e3 - 2/e 11 = 8 say and since HL is a subelement of Rep L , + we may apply Axiom 3 1B (8 > ~ 3 1 bmin, ; str' < b31). We obtain a word H L L E D L R ,(n)-bounded by Max(bmin,str+)+ q 3 1 = cp say. Now cp < k12 and s(HLL)> s(HL)- ~ 3 > 1 8 -~ 3 > 1 ro + e 3 . Similarly there is a word D R LFKRR where s(KRR)> 8 - r j l . Moreover ~

,

HA K

;HLLEDLRD MDRLF K R R = J , say,

where J is (n)-bounded by cp. Let S: = H L L ,S* = AS:, TO*= K R R , T* F TZC. Then (S,T ) = A ( S , , To)C ; AJC = (S*, T *) and (S*, T * ) is (n)-bounded by cp and ( n + 1)-bounded b y k,, + u4. Next

s(AS,)

< s ( A )+ s(So) + 2/e

where $ = ( R , + 2e3) + 2/e + r31+ u1 + y o + e3 + 2/e. Thus s(S*) > 1~~~ - $ = 2421 . By 6.14A, since k12 < b2,, u21 - e3 2 U 6 + u4 + € 3 and 1421 - J/ e3 > 146 we have Im s" = S. Hence ~

< s(S*) + u;2 + 2/e where 1fz2= $ + 1422 + 114 + e3. Similarly BJD = ( S t * ,T'*) where if X temporarily denotes St* or T'* then s ( X ) > u& and s ( X ) < s ( X ) + u22 + 2/e. Write 0" (S", T*),

J.L. Britton, The existence of infinite Burnside Groups

327

0'" = (Sf*, TI*). Although these are not preobjects in general, (S*, T * ) n ~ (S'*, l TI*),S'* = BS:, TI* = S:D. 2". We claim that it is sufficient to prove O* = Or*. Let 0" 2 O'*. Then A J C - BJD. Among the left subwords of J that are subelements of RepLn+l let U have maximal length. Then HLLC U . If U = J we would have = T* so Im s*= Im T*, i.e., T. Hence U J . Say J = UV. Thus A U V C - BUVD. A U is the

s=

+

largest left subword of A UVC which is a subelement of Rep Ln + l . H e n c e A U = B U . H e n c e C - D a n d O = 0'. 3". Case 1. 0 " ; 0'" Write kT2 for rp, kT3 for k13+ u4. Take j minimal such that O* = 0'". If j = 0 there is nothing to do so let j > 0. There are divisions jl,P,, ... and Q1, Q2, ... where some Pi has type j . We have O* = AJC, O'* = BJD, (S:, T t ) = J = S t X say. We first show (j-)S* contains a maximal j-normal subword of size > u75; by translation it is sufficient to show AS: or BS: does. Since k! < s(Pi) + s(QJ)< s(P,')+ kr2 < s(c.) + e3 + kr2 we have s(pi)> k! - e3 - kr2 > u75;thus we may assume Q AS:, hence Pi C SgXC and similarly Qi C SZXD. Now C or D is I ; say C is I. Then C S t X so the subword p p i of Of* exists. We prove p p j is not Qi. If p c . = Qi then by 6.22 since kT2 < k,, k7 we have QI c ppi =_ = EPfF where s ( E ) ,s ( F ) < u l . Hence s(QJ n E ) < Max(ro, u1 + e 3 )and

c.

s(Qi n Pi') 2 s(Qi) - 4/e - 2Max(ro, u1 + e 3 )

> k!

-

kT2 - 4/e - 2Max(ro, u1 + e 3 ) > ~ 4 0 .

This contradicts Axiom 40 since kr2 < b40 (and By the dual of 18.2A (with M = pi),since

s(Pi) > k:

-

e3 - kT2 > u73+ ~

9 and 0

X;:: QJX,'E Li).

kr2 < k,,,

a maximal i-normal subword of size > u74+ ~ 9 =0 u73 meets A or B and hence is contained in AS: or BS;. (Since S* or S'* is S: we have s(S:) > u& > rb.) Similarly if D is I. (t)follows since u73= u ? ~ .

J.L. Britton, The existence of infinite Burnside Groups

328

4". Without loss of generality assume B = 1. If A = I then C =1 D , i.e., C = I or I = D ; since kr, < b , we have by Axiom 10 that C = D = and O 4 = 0'" as required. Thus we may suppose that A $ I . For the next part of the argument we shall assume only that

f

(1)

AS:XC

7 S:XD

(instead of =). Let N , be the nice subword of the form (AS:)L I having maximal length. Since s(AS:) > u;, we have as in 6.8 s ( N , ) > u& - u2 - 2/e = y say. Then N, N; C S:XD. Let N, be the nice subword of S Z X D of the form StLhaving maximal length. Then s(N2)> y > s33 and N , N i c ASgXC. Let N , n N i . Then N N ' = N i n N,. Either N is N , or N' is N 2 x b y A x i o m 33 for 12 + 1 N = N ' . Hence s(N) > y > yo. Each of AS:, Sg has size > u& - e3 > u6 + u4 + e3 and k f , < b,, hence (cf. 6.14A) In1 = Hence

-

-

-

Ag ST.

s(sg) > s(As;) - u4

- €3

s(z).

also s(S:) + u;, + 2/e > Thus s ( A ) < u;, + 2/e + (yo + 2 e 3 ) + + u4 + c3 = u , say. Next A C N otherwise s ( N ) < s ( A ) + e3 < a + e3 + y , a contradiction. Thus N 3 AS:' say and s(SR') > y - a - 2/e = b , say. Now N ' is a left subword of S l , N ' = So*" say and has size > y. C, W exist such that AS,*W = Ce E Rep L n + l .By Axiom 19 and 13.21, L(S:')/L(Ce) > b - e3 - ( e r + 3/e) = b' say and b' > 5(a + + 3/e) + 3/e > 5 ( L ( A ) / L ( C e )+) 3/e so L(S,*)/L(C) > 5 L ( A ) / L ( C )+ 3 . But L(S:')/L(C) > b'e > 1 and since AS,*' S t L w e see that A =_ CKfor some k > 0. By (t)A S ; contains a maximal j-normal subword F', which may be assumed proper, of size > u75 = u73.Let F be the leftmost translate in AS: of F'. Where AS: = A'FB we have L ( A ' )< L ( C ) . For any integer N 2 1, F C Ce C CeN and we can consider the translates F, F 1 ,F 2 , ... of F in C e N ;here e.g., F 2 means F + + . NOW L((F, F 6 k ) )< 6k.L(C) + 2L(C) SO

ET

J.L. Britton, The existence of infinite Burnside Groups

329

L(A'(F, F 6 k ) )< ( 6 k + 3 ) L ( C ) < k L ( C ) + L(S,*') = L(ASz'). Thus A'(F, F 6 k )is a left subword of AS:'. With respect to ( 1 ) let uF = T, , p T , = F,, ..., uFi = T i ,pT, = Fi+,, ... . F , means F. Since u73> u and kr2 < kb, k $ , T , exists and T = F so s ( T , ) > u73.Since (F, Fdo ) has length k L ( C ) = L ( A ) ,F k C So, I'-so

vpFk exists; in fact it coincides with F. Thus F2 = pvF exists and coincides with F k . Similarly Fi+, exists and coincides with Fik (i = 1,2, ..., 6). Thus T, , T,, ..., T7 exist. By 8.15, which is available since kr, < k16, k17 and u73> ~ 2 9 (, T , , T 3 )2 ( F , , F 3 )= (F, F 2 k ) ; (since AS:' = StLwe have even ( F , , F 3 ) = ( T , , T 3 ) ) .Let p be maximal such that Tp exists; then p > 7. By 18.2 Tp meets D , hence (In O'*, T p )3 S:X = (S:, T:), and T,, ..., Tp have size > Using that p Ti = Fi + we have F , , F 2 ,..., Fp have size > u ? ~ . We prove by induction on s that if 1 < i < j < s

,

This is true for s = 3. Assume true for some s, 3 < s < p . Then TS+,exists and by 8.15, since u74> u29,if 2 < u < u < s + 1

In particular this holds when u = s - 1, v = s + 1 and when u = s - 1, u = s. By induction hypothesis ( T , , T,) fi (F, F ( s - l ) k ) By . 9.14, since kr2 < k,,, k,, and u74 > ~ 3 1 (, T , , Ts+,)fi (F, F"). Thus the result is true for s + 1. We now have

S t X D = A ' T , R for some R . Since L ( A ' ) / L ( C e )< l / e we have either s ( A ' ) < yo or by Axiom 19 and 13.21 s ( A ' ) < e3 + l / e + + E; + 3/e = {' say; hence s ( A ' ) < Max(ro, {') = { say. Tp meets D, so where S, = S: n ( T , , T p )we have s(S1) 2 u;, - 2/e - { > Max(ro + e3, ~ 9 5 ) .

330

J.L. Brittor;, The existence of infinite Burnside Groups

5". The word ( T , , T p )contains T,* which has size > u;,. The maximal normal subwords of ( T I ,T p )determined by S , , TO* are distinct and of size > 2195; we shall show later that, in view of (2), this is a contradictory situation. 6". Case 2. Not Case I . Here (S*, 7'") n ~ (S'*, l T'*). Using the same notation for the divisions, some Pi has type n + 1 . then by 6.24, since We claim that + s*.For if pi = k*,2 . < . k'8,. k *1 3 < k $ ;u;, - e3 > ul0, Qi = S!?. Let g = Max(ro, u1 + e3).

..

We have u;, - 4/e - 2(.$ + e2 + 2/e) > ~ 4 0 Since . s(Sg) > g, we can write S$ AB' where A is of minimal length such that s ( A ) > .$. Then where A = A a ( L ( a )= 1 ), s ( A ) < 5 + e2 + 2/e so s(B') >, s(Sz) - 2/e - (g + e2 + 2/e) > [. Thus S z = ABC where s(C) > g, s(B ) > s40.Now S: C Pi. = EP,! F where E, F have size < u l . Then A C E and C C F , so B C Pf. Similarly B C QI. B is (n)bounded by kT2 < b40; this is a contradiction by Axiom 40. Similarly # Tg. Hence C (S;, T:) and ppi exists. Now pPi # Qi otherwise we could obtain a contradiction by Axiom 40. Since S: < Pi < TG we have pS: < pPi < pTZ so pP # S'*,T S . Dually for Qi; in particular pQi exists. If ppi < Qi then < pQi; thus without loss of generality we may suppose Q j < ppi. Now 3'" < Qi; thus s'*and ppi are disjoint. Now < pQi < Pi so S*,pi are disjoint. Now kT2 < kb, kr3 < kb, k: - kT3 - e3 > ul0 and zi;, - e3 > tilo; thus we have S X < U < Qi where U is Im' pQi; it follows that s'*and Qi. are disjoint. Also s ( U ) > k: - k*13-'3-'4. Take the leftmost pi of type n + 1 . Then P,-P2- ... .Pi-, has the form S*J for some J . S*J meets o r touches pi, so S*'J meets o r touches p c . . Q i . ... . Qi-, is disjoint from ppi and contains S'*, hence it has the form S*'JL.Let U , be the part of U outside Qi. Then S*'JL contains S'* and U , and

,

c.

<

s*

s( U , ) >, s( U ) - 2 / e - (Yo + E 3 ) 2 k!

Now S*J

-

kT3 - e3 - 1l4 - 2/e - (yo

7S*'JL,i.e.,

+ e 3 )= q

say.

J.L. Britton, The existence of infinite Burnside Groups

(3)

331

A S , * J ~ J ~B S ; J ~

where J - J L J R and j is the maximum type of P,, ..., Pi-l; thus

j < n.

First suppose j = 0 or A = B. Since S*, S'* are maximum subelements of Rep L n + l ,j = 0 implies A f B. Hence A 3 B , so C n ~ Dl. Since C or D is Z we may use Axiom 10 t o conclude that C = D and 0 = 0'. Now let j > 0 and A f B . We show that S$ contains maximal j-normal subwords of size > u 7 3 .Consider a Q iof type j . If QiC BSZ then, by considering translates, S$ contains a maximal normal subword as required. If $ BSZ then QiC S $ J L and pQi exists and is different from (by Axiom 40) so 18.2A is ; if pi < p Q i available ( k & < k 4 8 ;k: - kr2 - e3 > u73+ ~ 9 0 ) hence then a maximal normal subword of size > meets A , but if p o i < then a maximal normal subword of size > u73 meets B . Now (3) has the form

oi c.

(3')

ASZJLJR7 S$JL

or ( 3" )

BS; J~

S; J~ J~

according as B or A is Z;(3') and (3") are of the same form as ( 1 ). Hence the argument of 4" is available; in particular we have " T p meets D" which shows that (3') is impossible, hence (3") holds. Hence Tp meets J R , so (In S t , T p ) 3S Z J L 3 ( S $ , U , ) . Hence ( T I ,T p )contains S , and U,. We have seen that s(S1) 2 u g 5 .Now note that s ( U , ) > r ) 2 u g 5 ,and S , , U , are distinct maximal normal subwords of ( T I, T p ) . 7 " . Thus in both Case 1, Case 2 we have (2), where p 2 7 and the right side is a subpowerelement of Rep L , + so is (j)-bounded by str+ < k S 9 .The left side is (j)-bounded by kF2 < k S 9 .Moreover ( T , , T p )contains two distinct maximal normal subwords, each of size > ug5 2 ug3.By 14, since u73 > u74 > ~ 3 0 the , right side of (2) contains t w o distinct maximal ( n + 1)-normal subwords. This is a contradiction.

J. I,. Britton, The existence of infinite Bitrnside Groups

332

§ 19

19.1. Note. All axioms have now been proved true for n + 1 except Axioms 5 and 29 which we shall discuss in this section. All previous results proved for n and not based on Axioms 5 or 29 are true for n + 1. 19.2. We shall consider a sequence

Y , , Y,,

..., Y,, k

Thus for i = 1, ..., k

-

2 2, Yi u n + ,Yi+,(i = I , ..., k - I).

1 we have Y i , Y j + lE J n + l and either

or some h f ( Y i )nyl some e q r ( Y i + , ) .

nzl

19.2A. We refer to 19.2 as Situation A. A special case of Situation A is when Y , Y,: call this Situation A+. Sitzdution B is as follows: Y , , Y,, ..., Y,E J n + l are given and for S. t. i = 1 ,..., k - l ( Y i l ) n T l ( Y ; + ; ) , s i > l , t i > 1. +

19.2B. We can find ei = f 1 ( i = 1 , ..., k ) such that X i = YFi and for i = 1 , ..., lc - 1 we have in Situation A 1. (x~?) = (x~!:,), 1 < si < t i , or 2. someahf(Xi)- some eqr(Xi+,) and moreover h f ( X i ) is a subword of X T for some positive m (of length < f e L ( X i ) ) and eq,(Xi+,)is a subword of X z l for some positive rn (of length > q'eL(Xi+,)).Similarly in Situation B S. t. 3 . ( X iI ) % ( X i ; ), si 2 1 , ti 2 1.

,

19.3. Choose an integer N > 0 to be kept fixed. We show that if, for some i, si and ti exist then they may be taken Z N and that if ti and si+lboth exist they may be taken to be equal; we shall therefore write s i + l instead of ti in 1 and 3. We need only remark that ( A ) s ( B ) implies (A') z (B') for any integer r # 0, by Axiom 1 1.

J.L. Britton, The existence of infinite Burnside Groups

333

Denote the left side of 1 , 2 , 3 by A i and the right side by B i + l By 5.1 1, A i and Bi+l are ( ( n + 1))-bounded by str+.

19.4. Note. If 1 or 3 holds, thenAi = B i + l , a maximal ( n + 1)normal subword S of one of A i , B i + l ,where s(S) > u & , determines by 16.3A a maximal normal Im* S = S’ in the other such that either Is(S) -s(S’)I < ui(:or Is@) + s(S’) - 11 < u;. Also u2 < u z . u3 < ui(: and uro Z ul0. Moreover a cyclic word Di exist such that A i n+l Di n+! B i + l , Im’(1m‘S) is S’ (where Im‘ is in the sense of 6.21) and D iis ( ( n + 1))bounded by Max(str+,k i 4 ) + ~ 4 < 0 k,,, k23, k16, k17. 19.5. Proposition. In Situation A or B , let B k contain a maximal ( n + 1)-normal subword S o f size > u67. Then A , contains a maximal ( n + 1)-normal subword of size > u68. Proof. Since us7 Z uro A k - 1 contains a maximal ( n + 1)-normal subword of size > u68 2 uro for either s(1m‘S) 2 1 - str+- u; > 2d67 > u68 or s ( h ‘ S )> u67 -Ui(: > u68. Assume Ai contains a maximal ( n + 1)-normal J of size > u68 for some j , 2 < j < k - 1. Then we show that Ai-l does. We have the following cases

where we have identified one of Bi, Ai with a subword of the other. Since J is a subword of a power of Xi we have L ( J ) < 2L(Xi) so in Case 111 some translate of J lies in Bi. In the other case J C Ai C Bi. Thus Bi contains a maximal ( n + 1)-normal subword

334

J.L. Britton, The existence of infinite Burnside Groups

of size > 2168 hence, by translation, at least si 2 N if Bi is cyclic and > -3 + q’e if Bi is linear. If Bi is linear call the two outermost of these maximal normals extrernal, we shall eventually identify this concept with the “extremal” in 10.3. Consider the sequence

For any linear B , in (1 ) except B , we define its extremal subwords similarly. For B , take the outermost maximal normals of size > 2467 as its extremal subwords. For any linear A i in (1) (then B i + l is linear) define its extremal subwords as the images (cf. 6.16) with respect to A i B i + l of the extremal subwords of B i + l ;they have size > - u; = u69. Each term of ( 1 ) except Ai-l contains a maximal normal subword of size > u68. A j - l contains a maximal normal subword of size > u69. Hence each cyclic term contains at least si 2 N > 3 maximal normals of size > u69. In Case IV we have for some A, B , C

-

e q , ( X j ) ABC,

Sf(Xj) B

L ( A ) 2 3L(Xi),

L ( B ) 2 3L(Xi)

since L ( e q , ( X j ) ) L(Gf(Xi)) > L(X,) (q‘e - f e ) > 8 L ( X i ) since 4’ > j + 8 / e . Thus without loss of generality we may suppose that the subword of Bi with which Ai was identified is of the form of B above. Hence B is central in e q , ( X j ) ,i.e., B lies between the extremals of eq,(Xi). For a cyclic word all subwords are central. Call a subword of one o!‘ ( 1 ) C.N. if it is central and maximal normal. Say a C.N. word is initial special if its size is > u69. Certain C.N. words are called special as follows: 1. Any initial special word is special. 2. If X is special in one of A i , B i + l and its image Y(lm‘ or Im*) in the other exists (hence Y is maximal normal and central) then Y is special. 3 . If X is special in one of B i , A i and is central in the other then it is special in the other.

J.L. Britton, The existence of infinite Burnside Groups

335

4. If X is special and X ' is a translate of X and X ' is central then X ' is special. We claim that any special word U has size > u33.There is a sequence C , , C2,... , c k where ck = U, C, is initial special and C,+ arises from C, as in 2 , 3 or 4 ( u = 1, ..., k - I ) . Thus C, is a subword of one of the words ( I ) say W,; thus W , is A f c u or , Rfc,,, say (u = 1 , ..., k ) . If C,+, is Im* C, then W , = W U + ]; let D, be chosen as in 19.4 such that W , - D,- W , + , . The assumptions of Section 10 apply t o the sequence obtained from W , , W 2 , ..., w k by inserting D, between W , , W,,+l whenever D , exists (str+< k 2 , , k23; 4 9 > ~ 3 2 ; Min(u69 - u 3 , 1 - S t Y + - u 2 ) 2 u32). If C,+, arises from C, as in 2, then if C,+, = Im' C, we may regard C, +. Cu+las a transformation 2a of 10.2; if Cu+l= Im* C, we may regard C, C,+, as a pair of transformations of type 2b. If C,+, arises as in 3 we may regard C, C U + ,as a transformation 3a, 3b, 3c or the trivial case of 2b (when ;is -). Now let CU+, arise from C, as in 4. We consider two examples. (i) C, is C.N. in Ai = hf (Xi) and Cu+l = Ci is central and Ai = PC,QC,+,R. Then W , = Ai,W u + l =_ Ai.We may regard C, +. C,+, as a transformation of the form 3a of Section 10 where for J , we take the word constructed from two copies of Ai by identifying the C, of the first copy with the C,+, of the second copy; thus J , is a subword of a power of Xi. (ii) C, C Ai = XisJ. Here we may regard C, -+ C,+, as a transformation of the form 2b; the X u of 2b is the linearization of Ai starting with C,, the Y,+l of 2b is the linearization of A i starting with C,+] (and = Y,+,). By 10.4, since s(C,) > u69 > ~ 3 2 we , see that c k =_ U has size > u 3 3 ,as required. Consider any term W of ( 1 ) and let its special subwords be S , , ..., S, (then Y > 2). If W is cyclic the translate Si is Sp+lsay and Sf = Sp+i for all i (if p + i > Y let S p + i mean Sq where y = p + i (mod Y)). If W is linear and Si is central then S; = S , + p say and +

+

x,

336

J. I,. Britton, The existence of infinite Burnside Groups

If however S; is not central (then W is of the form A , ) choose any p 2 Y and then (2) holds vacuously. Call p the period of W . 2". We prove that without loss of generality the periods of B i , Ai are the same. Case I is trivial. In the other three cases let the special subwords of W , where W is that one of B,, A i which is a subword of the other, be S , , ..., S,. These are central, hence special in the other word W' by the remark about Case IV so the special subwords of W' are ...,S , , S,, ..., S,, ... . If S; C ( S , , S,) then S; = S l + pand both periods are p . In the opposite case the period of W is arbitrary 2 a while the period p of W' is determined but 2 a. Hence the period of W can be taken to be p . 3". Let ni denote the period of Bi and A i . For the remainder of 3". We consider Situation A only. We prove

ni >

(i = j - 1 , ...) k - 1).

If 1 of 19.2B holds then the number of special words in each side is the same; for any special word S has size > u332 uro so Im* S exists, hence Im* S is special. Hence nisi = ni+lsi+l.But si < si+l hence ni > ni+,.Now let 2 hold. Let m be the number of special subwords in S,(Xi) and eqt(Xi+,). Then by 2.16A I m/ni - Zi/L(Xi)I < 3 , I r n / ~ ~ Zi+l/L(Xi+l)l + ~ < 3 where Zi is the length of the subword generated by the two extremal subwords of A , ; similarly for Zi+l. Thus

Hence m/ni < 3 + f e < q'e - 5 < m/ni+l so that ni > 7ri+,. Note: Since nk 2 1 we have ni 2 k + 1 - j . Also note that the number of special subwords in any Bi is > ni(q'e - 5 ) since n,N > n,(q'e - 5 ) (take N > q'e - 5 ) . 4". Next we show that, in Situation A or B, there exist three special subwords A , R, C of B , each of size > u67 and special subwords A ' , R', C' of Ai-,such that

J.L. Britton, The existence of infinite Burnside Groups

337

If follows that Ai-l contains a maximal normal subword of size

> U68 as required; to see this use 9.1 1 (str+ < k18, k19; 2467 2= u~~ 2 ~ 3 0 2467 ; 2 u j 0 ) and

where (A,B ) fi (A',B " ) and

S(B") > Min(1 - S t r + - U 2

we have that B"

-

U4 - € 3 ,

Us7 - U 3

C

(A',B', C') exists

-

Uq - € 3 )

> U j o , U68.

Hence (A',B ' ) 2 (A',B " ) . By 9.12 ( u 2~u j ~0 )B" isB'. In Situation B, if we take r consecutive special subwords in B i + l there are r corresponding special subwords in A i and, since I applies, for any r consecutive specials in A i there are r corresponding specials in Bi. Thus if we take r consecutive specials in B , including three A , B, C of size > us7 there will be r corresponding specials in Ai-, of which A', B', C' say correspond to A , B, C. Note that, in any case, if S, Tare distinct specials in some term of (1) then (S,T ) is an ( n + 1)-object, and moreover if S ' , T' are specials in some term of (1 ) and S' = Im' S, T' = Im' T then (S, T ) 2 (S', T ' ) ;this is by 8.1 5 since s t r + < k16, k17 and u~~ 2 ~ 2 9 . This remains true if Im' is replaced by Im* because, as we have seen, any D j has ( ( n + 1))-bound < Min(k16, k 1 7 ) . Now consider Situation A. We prove that to any rn consecutive specials of B , where rn < rk (q'e - 6), say s d + l , ..., s d + , , there exist rn consecutive specials u d + l , ..., U,+, of Ai-, such that ( s d + j , s d + j ) fi ( u d + j , u d + j ) for all i, j such that 1 < i < j < m. Any r consecutive specials in B j + l determine by 2 above r consecutive specials inAi. Take r consecutive specials a h + l ,..., in A j . In Cases I, 11, IV there are determined r consecutive specials of B i . In Case I11 denote the specials of A i by a , , ..., at without loss of generality and denote those of B j by a x ,..., a y where I < x < y < t and 1 < h + 1 < h + r < t. One of a x ,a , , , , ..., C Y , + ~ - ~ ( p = r i )is a translate of a h + l say a c + l .For a x ,..., a y to contain a c + l ,..., ac+ we require c + r < y which is so if y + 1 - x 2 p + r - 1. Now y + 1 - x > ri(q'e - 5 ) 2 ti+ r > p + r - 1 if r < nj(q'e- 6). Thus if we take r consecutive specials of B , where r < "k (q'e - 6) then since r k < rj (i = j - 1, ..., k ) there are r corresponding specials in Ai-l as required. Let T,, T2, ..., Tb be the specials in B,. Then b > rk(q'e - 5 ) > 3nk

338

J.L. Britton, The existence of infinite Burnside Groups

and at least three of T , , T,, ..., T3*, have size > 2167. Since 3 r k < 7rk(qfe- 4 ) , T , , T,, ..., T3nkhave images U,, U,, ..., U3*, in Aj-1.

19.6. Corollary. In Situation A , if B, contains properly a maximal ( n + 1 )-normal subword o f size > u67 then k is at most equal to the number of maximal ( n + l)-normal subwords o f size > u33 in XE. Proof. Let there be r special subwords in A 1 . Without loss of generality the period x, is at most r. But xl > k , hence k < r. Any special subword is maximal normal of size > u33.If A , is a f ( X 1 ) then A , C X p . If A , is Xi' then A , contains X;' and the number of specials, hence maximal normal subwords, in X;' is > x,,hence the number of maximal normals in XE is > 7r1 2 k.

19.6A. For any word M let p i ( M ) = L ( M ) / L ( X ; ' ) ( i = 1, ..., k ) 19.6B. In Situation A there are two cases: Case (a) j exists such that 2 of 19.2B holds (with i replaced by j ) ; Case (b) not Case (a). In Case (a) let i denote the largest such j . Then A i B i + , and B j + l has the form E ~ , ( X ~ It+ may ~ ) . happen that L ( B i + l )> 4 e L ( X i + l ) , i.e., B,+l also has the form E ~ ( X ~ if+ so ~ )define ; q" to be 4 ; if not define y r f to be 4'. Thus p i + l (B,+l) > 4".

-

19.7. Proposition. In Situation A, let B , contain a maximal ( n + 1)-normal subword ojsize > uS7. Then in Case ( a ) there are subwords A o.f A , , B c1.f B, such that p1( A ) < f + 61e, p k ( B ) > 4" - 16/e and (t)there is a word A LCBDAR,( ( n + 1))-bounded by Ic,, and equal to A in G , + , . In Case ( b ) we have ( b 1 ) if a subword B o j B k is given and 4 / e < p k ( B ) < p k (B,) - 12/e then there is u subword A o f A , such that p l ( A )< p k ( B ) + + 14/e and again (t)holds. Also Nle < p k ( B k ) . ( b 2 ) if a subword A o f A , is given and p , ( A ) > 9/e then there is a subword B o j B, such that ( B )> p1( A ) - 12/e and again (-1) ho Ids.

J.L. Britton, The existence of infinite Burnside Groups

339

Proof. ( b l ) Let b be the number of specials of B , in B . Then I b/7rk - L(B)/L(X,)I< 3. Now 4L(Xk) < L ( B ) < L(B,) - 1 2L(X,) so b > 7rk (L(B)/L(X,) - 3 ) 2 7rk 2 1 , hence b > 2. Now B , 1 XB Y where L(X) = L ( Y ) = 6L(Xk). The number of specials in X is at least four since 7rk(6 - 3 ) > 3. Thus there are specials V l , ..., V , of B , such that those in B are V s , ..., Vm-,; b = m - 8. By 4" above there are specials U , , ..., U, of A such that A = ( U , , U, ) -h ( V , , V , ). B y 9 . 1 5 , ~ i n c e u ~u39 ~ >a n d s t r + < k 2 4 , A n 5 1U f E ( V F , Vk-,)FU,", where the right side is ( ( n + 1))-bounded by Max(str+, k i 4 ) + ~ 4 -0 = k43. We have ( V F , V k - l ) > (V,, V m - 3 > ) B . Now I L ( A ) / L ( X , ) - m/7rl 1 < 3 so

,

p l ( A ) < m/e7r, + 3 / e < ( b + 8)/enk+ 3 / e

< b/e7rk + 1 l / e < p k ( B ) + 14/e

p ( A ) < m/e7rl + 3 / e < ( b + 8)/e7rk+ 3 / e

< b/en, + 1 l / e < p ( B ) + 14/e (b2). Let 8 = p, ( A ) . Let the specials of A , in A be say U , , ..., U,. Then m/7r, > L ( A ) / L ( X , )- 3 = Oe - 3 > 6 , so m > 7 . These m specials of A determine m specials V , , ..., V , of B,. Let A ' = (Ul, U,) B = (V,, V,-3). Then

,

L(B)/L(X,)

> (m-6)/xk-3

2 m/7rk-9

> m/nl-9 > Be- 12

(a) In the proof of 19.5 we saw that Y consecutive specials of A i determine Y of Bi, hence Y of A i - , , if Y < ni(q'e - 6). Since 7ri < ni-, < ... < 7rl it follows that if Y < 7ri(q'e - 6 ) then Y consecutive specials of A i determine Y of A

,.

340

J.L. Britton, The existence of infinite Burnside Groups

In the present case the total number t of specials in Ai = af(Xi) satisfies t/7rj < L ( h f ( X i ) ) / L ( X j+ ) 3 < ef + 3 < q'e - 6 so they determine specials U , , ..., Ut of A 1 . The number of specials in Bi+l is also t , and t/ri+I> - 3 + (L(Eq"(Xi+l))-2L(Xi+1))/L(Xj+l)> q''e-5-

If i < k - 1 , Bi+, c X , " s ' z ... z X i k , so the specials of Bi+l determine t specials of B, say Vl , ..., V , . This is also true if i = k - 1. Finally L( V 4 ,Vt-3)/L(X,) > ( t - 6)/7rk - 3 2 t/7ri+l - 9 > q"e - 14

L(U,, Ut)/L(X,)

< t/7r1 + 3 < t/ni + 3 < ef + 6

19.8. Consider Situation A. For i = 1 , ..., k - 1 let r(i) be the least integer q such that A i Bi+lor A i ; R i + l . Then 0 < r ( i ) < y2 + 1. If r ( i ) > 0 then A j and%j+l contain maximal r(i)-normals of size > 1 -str+ - u ; > q0. Call k - 1 critical; and if 1 < j < k - 2 call j critical if r ( j ) > Max(r(j + l ), ..., r ( k - 1)). Let the critical integers be h, , A, ..., h, in ascending order, where h, is k - 1. Put A, = 0. Let 0 < i < j < t , put a = hi, b = Xi, and consider

calling this sequence Eii. We shall consider two cases: ( l + ) A u r 7 u ) B v + l( u = a + l , ...,b ) i.e., 1 holds throughout Eij, and moreover we do not have that 1 holds throughout ti-lior that 1 holds throughout tii + l . (2') j = i + 1 and 1 does not hold throughout Eii. First consider (1 +): If i > 0 then a is critical; i < j < t so

J.L. Britton, The existence of infinite Burnside Groups

341

a = hi < A, = k - 1, hence r(a) > Max(r(a + l ), ..., r(k - 1 )) 2 0, so B,+, contains maximal r(a)-normal subwords of size > 2 ~ hence 7 ~ so does A , + , since A , + , is cyclic. Now Situation B holds for tijand Situation B is left-right symmetrical. By examination of the P.C.'s used, 19.5 holds if u67, u68 are replaced by uk7 = u 7 0 ,zik8 = u67 respectively, since u70 > u67 > u68; 1 -str+ - v g > uk-,; 1 -str+-uz-Uq - E 3 > ub8; Ub7 > u& + v:; uk7 > ub8 + u 3 + u4 + € 3 . Hence Bb+, contains an r(a)-normal subword of size > ubs = 2467. By 19.7 we have (*) if B C Bb+l and 4/e < p ( B ) < ( N - 12)/e then there is a subword A of A , + , such that A = ALCBDAR,which we abbreviate t o A = - B - , a n d ~ , + ~ ( A )

Max r(u) over u > h , which is 2 0. If 1 < u < A, then r(u) < r(X,) since X, is the smallest critical integer. We prove B h 1 + ,contains an r(X,)-normal subword of size > that Bb+, contains an r(hl)-normal subword of size > u67. If h, = b this is trivial since q02 u67. If h , < b then a + 1 = 1 < h , + 1 < b ; hence A k l + is in tijso is cyclic, and contains an r(hl)-normal subword of size > ~ 7 0 Arguing . as before, B b + , contains an r(hl)-norma1 subword of size > u67. Thus (*) holds. If i = 0 and t = 1 then a = 0, b = k - 1. If r(b) # 0 then B b + , contains an r(b)-normal subword of size > u70 and ( * ) holds. If r(b) = 0 then A , + , f Bb+,; also L(X,+,) 2 L(X,+,) since s,+~ < s ~ + Thus ~ . (*)holds takingA = B (then P , + ~ ( A ) < P~+~(B)). Consider (2+): None of r(a + l ) , ..., r(b - 1) is > r(b). If r(b) > 0 then B b + , contains an r(b)-normal subword of size > 1170 > u67, hence by 19.7 there are subwords A of A , + , , B of B b + , where

,

&+l

(A)
pb+,(B)> 4"- 16/e, and A

=

-B-.

This is also true if r(b) = 0.

19.9. Proposition. In Situatiorz A + either (i) XsinYl X l ! S 1 (i = 1 , ..., k - l ) , or (ii) A , contains a subword A ' a n d B , contains a subword B' where p k ( B ' ) > q - 28/e, p , ( A ' ) < f+ 20/e and A' AfLCB'DArR where the right side is ((n + 1))-bounded by k44.

nrl

J.L. Britton, The existence of infinite Burnside Groups

342

Proof. Assume (i) is false. Then some t i .satisfies (2+). Assume j chosen maximally. Thus either j = t or $it satisfies ( 1 +).Also j = i + 1. Moreover, for the largest u such that A,- B,,, we have

-

6 f ( X , ) eq ( X u + , )(where we have q instead of 4'). There is a subset k,, k , , ..., kp of 0, 1, ..., j such that k , = 0, k P = j , k p - , = i a n d [ k , k i + l ~ a t i ~ f i e ~ ( 1 + ) o r ( 2 +,..., ) ( i =p-1). 0 The following is true for u = p 1 and we prove by induction that it is true for u = 0, 1, ..., p.- 1. (**) For tk the first term contains A' and the last term conU P tains B' such that A' = -B'-, p ( B ' ) > 4 - 16/e and either (i) p ( A ' ) < f + 20/e and l k U k usatisfies +] ( i +), or (ii) p ( A ' ) < f + 6/e. Assume that (**) is true for some u , 0 < u < p - 1. First suppose that (i) holds. Then Ek,-, k , satisfies (2+). Hence its first term contains A" and its last term contains B" where A" = - B"- and p ( B " ) > 4' - 16/e, p ( A " )< f + 6/e. Since 4' > f + 37/e we have p ( R " ) > p ( A ' ) + lie, so L(B") > L ( A ' )+ L(Xd+,) where d = X k u , hence some translate of A' is contained in B " , since A' C A d + , , B " c B d + l . Hence A" = --B"- = -B'-. Now suppose that (ii) holds. If tk,-,k , satisfies (1 +)take the B of (2:) to be A' if p ( A r )Z 4/e while if p ( A ' ) < 4/e take B t o satisfy p ( B ) = 5/e (then B contains a translate o fA' ); this is possible since without loss of generality p ( R ) < f + 6/e < ( N - 12)/e. Then p ( A ) < p ( B ) + 14/c < f + 20/e. If tku-, k , satisfies (2') some translate of A' is contained in B" since p ( A ' ) + lie < f'+ 7 / e < q ' - 16/e < p(B"). Thus by induction, ( * * ) is true for u = 0. Thus Eii has first term containing A' and last term containing B' where p ( B ' ) > q - 16/e and p ( A ' ) < f ' + 20/e. If j = t we have finished since k , = k43. If j # t apply (b2) with A defined t o be B ' ; ( q - 16/e > 9/e). ~

19.10. Proposition. There is n o infinite sequence Y , , Y , , Y 3 ,... slrch that Y j u , , + ~Y j + ]( i = 1 , 2,3, ...), i.e., Axiom 5 holds f o r n Proof. Assume not. Again let r(i) be the least q such that

+ 1.

J. I,. Britton, The existence of infinite Burnside Groups

343

Yi uq Yi+,. Then 0 < r(i) < n + I . Let t = Max r(i) for i = 1 , 2 , 3 , ... . Then t < n + 1. If t < n + 1 then Yiun Yi+,(i = 1 , 2 , ...) contrary t o Axiom 5. Hence = n + 1. Consider Y pU , ( ~ ) , Y ~where v( p ) = E, and consider Y , , ..., YP+,.Now YP+lcontains a maximal t-normal subword so by 19.6 some N exists such that p + 1 < N then r ( i ) < t < n + 1, so Yiun Yi+, ( i = N ,N + 1, ...), a contradiction.

+,,

19.11. Proposition. Axiom 29 holdsjorn + l;i.e., i j X E J , Y E Cn+, and X >n Y then either (j-) some af(X)containsa subelement of L n + l of size > 4 0 , or (j-j-) X s ;U-' Y t . U for some U E II and some s, t such that 1 < s < (tl. Proof. 19.9 is available (with n instead of n + l ) , with X , = X ' , , X , = Y " . If (i) holds then (tt)holds. Now let (ii) hold. A subword A' of a positive power of X , and a subword B' of a positive power of X, exist such that pl(A') < f+ 20/e, p,(B') > q - 2 8 / e , and A' ; A ' L CB'DAlR. Taking a subword of B' if necessary we may without loss of generality suppose B' is a subword of Xi = Y t e E Ln+,. Now s""(B') > L ( B ' ) / L ( X z )- 3 / e - > 4 - 3 1 / e - E : > u g . Since 6.10 holds for n + 1, A' contains a subelem e n t o f L,+, o f ~ i z e > q - 3 1 / e - 1 ( ~ - ~ ~ ( k ~ < b , ~ ) . Let A ' = A I A , where L ( A 1 )= [ L ( A ' ) ] .T h e n A l or A , contains a subelement of 1 of size > ( q - 3 1/ e - ET - u4) - 1 / e = ~ 4 0N. o w L ( A 1 ) G L ( A , ) < i ( 1 + L ( A ' ) ) .

ET

,+,

1

:

+ L ( A ' )< L ( X , ) + L ( X , ) ( e f + 20) =

L ( X , ) ( e f + 21)< 2 e f L ( X 1 ) .

Hence A , and A , each have the form a f ( X , ) = a f ( X ) .

344

J.L. Britton, The existence of infinite Burnside Groups

20.1. We now know that all the axioms hold for n + 1 so it remains to complete the proof of the infiniteness of BS; as described in the introduction to Chapter 11. 20.2. Proposition. If f o r some W E TI we have

then fl;=o J , is not empty. Proof. We use the results of Section 17. For n = 1 , 2 , 3 , ..., choose W , such that W conj, W , E Jn and X conj, W * B,(W,) < B,(X) + E where E = l/e3. Since W conjn+l W , Bn+l(W,+l) < B,+,(W) + E . Since W conj, W,,B,+,(W) = B,+l(Wn). Hence

Since W , E J,, (W,) is (n)-bounded by str < k45. By 17.15 B,+1 ( W n ) < ~ 7 orB,(W,) 1 > 2B,+,(W,). Case 1 . B,(W,) > 2B,+,(W,) ( n = 1 , 2 , 3 , ...) Then

> 2(2(B3( W , )

-E

)

-E

)

> ...

Let s be the least integer such that B l ( W l ) / u 6 3 < 2s and s 2- 1. Let t>s;weprove(i)Bt+l(Wt)=O,(ii)B,+l(Ws+l)=O. Assume B t + l ( W t )> 0; then B t + l( W , ) > u63 and

B l (W,)> 2tB,+1(Wt)- ~ ( 2 ' - l +2t-2 + ... + 2 ) > 2tu63 - ~ ( 2 ' -2 )

J. L. Brifton, The existence of infinite Burnside Groups

345

Since U S 3 > 2/e3. Hence 2'< 2'-' < B,(W,)/US3. This proves (i). Now W , conj, W conj,,, W S + , hence W , conj, W S + , and 0 = B , + l ( W , ) = =Bt+l(W,+l) proving (ii). Now we prove W,+ E n J,. We have W s + l E JS+,c J , c ... c J,. Assume inductively that W s + l E J , for some t > s. Then B t + , ( W S + , ) = = 0 < U71 and ( W , + , ) is (t)-bounded by str = k46. Hence by 17.16 ( W , + , ) is ( ( t + 1))-bounded by str; thus W S + ,E J,+,. Case 2. Not Case 1. For some s = 1 , 2 , 3 , ... B,+,(W,) < ~ 7 1 We . prove B t + l ( W , ) < ~ 7 (t1 = s, s + 1, ...). This is true for t = s; assume inductively that B,+l ( W , ) < ~ 7 for 1 some t 2 s. Then B t + , ( W t + , ) < B t + l ( W t ) + ~ < ~ 7 1 +IfBt+2(Wr+l)> ~ . u~~ thenby 17.15 ~ 7> 1 B,+, ( W , + , ) - E > 2B,+, ( W , + , ) - E 2 2u7, - e . Hence E > ~ 7 21 l/e3, a contradiction. Hence B t + , ( W t + , ) < ~ 7 1 . Now for t 2 s W , conj, W conj, W , so W , conj, W , . Hence B , + l ( W , ) = B , + l ( W , ) < ~ 7 1 Now . W , E J , C JSW1 C ... C J,. Assume W, E J , for some t 2 s. (W,) is ((t))-bounded by str and Bt+, ( W , ) < ~ 7 hence 1 ( W , ) is ((t + 1))-bounded by str. Hence W , E J t + , and W , E n Jn by induction.

,

20.3. Proposition. n;=, Jn is empty. Hence by 20.2 we have B; = qc,,c,, c,, ... 1.

Proof. Suppose not and let U E n J,. Let F be the least integer Now UF has only a finite number of subwords X , , ..., X r . If X i is a subelement of some li of size > qo then j is unique. Hence p exists such that UF contains no subelement of of size > qo. Lp+l u Lp+, u Let rn 2 1. Then U E J , so U $ C, and U E Jm-l.By 5.7

> 1 +fe.

( 3 Y )Y E

c,, u>,-,

Y.

By Axiom 29 either (a,) some t i f ( U )contains a subelement of 1, of size > qo, or T-'. Y t . T , 1 < s < I tl. (b,) Us Let 1 < rn < rn; we prove that not both (b,), (b,,) hold. Y r t ' T', . Y ' E CmlY r eE L,. YIe = I in Otherwise Us'

J.L. Britton, The existence of infinite Burnside Groups

346

Hence in G,-l we have YtsCe = T . U S S r T-' e. = TIT-' = 1. Now Y E C, C J,-l so by 5.12 Y k # I in Gm-l ( k = 1 , 2 , ...), a contradiction. Hence (a,) holds for infinitely many m. Since U F 3 6f (U)' G,nvhence in

- TTI-1 y l t ' s e -

G,n-l.

we have the desired contradiction.

Apart from the consistency of the P.C.'s, this completes the proof of the infiniteness of B$ for all sufficiently large odd e (d 2 2), and we note that B$ is a Generalized Tartakovskii Group.

5 21. Consistency 21.1. Preliminary remarks. In the text, beginning with Section 5, the parameter conditions (P.C.), i.e., equations or inequalities involving parameters only, are emphasized oniy at first, since the reader will soon perceive that the P.C. in a given section of the text are easily identifiable, an understanding of the text being unnecessary for this purpose. The parameters not in the axioms but arising in the text are mainly denoted by the letters ti, k with numerical subscripts. Occasionally Greek letters are used for temporary abbreviations for expressions involving parameters. 21.2. The main parameters may be taken t o have values as in the following table, where E = 1 O-*. Parameter

.f'

Value

e-g

rb 7

zill

ro

r'

r"

r

-E

Moreover

4' = rb -O(l /c) y = Yo

~

O(rb)

q" = t i l l

r"

= Y"

-

-

O(rb) qo = q

O ( r o ) c" = r"

-

O(rb)

+ O(ro)

Here u l l and r" do not occur i n the axioms; they first occur in 7.3.

J.L. Britton, The existence of infinite Burnside Groups

347

21.4. The parameters in the :iiain text may be taken to have orders as follows. O(rb>:

u4

uifori=1,2,3,4,8,12,13,14,16,20,37,41,48,49, 60,62,63,64,65,81,91,92,94,96,97,98, 99 O ( r o ) : uj for j = 1 , 2 , 3 , 5 , 8 ui for i = 5 , 6 , 7 , 9 , 10,21,22,23,24,25,26,27,28, 29,30,31,32,33,39,40,42,43,44,45,46, 47,50,51,52,54, 56,57,59,61.66,67,68, 69,70,71,72,73,74,75,77,78,79,80,82, 83,84,89,90,93,95 u'. for j = 10, 15,28,30,40,43,83,78,79,84,67,68 * vi r d6 , i i i o , ufo, ti u2, 1-O(r): lci for i = 1 , 2 , 3 , 4 , 6 , 8 , 10, 12, 14, 16, 18,20,22,24, 25,26,27,28,29,30,31,34,36,43,44,47, 48,52,53,54,55,56,57,58,59,60,61 k', , kg , ki2, ki4 l-O(ro): ki f o r i = 5 , 7 , 9 , 13, 15, 17, 19,21,23,32,33,35,37 k y , k:, k:, k $ , k ; , k i 3 , ki5, 2i58 + O(ro):1ci for i = 39,40,41,42,45,46,50, 5 1

To,

O( 1 /el: O(r): O(r"):

"35

I

k;4

€1 €5 u36 9

r6 zi34/e

3

4.

348

J . L. Britton, The existence of infinite Burnside Groups

This completes the proof that for all sufficiently large odd e (d 2 2) B: is infinite and is a Generalized Tartakovskii Group.

References [ 1 ] M. Greendlinger, On Dehn's algorithm for the word problem, Comm. Pure App. Math. 1 3 (1960) 67-83. [ 2 ] J . Leech, A problem o n strings of beads, Math. Gaz. 41, No. 338 (1957) 277-278. [ 3 ] R.C. Lyndon, On Dehn's algorithm, Math. Ann. 166 (1966) 208-228. [4] M. Morse and C.A. Hedlund, Unending chess, symbolic dynamics and a problem in semi-groups, Duks Math. J . 11 (1944) 1 - 7 . [ 5 ] P.S. Novikov and S.I. Adjan, On infinite periodic groups, Izv. Akad. Nauk SSSR, Ser. Mat. 32,212-214, 251-524,709-731 (Russian). 161 V.A. Tartakovskii, Solution of the word problem for groups with a k-reduced basis fork > 6, Izv. Akad. Nauk SSSR, Ser. Mat. 13 (1949), 483-494 (Russian).

THE ALGEBRAIC INVARIANCE OF THE WORD PROBLEM IN GROUPS

F.B. CANNONITO” University of California, Irvine

This paper will be a sequel t o my paper [ 21 ; accordingly, I will lean very heavily on it for notation and concepts. Let K , and K , be two subclasses of the class of recursive functions such that K , C K , and K , # K,. Later, candidates will be proposed for K , and K , ; for the present the intuition is this: functions in K , \ K , are more costly to compute than those in K , . I will use K , and K , in connection with the word problem (w.p. henceforth) in finitely generated (f.g.) groups, in order t o discover sufficient conditions on K , and K , which will insure that the “cost” of the solution of the w . ~ is . an algebraic invariant of the group. That is, if K , and K , are properly chosen, then either all f.g. presentations of the group have w.p. solvable by a function in K , or they all have w.p. solvable only by a function from K 2 \ K , . When K , and K , are properly chosen 1 will call them “properly chosen.” The restriction t o f.g. presentations is not artificial since groups can be presented on infinitely many generators so that the w.p. for the infinite system is unsolvable even though this is not so for f.g. presentations. On the other hand, it makes sense to ask if the cost is an algebraic invariant of f.g. presentations since Kabin has shown that when one f.g. presentation has a solvable w.p. so must presentation has a solvable w.p. so must any other, although conceivably the cost might vary. Of course, some reasonable measures of cost may not turn o u t t o be algebraic invariants, neither must two distinct measures be compatible - - that is, assign the *This work was supported by AFOSR Grant No. 1321-67. 349

350

F.B. Cannonito, Algebraic invariance of the word problem

same relative cost on different groups. At any rate, in this paper I will only consider conditions on K , and K , which are sufficient t o insure the invariance of cost and leave the other questions t o ano t he r work . Hvnceforth, for any positive integer n , let the expression ( " n ) appearing in the margin indicate a condition o n K , (and K 2 ) , the integer to serve for bookkeeping purposes. Thus, assume a fixed Gadel numbering y of the words on the alphabet a l , a,, a3... such that the free groups of rank ns< 00, denoted Fn and presented as ( L ( , , ..., a, I ), have the property

Fn has a K , -admissible index y consisting of the set y(Fn ) of integers (indices), the (induced) multiplication rn : y(Fn), y(Fn), and the (induced) inverse function IN : y(Fn)+ y(Fn).

(* 11

--f

Note. In [ 2 ) the admissible indices were specifically ta-admissible, where C" is a member of the Grzegorczyk hierarchy. Here, K , (or K,) will replace the € a , mutatis mutandis. (Sometimes I will write Ki or K for either K , and K , .)

I t is possible to proceed as in [ 21 and define arbitrary Ki-admissible indices ( i = 1,2) for f.g. groups. But with certain exceptions, discussed below, the most natural way to index a group is from the index y of F,, given a presentation II of the group in the form ( a, , ..., a,, I R , S, T, ... ). Such indices are called standard indices in [ 2 1 . Intuitively considered, a standard index for a group with presentation n is obtained ny assigning each coset in Fn/N, ( N = the normal closure of R , S, T, .._in F,) its minimal index. From the proof of Theorem 5.1 of [ 21 we see when the K isatisfy (*2) K , contains the constant functions and K , is closed under the functions + , * , L , substitution, bounded sums: Z , and bounded u
then the propositional connectives, and the bounded quantifiers do not lead o u t of the class of K , decidable relations, and K , is closed under the bounded p-operator. Identical properties hold for

F.B. Cannonito, Algebraic invariance of the word problem

35 1

Note. From (*2) it follows that K , (hence K 2 ) contains the Kalmar elementary functions; cf. Kleene [ 101 p. 285, for la/b] is definable from the functions and operators given in (*2) given the K , definability of “<” and “<”. But these follow from L . In particular, when (*1) and (“2) hold, the condition K , C K 2 implies that for a group presented as Fn/N, the solvability of its w.p. by a K j function, (i = 1,2), implies that F,/N has a Ki-admissible standard index, for this is precisely the content of Theorem 5.1 of

[21.

Note. In order to define the relation E(Y,s) on the indices which holds when and only when the element with index Y, denoted by gr (likewise g,) determines the same coset as g s , it is necessary t o assume the logical conjunction of K , -decidable (resp. K 2 -decidable) relations is K , -decidable (resp. K2-decidable), (cf. p. 380, line 1 1 , of [ 21 ). If the characteristic function CR of a relation R is 0 on the extension of R and 1 otherwise, then C,,, = C, + C, I CR * C,. O n the other hand, CTR = 1 1 C, and so assuming (*2) gives the closure of the Ki-relations under all propositional connectives. It remains to be seen if under narrower assumptions on K , than those given, K , is sufficiently expressive so that ( * 1 ) is satisfied. In connection with this David Thompson, in a private communication, has shown that the Fn have E 2-admissible standard indices, but not E -standard indices. Conversely, when Fn/N has a Ki-admissible (standard) index i . we want t o conclude that the w.p. is Kj-decidable. For this t o be so, we must be able to deduce the Ki-decidability of y (N). From Theorem 5.2 of 121, this will be the case when the natural homomorphism cp : F, + Fn/N is Kj-computable with respect to y and i. At this point let m e recall the definition of the n t h iterate of a function f . That is, given f , a number-theoretic function, we define g(n) “the nth iterate off’” by the recursion

352

F.B. Cannonito, Algebraic invariance of the word problem

Since the operation in the set of indices corresponding to exponentiation of an element of the group is iteration of the induced multiplication function, and since the K-normal form property (cf. Definition 5.3 of [ 21 ) involves such iteration, we obtain the next condition on K, and K2 , namely ( " 3 ) the nth iterate of a K, (resp. K2) function is a K, (resp. K 2 ) function.

Note. Condition ( " 3 ) taken with (*2) implies K , contains the primitive recursive functions. Cf. Pkter [ 151 Section 7. When all conditions thus far assumed on Ki are satisfied we conclude, via Theorem 5.2 and Theorem 5.3 of [2] that the Ki solvability of the w.p. in Fn/N is a consequence of the existence of a Ki-admissible standard index for Fn/N. All that remains is to show the analog of Corollary 7.1 of [2] holds. That is, the Ki decidability of the w.p. is an algebraic invariant. Thus, assume Fn/N and F , /M are two f.g. presentations of the same group and IJ is an isomorphism Fn/N = F, /M. Further, assume F n / N has a Ki-admissible standard index for fixed i = 1,2. I want t o show F,/M has a Ki-admissible standard index. For this purpose it is sufficient t o show M is Ki-decidable in F, since F,,, has a Ki-admissible index by (* I). First, observe that if a group G has any Ki-admissible index i, then all its standard indices are Ki-admissible. For, suppose G is presented as F,/M. Since Fn has a Ki-admissible index for which it has the K , -normal form properly and ( * 3 ) holds, the natural homomorphism F , -+ F,/M is Ki computable, whence the assertion (cf. Lemma 7.1 of [ 21 ). To complete the argument, take Fn/N as G and invoke the equivalence of the Ki-decidability of the w.p. and the existence of a Ki-admissible standard index (cf. Theorem 7.1 of [ 21 ). Thus, I have shown that ( * l ) , ("2) and ( " 3 ) are sufficient t o insure the algebraic invariance of the cost of the w.p. with respect to f.g. presentations. This may be summarized in the

F.B. Cannonito, Algebraic invariance of the word problems

353

Result A . Let K , and K , be as above. In order that K , and K 2 be properly chosen, it is sufficient that K , (resp. K 2 ) be closed under the primitive recursive functions andlor relations and the free groups Fn have K , -admissible standard indices. To some extent the program of discovering how t o insure that K , and K 2 are properly chosen is incomplete, for I ought to show that there are groups whose w.p. is restricted t o either K , or K , \ K , . Succinctly put: that there is adequate pathology for K , and K,: As it turns out, for the choices I will propose for K , and K , the “pathological” groups exist. There is, however, some doubt

in my mind concerning whether or not one ought always t o strive for finitely presented pathology. Of course, before it is known that finitely presented pathological groups exist - and with the hindsight we presently have regarding how difficult it was, initially at least, to obtain such groups - it probably is very reasonable t o insist on producing finitely presented groups before agreeing that a (more or less) profound concept has been apprehended. However, with the exception, possibly, of the residually finite groups for which it is clear that finite presentability is fundamental (because the w.p. for f.g. residually finite (r.f.) groups may be unsolvable in general while the finitely presented r.f. groups have solvable w.P., cf. V.H. Dyson [ 4 ] , or A.W. Mostowski [ 141 ) it seems clear that for the purposes of algebra the more important concept is “finitely generated”. Hence, if one produces f.g. pathological groups one ought t o feel that the significance of the K , and K , is reasonably well established, especially if we can produce f.g. recursively presented pathological groups, for there is every reason to believe that having done so, the finitely presented pathological groups will be forthcoming through the Higman embedding technique (such reasons t o become apparent below). Indeed, for the choices of K , and K , t o be proposed, the Higman embedding does furnish the finitely presented pathological groups. Thus, I also want the condition (*4) there exist f.g. recursively presented groups with w.p. solvable by K , functions (resp. by K , \ K , but not K , functions).

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F.B. Cannonito, Algebraic invariance of the word problem

Note. The requirement that the group be both recursively presented and have a solvable w.p. is redundant since a f.g. group, which has a solvable w.p. with respect t o a finite system of generators, can be recursively presented with respect to these generators because the multiplication table of the group is then r.e. Of course, the converse does not hold. That is, not every f.g. recursively presented group has a solvable w.p. For the first candidates for K , and K,, I will take the primitive recursive (p.r.) functions and the recursive functions, respectively. The f*, then have admissible p.r. indexings (in fact, elementary indexings) by Lemma 4.1 of [ 21. Clearly ("n) for n = 2 , 3 are immediate and ( * 4 ) follows from Theorem 7.2 of [ 21, choosing, e.g. 01 = 4 and (Y = w + 1. My student R.W. Gatterdam has shown in his doctoral dissertation [ 71 that the Higman embedding technique prererves p.r. solvability of the word problem. To fix ideas, let G be recursively presented and let the Higman process, denoted by il, take G into finitely presented G, ; in diagrams G '1GH . Then, h i t t e r d a m has shown that H is a p.r. process and the set of indices of the image o f G in GI, is p.r. decidable in the solution t o the w.p. of Hence, for G possessing K , (= p.r. functions)-admissible index. we obtain GH with K1-admissible index; that is, a finitely presentable group with K , -decidable w.p. However for G possessing K 7 \ K , but not K , -admissible index we obtain f.p. G, with K ; \ K , decidable w.p. but not K , -decidable w.p. Because, if G, has K , solvable w.p. given, say, by function f , then the function g(.u) = ( . f ' ( x )x, E H ( G )

1*

, otherwise,

solves the w.p. in G and is K , contrary t o assumption; ("*" denotes any convenient value outside of H ( G ) ) .Hence, we have

Result B. Prirnitive recursiveness o j t h e solution of the w.p.is an ulgebruic invariant of'finitely generuted groups.

F.B. Cannonito, Algebraic invariance of the word problem

355

Note. Boone has announced in his paper [ 11, footnote 4, p. 50, the existence of a finite presentation with recursive but not p.r. solvable w.p. Applying the analysis above, J am able t o assert, for this group, the algebraic invariance of its word problem. But, in any event combining Result B, the case a = o + 1 of [ 21, Theorem 7.2, o r Theorems 7 and 8 of [ 31, and the construction of Gatterdam [ 71 gives

Result C. We can construct a finitely presented group G with algebraically invariant recursive but not p.r. solution to its w.p. A somewhat more provocative choice for K , and K , devolves as follows: If T(a, b, c) is the (intuitive) Kleene T-predicate, let 7 (x, y , z ) be a formula of the normal number theory N either as given in Mendelson [ 131 or in Kin0 [ 9 ] or Kleene [ 10,111, which expresses T . That is, if o is the standard model of", let

t 7(a, b, c ) when o I= T(a, b, c ) , N and N

7(a, b, c ) when not o t= T(a, b, c).

We say a (number theoretjc) function f : o -+ w is provably recursive in N if .f has an index e such that

Vx '7

3 y s ( e , x,y ) .

It is easy to see not every recursive f is probably recursive in N (cf. Fischer [ 61, Theorem 5.1). In fact, as Enderton has shown [ 51, n o recursive function need be provably recursive in N . I assume the formula 7 is chosen so that, at least, all p.r. functions are provably recursive; this is always possible as Kin0 has shown in [ 91 . Thus, let K , be the functions provably recursive in N and take K , t o be all recursive functions. Then ("1) holds because of the assumption on 7 and my Lemma 4.1 in [ 2 ] . Likewise ("2) and ("3) hold because of the assumption on 7, and because if g is a

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F.B. Cannonito, Algebraic invariance of the word problem

function p.r. in provably recursive functionsfi,f., ._.,f m , then g is provably recursive (cf. Property 4.1 in [ 51 ). Condition (*4) is

also immediate, cf. the discussion for the preceding choice of K , and K,, mutatis mutandis, and the Theorem 8.3 of Fischer [61 which shows there is a recursive but not provably recursive set of integers. (Alternately, the existence of such a set follows from the existence of a recursive but not provably recursive enumeration of the provably recursive functions, the construction given in my Lemma 3.1 of [ 21, and Property 4.1 of Fischer [ 61 ). Note. From (“4) and the fact (alluded to above) that the Higman embedding process preserves provable recursiveness we deduce the amusing consequence, given in Result D below: there exists a finitely presentable group with intuitively solvable w.p. such that it is not possible to prove in formal number theory that the solution is a recursive function. Consider this result in the light of objections given by some mathematicians (e.g. Wang [ 171 , p. 89, and Peter [ 151 ) to the converse of Church’s thesis: every recursive function is effectively computable. That is, a function is recursive if it has an index e such that (+)

w b V x 3 y T(e,x, y ) .

However, nothing is said, in general, about how y is determined by x. Alternately, for the machine interpretation of T(a, b, c ) given by Kleene in [ 101 p. 243, nothing is said regarding how the length c of the computation depends, a priori, on the input b. It is not altogether convincing that functions for which property (+) holds are effectively when y is not known to depend on x in some satisfactory way. The argument from experience which goes - all functions known on intuitive grounds to be effectively computable are, in fact, recursive - really does not insure that some proper subclass of the recursive functions might not already contain all such algorithms. Yet, if one chooses t o regard the class of functions provably recursive in (say) N (with respect t o a “realistic” formula 7 )as the true class of effectively computable functions we can deduce an anomalous version (Result D) of the Novikov-Boone

.

F.B. Cannonito, Algebraic invariance of the word problem

351

theorem in which the “bad” group has, on intuitive grounds, a “solvable” w.p. despite the fact that on formal grounds its w.p. is unsolvable! (However, cf. Kleene [ 101 , paragraph 3 of footnote 17 1 on p. 241 for a discussion of the converse t o Church’s thesis in which the converse is framed so that, in Kleene’s view, it is not contestible. Kreisel has written me, as well as published elsewhere, concerning his view o n Church’s thesis. He argues that it is necessary t o make a distinction between the concepts “mechanical” and “constructive” and, if one identifies these, then it is hard, if not impossible, t o doubt Church’s thesis. Presumably this means the converse is not in doubt whatever the view on mechanical vs. constructive.) In summary, then, we have by means of Gatterdam’s analysis [ 71 ,

Result D. We can construct a finitely presented group with algebraically invariant recursive but not provably recursive solution to its w.p. Note. The “effective” construction of the groups of Results C, D and E follows from the effectivity of Gatterdam’s construction and from the effective construction of sets of integers with characteristic functions of suitable subrecursive degree of solvability. Here, by subrecursive degree, 1 mean solvable by a K , or K , function for properly chosen K , or K,. The existence of suitable characteristic functions is not difficult to obtain. F o r example, refer t o Lemma 3.1 of [ 21. For the recursive vs. provably recursive case use the recursive but not provably recursive function which enumerates the provably recursive functions of N . The “effective” existence of such a function follows from the fact that N is an axiomatic formal theory, (cf. Mendelson [ 131, p. 29 (3)). Similarly one can show the Peter function is actually provably recursive and obtain Result E, mutatis mutandis. Another choice for K , and K , are the p.r. functions and the provably recursive functions. The discussion above carries over immediately, it only being necessary t o note that Rosza PCter’s recursive but not p.r. function cp enumerating the p.r. functions can

358

F.R. Cannonito, Algebraic invariance of the word problem

be shown, using Kino’s characterization of provably recursive functions, t o be provably recursive since cp is given by a transfinite induction up t o 0 2(cf. ; PCter [ 161, p. 125 f.) From this we obtain the existence of a set of integers which is provably recursive but not p.r. Such a set can be used t o satisfy (*4) in the standard riianner (cf. my Theorem 7.2 of [ 21 for the details). I state for the record the

R m i l t E. We can construct a ,finitely presented group with algehr-aicalh~invariant provably recursive but not p.r. solution to its \t’. p . At this point it is tempting t o try to carry o u t the above arguments for the finer degrees of the Grzegorczyk hierarchy. That is, take K , as 6 and K , as 8 a + k for fixed cy 2 3 and k 2 1 (here (Y > 3 since K , needs the elementary functions and probably a bit more). While t o a great extent this can be carried o u t , certain imperfections, which threaten to be of a permanent nature, persist at the present state of science. Since this represents a departure from the point of view in my paper [2] I will go into some detail. The proofs, which are more or less automatic once the difficulties are perceived, will, as usual, be omitted or else published elsewhere. The chief difficulty is encountered in the attempt t o obtain an M a analog of Theorem 5.3 of (21 which holds for suitably chosen K , ( b u t which does not hold precisely as stated in 121 ). This theorem provides the link whereby if we know that one of the quotients G / H of a K , group G has a Ki admissible index then H is K , decidable i n G . T ~ w the , equivalence between the word problem being solvable by Ki functions and the existence of a Ki index for G is established by this theorem (in conjunction with the K , analogs of Theorems 5.1 and 5.2 of [ 21 ). However, for the choice for K , the result of Theorem 5.3 does say. of E a for K , and not go precisely over (despite the incorrect claim in [ 21 that it does). The difficulty lies in showing that the natural homomorp h i m cp : G G / H is foa-computable, even under the assumption G has the E a normal form property. For, if we denote by i and j the & a indice\ in G and C / H respectively, then the index normal +

F.B. Cannonito, Algebraic invariance of the word problem

359

form decomposition of an element x in i ( G ) as x = 7,a! ~ o . . . o T @-r ,, a

goes over under (p to Cp(x) = k T l lo...o k , in j ( G / H ) and there is n o way in general to provide an a priori I%& bound on the iterated a multiplication in j ( G / H ) by which we obtain each kTl I . I t is for this reason that I required in (*3) that K , be closed under iteration. Thus, the best that can be said for the case under consideration is that Cp(x) is Z @ + l computable. Now, we usually apply this reault t o the special case G is a free group of rank n < and i is a standard index, so the iterated multiplications in i ( G ) which give T~~~ cause n o trouble since an 83 bound is available in terms of the Kleene concatenation function x * y . But there seems t o be n o way t o treat the iterations induced in j ( G / H ) by the index normal form owing t o the wholly arbitrary nature of the index j . If, in fact, j is a standard index then there is n o difficulty (but this would be tantamount t o assuming G / H hadan Ea! solvable w.P., the very property we hoped t o deduce from Theorem 5.3). As a consequence of this imperfection in the theory the algebraic invariance of the w.p. with respect t o the M a heirarchy applies only t o standard Ea! indices. It should be noted, however, that it is completely unreasonable t o consider the w.p. with respect t o any other kind of index because only the standard indices effectively relate words in the free group with their corresponding index through the technique of Godelization. In fact, n o way is presently known to construct an example of a non standard index despite the fact that allowance for the existence of these indices must be made. In summary then, all 1 can say is that p(x) is at most Ma!+l computable and accordingly, G/H has an E a + l solvable w.p. It is of minor comfort t o know that this is the worst that can prevail; for, given that G is an ga! computable group, we know that G has at most an standard index. Thus it is conceivable that a group G may have its word problem solvable precisely at level &a!+1 and not lower for any f.g. presentation and yet have an index for which it is an ga! group. It can not have an index at levels less than a! for this would, via the equivalence of w.p. and standard index, give rise to a standard index at ga! and contradict the algebraic invariance at level a! + 1 . For the sake of completeness 1 will list the easily demonstrated properties of 8“ indices for f.g. groups, a! > 3 (GH standing for “Grzegorczyk hierarchy”):

-

F.B. Cannonito, Algebraic invariance of the word problem

360

GH GH GH GH

1. f.g. subgroups of &a groups are

2. 3. 4.

GH 5.

-standard. -standard. G is an 8" group implies G is G is $"-standard implies all standard indices are at most Z a . G is &"-standard but not lower for standard indices implies G i s n o t a n & P g r o u p f o r a n y P < a - 1. &a solvable w.p. is equivalent t o &"-standard. This property is algebraically invariant for f.g. presentations.

A second difficulty occurs in building &" indices for free products with amalgamation of $" groups. Again iteration is the culprit. For, if G, and G , are &a groups with isomorphic subgroups H , < G , and H , < G , and with the isomorphism cp : H , tf H , such that both N, and H, are 8" decidable and cp and cp-l are computable, then the best we can say is G I *pG2 is at most This is because it is natural t o build an index for G,*,G2 from the indices of G I and G, . However the reduction of the product of two normal forms ( h x ,... x , )~( k y ,... y m ) = h ' z , ... z p involves repeated iteration of the process which "slips k past x,, ". In this process the index of the element in the amalgamated subgroup can grow arbitrarily large and hence there is n o a priori &a bound for the entire iteration. Hence G,*,G2 is at most &"+I is the best that can be said. The easily demonstrated revisions of [ 2 ] Section 6 are listed below ( A F P stands for "amalgamated free product"):

AFP 1 . If Gi is &%tandard and Hi < G , is &"-decidable for i = 1 , 2 and cp is an isomorphism of H I and H , such that both cp and g 1are & a computable then G1*,G2 is &"+l-standard. AFP 2. G, *G, is &"-standard when G , and G, are. A F P 3 . 1 f G l = G 2 a n d H 1 = H 2 andcp= 1(;IHthenG1*,G2 is&" standard (under the conditions in AFP 1). As a consequence of AFP 3 we can show the groups Ga+l presented in [ 2 ] as

for certain Ma:'

-decidable subsets Ucu+lof o,are, in fact,

-

F.B. Cannonito, Algebraic invariance of the word problem

36 1

standard and thus, have an &ff+l algebraically invariant w.p. (although conceivably they could have anomalous E f f indices). The next flaw of the &,a heirarchy turns up when we try to follow Gatterdam’s analysis of the Higman embedding. The difficulty is this: at a certain point in the construction it is necessary to pass from a group G and an isomorphism cp : H -+ H’ < G for some H < G t o a new group G, obtained from G by adjoining an infinite cycle t satisfying t k t - l =cp(k) for all k in H . It can be seen that G, is embedded in a certain free product with amalgamation. Consequently this construction preserves K , cost ( [ 71 Theorem 3.1). However, for the E f f hierarchy Gatterdam shows the passage from G t o G, possibly jumps two levels from E a to & f f + 2 and the Higman technique needs (at this state of science) a countable number of applications building a tower T , viz. G, G V l ,(G,, ),*, ... . Hence for the present it appears that the full class of p.r. functions is the minimum class which can accommodate the Higman construction. It remains to be seen if either the jump from gff t o E f f + in passing from G t o G, is necessary, or if (more likely) countably many applications do not force a p.r. index for the union of the tower T t o E f f + k for some k suffices. but rather, only a finite jump from A successful solution t o either problem will suffice to show the Higman embedding preserves the 8“ hierarchy (albeit possibly enThen because the groups tailing a finite shift from && t o %+I are recursively presented, we would conclude there is adequate pathology for the 8“ hierarchy. For the present all that we can say is for any p Z 3 there is a finitely presented (henceforth f.p.) group whose word problem can not be solved at level lower than 0.This is a consequence of Gatterdam’s proof that the Higman embedding preserves p.r. solution to the w.p. Since the groups G&+1 are recursively presented and p.r. they are embeddable in a p.r. group which a fortiori resides at some (unknown) level y of the &“ hierarchy. Then the word problem is solvable at level y + 1. Since the Gff+l have word problems at arbitrarily high levels we

* Cfr. this volume, “The computability of Group Constructions, Part 1,”

by the author and R.W. Gatterdam for a solution to this problem showing the “strong Britton extension” only jumps two levels at most (Theorem 5.3).

362

F.B. Cannonito, Algebraic invariance of the word problem

get the assertion, for the set of 7’s obtained by the embedding is necessarily cofinal in the natural numbers. Note. It may be possible through the use of techniques employed by Boone in [ 1 ] t o go directly from the sets Ua+l t o f.p. groups with w.p. solvable at level but not lower (for k < 00 depending uniformly on a ) . I d o not know whether o r not this program is feasible, but I1.J. Collins and C.F. Miller 111 have expressed belief in the potential fruitfulness of such an approach to finding adequate pathology for the 8 @hierarchy. Finally, with respect t o the g a hierarchy let m e observe that there is n o p.r. group which contains a copy of every f.p. &a group. For suppose such a group G exists. Then being p.r. G is g a for some a . But all f.g. subgroups of G a groups are at most E a + l and we know there are f.p. groups at arbitrarily high levels of the G a hierarchy. Thus the assumption on G is impossible. (I am indebted to Gatterdam for pointing this o u t to me.) Next, I want t o comment on the cost of the w.p. for small cancellation groups such as the “less than 1/6” groups studied by Greendlinger 181 and Lipschutz [ 121. If such a group, f.g. of course, has 3 presentation for which the relators may be enumerated by a K , function in order of increasing magnitude of, say, length or Cijdel number, then its w.p. must be K , solvable. This is because in small cancellation groups the algorithm for solving the w.p. is a monotonic reduction process consisting of examining finitely many subwords of a word W which is suspected to define the identity in order t o discover more than half of a symmetric relator. Because of the enumerating condition on the relators there are only finitely many candidates for comparison, all of which are obtainable by a K , function. Thus, if a fragment of a relator is found, it may be replaced in W by the inverse of its remaining part, which, being shorter than the part replaced, reduces (even before cancellations) the overall length of the original word. Thus, through iteration, which preserves K , by ( * 3 ) , a word W defining the identity may be brought to the empty word by means of a K , process. F o r example, finitely presentable small cancellation groups must have an

F.B. Cannonito, Algebraic invarirnce of the word problem

363

algebraically invariant K , w.p. O n the other hand it is possible t o construct an infinitely related recusively presented small cancellation group G which has a recursive but n o p.r. solution t o its w.p. (A typical construction for a recursively presented small cancellation group G with recursive but not p.r. word problem might go like this: let F , be the free group on u l , u 2 , ..., u7 and let H < F7 be the subgroup generated by a l l ~ , ~ u ~ ~ . for . . k E U, U a recursively decidable but not p.r. decidable set of integers. Take two copies of F , and amalgamate the subgroups H . Such a group is easily seen to be a “less than 1/6 group” and can be shown to have the required properties on its w.p. by imitating the technique in Cannonito [ 21 . N o claim is made for elegance or economy in this example.) G can be embedded in a finitely presented group which, by Gatterdam’s analysis, must have a K , \ K , decidable w.p. and which, accordingly, can not be a small cancellation group. The latter property could also be seen directly from the constructions required for the Higman embedding process.) I summarize this discussion as

Result F. A small cancellation group which has its relutors enumerable in order of increasing length by a K , function, hus a K , decidable w . ~ In . particular, finitely presented small cancellation groups have K , decidable w.p. (for suitable chosen K i ) . As a final comment, let m e note that direct products, free products and free products with amalgamation (satisfying suitable conditions on the amalgam such as those given in [ 21 Corollary 6.3), applied finitely many times, preserves the algebraic invariance of the w.p. I wish to acknowledge the many pleasurable and profitable discussions I have had o n this work with my student R.W. Gatterdam.

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F.B. Cannonito, Algebraic invariance of the word problem

References [ 1 I W.W. Boone, Word problems and recursively enumerable degrees of unsolvability. A sequel on finitely presented groups. Ann. of Math. 84 (1966) 49-84.

[2l 1.R. Cannonito, Hierarchies of computable groups and the word problem, Jour. Symbolic Logic 31, No. 3 (1966) 376-392. 131 F.B. Cannonito, On finitely generated groups with primitive recursive word probkin, TAO technical report 63-65, Hughes Aircraft Company, 1963, 21-pp. [ 4 ] V.H. Dyson, Notices Amer. Math. SOC.11 (1964), p. 743, Abstract 616-617. [ 5 I H.B. Enderton, On provable recursive functions, Notre Dame Journal of Formal Logic IX, No. 1 (1968) 86-88. [6l P.C. Fischer, Theory of provable recursive functions, Trans. Amer. Math. Soc. May (1965) 494-520. [ 7 j R.W. Gatterdam, Embeddings of primitive recursive computable groups, doctoral dissertation, University of California, Irvine, 1969. [8] M.D. Greendlinger, Dehn's algorithm for the word problem, Cornm. Pure and App. Math. 13 (1960) 67-83. 191 A. Kino. On provably recursive functions and ordinal recursive functions. Jour. Math. Soc. Japan, 20 (1968) 456-476. 1101 S.C. Kleene, Introduction to Metamathematics (D. Van Nostrand, Princeton, N.J., 1952). [ l l I S.C. Kleene, Mathematical Logic (John Wiley & Sons, 1967). [ 121 S. Lipschutz, On square roots in eight h-groups, Comm. Pure and App. Math. 15 ( I 962) 39-43. [ 131 E. Mendelson, Introduction to Mathematical Logic (D. Van Nostrand, Princeton, N.J., 1964). [ 141 A.W. Mostowski, On the decidability of some problems in special classes of groups Fundaments Mathematicae 59 (1966) 123-135. 1151 R. Pe'ter, Rekursivitlt und Konstruktivitat, Constructivity in Mathematics, ed. A. lleyting (North-Holland, Amsterdam, 1957) 226-233. [ 161 It. Pdter, Rccursive Functions, third revised edition (Academic Press, New York, 1967). 1171 11. Wang, A survey of Mathematical Logic (North Holland, Amsterdam, 1963).

I

THE COMPUTABILITY OF GROUP CONSTRUCTIONS, PART I F.B. CANNONITO” and R.W. GATTERDAM** University of California, Irvine, University o f Wisconsin,Parkside

The beginnings of computable algebra, especially as this applies to group theory, can be traced for all practical purposes to the fundamental paper of M.O. Rabin [ 91 . In this work Rabin showed the very natural equivalence between the existence of a “recursive realization” of the group in the natural numbers and the existence of a recursive solution to the word problem with respect t o any finitely generated presentation of the group. Of course, the property of having word problem solvable with respect t o a given presentation applies, in general, only to finitely generated (henceforth f.g.) presentations; there is n o problem in presenting even f.g. groups o n infinitely many generators so that the word problem with respect t o such a presentation is unsolvable. Subsequently Cannonito sharpened the concept of a computable group in [ 1 ] by superimposing the Grzegorczyk hierarchy, denoted “ ( & a ) a E w ”, and by showing that f.g. Ea-computable groups existed in profusion for infinitely many consecutive levels of the hierarchy. Although incorrectly stated in [ 11, it was shown in Cannonito [ 1,2] that with respect t o standard indices of &a groups, the word problem is an algebraic invariant; that is, every f.g. presentation of an &%omputable group has, with respect to a standard index, a word problem solvable by an gff function if one f.g. presentation of the group possesses this property. However, certain inelegant aspects of the theory of &a comput-

* AFOSR-1321-67 (Grant).

** AFOSR-70-1870 (Grant). 365

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F.B. Cannonito and R. W , Gatterdam, Computability of group constructions

able groups came t o light as a consequence of the discovery that the proof in [ 1 3 of the equivalence of $a computable realizations and G O solvable word problems only applies to standard indices. These aspects are discussed, among other parts of the theory, in Cannonito [ 21. Typical among the inelegancies is the inability, in general, t o say more than this: if a f.g. group has an &a realization then its word problem is not higher than & a + 1 .Furthermore, while the assertion in [ 1 ] claiming the groups G, presented as

o r highwhere U, is a set of natural numbers decidable only at er, had word problem solvable at level $, but not lower was true, the proof given was flawed in an essential detail; namely, contrary to the assertion in [ 1 1 , Theorem 6.1, the best that can be said for the free product with amalgamation of &a groups (under suitable & a conditions on subgroups and isomorphisms, etc.) is that its level of computability is n o higher than Thus applying the analysis in [ 1 ] , to the groups G,, we see the word problem is solvable no higher than 8a+l und n o lower than Za. However, in this paper we show the word problem of the G, actually resides n o higher than M a , owing t o the fact that the G, arise from a rather restricted type o f free product with amalgamation in which both factors are copies and both subgroups t o be amalgamated are also copies and the isomorphisms are the identities. This situation and its consequences are examined in detail in this paper with respect t o a “reIativiied Grzegorczyk hierarchy” which seems to be appropriate for all countable groups whether finitely generated or not and whether possessed of a solvable word problem for f.g. presentations or not. Owing t o the possible independent interest in this relative GrTegorczyk hierarchy we begin this paper with a discussion and proofs of this concept, which resembles the notion of relative computability as exposed in Davis [ 31 , the essential idea being to close a level 8, under the usual operations after first adding the characteristic function of some subset of the natural numbers. I n the next phase of the theory of 8, groups the Higman embedding [ 61 of recursively presented groups into finitely presented

F.B. Cannonito and R. W. Gatterdam, Computability of group constructions

367

(f.p.) groups was studied in the doctoral dissertation of Gatterdam [ 4 ] . It was shown there that the Higman embedding took f.g. groups with primitive recursive (p.r.) realization into f.p. groups which have a p.r. realization. Moreover, it was shown by Gatterdam that the actual embedding function was elementary (or, equivalently, Z 3 ) in the solution to the word problem of the receiver group. Hence, since the groups G , are all recursively presented, they can be effectively embedded in f.p. p.r. groups and so there is an infinite spread of complexity of f.p. groups with respect to the 8" hierarchy. But ideally one wants t o show the Higman embedding actually preserves the relative Grzegorczyk hierarchy because by so doing one by-product would be the possibility to locate precisely on the hierarchy the receiver groups of the G,. According t o Gatterdam's analysis all that could be said is that the receiver groups must slip arbitrarily high up the hierarchy, but conceivably there could be great gaps and n o method was at hand to locate the actual level of the hierarchy at which any particular receiver group resided. The reason for this dilemma can be found in a particular construction needed for the Higman proof which has come t o be known as the strong Britton extension. The situation is this: we start with a f.g. group G and an isomorphism 11 < G into G itself. If G has presentation, say,

( a l , ..., a , , ; R 1 ,...

)

then we obtain a new group G, which embeds G with presentation (a

l , ..., a,, t ;R , , ..., tht-l

= cp(h) for

all h E H

)

.

Thus, G, is obtained by adjoining a (distinct) infinite cycle t which gives the isomorphism cp by conjugation. Since Gp is a f.g. subgroup of a particular free product with amalgamation the computability of G, can be shown t o lie n o higher when G lies at level 8". But the Higman embedding uses a countable number of such extensions resulting in a manyfold extension G L P , p 2 p...3 or, more simply, G,. Thus the level of computability can slip extremely

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F.B. Cannonito and R. W. Gatterdam, Computability of group constructions

high since it seems t o require jumping possibly one level for each new infinite cycle adjoined. However, in this work we show (with respect t o the relativized hierarchy) that when G is E a computable then G, is at most at level & a + l Thus, . at this state of science the Higman embedding seems t o actually preserve the relativized hierarchy from some point on for the remaining constructions d o not appear to be troublesome at all. This will be the subject of Part I1 of this paper, t o appear later, and will be based on the analysis we give in the present paper. Additional constructions studied in this paper form the main stock in trade of infinite group theory. Among these are the “HNN extension” due to Higman, Neumann and Neumann. Also we show there can be n o universal primitive recursive group containing a copy, even, of all f.p. p.r. groups. For the convenience of the reader we give a flow chart showing the relationship the several sections bear to each other. T h e expressions “D9.3, L3.5, P2.6, and T2.7” refer respectively to Definition 2.3, Lemma 3.5, Proposition 2.6 and Theorem 2.7. We will observe the following conventions: “N” denotes the natural numbers 0,1,2, ... and unmodified “integer” usually means natural number, exceptions to be explicitly mentioned in the context. We omit parentheses as much as possible, particularly in forming the composition of functions; thus, “fg(x)”rather than ‘If(g(x))”. The characteristic function cA of a subset A of N is 0 on A and 1 off A . The notations “x I+ j ( x ) ” o r “x I+f y” give the actual value assigned t o x under the mappingf’. We usually omit the f when this will cause no confusion. Finally, the n-tuple (x,,..., x,~)is abbreviated t o when it is not necessary t o explicitly give the coordinates. The universal and existential quantifiers are respectively denoted by ‘‘V” and “ 3 ” . All other notation will be defined ill sitzr.

F.B. Cannonito and R . W. Gatterdam, Corllputability of group constructions

Definition of g n ( A ) computability: D2.1, D2.2, D2.3, D2.4

1

Technical aspects of Z n ( A ) computability: Pf. of T2.4, L2.5, L2.6

Z n ( ~groups: )

L3.1, T3.2, D3.3, P3.9 Finitely generated ,%'(A) groups: T3.4, 1.5, C3.6,C3.7 Primitive recursive groups: C3.8

Em beddings:

product: /D4-1 Direct

1

P4.2, P4.3

1

+ P4.4

/

Free product with amalgamation: D4.5, T4.6,C4.7,C4.8,

P4.15,P4.16,P4.17 Britton exten-

Subgroups D4.18, L4.19, 14.20

c5.4

Higman, Neumann, Neumann embedding: T5.5

Many-fold Britton ex tensions: L5 .lo, L5.11

.1

P5.6,P5.7

To Higman Thm. (Part 11) Fig. 1.

369

370

F.B. Cannonito and R. W. Gatterdam, Computability of group constructions

$2. Relative Grzegorczyk hierarchy

The purpose of this section is to define a relative Grzegorczyk hierarchy and t o verify certain properties of this hierarchy are retained in the relative version. Our point of departure is the hierarchy defined by Grzegorczyk [ 51, but we modify the definition and relativize with respect t o an arbitrary subset of the natural numbers A C N . The modifications are to use the characterization of the Crzegorczyk hierarchy due t o Ritchie [ 101 and replace recursion by iteration plus additional initial functions in the manner of Robinson [ 1 1 1 . We relativize by adjoining the characteristic function eA to the initial functions similar t o Davis' definition of relative recursion and relative primitive recursion, [ 31 . Essential use is made of Ritchie [ 101. Definition 2.1. A function f ' : N k + l + N is defined by limited recirrsion from functionsg : N k N, h : N k + 2+ N and j : N k + l + N if it is given by the schema +

subject t o the condition

The function 1':N N is defined from h : N -+ N and j : N lirriitcd itcrutioii if it is defined by the schema: -+

subject t o the condition

-,N by

F.B. Cannonito and R. W. Gatterdam, Covputability of group constructions

371

Of course the word "limited" in the definition above refers t o the bounding functions j . Omitting the bounds j one has the definition of primitive recursion and iteration. Following Ritchie, we use the "pairing" functions

J(x,Y 1 = ( ( x + Y )2 + X I 2 + y K ( z ) = E ( [ z " ] ) =[ z " ]

L ( z )= E ( z )= z

I [[z'/,]'/,I2

I[ 2 ' / , ] 2

where [2"3 = largest integer whose square is less than z. The important properties of these functions are KJ(x, y ) = x and LJ(x,Y ) = Y . = K , M 2 ( 2 ) = L we deInductively for n Z 2 and J = J ( 2 ) , fine

Then the relation between recursion and iteration is given by Theorem 3.2 of [ 101 :

Theorem. Let f(x("), y ) be defined b y primitive recitrsioiz

for

11

2

0. Then

where ,f' is defined b y the iteration

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F.B. Cannonito and R. W. Gatterdam, Computability of group instructions

.f"(z + 1 ) = H " f ' ( z ) fbr

H " ( w ) = J ( K ( w )+ I , H ' ( K ( w ) , L ( w ) ) ) EI"(z, w ) =

H ( 0 , g(O("))) if z = 0

(li(z. w )

otherwise

The importance of this theorem is that in a class of functions containing the pairing functions J , K, L , J @ ) ,M i @ ) and closed under substitution, closure under iteration implies closure under recursion. Moreover, J being monotone increasing f @ ( " ) , y ) < j(x("),y > implies f ' ( z ) < J ( Z , j ( ~ F + l ) ( z ..., ) , M;;')(Z))) so closure under limited iteration implies closure under limited recursion. We now define the relative Grzegorczyk hierarchy using the functionsf;, : N 2 + N of Ritchie. Definition 2.2.

Definition 2.3. Let A C N . Then the class of functions g n ( A )for n > 2 (with domain N k for arbitrary k and Range N ) is the smallest class of functions containing the initial functions Z(x)= 0

F.B. Cannonito and R. W. Gatterdarn, Computability of group constructions

Umn(x1,..., xn) = x,

for 1 < rn

313


f&

v)= x + 1 f' (x, v>= x + Y f n (x, Y)

E(x) = x CA

(x>=

[x"]

[ 1 ifi fxx e AA '0

E

and closed under substitution and limited iteration. Observe that E(x) E En(A) implies the pairing functions are in Gn(A) so by the theorem stated above, Gn(A) is closed under limited recursion. Ritchie's main result is that the usual Grzegorczyk hierarchy is g n = Cn(N). Clearly g n C C n ( A )for every A . Also, ~ , ( X , ~ ) E G G ~' +C' cg ' + l ( ~ ) i m p l i e s ~ " ( A ) c&"+'(A). The following theorem is useful.

Theorem 2.4. I f f 1 N k + N is defined by primitive recursion from functions in Gn(A)fi)r n > 2 then f E g n + ' ( A ) . Proof. By the previous theorem it suffices t o consider f : N -+ N given by iteration, f(0)= c = h o ( c )and f(x + 1) = h f ( x )= h X + l ( c ) (i.e., the composition h o h o ... 0 h ( c ) x + 1 times). We must show there exists j(x) E g n + l ( A ) such that f(x) < j ( x ) . The proof given is a modification of the techniques used in the proofs of Theorems 2.2 and 2.3 of [ 101. The modifications allow the application of these methods directly to the class Mn(A) for n 2 2. First we need some facts about f,, (x,y). The proofs which are not included may be found in [ 101 as indicated o r may be supplied by the reader using straightforward induction arguments. For n 2 2,fn(x, l ) = x . (pf: inductivelyJn+l(x, 1) = (I) fn(x, I,, (x,0 ) )= f, (x, 1) and f2 (x, 1 ) = XI. F o r n 2 1, x > 2, fjT (x,y ) is a strictly monotonic in(2) creasing function of y . (Lemma 1.1 of [ 101 ).

314

F.B. Cannonito and R. W. Gatterdam, Compatibility ofgroup constructions

(3)

(4) (5) (6) (7)

For y Z 1 and all n, f n (x,y ) is a strictly monotonic increasing function of x . (Lemma 1.2 of [ 101 ). Forx2 2,y> Oandn2 2,f,(~,y)Gf,+~(x,y). (Lemma 1.3 of [ 101. F o r p 2 2 , n Z 3 , x 2 2,andallq,fn(fn(x,p),q)< .fn ( x ,f, - ( p , 4)). (Theorem 1.1 of t 1011. For m > n 2 2 , x Z 2 and all y , z , f, (f, (x, y ) , f , ( x , z ) ) < f, (x, Y + z ) . (Theorem 1.1 of [ l o ] ). For any f ( x ( P ) E ) &.(A) with n Z 2, there exists k > 0, such that when x i 2 2 for all 1 < i < p , f ( x ( P ) < ) j,+, ( J ( P ) ( X ( P ) ) , k ) .

Proof of (7). We show that the desired property holds for the initial functions and is preserved under substitution and limited iteration. For x, y , x i 2 2 we have:

It is easy to verify by induction that f 3 ( x ,y ) = xy and that there exists Y , s > 0 such that for all x Z 0, J(4)(x,..., x ) < ( Y X ) ~(the Y and s depend on 4 ) . Then J ( 4 ) ( x ,..., x ) < ( Y X ) ~= f 3 ( Y X , s ) = f 3 (.f2( x ,Y ) , s) G .f3
F.B. Cannonito and R . W. Gatterdam, Computability of group constructions

315

Thus the desired property is preserved under substitution. If f ( x ) is defined by limited iteration from h ( x ) and j ( x ) , then f ( x ) < j(x) < fn+l(x,k ) for some k so limited iteration preserves the desired property. Thus we have verified (7). Returning now t o the proof of the theorem let f be defined by f ( 0 ) = c = h o ( c )and f ( x + 1) = hf(x) = h X + l(c) for h ( x ) E g n ( A ) and y12 2. Then by (7), there exists k such that h0.I) < f n + l ( y ,k ) for y 2 2. In particular hO(y) = y < max (2, v)= f n + l ( m a x ( 2 , ~ ) , f n + l (max (2, ~ ) , f , +( k ~ ,0)) for a l l y . Inductively assume h X ( y )< f n + l ( m a x ( 2 , y ) , f n + l ( k , xfor ) ) a l l y . Then

where the first line follows from (7), the second line from the inductive hypothesis a n d f n + l (max(2, ~ ) , f , + (~k ,XI) > 2, the third line from ( 5 ) , the fourth line from ( 1 ) and the fifth line from (6). Thereforef(x) = h X ( c ) < f n + l ( m a x ( 2 , c ) , f n + l ( k , - x ) ) E g n + ' ( A ) 0 as a function of x.

316

F.B. Cannonito and R. W. Gatterdam, Computability ofgroup constructions

Since by the theorem unlimited recursion leads from the class G p 2 ( At)o the class E n + l ( A ) ,the class U : = 2 E n ( A )is closed under iteration and hence primitive recursion. Thus we have

Corollary 2.5. The class U,,28n(A) is the class of A-primitive recursive functions, and f o r all A and each n > 2 , E n ( A ) is properly contained in E n + l ( A ) . The special case c ( A )E E m .for some m , (i.e., A is Z m decidable) is of particular interest. Here En(A)C 8 p for p = max { n, m } . However, suppose n < m so E f l ( A ) C E m . Then by the usual estimates an unbounded recursion leads from g n ( A ) t o g m + I . However we see from Theorem 4 that such an unbounded recursion leads from E n ( A ) to g n + l ( A ) C g m . It should be noted that c ( A ) E 8", c ( B )E g m does not in general imply Cin ( A ) = a n @ ) . The following lemma is useful in bounding certain recursive processes which appear later. I t is of considerable use in Part 11.

Lemma 2.6. Let f ( x )= g(x, h(.u))for g defined recursively by g(x, 0) = x and g(x, y + 1) = k ( x , y, g(x, y ) ) .Assume g(x, y + 1) < b(g(x,y ) ) and h ( x ) ,b ( z ) are E n computable f o r n > 2. Then f ( x ) is bounded by an g n + l computable jknction o j x . Proof. Define q ( x , 0) q ( x , 1)

=x

=b(x)

so b being g n computable, q is g n + ' computable. Clearly g(x, 0) < q ( x , 0) and g(x, 1 ) < q(x, 1). Inductively g(x, y + 1) < b(g(x,y ) ) < CPFdy)b ( z ) = q(x, y + 1) since g ( x , y ) < q(x, y ) . Thus f ( x ) < q(x, h ( x ) ) which is E n + ' computable. Notice that n o assumption is made above on the computability of k . A special case of note is n = 3 and k & " ( A ) computable for IZ > 4. Then f ' is ti'* ( A ) computable.

F.B. Cannonito and R. W. Gatterdam, Computability of group constructions

377

53.g n ( A1 groups and the word problem Following the definition given in [ 1 ] we say a countable group G is an “&“(A)group” (or is “ & . “ ( A computable”) ) if it has an “index” (i, m, j ) for i an injection of G o n t o an g n ( A ) decidable computable function m : i(G)X i(G)-+ subset of N , rn an i ( G ) where m is given by (i(gl), i ( g 2 ) )&! i(g1g2), and likewise for j : i ( G ) -+ i ( G ) given by i(g) & i(g-’). F o r G , and G, E n @ ) computable groups with indices (i,, m l ,j l ) and (i,, m 2 ,j,) respectively, we say a homomorphism f : G, G, is “ & “ ( Acomputable” ) if f:i , ( G , ) + i 2 ( G 2 )b y 3 , ( g ) p i2f(G) is an g n ( A )computable function. Note that the computability o f f depends on the indices for G, and G,. When not obvious from context, we will say “f is g n ( A )computable relative t o indices (i,, m , , j , ) and (i2, m2,j2)”. We freely use the results of $ 2 , the g 3 pairing functions, the g 3 computable functions ## 1-2 1 of Kleene [ 71 p. 222f (note in particular ## 16 through ## 2 1 , p. 230), the statements # A through # F of Kleene [ 71 p. 222f modified by replacing “primitive recursive” by “ E 3 ” and the concept of a group given by generators a , , a 2 , ... and relations R 1, R,, ... (see [ 81 ) which will be denoted G = ( a l , ...; R , , ... ). We begin with a lemma stated as Lemma 4.1 of [ 13 . In the proof we use a slightly different index which is more convenient to use later. -+

Lemma 3.1. A free group F = ( a , , ...; ) on finitely or countably many generators is $ 3 . Proof. F consists of freely reduced words of the form w = a j o a o... aira, for a o , ..., ar positive or negative (but non-zero) integers. Write 6 = 2a if a > 0 and (Y = -2a - 1 if a < 0. Then using the pairing functions of 5 2 we write for w A (the empty word) r

+

378

F.B. Cannonito and R. W. Gatterdam, Computability of group constructions

for p k the k th prime starting with p o = 2, p1 = 3 , ... . Denoting G N ( x ) V k < 1 hx((x)k 0) we see that x E i(Fn)for Fn = ( u o , ..., a,. ; ), finitely generated, iff x = 1 v [ G N ( x ) A V k < Ih.x((K((x),) < 1 2 ) A ( L ( ( X ) ~ )O))] and x E i(F,) for F , = ( u o , ... ;), countably generated iff x = 1 v [ G N ( x ) A V k < 1hx(L((x)k) O)] . To compute m(x,y ) one must “decode” x and y as words, freely reduce the concatenation of these words and then encode the result. It is clear that such a process can be interpreted by a recursion defined on G 3 functions. Moreover since m(x,y ) < x * y (a relation we use later), the recursion is limited and rn is an E 3 function. Similarly it is clear that inversion, j , can be performed by a recursion defined on g 3 functions and limited by j ( x ) < (x), + I)) so j is g 3 computable. plhwexp ( ( l h x ) ( o f n a x

+

++

+

+

\l
Using the above lemma we can show that the study of Z * ( A ) groups is non-emp ty. Theorem 3.2. For every countable group G there exists A that G is an E 3 ( A )group.

C

N such

Proof. G being countable (or finite) there exists a presentation, 1 + K + F + G 1 , as an exact sequence, for F free and at most countably generated hence E 3 . Let A = i ( K ) C i ( F ) C N. Following [ I ] , Theorem 5.1 we define the Z 3 ( A ) predicate +

Thus the predicate E says “x and y are in the same coset modulo K”. We define a unique index for each such coset, and hence for G , by r ( x ) = p.l’ < x ( E ( x ,v)).

Now i,(G) = ri(F) and x E i, ( G ) x E i ( F ) A r ( x ) = x . We define m G ( x y, ) = rm(x,y ) and note m&, y ) < x * y . Similarly j c ( x ) = rj(x ) . ++

F.B. Cannonito and R. W . Gatterdam, Computability of group constructions

319

Theorem 3.2 suggests that if G is finitely generated (f.g.), then A is the encoded version of the word problem. The relation between E n ( A )groups and the word problem is made precise in the following definition, which, for the sake of the discussion further below, will be given for countably generated groups. However, note that the relationship between B n ( A ) groups and the word problem depends on finite systems of generators, only.

Definition 3.3. A group G is said t o be E n @ ) standard relative to an index (i, rn, j ) if i is defined by minimalization from a presentation 1 + K F -+ G + 1 for F free on at most countably many generators, n >, 3 , and A C N , the minimalization to be performed on E(x,y ) of the proof of Theoyem 3.2 for E(x, y ) B n ( A ) decidable. -+

In Definition 3.3 we d o not require that A = i ( K ) but merely that i ( K ) be E n ( A ) decidable so that E ( x , y ) is E n ( A ) decidable. Clearly the index given in Theorem 3.2 is a standard index. Intuitively, a finitely generated G is given a standard index by solving the word problem for a presentation of G by an g n ( A )process. Theorem 3.2 says that the word problem for a countable group can be solved for any given presentation and some A . Our next theorem shows that for finitely generated groups the level of computability of the word problem is independent of the f.g. presentation.

Theorem 3.4. Zf G is fg. and g r z ( A )standard for n >, 3 then any standard index of G is g n ( A ) . Proof. Let 1 K + F $ G + 1 be a presentation of G for F finitely generated. We show K is g n ( A )decidable by showing u is g n ( A ) computable relative t o the index (i, rn, j ) on F and (i', ~ n j' ',) , the given g n ( A )standard index of G. Then x E i ( K ) tf x E i ( F ) A 6(x) = i'( 1). -+

380

F.B. Cannonito and R. W. Gatterdam, Computability of group constructions

F o r w E F , u(w) is computed as the product of images of generators of F corresponding t o the spelling of w . Thus since F is finitely generated, 6 ( x ) can be interpreted by a recursion on l h ( x ) involving m' where 6 is % " ( A )computable if the recursion is bounded. Here m ' ( y ,z)< y * z since ( i ' ,m ' , j ' ) is a standard index so 6(x) is bounded by * y 2 * ... * y y where each y i as well as r can be computed from x by an E 3 function. But *IzIy i = y l * ... * y r < r

p k exp

lhyi

21

(h c c i=l

n=O

(.pi),

1

for k =

5

i= 1

1hy,. Thus the recursion

is bounded and 6 is ?Z'I(A)computable.

0

Corollary 3.5. If G is f : g . and &'z(A) compiituhle f o r ti Z 3 then G is % . + l ( A ) s t u r ~ h r d . Proof. In the proof of Theorem 3.4 if G is % " ( A )but not E n ( A ) standard, the recursion defining u need not be bounded and therefore u and hence i( K> may be 8 n + l ( A ) rather than 8 ( A ) by Theorem 2.4. U Corollary 3.6. I f G i s f g . and & " ( A )computable for n 2 3 then G is 8 3 ( B )staridurd f h r B g n + ( A )decidable. Proof. In the case of Corollary 3.5, set B = i ( K ) . Then B is g n + l ( A ) decidable and G is E 3 ( B )standard by the usual construction of Theorem 3.2. 0 Corollary 3.7. I j G is an G n ( A )computable group f o r n Z 3 and H < G is a fiizitely generated subgroup then H is g n + l ( A )standard. I f G is ?Zn(A)standard then H is & " ( A )standavd. Proof. In the proof of Theorem 3.4 and Corollary 3.5 we did not need u surjective. Corollary 3.7 is then a restatement for u not surjective and N 2 Id'lK. 0

F.B. Cannonito and R. W. Gatterdanz, Coinputabilit-y of group constructions

Corollary 3.8. I f G is xg. and &Iz standard but not .forn 2 4, then G is not t;m j b r m < n - 1.

&Iz-

381

* standard

Proof. This is a restatement of Corollary 3.5 for A = N .

0

Theorem 3.4 above is a mild generalization of a result due t o Rabin, [ 91, that the computability of the word problem depends o n G and not any of G’s finitely generated presentations. Corollaries 3.5, 3.6, and 3.8 show the relationship between standard and non-standard indices, Corollary 3.8 is a special case for later use. ) is Corollary 3.7 shows that the property of being & ? ? ( Astandard inherited by finitely generated subgroups although in general a finitely generated subgroup of an M n ( A ) group may be Z n + l ( A ) . From the proof of Corollary 3.7 we also see that the embedding H - G is M”+I(A)computable since it ciin be computed by regarding an element in the index of11 t o be in i ( F ) and applying the &?Z+*(A) computable 6. As a companion t o the herditary result of Corollary 3.7 we see that under suitable conditions, quotient groups of E n ( A )groups are & ? ? ( A ) .

Proposition 3.9. I f G is an & : “ ( Agroup ) f o r 1 1 2 3 and K < G is an & ; “ ( Adecidable ) normal szibgrozip then GIK is Z n ( A )coniputable. Proof. In the proof of Theorem 3.2 replace E‘ by G and G by G / K and let (i, rn. j ) be the original index for G. The same definitions of E, c, 111 and j work as does the decidability criteria. 0 It is of course not true that all quotient groups of &’z(A) are E n ( A )for if that were the case all groups would be E 3 and, in particular, have a solvable word problem. Note however that if B = i ( K ) C i ( G ) in Proposition 3.9 then by replacing A by A join B = { .Y : 2 I .Y 3 [ s / 2 ]E A & 2 .x

we see G / K is t;17(Ajoin B ) computablc.

3

[(x

2

1)/2]

EB

382

F.B. Cannonito and R . W. Gatterdam, Computability of group constructions

$4. Free products with amalgamation

The free product with amalgamation is a useful construction when dealing with decision problems in groups since intuitively the normal form theorem yields decision procedures for such products modulo the decision procedures for the groups and the amalgamated subgroups. In the following we will study the free product with amalgamation-for g n ( A )groups, y 1 > 3 , where the subgroups amalgamated are themselves S n ( A )as is the isomorphism relating them. In this context recall that if one decision problem has an S ' ( B ) solution and another an &q(C)solution then they both have g n ( A )solutions for n = max ( p , q ) and A = B join C = { x : 2 I x * [x/2] E A & 2 : x 3 [(x I 1)/2]€ B l . In view of this our hypothesis will always involve a single computability class, g n ( A ) . In general we require not only the index of the product but also the manner in which the original factors are embedded. T h e following definition is used t o make such statements precise. Definition 4.1. Let C and L be g n ( A )groups with indices (i, m,j) and (i', m', j ' ) respectively. Suppose K : G + L is an embedding. Then we say K is an Iln(A)embedding o f G into L (or simply G is E n ( A )embedded into L ) i f (i) (ii)

(iii)

2 : i ( G ) - i ' ( L ) is &'*(A)computable with respect to ( i , nz, j ) and (i', m',j ' ) , (i.e. K is g n ( A ) computable). i i ( G ) c i ' ( L ) is s n ( A ) decidable, (i.e., G is an Z n ( A ) decidable subgroup of L ) . 2-l : 2 i ( G ) i ( G ) is C n ( A )computable with respect to (i', rn',j ' ) and (i, nz, j ) , (i.e., K - ' is g n ( A ) computable). -+

Notice that Definition 4.1 involves the particular choice of indices being used. In the following, the particular choice of indices

F.B. Cannonito and R. W. Gatterdam, Computability of group constructions

383

is specified only when not obvious from context. Also observe that in the case where G is f.g. and (i, m , j ) and ( i ’ , in‘,j ‘ ) are standard indices, conditions (i) and (iii) are superfulous. We demonstrate the type of result we desire for free products with amalgamation by first considering the more obvious situation for direct product, denoted G I X G,. Proposition 4.2. Let G, be E n ( A )groiips f o r n > 3 and a = 1,2. Then G = G, X G, is an $ “ ( A ) group and tlie embeddings G, -+ G are & ( A ) e m beddings. Proof. Let G, have indices (i,, nz,. j,) and give G an index by i ( g l ,8,) = J ( i , (gl), i, (8,)). I t is clear this index has all of the precribed properties. C We may ask about the universal properties of the direct product. Clearly the projections are E ’ l ( A ) computable. Proposition 4.3. Under the asszrrnptiorzs of‘ Proposition 4.3assiinie also K is an E n ( A )groiip and Q, : K G, are g n ( ( A ) cornpictable homomorphisms. Let 7 ~ , : G + G, he the projections and Q : K + G the unique homomorphism such that r a a = a,. Theri a is G n ( A ) cornpii table. -+

Proof:

G ( x ) = .l(Gl (x), G 2 (x)).

As we saw the index given G = G I X G, by Proposition 4.2 was the natural index with regard t o the universal property. However, if G I , G, (and hence G ) are f.g. then it is not the standard index. Next we relate this index t o the standard index. Proposition 4.4. Under the assimptioris of’Proposition 4.2 assiirne also that G, ure ,fg. and & “ ( A )stundurd. Then G is EI1(A)stundard and the idcntity isomorphism oii G is g n ( A )c o m p t u b l e jrorii G with standard index t o G with iride.Y ( i , m ,j ) oj’l’roposition 4.2 arid f r o m G with index ( i , rn, j ) to G with standard iride.u.

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Proof. Let G , be generated by a l , ..., a, and G 2 by u r + l , ..., a,. Consider F = ( a , , ..., ar+s;) and a 1 : F + G1 by ui p ui for i < r, ai p 1 for i > r and a 2 : F + G, by ai t+ 1 for i < r, ai I+ ai for i > I: Then Ga being & n ( A )standard the a , are I n ( A )computable and induce a : E'+ G which is 8 n ( A ) computable by Proposition 4.3. Thus k e r a is & " ( A )decidable so G is E n ( A )standard. This argument also shows that the identity isomorphism on G is g n ( A ) computable from the standard index t o the index of Proposition 4.2. Conversely since the Ga have standard indices it is an S n ( A ) process t o write ( g l , 8,) = (up1( P ~ )w,(ai)) , as words o n the generators (wlinvolves only u , , ..., u, and w2 only ur+,, ..., ur+J and so associate ( g l , g 2 ) with \v = w1 (ui)w2( u i ) E I;. Reducing w modulo kera, the identity isomorphism on G is E n ( A )computable from 0 the index of Proposition 4.2 t o the standard index. We will proceed along the same lines for free product with amalgamation. That is, first we will dcvelop an index which is natural, then verify the universal property and finally, restricting our attention t o f.g. groups show the relationship to the standard index. In this case the natural index will reflect the normal form representation of elements in the free prodiict. Informally this index will be called the normal form index. Since the normal form requires coset representatives in the factors modulo the amalgamated subgroup (amalgam for short) we use the following definition (compare [ 41 Definition 2.2).

Definition 4.5.Let G be an E'*(A)group and II < G an & " ( A )decidable subgroup for n 2 3 . An E n ( A )right coset representative syst e m for G mod H is an &'!(A)computable function k : i ( G ) + i ( G ) satisfying: (i)

x E i(G)

(ii)

m(x,j k ( x ) ) E i ( H )

(iii)

x

E

y

E

i(G)

m ( x , j ( j ? )E ) i(H)

++

k ( x ) = k(.y)

i(N)+ k ( x ) = i( 1 )

Intuitively k is a method of choosing right coset representatives for elements of G by a n Mn(A)process. There always exists G n ( A )

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right coset representative systems. F o r example for x E i ( G ) define k(x) = p y < x(y E i ( G ) A m ( x ,j ( y ) )E i ( H ) )if x $ i(H) and k(x) = i( 1 ) if x E i(H).Notice that any k as in the definition decomposes x E i ( H )as x = m ( h ( x ) ,k ( x ) )for h ( x )= m ( x ,j k ( x ) ) ,i.e. allows us to write g = hg' for h E H and g' a particular coset representative by an E n ( A )process. Our use of such representative systems is seen in the following.

Theorem 4.6. Let Ga be g n ( A )groups for n 2 3 and a = 1 , 2 . Assume Ha < G, are G n ( A )decidable subgroups and cp : H I + H , is an isomorphism such that cp and cp-, are g n ( A ) computable. Then for each choice o f E n ( A )right coset representative systems k, : ia(Ga)-+ i a ( G a )there is an & n + l ( A )index on G I *pG2 (the ,free product of G , and G, amalgamating H , and H , by cp). The natural embeddings G, -+ G are &"'+'(A)embeddings. Proof. This is a relativized version of Theorem 2.1 of [ 41 replacing the k, of that proof by the given k,. 17 Corollary 4.7. Let G, be g n ( A )groups f o r n 2 3 and a = 1 , 2 . Then G, * G, is an g n ( A )group and the natural embeddings G, G, * G, are g n ( A )embeddings. +

Proof. Relativize Corollary 2.1.1 of [ 41 . We now have the universal property.

Corollary 4.8. Under the assumptions o f Theorem 4.6 assume also that K is an & n ( A )or an En+1(A)standard group and r, : Ga + K are $ n ( A ) computable homomorphisms agreeing on the amalgam (i.e., if h E H , then r , ( h ) = r2cp(h)).Then the unique homomorphism r : G , *pG2 + K extending r 1 and r2 (i.e., such that r I G, = r 1 and r I G, = 7,) is , , + , ( A ) computable. Proof. Relativize Corollary 2.1.2 of [ 4 ] for K g n ( A ) .F o r K $ n + l ( A ) standard the bound on multiplication in K yields a bound 0 on the recursion for 7.

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We now can relate indices of G I *pGG2which arise from different choices of the coset representative systems (compare [ 41 Proposition 7 . 3 ) .

Corollary 4.9. Under the ussirinptions o f Theorein 4.6 let G,

*q

G,

liuile ir1tfc.v ( i , 1 1 1 , j ) with respect t o & “ ( A )coset representative sys-

tenis k a und (i‘* i n ’ , j ’ ) ltiitli respect to ( ; “ ( A )( m e t represerztative ,~>~steriis k i . Tlicn the identity isoniorphisni on G I *p G, is G n + l ( A ) c.oinpiituhle relatiw to ( i , rn, j ) and (it,In‘, i t ) .

Proof. The ernbeddings G, + G , *qGG2 are 8 ” ( A ) computable with respect to Cia,ina,j,) and ( i f ,i n ‘ , j ‘ ) so extend t o the identity isoniorphism on G I * q G 2 , & t l + l ( A )computable with respect t o (i. i n . 1 ) and ( i f , rn’,j ’ ) by Corollary 4.8 with K = G I *,+,G2,I % ~ * + ~ ( A ) computable using Lemma 2.6 t o bound the recursion for 7-. 0 We now restrict our attention t o f.g. groups and consider standard indices.

Theorem 4.10. Let G‘, he f.g. Grl(A)sturicfurd groicps f o r 11 Z 3 und ci = 1. 3. Assiriiie H , < Ga ure r t l ( A )decidable siihgroiips and cp : I ] , H , is un isornorphisrn such tliut cp und cp-‘ are & n ( A )cornpiituhlo. TIicri G ] :*$ G,- i,, [;)‘+] ( A ) staiidulef. +

Proof. We show that the word problem is g n + l ( A )decidable for a particular f.g. presentation of G I *pG2. The argument will be a “spelling” argument but it should be clear that it can be encoded as a recursion defined on G t 1 ( Afunctions ) and hence G I *pG2 is tlf7+1(A)standard. Let C , be generated by u , , ..., ar and G, by ..., Since G, are e f l ( A) standard there is an E n ( A )process for recognizing if a word o n the u , , ..., u, is in 11, and if so computing a word on u r + ] , ..., I(,.+, corresponding t o its image under p. Similarly one can compiite p1 of a word in H,. The statement that these processes are C t 7 ( A) requires that the original indices of the G, be standard. Let 11’ be any freely reduced word on the ui. If the first symbol in 11’ is ;I power of ak for 1 < h- < r , write HI = w 1 ... wI,for each

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381

wi+ 8 and so that j odd implies wiinvolves only symbols ak for 1 < k < r and j even implies wjinvolves only symbols “k for

r

Y

+ 1 < k < Y + s. Similarly if the first symbol is a power of ak for + 1 < k < Y + s write w = w 1.. w p as above interchanging the roles

of even and odd. In [ 81 such wiare called syllables and p is called the syllable length of w. Clearly the decomposition of tv into syllables is an E 3 process. We proceed by induction on the syllable length p of w . I f p = 1 then w = 1 in G , * q G 2 iff w = 1 in G , or w = 1 in G,, an g n ( A ) decision. I f p > 1 let 1 < q < p be the smallest integer such that either w q E H , or wq E H , . If there is n o such q, LV 1 in G , * q G,. The search for q can be interpreted as bounded ininimalization defined by the E 3 function wh::h decomposes w and the g n ( A )decision functions for H a . 11 tl-ere is a q and w q E H , , apply cp to w q as described above and replace w by w’ such that w = w‘ in G, * q G 2 but w‘ has shorter syllable length than MI. Similarly if w q E H , apply cp-l t o get w’.By induction we conclude that the word problem for G , * q G 2 is E n + l ( A )solvable. Thus if B C N is the encoded image of the kernel in the presentation of G , *,+,G2 on a,,..., ar+s, then G, * q G 2 is g 3 ( B )compuk) since B is E n + l ( A ) able by Theorem 3.2 and E r z + l ( Acomputable decidable by the above. 0

+

Corollary 4.1 1 . L e t G, be fg. g n ( A )standard groirps ,for n a = I , 2. Then G, * G, is G n ( A )standard.

> 3 and

Proof. In the proof of Theorem 4.10, the questions wk E Ha are replaced by w k = 1 in G, or G,. Since the index of w ’(and hence every succeeding reduction in the induction) is less than the index of w (there is n o cp or cp-1 t o possibly increase the index), the recursion used in an encoded version of the proof of Theorem 4.10 is bounded and the word problem E n ( A )decidable. 0 We now study the relationship between the “nornial form” index of Theorem 4.6 and the standard index of Theorem 4.10.

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Corollary 4.12. Under the assumptions of Theorem 4. I 0 let ( i f , in',j ' ) be the $ r z + l ( Astaridurd ) iiidex and (i,m, j ) be the (A) (norrnal f o r m ) iridex giver? b.v Tlieorein 4.6 urising froin some $ n ( A ) right coset represeiitutiw system. T1ieri the identity isomorphisirz oii G I *?G2 is $}?+l ( A ) cornpiitable fioiiz the index (i,In, j ) to tlie index ( i , in',J ' j. It is & } ? + I ( A ) conzputable from the index ( i ' , in', 1') to tlie iiitlc)s ( i ,t n , J ) . Proof. Since G, are f.g. and & } ? ( Astandard, ) there are $I1(A)quof.g. free groups. Moreover for + G, for tient maps u, : s E i a ( G a )x, E ia(l.'ca))with Zr,(s) = .Y. Let u : F + G , *pG2 be the & , + ' ( A ) quotient map for F = F ( l ) *F ( 2 )and ra : F ( a )+ F the E 3 embeddings. Then the embedding G, + G I *pG2 relative t o (ia,ina,j,) and ( i f ,i n ' , j ' ) is given by x p i r i ( x ) and so is g n + l ( A ) computable. By Corollary 4.8 the extension of the embeddings, which is obviously the identity, is , , + ' ( A ) computable, the K of Corollary 4.8 being G , *pG2 and having a standard index. The second statement of the Corollary is immediate from the fact that the quotient u : F G * G is & n + l ( A )computable relative to the ? 2. index ( i , i n , j ) (since I. is f.g.) using Lemma 2.6 t o bound the recursion. 0 Corollary 4.13. Under the asminptioiis of Corollary 4. I I let ( i f ,tn', J ' ) be the E1I(A)staidart1 index and let ( i , 111, j ) be tlw E r l ( A ) (norinul forin) index g i w i by Corollarj- 4. 7. Tlicw the identity isomorplzisrn on G I * G, is &'l(A)cornpiitable f r o m the index (i,m , j ) to (i', in', j ' ) arid also-froin the index ( i t ,in', j ' ) to the index (i,rn,j ) . +

Proof. In the proof of Corollary 4.12 the quotient map u is &!?(A) computable when the amalgam is trivial and so by the remainder of the argument the identity is tA1*(A)from the index (i,in, j ) t o ( i f , in', j ' ) . The technique used to bound the index of a product in the proof of Corollary 4.7 ( [ 4 j Corollary 2.1. I ) can be used to bound the recursion used in the computability of u : /.'+ G I * G, relative t o the index (i, i n , j ) . Then u is & ' ? ( A computable ) and hence so is the identity. 0

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By a standard technique Theorem 4.10 can be used t o construct f.g. 8 3 ( A )groups for any A C N in the following sense.

Corollary 4.14. For any A C N there exists u t : g . 8 3 ( A )standard group G, such that if G, is E n ( B )standard .for n 2 3 then A is G n ( B )decidable. Proof. Set F ' = ( a, b ; ) ,F" = ( c, d ; ) ,H ' < F' the subgroup generated by { U ~ D L I;-x~E A } and H" < F" the subgroup generated by {~ ~ d x c E- A~ }.; Then H ' is freely generated by the uXba-X for x E A and so is E 3 ( A )decidable. Similarly for H " . Let cp : H ' + H" be the E 3 ( A )computable isomorphism given by a.xbu-X p for x E A . We consider G, = F' *@F " . By theorem 4.10 GA is E 4 ( A )computable but since L) and L)-* are restrictions of thc identity on N, the index of each successive w' (in the proof of Theorem 4.10) is less than or equal t o p Ihx exp ( J (4, max L ( ( x ) , )). 1h x ) O
for .x the index of w and hence the recursion used in solving the word problem is bounded so GA is E 3 ( A )standard. Suppose GA is g n ( B ) standard for I I 2 3 . Then for any .Y E N, t o decide if x E A , compute the index ~ f u ~ b u - ~ in~ G,~ dby - ~ an g n ( B ) process. This process is a decision procedure for A since x E A i f f u X b a - X c X d - l c -=X 1 in G, . o The above constructions for f.g. G t z ( A )groups yield interesting corollaries for 8'1 = &t1(N)groups. In particular we consider ~ 3 3 A ) groups for A &Iz decidable with y1 > 3 .

Proposition 4.15. Under the assumptioris of Theorem 4.10 let 1 1 2 3 und A be & m decidable f o r m 2 n + 1. Then G , *@G 2 hus a n G m iridex f o r each &" ( A ) coset reI,resciitati,~e.yj'steni. Moreow,- G', *@ G, is 8" stundurd arid f o r in 2 pi + 1 the identity isomorphism is G m computable with respect t o arij, o.f the uhove indices. Proof. Since m 2 n + 1 and A is 1;" decidable & " + ' ( A )C 8" so the first part of the statement is immediate from Theorems 4.6 and 4.10. Similarly g n + l ( A )c grn f o r m 2 n + 1 so the second part follows from Corollaries 4.9 and 4.12. 0

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F.B. Cannonito and K. W. Gatterdam, Computability of group constructions

Of course we could specialize other statements t o the case of 8)*groups, e.g. Corollary 4.8. Of special interest is the following specialization of Corollary 4.14.

Proposition 4.16. For 11 > 4 there exist $ I 1 standard groups which arc>not E l ' standard. For I? > 5 there exist E n standard groups whicli ure riot Z ~ I2 . ~

Proof. I n [ 1 ] it is shown that for n 2 4 there exists sets g n decidable but not & ' I - l decidable. If A is such a set then the group G, of Corollary 4.14 is 8 3 ( A ) C M f 2 standard but not 8 n - 1 standard since that would imply A 8lZ-l decidable. If n > 5 then G A is not 8'*-2 since by Corollary 3.8 that would imply GA is Zn-' standard. 0 We say a group is primitive recursive (p.r.) if it is E n for some say a group is finitely presented (f.p.) if it has a f.g. presentation involving only finitely many relations. C.F. Miller I11 raised the question of whether there could exist a p.r., f.p. group containing as subgroups a copy of every p.r., f.p. group. As reported in [ 2 I the answer is strongly negative.

I E . We

Proposition 4.17. There does not exist a p.r. group containing every f.g., p.r. group a s u subgroup. There does not exist a p.r. group containing e w r t - J p . . p.r. group us u subgroup. Proof. By Proposition 4.16 there exist f.g. E n groups not En-* for n 2 5. Applying Theorem 5. I of [ 4 ] there exist f.p. Ern groups for t11 2 I I 2 and every n > 5 . Now suppose there was a p.r. group C containing a copy of every p.r., f.g. (respectively f.p.) group as a subgroup. Then G is & p for some p so by Corollary 3.7 each of its f.g. subgroups is $P+l which is a contradiction since we have just seen there exist f.g. (respectively f.p.) g m groups for arbitrarily high 111. 0 ~

The remainder of this section is devoted t o a technical devise for finding the computability of certain special subgroups of a free product with arnalganiation. The computability of these subgroups

F.B. Cannonito and R. W. Gatterdam, Computability qf‘xroup constnictions

39 1

is of importance for some later applications so the statements arc included here for completeness. The casual reader may choose to ignore this material. We present a generalization of Definition 2.3, Proposition 2.3 and Lemma 2.1 of [41.

Definition 4.18. Let G be an g n ( A )group for 11 2 3 and H < G an E n ( A )decidable subgroup. A subgroup K < G is said to be H, & n ( A )compatible if it is g n ( A )decidable and there exists an & “ ( A )right coset representative system for G mod H satisfying in addition t o conditions (i) through (iii) of Definition 4.5: (iv)

x

E

i ( K ) + k ( x )E i ( K ) .

What we require in Definition 4.18 is the existence of an g f l ( A ) right coset representative system for G mod H so that whenever a coset Hg satisfies Hg n K flthen the representative of Ilg is in K . Notice condition (iv) is trivially satisfied when H < K < G , all &n(A)decidable, for then g E K , 2 E G, gg-1E H implies gg-1 E K so S E K and any coset representative of Hg is in K . The following lemma characterizes compatibility and is more convenient than the definition for applications.

*

Lemma 4.19. Let G be g n ( A )with index ( i , m , j )stich thut 0 $! i ( G ) . Let H < G, K < G he & n ( A )decidable siihgroirps. The fdlowing are equivalent: there exists un G n ( A )compiituble ,fzinctiorz (1 ) d : i ( G ) + i ( K ) u ( 0 ) m c h that d ( x ) = 0 1 3y(.v E i ( K )A m ( x ,j ( y ) )E i ( H ) )und d(x) 0 * m ( . ~j d, ( x ) )E i ( H ) (2) K is H, &!‘(A)compatible (3) His K , E n ( A )compatible.

*

++

Proof. Replace every occurrance of “p.r.” in the proof of Proposi0 tion 2.3 of [ 4 ] by “ & t * ( A ) ” . Observe that the condition 0 sf i ( G ) in Lcmma 4.19 is not critical since if 0 € i ( G ) ,i ( g ) can be replaced by i ( g ) + 1. Assume H < G, K < G all &:”(A) and also that { H , K } < G , the subgroup of G gen-

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F.B. Cannonito and R , W. Gatterdanz,Computability of group constructions

erated by H and K , is &?.“(A) decidable. Then we may take d ( x ) = 0 for x ($ i( { H , K } ) since if for g E G there exists g E K such that gg-1 = h E H then g = h i E { H , K } . Thus t o conclude K is H , g n ( A )compatible in G it suffices to show K is H , E ” ( A ) compatible i n { H, K } . The use of the notion of compatibility is found in the following lemma.

Lemma 4.20. Under the conditions o j Theorem 4.6 let Ka < G, be S“(A) ttcwidable szibgroirps satisfying K , is H a , & ) ’ ( Acompatible ) for a = 1 , 2 (I) (11) p ( K 1 n H i ) = K2 n H 2 . Tlieri K = { K , , K 2 } < G = G I *q G, satisfies K is E n + l ( Adecidable ) in G with respect to some normal ( 1) forrn index on G hence E n + l ( A )decidable in G with respcwt to a n y normal f o r m index or a n y standard index O F ZG ( i e the embedding K < G is an & “ + ‘ ( A embedding) ) f o r cp‘ = cp I K , n H , , K = K , *@,K2 (2) G u n K = K, j o r a = 1 , 2 . (3)

Proof. Since by Corollaries 4.9 and 4.12 all normal form indices and all standard indices are related by an & I 7 + , ( A )computable identity isomorphism on G it suffices to show K is E t7 + l(Adecidable ) re I a t ive t o thc particular G ( A ) right cose t rep resen ta tive sy sten1s with respect t o which the K , are Ha &Il(A)compatible. To show this ;IS well as ( 7 ) and (3) replace every occurrance of “p.r.” in the proof of Leniina 2.1 of [ 4 ] by “Mn+l(A)”except that “p.r. right coset representative system” should be replaced by “ & l 2 ( Aright ) 0 coset representative system”.

35. Strong Britton extensions I n this section we consider a construction closely related t o the free product with amalgamation, the so called strong Rritton extension. We will refer to presentations of groups on generators and relations, vir. C; = ( N , , . _ ;_R _..) and, when the meaning is clear, interpret this notation with liberty. For example, we may write

393

F.B. Cannonito and R . W. Gatterdam, Computability of group constructions

G, r ; ) to mean ( a l , ..., r ; R , , ... ) = G * ( r ; ) ;i.e. when the set of generators has some implied relations, we assume they are included in the relations of the presentation. We also use the notation { B } < G for B C G, { B }the subgroup of G generated by B. The following definitions are relativized versions of Definitions 3.1 and 3.2 of [41. (

Definition 5.1. Let G be an E n ( A )group. An E n ( A )isomorphism in G is an isomorphism cp : H + H ' such that H , H' are E n ( A )decidable subgroups of G and cp, cp-' are E n ( A )computable (relative to the inherited index on H, H ' ) . Definition 5.2. Let G be an E n ( A )group and cp an E n ( A )isomorphism in G. The strong Britton extension o f G by cp, denoted by G,, is the group with presentation ( G, t ; tht-l = p ( h ) tlh E domain cp ).

The E n @ ) computability of G and cp induces a computability structure in G, according to the following theorem and corollary.

Theorem 5.3. Let G be an E n ( A )group and cp an E n ( A )isomorphism in G, f o r n > 3. Then G, is an E n + l ( A ) group with index (i', m', j ' ) inherited as u subgroup o f a (particular) free product with amalgamation and its associated normal form index. The embedding G + G, is an E n + l ( A ) embedding relative to ( i ' ,m', j ' ) . Proof. This is a relativized version of Theorem 3.1 of [ 41 . The proof is identical replacing gn by g n ( A ) using Lemma 2.6 t o bound the recursion defining 7. Corollary 5.4.I f G is fig. and E n ( A ) standard in the hypothesis o f Theorem 5.3, then G , is E n + l ( A )with index (i', m', j ' ) and G G , is an $ n + ' ( A ) embedding. In addition, G, is E n + l ( A ) standard with The identity isomorphism on G, is g n + l ( A )comindex ([ I&, putable from index (i', m', j ' ) to index ([ and E n + l ( A ) computable from index (t h,A to index ( i t ,m', j ' ) . With respect to (l,&, fi the embedding G + G, is E"+I(A)computable und an g n + l ( A) embedding. +

a.

&,a

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F.B. Cannonito and R . W. Gatterdam, Computability of group constructions

Proof. We modify the proof of Theorem 3.1 of [ 41 . Since G is f.g. and 8I1(A)standard, L has an g n + l ( A )standard index, say (b &, i">, by Theorem 4.10. We compute 7 : L + L from index (i', m ' , j ' ) to the index r%, where the latter index is an g n + l ( A )standard index. Corollary 4.8 shows 7, induced by T~~ and 712 is automatic) as is T~ induced by 721 and 7 2 2 . Silnilarly ally Z r 1 + l ( Acomputable and T~ is Mrl+l(A)computable. Thus G, < L is T induced by Mrr+l(A) decidable. Since the embedding G -+ L is an &.+'(A) embedding and G < G,, this shows G + G, is an embedding. To see G, is 8Ir+1(A)standard, observe that it is a f.g. subgroup of the , . + ] ( A ) standard group L and so Corollary 3.7 applies. To see that the identity isomorphism on G, is computable from index (i', m',j ' ) t o index (2rfi, 2 observe that 7 can be defined from L to G,, the former with index (i', m ' , j ' ) and the latter with index 172, and that relative t o these indices T is M i f + l ( Acomputable ) by applications of Corollary 4.8 as above. The identity isomorphism in question is a restriction of T t o G, < L so is G r 1 + l ( Acomputable. ) The identity isomorphism from index (( i $ , to (i', m',j ' ) is g n + ' ( A )computable since if G, = F / K for F f . ~ free, . the quotient map E'+ G,, where C, has index ( i f ,m',j ' ) , is G"+'(A) computable using Lemma 2.6. Then x E i(C,) can be viewed as an element in F and so the identity is g n + ' ( A ) computable. This also shows that C + G,, with respect t o (i, m , j ) , is an & l 7 + I ( Aembedding ) since with respect t o ( i f ,FFZ',~') it is an &,+'(A) embedding which may be carried over t o the new index by the (;Iz + I ( A )computable identity isomorphism. 0

(z a,

(c A,

* , . A

Using the above wc obtain a version of the Higman, Neumann, Neumann Theorem.

Theorem 5 . 5 . Let G' be ail g " ( A ) group for ~i >, 3. Theii G can be ( A )standard group having two generators embctlded irito mi 6 The ernbeddirig is ari G'"' ( A ) embedding.

"+'

Proof. By the rclativized version of Theorem 3.2 of [ 4 ] G can be embedded into an & " + ' ( A ) group on three generators. Following the construction given in the proof of Corollary 3.2.1 of [ 4 ] , G'

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can be embedded in a two generator group of the form Gb for $ an E n + ' @ ) isomorphism in G'. By Theorem 5.3, G i is G "'*(A) but a simple application of Lemma 2.6 shows the recursion involved can be bounded and so G i is G ' + ' ( A ) . By Corollary 5.4, G i is E standard and the embedding G + G ' + G $' is an 6 n+' ( A ) embedding.

'+'

As before, the specialization of the above results t o the case of 6 3 ( A )groups for A g n decidable is o f interest. Proposition 5.6. I f G is G 3 ( A )f o r A G n decidable, n Z 4, and cp is an & n - l ( A ) isomorphism in G, G , is g n . I f G is ,fg. and G3(A) standard f o r A g n decidable, n 2 4, and cp is an g n - l ( A ) isomorphism in G, G , is E n standard. Proof. Immediate from Theorem 5.3 and Corollary 5.4. Proposition 5.7. I f G is an G 3 ( A )group f o r A E n decidable, n Z 4, then G can be embedded into an G n standard group on two generators. The embedding is an &'' embedding. Proof. Immediate from Theorem 5.5. 0 In the remainder of this section we develop technical ideas for later application. Again, the casual reader may not wish t o read further. First we obtain an analog of Lemma 4.20 for the strong Britton extensions (see [ 4 ] Definition 3.3 and Lemma 3.1). Definition 5.8. Let G be an Z n ( A )group, cp an G n ( A )isomorphism in G with domain M ,and K < G an g n ( A ) decidable subgroup of G for n k 3. We say K is a t 2 ( A invariant ) under cp if (i) K is H , &-"(A)compatible (ii) K is cp(H), compatible cp(H n K ) = cp(H) n K. (iii) Condition (iii) of the above definition says h E H n K iff cp(h)E K. We can locate &'I+l(A)decidable subgroups of G, (with normal form index) according to the following

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Lemma 5.9. Let G be un g n ( A )group for n 2 3 , cp an E n ( A )isomorphism in G, K < G un g n ( A ) deciduhlc subgroup 8 n ( A ) invariatit wider cp. Define II = domain cp und cp' = cp I H n Then the embcdciing of the t ; l 1 + l ( A ) group K,, into the t ; t z + l ( A ) group G, bv k E K, k P k E G und t b t is an E n + l ( A )embedding relative to the iritlices given by Theoreill 5.3.

Proof. We proceed exactly as in the proof of Lemma 3.1 of [ 4 ] . Using the notation of that proof, we see K,, < L' is an g n + ' ( A ) group L' corresponding to decidable subgroup of the some choice of & n ( A )coset representative system, L' < L is E n + l ( A )decidable for an index on L given by a specific choice of coset representative system and by Corollary 4.9 the identity isomorphism on L is g n + 1 for any choices of g r 2 ( A coset ) representaid tive systems. Thus K,, < L' < L z L > G, is an Z n + ' ( A ) embedding. The verification of the appropriate compatibility conditions for L' < L is identical t o that given in the cited proof noting that the decisions involved are all &',"(A) decidable. 0

We observe that when the conditions of Lemma 5.9 hold, K,, = K . t } < G , and, by Proposition 3.3 of [ 4 ] , { K , t } n G = K in Gp. Next, we consider a many-fold application of the strong Britton extension. In particular we allow countably many such extensions corresponding to isomorphisms c p l , c p 2 , ._.in G . Observe that G,I,,Z ._.may be finitely o r (countably) infinitely generated. Here each pi is to be an g n ( A )isomorphism in G and not (more generallY) i n (;PI,... , " j - [ so our result does not apply as it stands t o the general case of 'Britton towers". Lemma 5.10. Let G he an & " ( A )group f o r n 2 3 and cpl, cp2, ... be an ordered sequencc] oJ' g n ( A) isomorphisms in G with domain pk = Hk < G. Define Gia = G,l,,2, .,. = ( G , t i , t2, ...; t k h k t k ' = Vk ( h k ) f o r all k , and hk E Ilk). Then G, is an E n i l ( A )group and embedding. the embedding G < G, is an &."+'(A)

F.B. Cannonito and R. W. Gatterdam,Computability of group constructions

391

Proof. Our proof is similar t o that of Lemma 3.2 of [ 41 but with two modifications (note that the functions u, u l , u2 of [ 41 are our J , K , L ) . Let G = Ga Gql,..., qrn = G, . We first show that each G, is an group (clear for Go). Let G ' be a copy of G with g' E G ' , the copy o f g E G. Define L , = ( C * ( r l , _..,r , ; )) * $ ( C ' * (sl, ..., ,s 2) for $ : G * Y, H , r l - l * ... * T, H , r,-l -+ G ' * s1 p1( H 1 ) ' s l - l* ... * ,s p , (H,)' sm-l by g t+ g' and r i h r i 1 t+ si cpi(h)'si-l for all h E Hi and i = 1, ..., rn. The factors of L , are g n ( A )groups by Corollary 4.7 and the domain and range of $ are g n ( A )decidable by repeated application of the relativized version of Proposition 2.1 of [ 4 ] . I t is clear that the individual homomorphisms g t+ g' and rihril t+ s i p ( h ) ' sil are E n ( A )computable (as homomorphisms from G or riHiri into G ' * (sl, ..., ,s ;)). Moreover since the normal form of $(w) for w E G * ( r l , ..., r n ; )is the same as that for w with only the symbols changed, it is clear that the recursion used t o define $ as an extension of these individual homomorphisms by Corollary 4.8 can be bounded so is g n ( A ) computable. Similarly +-I is group by Theorem 4.6. Z n ( A ) computable so L , is an L , given by g !+ g, g' t+ g ' , The homomorphism 7 : L , ri t+ s i 1 r i and si t+ { 1 } is Z n + I ( A )computable, fixing only G, < L , by an argument entirely analogous t o that used in showing r gn+l computable in the proof of Theorem 3. I of [ 4 ] using Lemma 2.6 to bound the recursion. Thus G , is an E n + ' ( A ) decidable subgroup of L , . Let L , (and hence G,) have index (i,, m,n, j, 1. Next we show that for k < rn, the embedding Gk < G , is an E n + l ( A )embedding. This is immediate f o r k = 0 and for k > 0 it suffices t o show L k < L , is an g n + l ( A )embedding. Observe that G * r 1H l r l - ' * ... * r k H k r k - l < G * r1H l r 1 - ' * ... * r, H , r - l and G' * S i p 1 ( H i ) ' S 1 - ' * ... * S k p k ( H k ) ' S k - l < G ' * S 1 ' @ 1 ( H l ) ' S 1 - ' * ... * ,s p , (H,)' s,-l are 8 3 decidable and so the conditions of Lemma 4.20 are satisfied and L k < L , is an &'?+'(A)embedding. T o form i(G,) let n ( g ) = minimum rn such that g E G , and set i(g) = J(n(g)7 in(g)(g)). Then 9

+

--f

F.B. Cannonito and R. W. Gatterdam, Computability of group constructions

398

so i(G,) is &IZ+1(A) decidable. The inverse operation on the group G, is encoded by j ( x ) = J ( K ( x ) , iK(,)(L(x))) so j is g n + I ( A )computable. T o define the encoded group multiplication, let K k , , : i k ( G k )+ ,i (G,) be the gn+1(A)embedding for k < m. Since g E G, and g E G, for k < m implies gg E G, and gg 4 G, -I we define

so in is P i l ( A )computable. Clearly the embedding G = Go < G, is an

gn+l(A)

embedding.0

In the above proof observe also that the embeddings G, < G , are ~ / Z +( IA )embeddings since the Gk < G, for k < m are Gn+qA) embeddings and for x E i, (G,) the computation of the minimal k such that x E ik(Gk) can be performed by bounded minimalization and hence is 8 T z f l ( A ) computable. We have the following extension to Lemma 5.9.

Lemma 5. I I . Under the assumptions of Lelizma 3.10, assume also that K < G is an g n ( A )decidable subgroup which is g n ( A ) invqiant under the pk f b r all k. Then i)

ii)

K, = ( K , t l , _ _; tkhtk-l . = p k ( h )f o r all k = 1, ... and all h E f l k n K ) = { K , t , , ... } < G, is an gn+'(A)embedding K , n G = K.

Proof. We use the notation in the proof of Lemma 5.10. It suffices to show the embedding K, = ( K , t l , ..., t , ; tkhtk-' = p k ( h ) for k = 1 , ..., rn and all h E H , n K ) < G, is an Z*+'(A)embedding. Define L k = ( K * ( r l , ..., r, ; >)* + I ( K ' * ( sl, ..., s, ;>)for K' a copy of K , by k t+ k' and

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399

k' and rihr;l p sipi(h)'s;l for k E K and h E Hi n K. Then by k as in the case of L,, L k is an $n+'(A) group. It suffices to show the embedding L k < L , is an g n + I ( A )embedding. This is immediate from Lemma 4.20 observing that

K * r I H l n Krl-l

r , H,

*...* r,H,n

Kr,-l

< G * r l H I r l - l*...*

r,-l

and

are E n ( A )decidable and $' is the restriction of $. It is obvious that K, = { K, t l , _..,t,} < G, and so K, = { K , t l , ... } < G, proving (i). To prove (ii) it suffices to show K, n G = K for all rn. By lemm a 4 . 2 0 , K m n G < L k n G = L k n G n ( G * ( r ,,..., r , ; ) ) = ( K * ( r , , ..., r , ; ) ) n G = K . SinceK,> K a n d G > K , K , n G G > K proving K, n G = K for all rn. References [ 1 ] F.B. Cannonito, Hierarchies of computable groups arid the word problem, Journal of Symbolic Logic 31 ( 1 966) 376-392. 121 F.B. Cannonito, The algebraic invariance of the word problem in groups, this volume. [ 3 ] M. Davis, Computability and unsolvabilitv (McGraw-Hill, New York, 1957). 141 R.W. Gatterdam, Embeddings of primitive recursive computable groups, doctoral dissertation, University of' California, Irvine, 1970. Submitted for publication. [ 5 ] A. Grzegorczyk, Some classes of recursive functions, Rozprawy Matematyczne 4 (1953) 4 6 pp. 161 G. Higman, Subgroups of finitely presented groups, Proceedings of' the Royal Society, A 262 (1961) 455-475. [ 7 ] S.C. Kleene, Introduction to metamathematics (Van Nostrand, Princeton, New Jersey, 1952). [ 81 W. Magnus, A. Karrass and D. Solitar, Coinbinatorial Group Theory (Interscience Publishers, New York, 1966).

400

F.B. Cannonito and R . W. Gatterdam, Computability ofgroup constructions

[ 9 ] M.O. Rabin, Computable algebra, general theory and theory of computable fields, Transactions of the American Mathematical Society 95 (1960) 341-360. 1101 K.W. Ritchie, Classes of recursive functions based on Ackerman’s function, mimeographed lecture notes, University of Washington, 1963. [ I 1 ] K.M. Robinson, Primitive recursive functions, Bulletin of the American Mathematical Society 5 3 (1947) 925-942.

THE WORD, POWER AND ORDER PROBLEMS IN FINITELY PRESENTED GROUPS Donald J . COLLINS Queen Mary College, London

3 1. Introduction We consider some generalizations of the word problem. In particular we consider their relationship t o the word problem and also the relationship of any of these more general problems t o another. By the power problem for a group presentation G we shall mean the problem of determining of any two words u and u of G whether or not u is a power of u *. This is a true decision problem and when we say that the problem is (recursively) soluble we shall, of course, mean that the set of ordered pairs (u, u ) of words of G such that u is a power of u is a recursive set. By the order problem for a group presentation G we shall mean the problem of computing, for any word w of G, the number ord(w) where ord is the function from the set of words of G to the natural numbers defined by ord(w) =

order of w if w has finite order 0

if w has infinite order.

This problem is not a decision problem in the usual sense of the phrase. We shall say that the problem is (recursively) soluble if the function ord is a recursive function. The solubility of either * We adopt the convention that 1 is the 0th power of any word u, i.e.,

1 = uo in G. A perhaps more natural but rather clumsy name for this problem might be the generalized word problem for cyclic subgroups.

401

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D.J. Collins, The word, power and order problems in groups

problem implies the solubility of the word problem. F o r w is a power of 1 if and only if w = 1 in G and ord(w) = 1 if and only if w = 1 in G. These problems have been studied by Lipschutz [6,7] and McCool [ 81 for various types of ‘small cancellation’ groups and have been shown t o be soluble in these cases. Here we are more concerned with the question of the existence of groups for which these problems are not soluble. Granted the existence of groups with insoluble word problem, this question is trivially answered since any such group will have insoluble power and order problems. If, however, we require that the examples we desire must have soluble word problem, then the answer is less immediate, particularly if we also insist that the groups be finitely presented. As a further question, we might ask whether or not we can construct examples in which one of these two problems is soluble but the other is insoluble. Gathering these together we can formulate a more general question, which includes the previous ones, as follows: if a, b and c are any three recursively enumerable (r.e.) degrees of insolubility with a < b and a < c, does there exist a finitely presented group whose word problem is of degree a, whose power problem is of degree b and whose order problem is of degree c.* Our Theorem A below gives an affirmative answer to this quest ion. The first step forward answering these questions was taken by McCool [ 81 who constructed two infinite presentations such that in one the power problem is soluble but not the order problem while in the other the reverse is the case. The existence of such examples is not unexpected since when we permit infinite presentations, we allow much more scope for establishing that certain things are not recursive. Interestingly it turns o u t that despite the scope thus afforded, the construction of the example mentioned first above, although requiring only a few lines, is quite delicate. In attempting to move from an infinite presentation to a finite presentation, there is one obvious way for us to go. We can firstly embed in a finitely generated group using, say, the technique of * The order problem i\ of degree c if the function ord is c-recursive and is not d-recursive for any degree d


D.J. Collins, The word, power and order problems in groups

403

Higman, Neumann and Neumann [ 51 and then use Higman’s remarkable method [ 41 to embed in a finitely presented group and hope that the degree of insolubility of the problems will not be raised as a result of the embeddings. Since Clapham [ 1,2] has shown that the Higman embedding preserves the degree of the word problem, it is reasonable t o hope that this approach will be fruitful and indeed it turns o u t that this method gives exactly what we require. We now state our results formally.

Theorem A. Let a, b and c be any r.e. degrees with a < b and a S c . Then there exists a finitely presented group L such that the word problem for L is o f degree a; (i) the power problem for L is o f degree b; (ii) the order problem for L is of degree c. (iii) Theorem A is an immediate consequence of the following two theorems.

Theorem B. Let a, b and c be any r.e. degrees with a < b and a < c. Then there exists a finitely generated group G with a recursively enumerable set of defining relations such that the word problem for G is o f degree a; (i) the power problem for G is o f degree b; (ii) the order problem for G is of degree c . (iii) Theorem C. Let G be a finitely generated group with a recursively enumerable set of deji‘ning relations and suppose the word problem for G is o f degree a; 1. the power problem for G is o f degree b; 2. the order problem jbr G is o f degree c . 3. Then there exists a finitely presented group L in which G can be (recursively) embedded such that the word problem for L is of degree a; (i) the power problem for L is o f degree b; (ii) the order problem for L is o f degree c . (iii)

4 04

D.J.Collins, The word, power and order problems in groups

Theorem C is a generalization of Clapham's theorem in [ 1,2] concerning the Higman embedding. We prove Theorem C by simply following Clapham and verifying that the additional requirements are satisfied.

8 1. Theorem C It is convenient to begin with Theorem C. Unless otherwise specified our notation and terminology are taken directly from [ 1,23. Initially we concentrate on the power problem. Clapham uses three main constructions and we therefore establish some lemmas concerning these constructions. Lemma 1 . 1 *. Let G and G' be groups with recursively enumerable sets ojgenerators, let U be a subgroup of G which is d-soluble in G, U ' a subgroup of G' which is &-soluble in U' and 4 an d-recursive isomorphism between U and U'. I f G and G' have d -soluble power problems with respect to these sets o f generators, then { G, G ' ; u = @ ( u )ii, in U } has 94-soluble power problem with respect to the set of generators consisting o f the union of the two g i w n sets.

Proof. Our hypothesis imply that we can d -recursively compute reduced forms for elements of the generalized free product { G, G ' } . Without loss of generality we may therefore assume that our arbitrary u and u are in reduced form. Let these reduced forms be zi, ii2 ._ .u,,, and u1 u2 ... u, where successive u iand ui come alternately from G and G ' but not from the amalgamated part (except when rn = 1 o r n = 1 ). Case (i). Let us suppose that u is cyclically reduced, i.e., either n = 1 or u1 and u, belong t o different factors of the generalized free product. If n = 1, then ( 3 r ) u = ur implies that u belongs t o the same factor as u so that in this case we may use the d - r e c u r sive algorithm for that factor. If n > 1, we use the fact that for any * This IS dn analogue of Clapham's numbers.

Lemmas 3.5 and 9.1. d denotes some set of natural

D.J. Collins, The word, power and order problems in groups

405

r, ur is in reduced form and hence u = ur only if I r I < m . Thus the power problem reduces to the word problem which, by Clapham's Lemma 9.1, is 4-soluble. Case (ii). Suppose that u is not cyclically reduced. Then there is an 4-recursive procedure to express u as y-1 u o y where uo is cyclically reduced. Moreover, (3r ) u = ur if and only if (3r)yuy-l = uor Since we can put y u y ' in reduced form we can apply the procedure of case (i). Lemma 1.2". Let G be a finitely generated group with d-soluble power problem and let $ 1 ,$ 2 , ..., be an d-recursively enumerable set of &-recursive isomorphisms o f the 4 -recursively enumerable set A , , A,, ..., ofsubgroups of G, into G , such that for i = 1 , 2 , ..., A i and $ i ( A i )are d-soluble in G. Then { G, t l , t 2 , ... ; t;'aiti = Gi(ai), ai

i n A j , i = 1 , 2 , ...}

has 4-soluble power problem with respect to this set ofgenerators. Proof. Let G, = { G, t l , t 2 ,..., t,; t ; l a i t i = Gi(aj),ai in A i , i = 1 , 2 , ..., n } ; then G = U, G, . The lemma will follow if we can prove that each G, has d-soluble power problem and that the mapping from arbitrary G, to the corresponding algorithm is &-recursive. The former can be established by a repetition of the argument given by Clapham for his Lemma 9.2 but replacing applications of his Lemma 3.5 by applications of our Lemma 1.1. The 4-recursiveness of the mapping from a G, t o the corresponding algorithm follows obviously from the general character of the argument**. In addition t o the two constructions just considered, Clapham also makes use of direct products. For our purpose it would be

* This is an analogue of Clapham's Lemmas 3.6 and 9.2. ** In the proof of his Lemma 9.2, Clapham neglects to verify

that the mapping from a G, to the corresponding algorithm is d-recursive. It is clear that this is indeed the case but the point should be mentioned since it is crucial to the argument.

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D.J.Collins,The word, power and order problems in groups

most convenient if we could show that whenever G and G ' have d-soluble power problems, then G X G ' has d-soluble power problem. Unfortunately, we have been unable to establish this and it may well be false - we return t o this question in 9 3 . However, it turns o u t that the following lemma is sufficient for our present purpose. Lemma 1.3. Let G und G' be groups with & -soluble power problerns relative to some generatqrs. I f either G or G ' is torsion-free, then the direct product G X G' has &-soluble power problem relative to the iinion of the two sets of generators.

Proof. Let us assume that G is torsion-free. If zi and u are arbitrary elements of G X G ' , we may write zi = zil u2 and u = u , u2 where t i 1 , u1 E G and u 2 , u 2 E G'. We consider two cases according as u l = 1 or u 1 f 1. Since the word problem for G is &-soluble this distinction can be made &-recursively. If i l l = 1 then zi = ur if and only if u1 = 1 and u2 = u Z r . The problem thus reduces t o the word problem for G and the power problem for G'. Now suppose u1 # 1 ; then u = ur implies u 1 = u I r and there can be exactly one such value of r since G is torsion-free. We can 94-recursively determine whether or not ( 3 r ) u 1= u , and ~ if such an r exists we can 94-recursively determine what this unique value is. It follows that the problem is reducible to the power problem for G and the word problem for G ' . Initially we assumed that G was torsion-free and then gave a procedure for d-recursively solving the power problem for G X G ' The assumption that G ' was torsion-free would also lead t o a procedure. I n general we may not know which of G and G ' is torsionfree. From the point of view of recursive function theory, this is quite irrelevant - all that is required is that one establishes that a procedure exists. The lemma is therefore proved. The following are well known; we state them as lemmas for ease of reference.

D.J.Collins, The word, power and order problems in groups

407

Lemma 1.4. Let G, G ' , U, U' and @ be as in Lemma 1.1.I f G and G' are torsion-free, then { G, G ' ; u = @ ( u )u, in U ) is torsion-free*. Lemma 1.5. Let G , @, , @, is torsion-.free, then {

..., A , , A , , ..., be as in Lemma 1.2. I f G

G, t l , t,, ...; t i 1 a j t j= & ( a j ) , a j in Ai,i = 1 , 2 , ...}

is torsion-free" Lemma 1.6. Let G and G' be torsion-free. Then G X G ' is torsionfree. The fundamental aspect of Clapham's work is the relationship between the concept of an &-recursive, recursively enumerable set and his notion of an d-strongly benign subgroup of a finitely generated group. We parallel this by defining the notion of d-completely benign based on the following lemma.

Lemma 1.7. Let G be a f : g . torsion-free group and H a subgroup of G. Then the ,follotvirzg are equivalent: G can be embedded in a c p . torsion-free group K with (i) &-soluble power problem which has a f g . subgroup L such that G n L = Hand G, L and G L are d-soluble in K ; the group { G, t ; t-l ht = h, h in H } can be embedded in (ii) a f : p . torzionyfree group K ' with 4-soluble power problein and G and { G, t } are d-soluble in K ' ; thegroup { G, G , ; h = h , , h in €1) can be embedded in a (iii) f : p . torsion-free group K" with d-soluble power problem and G , G , and { G, G , ) are &-soluble in K " .

-

Proof. The lemma is. of course, very similar t o Clapham's Lemma 4.1. Moreover, it is proved by a repetition of the argument given by Clapham for his lemma but replacing applications of his Lem* The assumptions concerning &-recursiveness

are of course redundant here.

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D.J. Collins, The word, power and order problems in groups

mas 3.5 and 3.6 by applications o f our Lemmas 1. I and 1.2 and noting that by our Lemmas 1.4 and 1.5 all groups involved are torsion-free. A subgroup H of a f.g. torsion-free group G is called d-cornpletely benign (in G ) if any, and hence all, of conditions (i), (ii) and (iii) of Lemma 1.7 are satisfied. We emphasize that this definition applies only to torsion-free groups. We now have our main technical result.

Theorem 1.8. A subgroup of a f g . free group is A-completely benign if and only i f it is &-recursive and recursively enumerable *. Proof. Yet again we mimic Clapham’s argument with some alterations and additions. Throughout his paper, Claphani is concerned to show that certain word problems are 94-soluble. He always does this in one of three ways. These are: I . by using Lemma 9.1 o r a finite version of it; 11. by using Lemma 9.2 or a finite version of it; 111. by using the fact that if two groups have 94-soluble word problem, then so does their direct product. It follows from this that our theorem will be proved if we repeat Clapham’s paper from Corollary 4.2.1 t o Lemma 1 1.2 with the following alterations and additions: 1 . replace ‘ &-soluble word problem’ by ‘ &-soluble power problem’; 2. replace ‘ &-strongly benign’ by ‘ &-completely benign’; 3. whenever any group is given, add the hypothesis that the group is torsion-free; 4. replace any proof by I by applications of our Lemma 1.1 ; 5. replace any proof by I1 by applications of our Lemma 1.2; 6. replace any proof by 111 by applications of our Lemma 1.3; 7 . use our Lemmas I .4- 1.6 to verify that any groups constructed arc torsion-free. * This is the analogue of Clapham’s main technical

result, his Lemma 11.2.

D.J. Collins,The word, power and order problem in groups

409

Although we have not yet mentioned the order problem, the bulk of the proof of Theorem C is now complete. For the final stages we introduce the order problem and, firstly, obtain another trio of lemmas. Lemma 1.9. Let G, G ' , U, U' and 6 be as in Lemma 1.1 save that we assume the &-solubility of the order problems rather than the power problems. Then { G, G ' , u = 4 ( u ) ,u in U ) has d-soluble order problem.

Proof. I t is well known that in a free product with amalgamation an element w has finite order if and only if it is a conjugate of an element x which belongs t o a factor and is of finite order. We can &recursively determine if such an x exists and compute it if it does. Then ord(w) = ord(x).

Lemma 1.10. Let G be a fig. group with d-soluble order problem and let $, ..., and A , , A , , ..., be as in Lemma 1.2. Then {

G, t l , t,, ...; t;laiti

= @i(ai),ai

in A i ,i = 1 , 2 , ...}

has 94-soluble order problem. Proof. This can be derived from Lemma 1.9 since this construction is essentially a special case of a free product with amalgamation. Lemma 1.1 1. Let G and G' have d-soluble order problems. Then G X G' has 94-soluble order problem.

Proof. Let w = w,w 2 where w1 E G and w,E G'. Then the order of w is the lowest common multiple of the orders of w , and w , (where 1 .c.m. ( 0 , n ) = 0 for any n ) . Proof of Theorem C We are given a f.g. group whose word, power and order problems are of degree a, b and c respectively. Let us suppose that G = F / R where F is a f.g. free group and R is an d-recursive, recursively

410

D.J. Collins, Tlie word, power and orber problem in groups

enumerable normal subgroup (where d is some representative o f a). Then of course R is 4-soluble in F' and so by Theorem 1.8, is &-completely benign in F , i.e., H = { F',, F Z ;y 1 = y 2 , Y in R } can be embedded i n a f.p. torsion-free group K whose power problem is &-soluble and F , , F , and { F,, F , } are 4-soluble i n K . The required f.p. group L is then ;btained in the following nianner. Firstly we construct the direct product K X G. Now there is a mapping X : F, + K X G defined by sending j ' , 3 ( f ; , f l R ) and a mapping v : b', K X G defined by sending j ; 4: (f' . 1 ) and clearly X and v agree i n the amalgamated part of H . Metice there is a homomorphisni c$ : H + K X G which extends h and v and in fact @J is a monomorphism. Thus we may form L = { G, t ; t-lht = @ ( h )zi , in H } which contains G and is in fact finitely presented. All asser+

tions niade in this paragraph are verified i n Hignian [ 41 . We claim further than L satisfies the conditions (i), (ii) and (iii) of the statement of Theorem C. Part (i). The power problem for K is d-soluble and hence the word problem for K is d-soluble. The word problem for G is d soluble and thus the word problem for K X G is 4-soluble. Clapham verifies that H and @ ( H )are d-soluble i n K X G whence L has &-soluble word problem. Since G is (recursively) embedded i n L the word problem for L is of degree a. Part (ii). Let 'M be a representative of b. The power problem for K is &-soluble and hence %-soluble since SQ < '8. By hypothesis the power problem for G is %-soluble and. since K is torsion-free. the power problem for K X G is 9-soluble ( b y Lemma 1.3). Since H a n d @ ( I / )are 9-soluble i n K X G, Lemma 1 .? gives 11s that the power problem for L is %-soluble. The fact that G is embedded in L ensures that the power problem for L is of degree b . Part (iii). Since K is torsion-free, the order problem for K is equivalent t o the word problem and thus is C-soluble (where e is some representative of c ) . By Lemmas 1 . 1 1 and 1.10, the order problem for L is C-soluble. Since G is embedded in L , the order problem for L is of degree c . One final point in connection with Theorem C should be noted. We have shown that the degree o f the order problem is preserved in 21 rather indirect sort of way by utilising the torsion-freeness of

D.J. Collins, The word, power and order problems in groups

41 1

K. I t is clear, however, that if we were concerned solely with the order problem, then on the basis of Lemmas 1.9 - 1.1 1, an analogue of Theorem 1.8 could be established directly and then the remainder of the proof carried o u t as in part (iii) above.

92.Theorem B We construct group presentations with infinitely many generators and defining relations and with word, power and order problems of varying degrees of insolubility. Then we use the technique of Higman, Neumann and Neumann t o obtain the f.g. group G of Theorem B. Throughout this section N denotes the natural numbers. The following lemma is well known. Lemma 2.1. Let b be any r.e. degree. Then there exists a one-toone recursive fiinction f of one variable such that (i) 0 f(N); f ( N ) is of degree b. (ii)

+

Corollary 2.2. Let f be as in the above lerizma and k t s b = { ( f ( n ) , n + 2); n E N } . Then (i) s b is recursive; { m E N; ( 2 r ) ( m ,r ) E s b ) } is ofdegree b; (ii) $ ( m , r ) E s b and (m,r') E S,, then r = r ' ; (iii) if(in,r ) E s b then m > 0 and r > 1 *. (iv) Proof. This follows from the lemma and the fact that f is one-to-one. Let T be any r.e. set of degree a. Let M , = (a, 6 , c, d ; am bam = cm dcm , in E T ) . Lemma 2.3. The word, power and order problems .for M , are of

degree a . * Wc require conditions (iii) and (iv) for later technical use. Condition (iii) is essential whereas (iv) is a inere convenience.

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D.J. Collins,The word, power and order problems in groups

Proof. Clapham shows that the word problem is of degree a by observing that M , is the free product of two free groups amalgamating two subgroups which are a-soluble * in the free groups but are not d-soluble for any d < a . By Lemma 1.1 , the power and order problems are also a-soluble and hence of degree a. Let b be an r.e. degree and let Sb satisfy conditions (i)-(iv) of Corollary 2.2. Then let

Lemma 2.4. The word and order problem for M , are soluble. The power problem f o r M 2 is of degree b. Proof. This group presentation is due, essentially, t o McCool [81 who shows that the word and order problems are soluble and that the power problem is insoluble. For the sake of completeness we shall repeat his arguments. I f we let G, = ( x , ~y ,m ; x,y, = y m x m )when for all r , (nz,Y) $ S b and let G,,, = (x, , ?., ; x , , ~ , , ~ Y , x , ~ ,x , = y m r )when ( m ,r ) E S,, **, then M 2 is the free product of the Gm . Thus M 2 is certainly torsion-free and so the word problem and order problem are equivalent. We solve the former by showing that we can compute the normal form of an arbitrary word. It clearly suffices t o consider a word w in the form x z 1y m01l xaz m y mz"' b xat mt y P mtt where crl and 0 1 ,i = 1 , 2, ..., t are not both zero. 7'0 compute the normal form of w we must be able t o tell of an arbitrary x m a y m P ,cr and p not both zero, whether or not this equals 1 in G, . Clearly xmaymO = 1 in G, if and only if there exists r such that ( m , r ) E S b and ra + = 0 , i.e., x , , ~ I',P = 1 in G, if and only if -P/a is a natural number such that ( m , - p / a ) E s b . Since S b is recursive we can test whether or not this is the case. The insolubility of the power problem is obvious since ( 3r ) x , =

* Since we n o longer follow Clapham's **

notation quite so closely, we write a-soluble directly instead of &soluble with G? some representative of a. tlere, and subsequently, we use condition (iii) of Corollary 2.2.

D.J. Collins, The word, power and order problems in groups

413

ymr if and only if ( 3 r ) (m, r ) E S b . It remains for us t o establish the degree of the power problem. Since we can compute normal forms, we may assume, possibly after some conjugation, that u and u are in normal form and that r is, in addition, cyclically reduced. If u has length greater than 1, then by the same length argument as used in Lemma 1.1, the problem reduces t o the word problem. If u has length 1 we need only consider u if it also has length 1 and is in the same G, and thus we have the situation where we are examining two words xm@y,P and x,Y ym6. We can b-recursively decide whether o r not ( 3 r ) (m,r ) E S b . If not, then u = u f if and only if a = t y and /3 = t 6 and this is clearly decidable. If on the other hand (m, r ) E S b then u = u t if and only if there exists s such that x;-'Y yk-t6 = x,-sy,rs in the free abelian group on x , and y m . The latter occurs if and only if CY -ty = --u and /3 - t 6 = -rs and the existence of these two equations is equivalent to t = ( r a + P)/(ry-t 6 ) . (Notice that since x,Yym6 can be assumed not t o be 1 in G, , r y + 6 # 0). The integer t is thus uniquely defined and computable (if it exists) and the problem reduces t o the word problem. Let S, satisfy conditions (i)-(iv) of Corollary 2.2 but with degree c instead of degree b. Let

M,

=

( x l ,x 2 , ...; xmr! = 1 , (m,r ) E S,).

Lemma 2.5. The word and power problems for M , are soluble. The order problem for M , is of degree c . Proof. This presentation is again essentially due to McCool [ 81 , and he solves the word and power problems and shows the order problem insoluble. F o r completeness we repeat his arguments. I f G, = ( x , ; @I) when for all r , (m,r ) 4 S, and G, = ( x , ; xmr!= 1) when (m, r ) E S,, M , is the free product of the G, . To solve the word problem it therefore suffices t o determine if an arbitrary xma,a # 0 , whether o r not this is 1 in G, . Clearly, xma = 1 in G, if and only if there exists r such that (m,r ) E S, and r ! divides a . Since S, is recursive we can determine whether or not such an r exists.

D.J. Collins,The word, power and order problems in groups

4 14

The solution of the power problem is, in the present author's view, surprisingly delicate. We may assume that we are given two words u and u in normal form with u in addition cyclically reduced. If the length of u is greater than 1 , we reduce to the word problem. I t then suffices t o consider two words xma and xmP where a 7P # 0. We want t o test whether o r not x m a = xmfP in G, for some t . As a first test we check t o see if a = t o for some t. If so then the problem is settled; so suppose that a is not an integral multiple of 0. If x,na = x,*P in G, then there exists r and k such that a = t o + k r ! and ( m ,r ) E S,. We claim.that this equation implies that r < 10 I. F o r if r >, I 0 I then tP + kr ! is an integral multiple of 0 and we have assumed that this is not the case*. The insolubility of the order problem follows from the fact that x, has finite order if and only if ( 3 r ) ( m ,r ) E S,. To determine the degree of the order problem we argue as follows. It suffices to consider elements whose normal form is x , f f , a # 0 since any element whose cyclically reduced normal form has length greater than 1 is not of finite order. We can c-recursively determine whether o r not ( 3 r ) ( m ,r ) E S,. If (m,r ) E S,, then the order of xma can be computed from the order of x, which is r ! If n o such r exists, then ord ( x , ~ )= 0. The order problem is therefore c-soluble and hence of degree c. Now let M = M , M3

*M, *M,

be the free product of M , , M , and

*

Lemma 2.6. The word problem for M is o.f degree a. (i) The power problem for M is ojdegree b. (ii) The order problem f o r M is ojdegree c. (iii) Proof. This is trivial from Lemmas 1.1 and 1.9. Let F ( p , 4 ) be the free group on the symbols p and 4. Also let M be rewritten as (gl , g 2 , ._.I R , = 1 , X = 1,2, ...) and let * The delicacy of the argument is in the use of the factorial which seems to be quite crucial.

D.J. Collins, The word, power and order problems in groups

415

E = M * F ( p , 4 ) . If F(s, t ) is the free group on the symbols s and t then let G = ( E * F(s, t ) ; gipiqpi = sitsi, i = 1,2, ...)

Proof of Theorem B. I t clearly suffices t o prove that the groups ( gipiqpi ) and ( sitsi ) where i = 1 , 2 , ..., are a-soluble in E and F(s, t ) respectively. This is easily shown using the standard Nielsenstyle argument and the normal form theorem for free products. Details of the required argument are given in Lemma 2.1 of Collins [ 31 (note that n o t w o generators of M are equal). 53. Direct products

In 5 1 , we mentioned that certain difficulties seem t o occur in connection with the power problem and direct products. We examine the matter a little more closely here. We firstly establish the following lemma.

Lemma 3.1. Let G and G ' each have soluble power problem and soluble order problem. Then G X G' has soluble power problem. Proof. Let u and u be arbitrary elements of G X G ' and suppose u = u l u 2 ,u = u I u 2 w h e r e u l , u l E G a n d u 2 , u 2 E G ' . As a first test we check whether or not there exists y 1 and r2 such that u l = u1 '1 and 1i2 = ~ ~If n~o such 2 r 1. and r2 exist then u is not a power of u. Let us suppose that they d o exist; we can certainly then recursively compute such a pair of integers. Also we can recursively compute p 1 = o r d ( u l ) and p 2 = ord(u2). Then ti is a power of u if and only if there exist integers k and 1 such that r1 + kpl = r2 + 1p2. Such a k and 1 can exist if and only if r1 - r2 is a multiple of the greatest common divisor o f p , and p 2 (where g.c.d. (0, t ) = t = g.c.d. ( t , 0) for any t ) , and we can determine whether o r not this is so.

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The above argument breaks down if we remove the hypotheses that the order problems for G and G ' be soluble. Without these hypotheses we run into difficulties if the values of r1 and r2 which we obtain are not equal. For it is conceivable that there could be other values satisfying the equations i l l = ulr1 and 2i2 = u2'* which would be equal. This will not occur if either u1 o r u2 is of infinite order but without the solution t o the order problem we have, in general, no way of knowing when other values exist and what they are when they d o exist. All this leads t o the conjecture that there must exist groups G and G ' , each with soluble power problem, but such that the power problem for G X G ' is insoluble. By our lemmas neither of these two groups can be torsion-free and at least one must have insoluble order problem. Unfortunately the author is at present unable to settle this question and it therefore appears in the list of problems posed during this conference. In attempting to deal with the problem the first piece of information that appears useful is knowing what kinds of groups exist which have soluble power problem and insoluble order problem. As matters stand at present, the only examples known are those which can be derived from McCool's fundamental example by embeddings. We seek to show that these examples are of n o avail with regard t o this question.

Example. Let S and T be two sets satisfying conditions (i)-(iv) of Corollary 2.2 (for the moment we are not concerned with degrees and insist only that in (ii) we have non-recursiveness). Let

We consider the power problem in M X N . Let ii = zi2 and u = u 1 u 2 , as usual, where u l , u l E M and u l , u2 E N. If either u1 or u2 is not a conjugate of a power of a generator, then the problem is easily solved since in this case the u1 or

D.J. Collins, The word, power and order problems in groups

417

u2 must be of infinite order and we can apply the uniqueness argu-

ment of Lemma 1.3. It should, of course, be observed that we can determine when an element is a conjugate of a power of a generator. In our present situation we can always remove conjugating elements without loss of generality and it therefore suffices t o consider the situation in which u1 = x m P , u2 = y p s and ZL] = xmQ, u2 = y p y . The first step is t o test whether or not ( 3 rl)xrna= xrnrlD and ( 3 r2)ypY = ypY,6 . A negative answer here means a negative answer overall. So suppose we have found r1 and r2 satisfying the equation. We now subdivide into cases according as the following conditions are satisfied: l(a) Q =orl; ( k i t ) @# P t and ( 3 n, k ) ( x m n != 1 and Q = orl + k n ! ) ; l(b) 2(a) y= 6r2; (V t ) y # 6 t and ( 3 q, 1 ) (y:! = 1 and y = 6 r2 + Zq!) 2(b) I t is clear that l(a) and l ( b ) are mutually exclusive as are 2(a) and 2(b) and that 1 and 2 are quite independent. Moreover, our use of the algorithms for M and N will tell us exactly which of these hold - and when I(b) o r 2(b) holds, the algorithm will compute the value of n o r q respectively. Case l(b), 2(b). Here we have u1 and u2 both of finite order and we can compute these orders since we know the values of n and q . The argument of Lemma 3.1 can therefore be applied. Case l(a), 2(a). On account of the very simple way in which xmQ = xrnPrl and y p y = yP6'2 we have n o information as t o whether xmP and y p s are of finite order. As a result this case is more complex. We firstly check whether any of the equations r 1 = r 2 ,xmQ = xrnPr2 in M and y p y = yP6'1 in N hold. If any one is true, then u 'is a power of u. So suppose all three fail. We want to know whether or not there exists r such xma = xmPr and ypT = y P s r .Since r1 # r 2 , this can happen only if there exist n, q, k and Z such that a = or + k n ! , y = 6r + I n ! ,x m n != 1 and y P 4 ! = 1. We want t o examine what conditions are imposed on n and q if this does indeed occur. Recalling that Q = o r , and y = 6r2 we obtain, after a little manipulation, the equation

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D.J. Collins, The word, power and order problems in groups

p6(rl - r 2 ) = 6 k ( n ! ) - p l ( q ! )

("1

Without loss of generality we may suppose that n < 4 . Then by McCool's divisibility argument (see Lemma 2.5) it follows from * that n ! < IPS ( r l - r 2 ) I and hence a fortiori n < 106 ( r l - r 2 ) I . Now we assumed that xmQ # xmPQ ; this means that Q - p r 2 is not a multiple of n ! , i.e., P(rl - r 2 ) is not a multiple of n ! Then clearly /36(r1- r 2 ) is not a multiple of 6 n ! . Using divisibility again, we deduce from * that 4 < I6n! I < IPF2(rl -r2>1. The net result of the above argument is that if r does exist then xmP and yp6 are elements of finite order and that these orders have upper bounds which can be recursively computed from the values of 0,6, r 1 and r 2 . This enables us t o determine whether o r not r exists. Case I(a), 2(b) and l(b), 2(a). These two are obviously similar. We examine the latter in detail. Let Q = orl + k , n ! and y = 6 r 2 . If xmQ= x , P r and ypY = y p S rthen there exist I, 4 , k, such that a = p r + k n ! , y = 6 r + f q !a n d y g ! = 1.Thisyields P

PS ( r l - r 2 ) = 6 ( k - k l ) n ! - plq! .

(t)

We can then argue exactly as in the previous case since it follows from i that y < I6n! I. The above argument shows that the power problem for M X N is soluble. Although we have here considered only McCool's fundamental example rather than the supergroups in which they are embedded, it should be clear that for the supergroups, essentially the only time that we do not have an element of infinite order is when we have an element which is an image of some xmP so that the situation is not altered in any way by considering these supergroups. We conclude by proving a lemma which shows that groups with soluble power problem and insoluble order problem have an unexpected algebraic property. Lemma 3 . 2 . Let G h e u group with solrible power p r o b l e m and irisolztble order problem. Then G lius elenzerits o j ' e w r y order.

D.J. Collins, The word, power and order problems in groups

419

Proof. Firstly we remark that G must have elements of infinite order. For if every element is of finite order then given any w we can compute ord(w) by successively comparing w ,w 2 , w 3 , ..., with 1 until we find the least n such that wn= 1. To complete the proof we shall establish the following claim from which the lemma then follows easily. The claim is: for every n > 2 there exists w E G such that ord (w)is a multiple of I? *. Suppose not; then there exists no > 2 such that for every w € G, either ord(w) = 0 or there exist k and Y such that ord(w) = kno + Y with 0 < Y < n o . In this situation we assert that an element w of G has finite order if and only if the following disjunction holds: (31c)w-l =wkn0 o r ( g k ) w - 2 = w h o o r ... or(3k)w-(no-1) = w h o . If the disjunction holds then certainly w has finite order. But the converse is immediate from our supposition. This means that by a fixed finite number of applications of the algorithm which solves the power problem, we can determine whether or not w has finite order. If w has finite order we can then go on to compute ord(w) by comparing w ,w 2 , w 3 , ..., with 1. If w does not have finite order, then ord(w) = 0. We have thus established the existence of a procedure for solving the order problem which is contradictory**. * The author

is indebted to C.F. Miller, 111, for pointing out that the apparently more sweeping assertion that G has elements of every finite order is a trivial consequence of the claim. ** We may not know what no is, but this is irrelevant. It is enough that it exists in order for the existence of the procedure to be established.

References [ 11 C.R.J. Clapham, Finitely presented groups with word problems of arbitrary degrees of insolvability, Proc. Lond. Math. SOC.,Series 3, 14 (1964) 633-676. [2] C.R.J. Clapham, An embedding theorem for finitely generated groups, Proc. Lond. Math. SOC.,Series 3, 17 (1967) 419-430. [ 3 ] D.J. Collins, On embedding groups and the conjugacy problem, to appear in J . Lond. Math. SOC. [4] G. Higman, Subgroups of finitely presented groups, Proc. Roy. SOC.,Series A, 262 (1961) 455-475. IS] G. Higman, B.H. Neumann and Ilanna Neumann, Embedding theorems for groups, J . Lond. Math. SOC.,25 (1949( 247-254.

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D.J. Collins, The word, power and order problems in groups

[ 6 ] S. Lipschutz, An extension of Greendlinger’s results on the word problem, Proc. Amer. Math. SOC.,15 (1964) 37-43. [7] S. Lipschutz, On T-fourth groups, in preparation. [8] I. McCool, On free product sixth group, Proc. Glas. Math. SOC.10 (1969) 1-15.

THE HIGMAN THEOREM FOR PRIMITIVE-RECURSIVE GROUPS - A PRELIMINARY REPORT

R.W. GATTERDAM”

University of Wisconsin, Parkside

An extensive study of g n ( A )groups can be found in this volume,

[ 1 1 . A group is an A-p.r. group (primitive recursive relative t o

A C N ) if it is g n ( A ) for some (finite) n ; it is a p.r. group if it is g n ( N ) for some n. As mentioned in [ 1 I the following version of the Higman theorem, [ 51, holds for A-p.r. groups:

Theorem. L e t A C N be a recursively enumerable set. Then an A p.r. group can be embedded as an A-p.r. decidable subgroup of u finitely presented A-p.r. group. The embedding and its inverse, where defined, are A-p.r. computable. The complete proof of this theorem is contained in [ 41 . Also observe the similarity t o the Clapham result, [ 21 and [ 31 , which can be viewed as the same statement replacing “A-p.r.” by “ A recursive”. In [ 4 ] both the theorem and our version of the Clapham result are proved by a technique similar to that of Schoenfield [ 6 ] using the computability of the group constructions free product with amalgamation and strong Britton extension as discussed in [ 1 1 . The purpose of this report is t o briefly outline the proof given in [ 41 considering two crucial facets in some detail. First we see how countably many applications of the strong Britton extension (as in Lemma 5.10 of [ 1 ] ) are used in the proof. The proof is then completed by a direct induction based on the A-p.r. computable * AFOSR-I 321-67

and AFOSR-70-1870 (Grants).

421

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R. W. Gatterdam, The Higman theorem for primitive-recursivegroups

decision function for the word problem of a finitely generated, A-p.r. group. This report should be viewed as preliminary in that it does not utilize the full information o n the computability of the strong Britton construction as found in [ 11 b u t rather the weaker results of Chapters 2 and 3 of [ 41 . With these newer results it is hoped that the Higman construction can be shown to induce at most a jump of a few computability levels of the relativized Grzegorczyk hierarchy. An embedding of an A-p.r. group into another A-p.r. group is an A-p.r. embedding if it is an Gn(A) embedding for some n as in [ 1 1 . Similarly an A-p.r. isomorphism cp in G is an g n ( A ) isomorphism and we consider the A-p.r. group G, = ( G, t ; tht-l = ~ ( h ) , ii E H = domain cp ) as in [ 1 3 . In view of Theorem 5.5 o f [ 11 we restrict o u r attention t o finitely generated (f.g.), A-p.r. groups and following [ 61 say such a group is A-p.r. Higman if it can be A-p.r. embedded in a f.p., A-p.r. group. Again following [ 61 an A-p.r. isomorphism cp in an A-p.r. Higman group G is said t o be A-p.r. benign if G, is A-p.r. Higman and, in particular, an A-p.r. decidable subgroup H < G is A-p.r. benign if G, = G‘, l H is A-p.r. Higman. Then paralleling [ 61 the following are proved in [ 41 (replacing p.r. by A-p.r.1: 1 . H < K < G for G A-p.r. Higman, K f:g. and A-p.r. decidable, tiien I€ is A-p.r. benign in G ijf it is A-p.r. benign in K.

3. The intersection 0j.A-p.r. benign subgroups is A-p.r. benign.

3. Tile (group theoretic) union of A - p r . benign subgroups is A-p. r. benign if A-11.r. decidable. 4. The image of an A-p.r. benign subgroup under an A-p.r. computable Iiornoniorphism is A-p.r. benign if A-p.r. deciduble.

5 . The preimage of an A-p.r. benign subgroup under un A-p.r. computable homomorphism is A-p.r. benign. 6. The restriction of a n A-p.r. isomorphism in G with domain G to an A-p.r. benign subgroup is an A-p.r. isornorphism iii G.

R . W. Gatterdam, The Higman theorem for primitive-recursive groups

423

As might be expected the proofs of 1 . through 6. given in [ 4 ] are similar t o proofs in [ 6 ] with attention given t o the decidability of the various subgroups involved and the computability of the homomorphisms (both in the statements and internal t o the proofs). Using the above facts the link between A-p.r. Higman groups and A-p.r. benign subgroups is established by

7 . Let G be j:g., A-p.r. and let 1 + K + F + G + 1 be a presentation of G f b r F free and f g . Then G is A-p.r. Higrnan iff K is A-p.r. berz ign. In view of 7. we turn our attention t o A-p.r. decidable subsets of a f.g. groups F . F o r P C F , an A-p.r. decidable subset we define the subgroup E p < F * ( z ; ) t o be that subgroup generated by all words of the form XzX-'for X E P. We say P is an A-p.r. benign subset of F if E p < F * ( z ;) is an A-p.r. benign subgroup. We are now in a position to prove the crucial lemma (Lemma 5.1 of [ 4 ]). It is the proof of this lemma which requires countably many applications of the strong Britton extension and we consider it in some detail.

Lemma. I f P < F is an A-p.r. decidable subgroup o f the f : g . f r e e group F which is A-p.r. benign as u subset then it is A-p.r. benign as a subgroup.

To prove the lemma, set G = F * ( c, d ; ) and for every word X E F define px : { c } -+ { dX } an isomorphism of infinite cyclic subgroups of G. Let X I , X,, .. enumerate the words of F and consider GIFl = ( G, t X 1 ,t X 2 ,...; t Xc t X - l = d X b' X E F ) . The proof proceeds in the following steps:

1 . GIFl is an A-p.r. group (by Lemma 5.10 of [ 1 1 ; it is here that the strong form of Lemma 5.10 as compared to Lemma 3.2 of [ 41 may be used t o strengthen the result). 2. G I F ~is A-p.r. Higman (by a construction). 3. GIPi = { G, t X for all X E P } < GIFl is A-p.r. decidable (by a n indwtive argument during the construction of GIFl ).

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R. W. Gatterdam, The Higrnan theorem for primitive-recursive groups

4. { c, d, t, for all X E P } = { c, d, t X for all X E P, P } < G p l is A-p.r. decidable (an application of Lemma 5.1 1 of [ I ] ).

5. { c, d, t , for X E P } is A-p.r. benign (as the image of { c, a', XzX-I for all X E P}< F * ( c ,d, z;) under an A-p.r. homo-

morp hism).

6 . P = F n { c, d, t, for all X E P } completing the proof of the lemma.

Next we consider certain subsets of F = ( a , b ; ) .We say an A-p.r. decidable subset B C N k (k-tuples of natural numbers) is A-p.r. benign if the set of all words in F of the form, ax' baX*b ... baXk for ( x l , _..,x k ) E B is an A-p.r. benign subset of F.A n A-p.r. computable function J' : N k + N is A-p.r. benign if the subset of k+ I-tuples of the form ( x l , ..., xk, f ( x l , ..., x k ) ) is A-p.r. benign. We then establish the following:

Lemma. If A C N is recursively enumerable and f:N k + N is A-p.r. computable then J'is A-p.r. benign. The proof of the above in [ 4 ] differs somewhat from the proof given in [ 61 in that the characterization of A-p.r. computable functions as the smallest class of functions containing the initial functions Z , U m n , f o ,E and cA and closed under substitution and (unlimited) iteration (see [ 13 ), is used directly. In particular we show that cA is in the class of A-p.r. benign functions since x E A ++ 3 yQ(x,y ) for Q a p.r. (actually g 3 )predicate related to the Kieene T predicate. Also the closure of the class of A-p.r. benign functions under (unlimited) iteration is shown by a direct construction (Proposition 5.10 of [ 41 ). The proof of the theorem is now completed by proving

Lemma. Let A C N be recursively enumerable and P C F be an A-p.r. decidable subset o f a j:g. free group F. Then P is A-p.r. benign.

R. W. Gatterdam, The Higman theorem for primitive-recursive groups

425

The proof uses the previous lemma applied t o the characteristic function of P. The construction given in [ 41 follows that of [ 61 closely verifying A-p.r. computability at each step. References [ 11 F.B. Cannonito and R.W. Gatterdam, The computability of group constructions, part 1, this volume. [ 2 ] C.R.J. Clapham, Finitely presented groups with word problem of arbitrary degrees of insolubility, Proceedings of the London Mathematical Society 14 (1964) 633-676. [ 3) C.R.J. Clapham, An embedding theorem for finitely generated groups, Proceedings of the London Mathematical Society 17 (1967) 419-430. [ 4 ] R.W. Gatterdam, Embeddings of primitive recursive computable groups, doctoral dissertation, University of California, Irvine, 1970. Submitted for publication. 1.51 G. Higman, Subgroups of finitely presented groups, Proceedings of the Royal Society, A 262 (1961) 455-475. [ 6 ] J.R. Schoenfield, Mathematical Logic (Addison Wesley, 1967).

CONNECTIONS BETWEEN TOPOLOGICAL AND GROUP THEORETICAL DECISION PROBLEMS Wolfgang HAKEN University of Illinois, Urbana

The main purpose of this article is t o provide the logician with a brief (and thus rather incomplete) survey concerning results and open questions on decision problems in topology and their relations t o group theoretic decision problems.

5 1. Description of some basic decision problems in topology Usually topological decision problems are concerned with finite simplicia1 complexes. Such complexes can be described (up t o isomorphism) by their incidence matrices which are finite matrices with integral entries. (For definitions see for instance [ 201 or [ 31 .) If the questions of a decision problem deal with more general topological spaces, as for instance topological manifolds, or with spaces with more structure, as for instance differentiable manifolds, then a detailed discussion is required as t o how such a question can be posed in finite terms (as is required for a decision problem). A method of presenting a differentiable manifold in finite terms is discussed in [ 31 . In this article we restrict ourselves t o decision problems of finite combinatorial topology, i.e., the spaces considered are finite simplicial complexes, in particular, the n-manifolds are compact combinatorial n-manifolds (= finite complexes such that the simplices incident t o a vertex form a combinatorial n-ball, i.e., a complex which is isomorphic to a piecewise linear subdivision of a standard n-dimensional simplex). All mappings are 427

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simplicia1 (= piecewise linear, continuous) mappings; thus all homeomorphisms are cornbinatorial equivalences. F o r dimensions I I < 3 this means n o restriction of generality since Moise [ 131 has proved that every topological 3-manifold can be triangulated into a conibinatorial 3-manifold and that all its triangulations are combin ;it oria 11y equivalet i t .

1 .a. Homeomorphism problems The ( piecewise linear) homeomorphism problem for n-dimensional cornplcses is the problem to decide for any two given ncomplexes K ; ‘ , K$ whether or not K ; and K ; are (piecewise line arl y ) honi e om orp h ic . The most interesting sub-problem is the homeomorphism problcin for n-manifolds where the given complexes are assumed t o be n-manifolds M ; ’ , M ” Usually one considers further subpr3blems 2’ by qxcifying the dimension n , e.g., n = 2, n = 3 , n = 4. A n extremely specialized (but especially interesting and for / I > 2 still open) sub-problem is the horrieomorphism problem with tlzc n-sphere where one of the given manifolds is fixed t o be the n-sphere 5’”. A more general problem is the homeomorphism problem for paws of coiizplc~xes(one being a sub-complex of the other). This is t o decide for any two given pairs ( K , , L , ) , ( K 2 ,L,) where L , C K , , L , C K , whether or not there exists a (piecewise linear) homeomorphism f ’ from K , onto K , which maps L , o n t o L,. A well publicized sub-problem of this is the equivalence problem of (chssical) h o t theory, where both K , and K , are fixed t o be the 3-sphere S 3 and the given sub-complexes L , , L , are polygonal 1 -\plieres (“knots”). What is sometimes called the knot problem is the equivalence problem with the trivial knot, i.e., the sub-problem o f the dbove where L , is fixed to be the boundary of a plane tri;ingle. I n general, if K , and K , are homeomorphic t o some fixed complex K , and L , , L 2 are homeomorphic t o a fixed complex L , we have the knotting problem of L in K.

1.b. Isotopy and hoiiiotopy problems

A modification of the knotting problem of L in K is the isotopy

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problem of L in K. Here K , and K , are not only homeomorphic but identical to K and the question is whether or not L and L , are isotopic in K (i.e., L , can be deformed into L , within K in such a way that n o self-intersections occur at any stage of the deformation). Another modification is the homotopy problem of L in K where instead of L , , L , we consider two given (continuous, simplicial) mappings I, : L + K , I, : L + K, and the question is whether or not I , and I , are homotopic (i.e., I , can be deformed into I, where self-intersections of the image are permissible at all st ages). An interesting special case of the homotopy problem is obtained if L is chosen t o be the I-sphere S1. We call this the homotopy problem of closed curves in K. An important sub-problem arises if one of the mappings, say 1, : S1 + K , is fixed t o be constant (i.e., maps all of S1 into one point in K ) . Let us call this the contractibility problem of closed curves in K (since a closed curve I, : S1+K is called contractible in K if I, is homotopic t o a constant map). The importance of the two problems mentioned is mainly due t o the fact that the homotopy problem of closed curves in a connected complex K is logically equivalent to the conjugacy problem in the fundamental group 7r,(K) of K whereas the contractibility problem of closed curves is equivalent to the word problem in 7rl(K). This can be seen as follows. The elements of n l ( K )are the homotopy classes of loops in K which originate and terminate at some fixed base point 0 in K . (Two loops are in the same class if and only if they are homotopic in K relative t o the restriction that the initial and terminal points must be kept fixed at 0 at all stages of the deformation.) The identity element of 7rl(K)is the class of all contractible loops. Now an oriented closed curve C in K corresponds to a conjugacy class [c] of an element c of 7r1 ( K ) . In order t o produce a loop that represents c we choose an auxiliary path A from 0 to a point P on C and we compose A with a path from P to P around C (in the direction of the orientation of C) followed by A in the direction from P to 0. The freedom we have in choosing A allows us to obtain representatives of all conjugates of c in 7r1 ( K ) . Two closed curves correspond t o the same conjugacy class if and only if they are homotopic in K . On the other hand a presen-

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tation of n l ( K ) can be obtained from K by a n easy procedure (see for instance [ 201 ) where the generators are represented by certain loops in the 1 -skeleton of K and the defining relations are read from the triangles of K . For a given word c (in the generators of ‘$ ) it is then easy t o construct a closed curve C in K that corresponds to [ c ] ; and to a given closed curve C one can obtain a corresponding word c‘. With this one can easily establish the asserted eqii ivalences. The equivalences mentioned in the above paragraph are so obvious that topologists usually identify the equivalent decision problems and thus, for instance talk about the word problem in 7r,(K) without ever mentioning a presentation ‘J ofl ?r,(K)but just dealing with closed curves in K . I f K is an n-manifold M n with yt > 2 then to every closed curve in M “ one can find a homotopic simple closed curve. Thus instead of presenting a closed curve by a mapping 1 : S1+ M n we can consider a homotopic simple closed curve in M n , presented by its image. which is just a closed polygon L in M n . Then the questions which make u p our decision problems can be formulated as follows: Given two (polygonal) simple closed curves L , L , in M n , do L and L , bound a singular annulus in M N ?(where a “singular annulus” is the continuous, simplicia1 image of an annulus; the existence of such a singular annulus between L , and L , in hi” is obviously equivalent t o the fact that L , and L , are homotopic in A!’?). And: Given a simple closed curve L in M ” , does L bound a singular disk i n M “ ‘?

A sirigirlur dish- means a disk that may have self-intersections (being just the continuous image of a disk). This definitely distinguishes the word problem in the fundamental group of a 3manifold M 3 from the knot problem in M 3 . For, L is a trivial knot in M 3 if and only if it bounds a non-singular disk in M 3 .

RcJlll(Irli:

1 .c. Isomorphism problems of fundamental groups The questions of the isomorphism problem are whether o r not two given complexes K , , K? have isomorphic fundamental groups. An important sub-problem ;a solution of which would be implied

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by a solution of the word problem) is the siinplv connectedness problem where the questions are whether o r not a given complex K has trivial fundamental group (i.e. is simply connected).

82. Unsolvability results All unsolvability results about topological decision problems which have been obtained so far have been derived from the unsolvability of the word problem [ 2,4,14] , o r triviality problem [ 1,l 1,18 1 of group theory.

2.a. n = 2 To a given presentation ’$ of a group G we can construct a 2dimensional complex K 2 ( ‘$) which corresponds t o where the is isomorphic t o G. We construct fundamental group .rrl(K2( K’( T ) as a cell-complex (but it could be easily subdivided into a simplicia1 complex). A group presentation ‘$ = ( { g l , ..., g,,}, { r , ..., r s}) consists of a set o f Y generatorsgl, _ _g,, _ .and a set of s relators i l , ..., 1, where the relatorsri are words i n the symbols gi’’ . We d o not demand that all the trivial relators g j g j - I and gr1gi be included among the rj’s. However. we allow that two (or more) o f the relators be equal (ri = i k although j k ) o r that some of the relators be cnipty words (the empty word being denoted by *). I n order t o construct K 2 ( V ) we choose ;I point 0 for the only vertex of K 2 ( ). Corresponding t o each gencrator g i in ’$ we choose a ]-dimensional cell G j of K 2 ( ‘p) which is an arc that originates and terminates a t 0. Except for their end points. the G iare chosen pairwise disjoint. Thus UT= G iis a wedge of r loops with fundumental group the free group of rank Y. Corresponding t o each relator ri in ‘$ we choose a ’-dimensional cell Ri o f K ’ ( V ) which is a disk whose boundary aRi is identified t o a closed curve i n U:=l G iwhich runs through the G, in the same order in which the gi occur in r, (such that one direction of Gi corresponds t o gi and the opposite direction to gl:’ ). We say that “the reading of aRi in the G j is equal t o rj”. (If ri = * then we identify all of aRi t o Q . ) Except for their boundaries, the Ri are chosen pairwise disjoint.

v))

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This finishes the construction of K2($). Obviously, if we add a trivial relator to $ we do not change the group G presented by $!3 ;however we change the homeomorphy type of K 2 ( $ ) by adding a 2dimensional cell (in such a way that the 2nd Betti number of K2(’lp) is increased by 1). Now the unsolvability of the triviality problem of group theory implies immediately the unsolvability of the simply connectedness problem for 2-complexes (and thus of the word problem, the conjugacy problem, and the isomorphism problem of the fundamental groups for 2-complexes).

2.b. n > 4 In order to construct an n-dimensional manifold M n ( $) corresponding to a group presentation $ with n l ( M n ( $ ) ) 4 G (where G is the group presented by ‘$3 ) we may embed the 2-complex K 2 ( !@)into the ( n + 1)-dimensional Euclidean space E n + l (provided that n > 4); then we choose a simplicial, regular neighborhood N of K 2 ( ! @ )in E n + l (which is an (n + 1)-manifold with boundary). It is not difficult to show that not only the fundamental group of N but also the fundamental group of its boundary is isomorphic to G. Thus we may choose the boundary of N for M f z (q). This construction of Mn(’$) is essentially the same as that indicated in the textbook of Seifert-Threlfall [ 201 as an exercise (see also [3]). Again, we conclude immediately the unsolvability of the triviality problem, word problem, conjugacy problem, isomorphism problem o f the fundamental-groups for n-manifolds with n 2 4. It was the great discovery of Markov [ 121 that one can also conclude the unsolvability of the homeomorphism problem for n-munl/blds with n 2 4. His proof is based on two observations (for details see also [ 31 ): i) If ’$ presents the trivial group then the presentation !+3* which is derived from by adding r times the empty word as a new relator can be transformed into the “standard presentation” C r, r + s - ( { g l , .... gr 1 , .C g , , ..., gr, * s } ) (where r of the relators are equal to the l-letter-wordsgl, ..., gr and s relators are empty) by certain especially simple Tietze transformations. We call them Markov oper‘‘I t‘ions.

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ii) If a group presentation is derived from a group presentaby a Markov operation then M n ( is homeomorphic t o tion

MY

!$I

1.

v2)

Consequently, if a given presentation ‘10 presents the trivial group then M n ( V *) is homeomorphic to M n ( C r, r + s ) . Thus a solution of the homeomorphism problem for n-manifolds (for some n > 4) would imply a solution of the triviality problem of group theory. Moreover, since M n ( C r, r + s ) can be seen t o be an “n-sphere with s handles of index 2”, it is an unsolvable problem to decide whether or not a given n-manifold with n > 4 is an n-sphere with handles of index 2. However the homeomorphism problem with the n-sphere itself is still open (see Section 4.a).

2.c. n = 3 It seems to be rather difficult to obtain unsolvability results about 3-dimensional manifolds. Not every finitely presented group is isomorphic to the fundamental group of a 3-manifold. Moreover, some progress has been made in solving decision problems for 3manifolds (see Section 3.b). In this section I shall consider an unsolvable decision problem on 3-manifolds the questions of which are somewhat more sophisticated then those of the basic decision problems described in Section 1. However, the questions of this unsolvable problem have a certain similarity t o questions of solved decision problems which we shall discuss in Section 3.b. So it appears that in dimension 3 solvable and unsolvable decision problems come relatively close to each other. Let G be a finitely presented group with unsolvable word problem and let !/3 be a presentation of G (as for instance exhibited in [ 21 ). Now we construct a 3-manifold M: which is rather simple in so far as it is a subspace of the 3-sphere S 3 which is bounded by some tori. The questions of our decision problem will be concerned with (polygonal) simple closed curves in this fixed 3-manifold. Corresponding t o each generator g i (i = 1 , ..., r ) in we choose a (non-singular, polyhedral) disk Gi2 in the 3-sphere S 3 so that the G’ are pairwise disjoint. We choose small (regular, polyhdedral) neighborhoods N iof the boundary curves of the G iin S3 and we

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remove the interiors of the Nifrom S 3 . This yields a 3-manifold F3 which is bounded by Y tori T: (the boundaries of the N i ) and the fundamental group of which is the free group of rank Y . Corresponding t o each relator r, in '$ ( j = 1 , ..., s) we choose a (polygonal) simple closed curve C, in the interior of F 3 which pierces the disks G; in the same order in which the corresponding letters g i occur in Y, (so that to a symbol g i - l there corresponds a piercing in the opposite direction t o a piercing corresponding togi). In order to obtain M:; we remove from F 3 the interiors of small (regular. polyhedral) neighborhoods of the Ci. Then M i is bounded by the tori Ti' and by s tori, say U f . Now our decision problem consists of the following questions. Given a (polygonal) sitnple closed curve L in the interior ofhi:;, does L bound a singular disk with holes in M i such that the boundary curves of the holes lie in the tori U t ? This decision problem is unsolvable since it is logically equivalent to the word problem i n G. F o r each word c in the generators g j one can construct a polygonal simple closed curve L in M i which pierces the G; corresponding t o c (and from each polygonal simple closed curve in M i one can read the corresponding word u p to cyclic permutations). Then c represents the group identity in G if and only if'L is contractible in a complex K which is the union ofM:; and s cones over the tori U2 (the interiors of the C" I being pairwise disjoint and disjoint from hi:;).K is not a 3-manifold since i t contains precisely s exceptional points (the vertices of the cones CF) the neighborhoods of which are not 3-cells. But L is contractible in K if and only if it bounds a singular disk in K , and this again is the case if and only if L bounds a singular disk with holes in Ad:,, as demanded.

";'

$3.Solvability results

The 11O I I I c o i l 10 rpli iw I pro h k i n jOr 2-cliineir sio iial ~*omplex es has bccn solved by Papakyriakopoulos [ 171 .

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The fundamental groups of the 2-manifolds without boundary are one-relator groups and thus the corresponding word problems are solved by Magnus [ 101. The 2-manifolds with boundary have free fundamental groups.

3.b. n = 3 Solutions have been obtained for some special cases of the homeomorphism problem of 3-manifolds and of the word problem in their fundamental groups. These solutions apply only t o 3-manifolds with or without boundary which contain so-called incompressible and boundary-incompressible surfaces. So it appears that the more complicated 3-manifolds are more easily accessible for solving the basic decision problems. In a 3-manifold M 3 we consider surfaces (i.e., polyhedral 2manifolds) M 2 so that the interior of M 2 lies in the interior of M 3 and the boundary of M 2 lies in the boundary o f M 3 (but may be empty). The concepts of inCOFT.?preSSibi/it.Vand botindari,-irzcornpressibility for such a surface M 2 in M 3 were originally defined geometrically [ 51 by conditions that no handle of M 2 can be “cut o f f ’ by a (non-singular) disk in the complement of M 2 in M 3 , etc. But in the case that M 2 is orientable the loop theorem and Dehn’s lemma, as proved by Papakyriakopoulos [ 15.161 , permit definitions in terms of homotopy as follows. “Incompressible” means, if a closed curve C in M 2 is contractible in M 3 then it is also contractible in M 2 . “Boundary-incompressible” means, if a curve C in M 2 that originates and terminates at the boundary of M 2 can be deformed within M 3 into the boundary of 1143 without moving its end points, then such a deformation of C can also be carried out within M 2 . The class of 3-manifolds which contain incompressible and boundary-incompressible surfaces is rather large: i ) If an orientable 3-manifold M 3 has a boundary which does not only consist of 2spheres then it contains an incompressible and boundary-incompressible, orientable surface. ii) If an orientable 3-manifold without boundary has an infinite first homology group or has a fundamental group which is a free product with amalgamation then and only

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then it contains an orientable, incompressible surface. (The latter part of ii was observed by Waldhausen [231.) We can solve (as indicated in [ 61 ), the special case of the homeoniorphism problem f o r 3-manifolds which we obtain by restricting the questions as follows.

( I ) The given 3-manifolds M i , M ; are given in such a way that they are known t o be “irreducible”, i.e., that every polygonal 2sphere in 1143 bounds a 3-ball in Mi3 (i = 1,2).

(TI) M; is given in such a way that it is known to contain orientable incompressible and boundary-incompressible surfaces. (111) M: does not contain a sub-manifold z3 with the following properties. i) is a fibre bundle over S1 with fibre a surface F2 (with or without boundary). ii) The boundary of G3 (if n o t empty) consists of incompressible surfaces in M 3 . iii) The first Betti number of $3 is 1.

a3

The main tools for solving the problem described are algorithms for determining incompressible and boundary-incompressible surfaces of minimal genus in given 3-manifolds. The basic theory for deriving such algorithms has been developed in [ 5 ] (see also [ 191 ) one of the simplest applications was an algorithm for the knot problem: to determine whether or not a given polygonal simple closed curve in a 3-manifold M 3 bounds a (non-singular) disk (= a surface of genus 0) in M 3 . In general it is possible to determine for a given 3-manifold M 3 and for a given integer g whether or not M 3 contains an incompressible and boundary-incompressible surface of genus < g. I f such a surface exists then the algorithm constructs one which is of minimal genus. Moreover, as indicated in [ 61 , in many cases the algorithm yields all possible types of such surfaces in 1113.

Reinurk: If in the questions of the unsolvable decision problem stated in 2.c the word “singular” is replaced by “non-singular” then we get a decision problem that can be solved (by some modification of the algorithm for the knot problem). This shows how much easier it is t o find algorithms for determining non-singular

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surfaces than for determining singular ones, or, in other words, how much easier it is to solve isotopy problems than homotopy problems. The algorithms mentioned above can be applied t o split the given 3-manifold M : by incompressible and boundary-incompressible surfaces of minimal genus into sub-manifolds, provided that M : contains such surfaces and that we know some upper bound for their genus. For this reason we need the restriction (11) above. Then we split the sub-manifolds again by simplest possible surfaces into sub-manifolds, and so on. We can continue this process as long as not all boundary surfaces of the sub-manifolds obtained are 2-spheres. I t can be proved that this process must eventually terminate and thus yield a finite decomposition of M : the 3-dimensional pieces of which are bounded only by 2-spheres. Here we need the restriction (I) in order to conclude that these pieces are 3-balls, i.e., that we actually arrived at a cell-decomposition of M : . We try to arrange this splitting procedure in such a way that at each stage there are only finitely many possibilities of splitting and that all the possible splitting surfaces can be algorithmically found. If this succeeds then there exist only finitely many (isomorphism types of) cell-decompositions of M : we can end up with, and all of them can be constructed. F o r this purpose we need the additional restriction (111) (which was not recognized in [ 61 ), Then the other given 3-manifold, M : , is homeomorphic t o M : if and only if the same procedure can be applied t o it and yields cell-decompositions which are pairwise isomorphic t o those obtained for M?. Waldhausen [ 241 found a solution for the word problem in the fundamental groups o f 3-manifolds under the restrictions (I) and (11) stated above. The first part of his algorithm is a splitting procedure applied t o the given 3-manifold M 3 as described above. However, it is sufficient to obtain some one of the special cell-decompositions of M 3 and it is not necessary that there are only finitely many of them. Therefore the restriction (111) is not required. Then the given closed curve L in M3 is deformed into a nicest possible position rel-

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ative t o the special cell-decomposition which permits t o decide whether or not L is contractible in M 3 . (This solves the contractibility problem for closed curves in M 3 which is, according t o our discussion in Section 1, equivalent t o the word problem in ~1

(M3).)

From another result of Waldhausen [ 231 it follows that the hor?ieomor~J/iisrvi problem jor orientable, irreducible 3-manifolds without boundary that contain orientable, incompressible suyfaces is eyziivalent to the isomorphivn problem o j their fundamental groups.

5 4. Some particular open decision problems In this section we mention some topological decision problems which are still open but are so closely related t o group theoretic decision problems that one might attack them with methods of combinatorial group theory as well as with topological methods.

4.a. The homeomorphism problem with the n-sphere for n > 5 We call a group presentation balanced if the n u m b e r s of relators in !$ equals the number r of generators in !$ . I f the trivialiti- problem f o r balanced presentations could be proved unsolvable then this would imp1.y the imdvability o f the homeomorphism problem with the n-sphere for each n > 5 . This can be seen as follows. If a presentation of the trivial ' ), as constructed in group is balanced then the n-manifold M n ( !$ Section 2.b for any 11 > 4, is a homotopy rz-sphere (i.e., its fundamental group and its homology groups are isomorphic t o those of the n-sphere S n ) . But the generalized Poincare conjecture for dimensions > 5 , as proved by Smale 121 I and Stallings [ 2 2 1 , says that a homotopy n-sphere is an n-sphere if I Z 2 5 . Thus a given balanced group presentation S$ presents the trivial group if and only if M n ( !$ ) is homeomorphic t o S n for any n 2 5. This would hold for I I = 4 also, if the generalized Poincark conjecture were proved for the (very special 1 case of Seifert-Threlfall-Markov-4-manifolds as constructed in Section 2.b.

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4.b. The homeomorphism problem with the 3-sphere, the simply connectedness problem for 3-manifolds, and the Poincar6 conjecture The (still open) PoincarC conjecture states that a compact, simply connected 3-manifold without boundary is a 3-sphere. This does not constitute a decision problem. However, the truth o f the Poincark conjecture would imply the equivalence of the two decision problems mentioned in the headline. Some partial results have been obtained regarding the two decision problems which has been discussed in Sections 1I.b and 1I.c of [ 8 ] . Recently Jaco [9] obtained the result that the simply connectedness problem for 3-manifolds is equivalent to a decision problem the questions of which may be formulated as follows. Given 2s elements x l , ..., x,, y l , ..., y , in a direct product F, X F, of two free groups of rank s such that (i) the projections of x l , ..., x,, y l , ..., y , into the factors of F, X F, generate the factors, (ii) the relation ~ l y l ~ ~ y i ~ ~...... 2 x,y,x,'y;' ? / 2 ~ ~= 1y holds. ~ Question: Do x l , ..., x,, y l , ..., y , generate F, X F,? In Jaco's theory the x l , ..., x,, y l , ..., y , are generators of the fundamental group 7rl (M2) of a handle surface M 2 of genus s, where (ii) is a defining relation for 7rl ( M 2 ) .Given is a homomorphism 7r1 ( M 2 ) + F, X F,, the image of which projects o n t o the factors, and the question is whether or not this homomorphism is onto. So we may call this an epimorphism problem. Regarding this problem it is remarkable that C.F. Miller found a simple proof that if the condition (ii) is omitted then the questions form an unsolvable problem.

4.c. The sufficiently largeness problem for 3-manifolds Waldhausen calls an orientable 3-manifold sufliciently large [ 23 1 if it contains an orientable, incompressible surface. The restriction (11) in Section 3.b would be weakened if we could algorithmically decide whether or not a given 3-manifold M 3 is sufficiently large. If the first homology group H 1 ( M 3 )is infinite then M 3 is known

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to be sufficiently large. However, if H1(M3) is finite we have an open problem. This problem is equivalent t o the problem t o decide whether or not the fundamental group of M 3 is a free product with amalgamation.

4.d. The homeomorphism problem for fibre bundles over S1with fibre a 2-manifold A fibre bundle Over Sl with fibre a 2-manifold M 2 (with o r without boundary) is a 3-manifold obtained from the Cartesian product M 2 X [0,1] of M 2 with the unit interval [0,1] b y identifying the top, M 2 X ( l ) , t o the bottom, M 2 X (0), according to a homeomorphism cp from M 2 X (1) o n t o M2 X (0). Thus the 3manifold obtained is determined by M 2 and cp. If (11: M 2 + M2 X (0) and p : M 2+ M2 X ( 1 ) map a point p o f M 2 t o p X (a) and p X (11, respectively, then cp is determined by a self-homeomorphism g = a-1 cp p of M 2 .Two 3-manifolds obtained in this way from the same M2 by two self-homeomorphisms (pl and t@2, respectively, are homeoand (p2 are conjugate elements in the morphic if and only if group of self-homeomorphisms of M 2 . Thus the homeomorphism problem f o r these 3-manifolds is equivalent to the conjugacy problem in the group o f self-homeomorphisms of M 2 . A solution of this problem would remove the restriction (111) in Section 3.b. 0

References [ 11 S.I. Adjan, The algorithmic unsolvability of checking certain properties of groups, Dokl. Akad. Nauk SSSR 103 (1955) 533-535 (in Russian). [ 2 ] W.W. Boone, The word problem, Ann. Math. 7 0 (1959) 207-265. [ 31 W.W. Boone, W. Haken and V. P o h a r u , On recursively unsolvable problems in topology and their classification, in: A. Schmidt e t al. (eds.), Contributions to mathematical logic (North-Holland, Amsterdam, 1968) 37-74. [ 4 ] J.L. Britton, The word problem, Ann. Math. 77 (1963) 16-32. [ S ] W. Haken, Theorie der Normalflichen, ein Isotopiekriterium fur den Kreisknoten, Acta Math. 105 (1961) 245-375. [ 61 W. Haken, Uber das Homoomorphieproblem der 3-Mannigfdltigkeiten, 1, Math. Zeitschr. 80 (1962) 89-120. ( 7 J W. Haken, Some results on surfaces in 3-manifolds, in: Studies in modern topology, MAA Studies in Mathematics, vol. 5 (Prentice-Hall, Englewood Cliffs, 1968) 39-98.

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44 1

[8] W. Haken, Various aspects of the 3-dimensional Poincare problem, in: Topology of manifolds, (Markham, Chicago, 1970) 140-152. [9] W. Jaco, Stable equivalence of splitting homeomorphisms, in: Topology of manifolds, (Markham, Chicago, 1970) 153-156. [ 101 W. Magnus, A. Karras and D. Solitar, Combinatorial group theory, (Wiley, New York, 1966). [ 11] A.A. Markov, Impossibility of algorithms for recognizing some properties of associative systems, Dokl. Akad. Nauk SSSR 77 (1951) 953-956 (in Russian). [I21 A.A. Markov, Insolubility of the problem of homeomorphy, Proc. Intern. Congress of Mathematicians, 1958 (Cambridge University Press) 300-306. [ 131 E.E. Moise, Affine structures in 3-manifolds, V, The triangulation theorem and Hauptvermutung, Ann. Math. 5 6 (1952) 96-114. [ 141 Novikov, P.S. On the algorithmic unsolvability of the word problem in group theory, Trudy Mat. Inst. Steklov, no. 44, 1955 (in Russian). [I51 C.D. Papakyriakopoulos, On solid tori, Proc. London Math. Soc. (3), 7 (1957) 281-299. [ 161 C.D. Papakyriakopoulos, On Dehn’s lemma and the asphericity of knots, Ann. Math. 66 (1957) 1-26. [17] C.D. Papakyriakopoulos, A new proof for the invariance of the homology groups of a complex, Bull. Soc. Math. Gr&e 22 (1943) 1-154. (1946) (in Greek). [ 181 M.O. Rabin, Recursive unsolvability of group theoretic problems, Ann. Math. 67 (1958) 172-194. [ 191 H. Schubert, Bestimmung der Primfaktorzerlegung von Verkettungen, Math. Zeitschr. 76 (1961) 116-148. [ 201 H. Seifert and W. Threlfall, Lehrbuch der Topologie, (Akademische Verlagsgesellschaft B.G. Teubner, Leipzig, 1934). [21] S. Smale, Generalized Poincark’s conjecture in dimensions greater than four, Ann. Math. 74 (1961) 391-406. [22] J.W. Stallings, Polyhedral homotopy spheres, Bull. Am. Math. Soc. 66 (1960) 485488. 1231 F. Waldhausen, On irreducible 3-manifolds which are sufficiently large, Ann. Math. 87 (1968) 56-88. [24] F. Waldhausen, The word problem in fundamental groups of sufficiently large irreducible 3-manifolds, Ann. Math. 88 (1968) 272-280.

ON THE WORD PROBLEM AND T-FOURTH-GROUPS Seymour LIPSCHUTZ Temple University,Philadelphia

5 1. Introduction Let G be a finitely presentnd group with generating elements ..., R , = 1. We assume without loss in generality that the R i , called relators, form a symmetric set, i.e. that the R i are cyclically reduced and are closed under the operations of taking inverses and cyclic transforms. We call G a T-fourth-group if the following two conditions hold: (i) One-fourth condition: If R i z X Y and Ri = X Z are distinct relators, then the length of the common initial segment X is less than '/J the length of either relator. (ii) Triangle condition: If each of three relators Ri, Ri and R , is written along one side of a triangle then cancellation cannot occur at all three vertices, i.e. at least one of the products RiRi, RiRk, R,Ri is freely reduced without cancellation. The class of T-fourth-groups is of interest because Greendlinger [ 4 ] solved the word and conjugacy problems for these groups. Also Lyndon [ 6 ] solved the word problem and Schupp [91 the conjugacy problem for a class of groups with similar conditions. Our main theorem follows.

a,, _..,a, and defining relations R , = 1,

Main Theorem. Let K be the free product o f T-fourth-groups with an in,finite cyclic group amalgamated. Then K has a solvable word problem. Results identical t o that above and others below have been proven by the author [ 5 1 for Greendlinger sixth-groups, groups 443

444

S. Lipschutz, On the word problem and T-fourth-groups

which satisfy condition (i) above with $ replaced by &. We also note that McCool [S] has proven similar results for Britton freeproduct sixth-groups.

$2. Notation Capital letters, V, W , A , B, ... will denote freely reduced words unless otherwise stated or implied. We use the following notation for words V and W in the group G: 1(W) for the length of W, V = W means V and W are the same element of the group G, V z W means V and W are identical words, V A W means V does not react with W , i.e. U = VW is freely reduced, V C W means V is a subword of W , i.e. W 2 A V B . The letter R with o r without subscripts shall always denote one of the relators R , , ..., R,. We say W is fully reduced if it is freely reduced and does not contain more than half of a relator. We say W is cyclically fully reduced if every cyclic transform of W is fully reduced.

§3.Preliminary lemmas We shall require the following obvious consequence of Theorem 2 in [41. (Also stated in 131 .)

Lemma 1 (Greendlinger). Let W be a freely reduced word in a Tfourth-group G , and suppose W = 1. Then W contains more than of c( relator, or W contains disjointly two subwords, each containing more than o f u relator.

3

Remark. The generalized word problem for a group G modulo a subgroup H is t o decide whether or not an arbitrary element of G is also in H . The generalized word problem reduces to the word problem when H = 1 ; hence is unsolvable in general (c.f. Boone [ 1, § 35 1 1.

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445

However, the following result is easily shown using properties of free products with amalgamations (c.f. [ 5 , Lemma 21 or [7, Section 4.21 ):

Lemma 2. Let K be the free product of groups G , with a subgroup H amalgamated. If the word problem for the Gi and the generalized word problem for the Gi modulo Hare solvable, then K has a solvable word problem. The next t w o lemmas are about T-fourth-groups G. As the proofs are relatively long and combinatorial, we give them in Sections 5 and 6.

Lemma 3. Suppose W = U where U is fully reduced. Then l(U) < rl(W),where r is the length of the largest defining relation in G. Lemma 4. Suppose W is cyclically fully reduced, and suppose there is no relator R 2 U t such that W is conjugate to a power of U. If W 2 is also cyclically fully reduced then W n is fully reduced for all n. If W 2 is not cyclically fully reduced then there is a relator

where AB is a cyclic transform of W , and (TB)" is fully reduced for all n. In either case, W has infinite order. 94. Main results Our first theorem is similar t o one proved by Greendlinger [ 2, Theorem VIII] for sixth-groups.

Theorem 1 . An element W in a T-fourth-group G has finite order if and only if there is a relator R 2 U k and W is conjugate to a power o f U.

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S. Lipschutz, On the word problem and T-fourth-groups

Proof. Clearly, if there is a relator R s U k and W is conjugate t o a power of U , then W has finite order. On the other hand, suppose W # 1 and suppose there is n o relator R z U k such that W is conjugate t o a power of U. By taking cyclic tranforms of W and substituting whenever more than half of a relator appears, we can ob tain a cyclically fully reduced word V which is conjugate t o W. By Lemma 4, V has infinite order. Therefore W also has infinite order and the theorem is proved. The order problem for a group G is t o decide the order of any element in G. If k is the maximum positive integer such that there is a relator R = U k in a T-fourth-group G then, by Theorem 1 , n o element has finite order greater than k . This fact together with the solvability of the word problem for G imply

Corollary 1 . The order problem for a T-fourth-group G is solvable. Theorem 2. Let G be a T-fourth-group. Then the generalized word problem for G modulo any cyclic subgroup H is solvable. Proof. Let V be any element in G. By Corollary 1 , we can determine the order of the generator W of H . Suppose the order of W is finite. Then V belongs t o H iff V = W n for 1 < m < order (W). Since the word problem for G is solvable, this case is decidable. Now suppose W has infinite order. Note that V belongs t o gp(W) if and only if A-I V A belongs t o gp(A-1 WA). Thus by taking cyclic transforms of W and substituting whenever more than half of a relator appears, we can reduce our problem t o the case where W is cyclically fully reduced and has the properties of Lemma 4. In fact: we can assume in Lemma 4 that W z A B . Suppose W n .is fully reduced for all n . By Lemma 3, V = W m only if I m I < I(Wm ) < rl( V) where r is the largest defining relation in G. Thus we can decide if V belongs t o gp(W). On the other hand, suppose Wn is not fully reduced. Then, by Lemma 4,

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where E = 1 o r E = 0. Hence V is in gp(W) iff V or VW-1 is in gp(TB). But (TB)k is fully reduced for all k . So the theorem is true for this last case also.

Remark. Our Main Theorem now follows directly from Theorem 2 and Lemma 2. In fact, we can generalize even further. Let 2,denote the class of T-fourth-groups. Let Z i , i > 1 , be the class of groups which are the free product of groups in Zi with an infinite cyclic group amalgamated. Let G belong t o Z j for some i. Then by Theorem 5 in [ 5 1 , the word problem for G and the generalized word problem for G modulo any infinite cyclic group are solvable. ~

Theorem 3. Consider a T-fourth-group

G = gp(al, ..., a n , 61, ..., bm ; R1 = 1 , .., R , = 1 )

such that every freely reduced word W = W(a,) is fully reduced. Then H = gp(u,, ..., a,) is a free subgroup o f G, and the generulized word problem for G modulo H is solvable. Proof. By Lemma 1, every non-trivial word W(a,) f 1 ; hence H is free. Let V belong t o G. If V is in H , i.e. if V = W(a,) then, by Lemma 3 , we know the maximum length of W . Since there are only a finite number of words W(a,) of any given length, and since the word problem for G is solvable, the theorem is true.

55.Proof of Lemma 3 Suppose the lemma is not true and suppose W is a word of minimum length for which the lemma does not hold, say I( W ) = n. Then there exists a fully reduced word V such that W = V-1 and I( V )> rn. So

44 8

S. Lipschutz, On the word problem and T-fourth-groups

The minimality of W guarantees that W is fully reduced, and that (1) is freely reduced. Thus (1) satisfies Lemma 1. Moreover, since W and V are fully reduced, W V must contain more than f of a relator R . Say W = AB, V 2 CD, S = BC; where R g SE-' g BCE-I and I(S) > Z(R). Now, substituting in ( I ) , we have

WV

= ABCD s ASD = AED = 1.

Furthermore,

D is fully reduced and I(D) > r(n - l ) , I(C) <

+ <+

Z(R) since V is fully reduced,

I ( B )>

I ( R )since I ( S )> $ I ( R ) ,

I(E)

I ( R )since I C S > )

t z(R).

Thus, I(E) < Z(B), which implies I(AE)< I(AB)= I(W). Let W' = AE, V ' = D . Then W'V' = 1 and V ' is fully reduced. Moreover, I(W') = I(AE)< I(W) = n , so I ( W ' ) < n - 1. But I( V ' ) = I(D) > r(n - 1 ) > rI(W'), so that W' and V ' violate the lemma. This contradicts the minimality of W , so our lemma is true. $ 6 . Proof of Lemma 4

+

Suppose W n contains more than half of a relator; say W contains S where R z ST-' and f ( S )> I(R).Since S C W n , we say S 2 , V'A where V = AB, B $ 1, and V is a cyclic transform of W . Note t f 0 because V and so A are fully reduced. We claim that t = 1. Suppose not, i.e. suppose t > 1. Then V t - ' A is a common initial segment of the relators R z V'AT-1 and R' = V t - 1AT-1 V. But

I( V t - ' A ) 2

I(S) > $Z(R);

hence, by the one-fourth condition, R

2

R ' . Consequently, V and

S. Lipschutz, On the word problem and T-fourth-groups

44 9

V t - 'AT-' commute as words in a free group, and so are powers of the same word U. Therefore, W is conjugate to a power of U and R z V ( V t - 1 A T - ' ) is a power of U. However, this contradicts our original hypothesis. Thus t = 1 and the first part of the lemma is proved. We can now assume that S = ABA, V = A B and R z ABAT-,. We also have B 1. Moreover, since I( V ) < I(R),we have A 1 and T 1. Furthermore, by choosing the largest possible S , we can assume that B A T and T A B . (That is, if say T % T'x-' and B = x B ' , then R z ST-1 z SxT'-1 and P = ABAxB' and so we could have chosen Sx in place of S.) We also have that A A B and B A A because W is cyclically fully reduced, and we have A A T-' and T-l A A because R is freely and cyclically reduced. In addition, we have I ( T ) < I(R) because I(S) > Z(R),and we have I ( B ) < I(R) because I ( V ) < I(R) and A $z 1. We are now ready to show that (TB)n is fully reduced for all n. Suppose (TB)n contains Q where Q is more than half of a relator, say R' z QP and l(Q) > Z(R'). There are seven possibilities.

*

+

+

3

4

Case 1. Q C B . Impossible because V and so B are fully reduced. Case 2. Q C T. Say T = MQN. Then R z ABAN-1 Q-lM-' . By the

one-fourth condition, the relators R ' = QP and R * z QNA-l B-' A-' M are identical; hence I(R') = Z(R*)= Z(R). Accordingly, 'I(R) 2

=

$ I(R') < I ( Q ) < Z(T)<

I(R),

which is a contradiction. Hence Q is not contained in T.

Case 3. Q C TB but Q @ Tand Q (L B. Say T z T 2 T , , B = B,B2

a n d Q z T , B , . T h e n R = A B l B 2 A T 1 - l T 2 - la n d R ' z T,B,P. Note that Q C T implies B , P B and that Q B implies T , $ 1 . Since I ( Q ) > $ Z(R'), either Z(Tl)> Z(R') or Z(B,) > I(R'). (a) Suppose I(T,) > :Z(R'). By the one-fourth condition, the relators R ' z T , B I P and R" z T,A-lB-lA-l T2 are identical. Since B , 1, this contradicts the fact that A A B z B , B 2 . (b) Suppose I(B, ) > Z(R'). Then the relators R" z B,PT, and R * 2 B,B2AT-1A are identical. Since T , $ 1, this contradicts the fact that A A T-l zz T,-l T 2 - l . Thus Case 3 cannot occur.

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Case4. Q C B T b u t Q $ B a n d Q $ T . S a y B = B 4 B 3 , T = T 3 T 4 and Q z B, T,. Then R 1 AB4B,AT4-I T3-l and R' 2 B,T,P. Note that Q $ B implies T , Q 1 and that Q $ T implies B , 1. Since I ( Q ) > I(R'),either Z(B,) > &R')or Z(T,) > f Z(R'). (a) Suppose I(B,) > Z(R'). By the one-fourth condition, the relators R' s B, T,P and R" = B,AT4-' T3-lA B , are identical. Since T , 1 , this contradicts the fact that T4-' T3-l E T-' A A . (b) Suppose I(T,) > Z(R').Then the relators R" = T,PB, and R-' s TA-1B-'A-1 are identical. Since B, p 1 , this contradicts the fact that B , B , s B A A . Thus this case cannot occur.

+

+

= MN and Q s FBM. Then R' s FBMP and R z A B A F - ' E - ' . Note that F Q 1 because Q $ TB. We also have that A Q 1 and B Q 1. Thus the three relators Cusp 5. Q C TBT but Q $ TB and Q $ BT. Say T s EF

Ri s BMPF,

Ri z F-'E-IABA,

R,

= A-lTA-lB-I

violate the triangle condition. Hence this case is impossible.

Case 6. Q c BTB b u t Q $ BT and Q $ TB. Say B = EF E MN and Q z FTM. Then R: = FTMP and R AMNAT-'. Note that M 1 because Q $ BT. We also have that A 1 and T 1. Thus the

+

three relators

R i = MPFT,

R, s T-'AMNA,

+

+

R,

1A-lTA-lN-lM-l

violate the triangle condition. Hence this case cannot occur.

Case 7. Q $ TBT and Q $ BTB. Then Q = EBF s MTN where F and T have a common nontrivial initial segment, and where N and B have a common nontrivial initial segment. Now the relators R s ABAT-' and R* 1 AT-1AB are distinct because B A T. Hence by the one-fourth condition, &A)< I(R). Accordingly, either I ( B )> I ( R )or Z ( T )> I ( R ) . (a) Suppose I(B) > I(R).Now R z A B A T - ' and R' z QP 2 EBFP. By the one-fourth condition, the relators R i 2 BAT-' A

+

+

S. Lipschutz, On the word problem and T-fourth-groups

45 1

and R . BFPE are identical. Since F and T have a common nonJ.-. trivial initial segment, this contradicts the fact that T-’ A A . (b) Suppose Z(T) > Z(R). Now R z A B A T-’ and R‘ = QP = MTNP. BY the one-fourth condition, the relators R, z TNPM and R-l = TA-lB-lA-’ are identical. Since N and B have a common nontrivial initial segment, this contradicts the fact that A A B. We have shown that this last case is also impossible. It remains to show that W has infinite order. By Lemma 1, a fully reduced nonempty word cannot be 1. Thus we have proved that Wn # 1 for all n or that W2 is conjugate to ABAB = TB such that (TB)n # 1 for all n . In either case, W has infinite order. Thus the lemma is proved. 5

References [ l ] W.W. Boone, The word problem, Ann. of Math. (2) 70 (1959) 207-265. [2] M. Greendlinger, On Dehn’s algorithm for the conjugacy and word problems, with applications, Comm. Pure Appl. Math. 13 (1960) 641-677. [ 31 M. Greendlinger, Solutions of the word problems for a class of groups by means of Dehn’s algorithm, and of the conjugacy problem by means of a generalization of Dehn’s algorithm, Doklady Akad. Nauk SSSR 154 (1964) 507-509. [4] M. Greendlinger, On the word problem and the conjugacy problem, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965) 245-268. [S] S. Lipschutz, An extension of Greendlinger’s results on the word problem, Proc. Amer. Math. Soc. 15 (1964) 37-43. [6] R.C. Lyndon, On Dehn’s algorithm, Math. Ann. 166 (1966) 208-228. [7] W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory (Wiley, New York, 1966). [8] J . McCool, The order problem and the power problem for free product sixth-groups, Glasgow Math. J. 10 (1969) 1-9. [9] P.E. Schupp, O n Dehn’s algorithm and the conjugacy problem, Math. Ann. 178 (1968) 119-130.

ON A CONJECTURE OF W. hlAGNUS James McCOOL and Alfred PIETROWSKI University of Toronto

Let F, be the free group with free generating set X,,..., X,, and let G be a group with one defining relation given by the presentation (X, , ..., X, ; R = 1 ). We denote by P ( G ) the set of all presentations of G of the form ( X,,..., X, ; S = 1 ). Two elements ( X,, ..., X , ; S = 1 ) and ( X I, ..., X, ; Q = 1 ) of P ( G ) are said to be N-equivalent if there exists an automorphism 4 of F , such that 4,s = Q L l . This defines an equivalence relation on P(G); the number of equivalence classes under this relation is denoted by I P ( G ) I. It has been conjectured by Magnus ( [ 1 1 , page 401) that I P(G) I = 1 for every group G with one defining relation. In theorem I we show that this conjecture is false. It was shown by A. Shenitzer [6] that if G has presentation (X,, ..., X,;R = 1 ), and G is a non-trivial free product, then there exists an automorphism q5 of F, such that q5R does not involve all of the free generators X,,,.., X,.By combining this result with a result due to J.H.C. Whitehead [ 7 ] , Shenitzer gave a constructive method for deciding, given a group G with one defining relation, whether or not G is a non-trivial free product. In an attempt to generalize this result, one might conjecture that if G has presentation (X,, ..., X,;R = 1 ), and G is a non-trivial free product of two groups with an infinite cyclic subgroup amalgamated, then there exists an automorphism q5 of F, such that @R= W , W,, where W , and W, are non-identity elements of F, which have no free generator X i in common. We show in Theorem 2 that this conjecture is false. Let k and 2 be integers. We denote by G k , [the group with presentation ( X I ,X,;X , k = X,' ). 45 3

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J. McCool and A . Pietrowski, On a conjecture of W. Magnus

Lemma 1. Let 1 = pt + 1 for some integers p and t. Then presentation ( X , , x,;x, = ( X , k X 2 - t ) P).

Gk,l

has

2!

(Xl,YJ,; X,kY@= Y , X , = YP)

2

( X I ,x,,Y ; x, = ( X , k X 2 - t ) P , Y = X 1 k X , 9

2

( X 1 ,x,;x, = ( X l k X 2 - ' ) P )

Definition. Two elements W, and W , of Fn are said to be equivalent if there exists an automorphism 4 of F, such that $Wl = W,. The element W of Fn is said to be of minimal length if the length of W is less than or equal to the length of any element equivalent to w. Theorem 1. Let li, I, p , t be positive integers such that k 2 2, p t > 2 a t l d / = p t + 1 . ThenIP(Gkl)I> 1.

> 2,

Proof. It follows immediately from Theorem 3 of [6] that the elements X , k X 2 - 1and ( X , k X 2 - t ) P X,-l of F , are of minimal length. Now X l k X , - l has length k + 1, while ( X l k X 2 - r ) P X 2 - has 1 length k p + I, and k + 1 f kp + I , since p > 2. Since these elements are of minimal length, and their lengths are different, it follows that ( X I k X , - ' ) p X 2 - l is not equivalent t o ( X 1 k X 2 - 1 ) ' 1 .Hence the two presentations ( X , , X, ; XlkX,-' = 1 ) and ( X , , X,; ( X , k X 2 - t ) p X,-1 = 1 ) of G k , [ are not N-equivalent, and so I R G k , ' ) I > 1. Corollary. Let m be any positive integer. Then we can choose k I S O that I P(Gk,,) I > m.

aid

Proof. Choose k > 2 and 1 = p 1 p 2 , . . p m + 1 , wherepl, ..., p m are the first m positive primes. Let ti be such that 1 = piti + 1 ( i = I ,2, ..., m).Then Ri = ( X 1 k X 2 - t i ) PXi 2 - l is of minimal length in F,, and has length k p i + I, so that R iis not equivalent to R j f l if i # j , since kpi + 1 f k p j +l. Now (X,, X, ; Ri = 1 ) is a presenta-

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455

tion of G k , [ ( i = 1,2, ..., m ) , so that G k , l has at least m non-equivalent presentations. A more complete list of G k , l groups for which I P(G,,,) I > 1 is given in [ 3 1 , together with some additional results which may be useful in determining the number of non-equivalent presentations of such groups. It is interesting to note that 0. Schreier [ 51 has shown that every automorphism of G k , l = ( X , , X , ; X l k X 2 - ' = 1 ) is induced by an automorphism of F , ; this may be contrasted with the results of E.S. Rapaport [4], that the group L = ( X , , X , ;X , 3X2X1-'X 2 - 2 X ; 1 X 2= 1 ) has an automorphism which is not induced by any automorphism of F , , and that IP(L)I = 1. We note that the method of the proof of Proposition 6.3 of [ 21 can be used to give the following result. Lemma 2. Let k, 1, k , and I , be integers different from 0, f 1. Then G k , l is isomorphic to Gk,,Ll iA and only iA either I k I = 1 k , I and I1 I = Il,I, or I k l = 11,l and 11 I = I k , I.

Theorem 2. Let k, I, p , t be as in Theorem 1. Then ( X l k X 2 - ' ) PX2-I is not equivalent in F , to any element o f the form X , ' X 2 - ' . Proof. Suppose ( X l k X 2 - ' ) P A',-, is equivalent to X,'X,-s. Then G k , l is isomorphic to Gr,$,and so, by Lemma 2, either I r I = I k I and Is1 = I 1 I , or IrI = I 1 I and I s 1 = I k l . It follows from this that X l k X , - ' , and so ( X 1 k X 2 - f ) p X 2 -is1 equivalent to X l k X , - ' . This is a contradiction. Thus Theorem 2 gives examples of groups ( X , , . .., X , ;R = 1) which are non-trivial free products of two free groups with an infinite cyclic subgroup amalgamated, such that no automorphic image of R in Fn is of the form W, W, , where W, and W, are nonidentity elements of Fn with no free generator X i in common.

456

References [ 1 J W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory (Interscience, 1966). [ 21 W.S. Massey, Algebraic Topology, An Introduction (Harcourt, 1967). [3] J. McCool and A. Pietrowski, On free products with amalgamation of two infinite cyclic groups, to appear in Jour. Alg. [4Jb.S. Rapaport, Notc on Nielsen transformations, Proc. Amer. Math. SOC. 10 (1959) 228-2 35. (51 0. Schreier, Uber die Gruppen A Q E b = 1, Abh. Math. Sem. Univ., Hamburg. 161 A. Shenitzcr, Decomposition of a group with a single defining relation into a free product, Proc. Amer. Math. SOC.6 (1955) 273-279. [ 7 ] J.H.C. Whitehead, On equivalent sets of elements in a free group, Ann. of Math. 37 (1936) 782-800.

AN ELEMENTARY CONSTRUCTION OF UNSOLVABLE WORD PROBLEMS IN GROUP THEORY* Ralph McKENZIE and Richard J. THOMPSON University of California,Berkeley

In this paper we introduce a new approach to constructing finitely presented groups with unsolvable word problems. The argument has a concrete motivation which should make it easy to follow. A brief intuitive sketch of the procedure is the following: Let o denote the set of all numerical functions (i.e. infinite sequences of natural numbers) and let 8 denote the group of all permutations of a.Behind our arguments is the consideration of a certain primitive recursive set of relations 92,involving altogether only a finite set of letters correlated with elements of (3, which has the property that the subset cR,,j C 92 consisting of the relations of 72 which are valid in (8 is non-recursive. After some preliminaries we will be in a position to define 92 and to prove: there exists a finite set of relations U,which are true in (8 and which imply 32., Then U will be constructed from a set of letters L o , consisting of the letters which appear in92 and new letters correlated with additional elements of (8. A member of 92 will then be implied by U if and only if it belongs to %!c,j - since the relations U are true in 8 - hence the finitely presented group ( L o;c2c ) will have an unsolvable word problem. The novelty (and the virtue) of this approach is that after one understands the underlying notions, the argument itself is nearly trivial, and combinatorial (in a broad sense) rather than grouptheoretic in nature. * This work was aided by National Science Foundation grants GP-7578 and GP-6232x3. 457

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5 0. Preliminaries The basic notation we will use is as follows: The set of all natural numbers is denoted by 0. Given two sets X and Y , y X denotes the set of all functions from Y into X . We consider every natural number I ? t o be identical with the set { O,l, ..., n 1 } and therefore is a set of functions, which we call n-termed sequences: a = ( a o , ..., a , - ) is the function such that a(i>= ai. w is the set of all infinite sequences of natural numbers; for these we may use the notation a = ( a o ,a l , a 2 , .. ). We put g w = U n E w i.e. the set of all sequences of natural numbers having an arbitrary finite length. Given u E Ww and a E s w U w w ,we can define, of course, the concatenation of the two sequences: this is the sequence una = ( u0, .._,urn , a o , ..., at* 1 ) if u = ( uo, ..., um - and a = ( a o , ..., or una = ( u o , ..., um - 1 , a o , a l , ... ) i f a E w w . The formula u C asserts that u is an initial part of p (or p begins with a), i.e. 3 a u^a = p. We use the notation d n )for the sequence obtained by concatenating u with itself n times (where u E s w ) . Thus ( 0 ) ( nis ) a sequence of n zeroes. The irztervul in w w determined by u , where u E ,LJw,is the set I. = { a E w I u C a } . It is easily seen that when u , T E w w , .I C I , iff T c u. We shall sometimes denote the members of w w by j ; g, h, ... E w o if it seems desirable to emphasize their character as numerical functions rather than as sequences. Our functions are often written on the left so that, for instance, if L g E w and i € w then fg(i) = f { g ( i ) ) ;interchangeably, g ( i ) , gi and gi then denote the same thing. Functions being considered as elements of permutation groups, however, will always be written on the right. Then of course t:G will denote the function such that xFG = ( ( x ) F ) G . Let Y and j’ be two elements of a given group. Their commutator is the element [x,1.1 = xyx-l y - l . We have [x,y ] = 1 if, and only if 1’ * Y = where we write y * x for x - l y x (for reasons which will become apparent later). Here of course “1 ” denotes the identity element of a group. Elsewhere it may denote the second smallest natural number. The most important concept of this paper is contained in ~

-,

1 3 ,

R. McKenzie and R.J. Thompson, Unsolvable word problenis

459

Definition 0.1. GI is the group of permutations of w w whose members are the permutations F satisfying

va E ww

3 0 , T E %J

[u c a &

vp E ww [(or'P>F=T , 3 ] 1 .

We shall not bother to justify the definition; it is not hard to show that if F and G satisfy the defining condition then so do the permutations FG and F-'. A more useful working definition of 0 is the following. Let us say that a set C C 9 w is a partition set for w provided that every sequence a E w has exactly one initial part belonging to C (in other words, provided the intervals l o , u E C , form a partition of the set of infinite sequences). For example, 2w is a partition set. Every partition set (if it fails to contain the empty sequence) must clearly be denumerably infinite. Theorem 0.2. Let C,, C, , cp be any triple where C, und C , ure partition sets and cp is a one-one map from 2, onto C , . Then the formulas (u-P)F = p(u)^P ( f o r u E C, and E " w ) define a permutation F which belongs to 6.Moreover, every member o f Q can be put in this form. Proof. The first statement is clear from the definitions. Let us suppose, on the other hand, that F E &. Then we let cp'(o,T ) (for U , T E w w ) mean that ( Q ) F = 7-0 whenever P E w . We put

As cp' is a function (note that cp'(u) = T and cp'(o, T ) have the same meaning), if we let cp be the restriction of cp' to E,, and C be the range of cp, C,, C , cp and F will be seen to be related as required by the theorem.

5 1. The arrow notation It now appears from Theorem 0.2 that the elements of 6 are, in essence, certain transformations of finite sequences. We shall

R. McKenzie and R.J. Thompson, Unsolvable word problems

460

employ an abbreviated system of notation which reflects this fact and makes no mention of infinite sequences; however, the reader should always remember that the objects being discussed are really permutations of w . The arrow notation is best defined by giving an example from the next section. There we shall write

This formula is understood to define the function E E Q given by the partition sets Co = { ( 0 ) } U { ( 1 , i ), ( i + 2 ) I i E w } and C , = { ( 0,O ) } U { ( 0,i+ 1 ), ( i + 1 ) I iE w } , and the map cp : C o C specified by the arrows; in other words, if n o , n l , ... E o then +

*

The permutation E (= EE) is specified in this notation by

The reader may formulate for himself the rules for computing arrow notation for products and inverses of given permutations. We remark that, in general, many arrow notations define the same function: e.g., E above is also given by ( 0, j ) -+ ( O,O, j ); ( 1, i , j ) ( O , i + 1 , j ) ; ( i + 2 ) + ( i + 1 ). However we shall always -+

R. McKenzie and R.J. Thompson, Unsolvable word problems

46 1

use the unique, “irredundant” notation stemming from the partition sets constructed in proving Theorem 0.2. We will take uF = T , where u and 7 are given finite sequences, to mean that F acts on any (Y E w which begins with u by deleting u and replacing it by 7.Thus uF = ~ ( u7, E M u ) implies 07j ) F = 7Yj)foralljEw.

$2.The groups .$ and !$ The elements of whose relations we shall study can be exhibited as the generators of a certain subgroup. In fact, we shall work with the subgroup 8 c 0 generated by three basic permutations D,E, R - and a system of permutations Af,correlated with every function f € w . These are defined below.

462

R. McKenzie and R.J. Thompson, Unsolvable word problems

Definition 2.1. (i) .$ = Gp(D, E, R , A f ( f E (ii) !$ = Gp(D, E ) . It will be useful subsequently to remark that when we set A4 = E - I D and Q = D2RD-l then ( 1,i ) M = ( 1, i + 1 ) and ( 1, i > Q= (l,i, 0).

53. Letters, words and relations Since our readers are assumed to know the basic facts about groups defined by generators and relations (see, for instance, [41) we can be very brief here. Let 5 = FG(L) be a free group freely generated by a set L = { D, E, R, Af(.fE “0)) in one to one correspondence with the permutations defined in 5 2. The elements of L will be called the basic letters, and the elements of will be called words; expressions denoting words will be printed in italics. (Note that we d o not distinguish between a word and its reduced form.) A relution is either an ordered pair of words, written W, = W , , or a word W, thought of as the pair W = 1. The formula I& asserts that the relation 8 is a consequence of, or is derivable from, the set of relations J. Each group 0 which we discuss will have a set of generators correlated with some set of basic letters L o . Implicitly, we have an epimorphism Y ‘ ’ ~from FG(L,) onto 0) which maps the letters to the corresponding generators. In this context the expressions “W, = W , in (\j,’’ “W, = W, is true (or valid) in (9,”‘‘r(\)(W,) = roi (W I 1’’ all have the same meaning (where W,, W , E FG(L,)). Let J be a set of relations in the letters L o . We say that W has the prescwtation ( L o ; 6 ) iff the kernel of the map rCsiis identical with the normal subgroup of FC(L,) generated by { W,W,-I 1 W, = W, E 6 }. If this is true then we have, for W E FG(L,) : W = 1 in Oj if and only if I-& W. Theorem 3.1.

has the presentation

(i) ( D , E ; [E-ID, EDE-’1, [E-lD, E2DE-21).

R. McKenzie and R.J. Thompson, Unsolvable word problems

463

Proof. First we must show that the two commutator relations are valid in '$ (check Definition 2. I(ii)). This is so because E-' D acts only on I ( 0 ) U I( ) whereas EDE-l and E2 DE-2 leave the points of this set fixed. Next we must show that every relation valid in is derivable from the two given relations. Let (9 be the finitely presented group (i) and let us put W, = E, Wn+, = E"DE-". From the given relations we have that E-1 D W, = W, * E-l D in (9 for a = 2,3. Using this we prove

-

In fact, ( a l ) is trivial f o r m = 0 and follows from (a?) for m

> 0:

Thus we have (al), (a2) for k = 1,2. But if (al), (a2) are true for k = 1, 1 + 1 (12 1 ) then we have W1+3 = WI+, - W[+2 * W I + , - l by the case k = 1 , and so (a2) f o r k = 1 + 2 follows, and then (al). Hence the proof of (a) can be given by induction. Now ( a l ) leads to a normal form for elements of N: Let W E FG(D,E). In (9 we have

(b)

W = (Wyt - ... - W::)-'

(bl)

*

-

(WpI' ... W z )

for some k, 12 0, where 0 < i, < i, < ... < i, and 0 < j o < jl < ... < j l and the p s and 4, are positive. (The proof of (b) is by induction on the reduced length of W : D = W;' W I E = W 0 - WO2 can be put in this form, and ( a l ) implies that prod-

-'

-

5

R. Mctienzie and R.J. Thompson, Unsolvable word problems

464

ucts and inverses of words in this form are equivalent in csf, to words which have the form.)

To complete the proof, let W E FG(D, E) such that W = 1 in '$3. Then the expression ( b l ) for W in (Y will also hold in '$3 (by the first paragraph of the proof); therefore it is enough to show that if WPk 'k

*

... *

wi"," in q7

with positive exponents and strictly decreasing indices, then this relation is also valid in (d. In fact, (b2) implies that k = 1 and is = j , and p , = y S for all s < k , as one will see by checking the definitions of the permutation: observe that the permutation denoted by WjP satisfies ( i ) WiP = ( i )? O)@), and ( k ) Wip = ( k )for all k < i. Thus (b2) implies i, = j , and p o = 4,. Then one can cancel Wi"," from both sides and repeat the argument. 54. Deferred action; $-computable functions

Definition 4.1. (See Definition 0.1) Let F E 6 and let CJ E Z w . We define F,, the 0-deferment of F , as follows: (o"p)F, = o-(PF) for all p E -CJ; and CYF,= CY if CY E w w - I,. F, acts on I , exactly as F acts on I* = w , and acts trivially elsewhere. Thus one can easily show that if F, = G, and neither u C T nor T C u,then E ' = G = 1. We proceed to show that the group contains the deferments of all its members. Definition 4.2. We correlate with every word W in the letters D and E and with every finite sequence of natural numbers u = ( o,, ..., uk ) a new word W(o), or W(o,, ..., ak): (1)

W(0)EW;andfornEw:

(2)

E(0) E-lD-lE*,E(l)= D-1ED-1E-1D2,E(n+1)= EnE(l)E-";

(3)

D(0)- E-lE(l)E7D(l)ZD - I E ( ~ ) D , D ( ~ z + I ) = E ~ D ( ~ ) E - ~ ;

R. McKenzie and R.J. Thompson, Unsolvable word problems

(4)

465

W(uo, ..., a k ) 3 puo... puk (W) where p m is the endomorphism of FG(D, E ) such that p m ( D ) = D(m) and Pm ( E l = Wm).

Theorem 4.3. Suppose that W E FG(D, E) and W defines the eleThen for u E W w , W(u) defines F,. Therefore F, E ment F E

v.

v.

Proof. Let p m be as in Definition 4.2, let r be the canonical epimorphism FG(D, E ) -+ !#,and for each finite sequence u let v, denote the endomorphism of Q which maps F + F, . The reader may check Definition 4.2 and $ 2 to see that for each one-term sequence u = ( m), v,r(E) = r p m ( E ) ;v,r(D) = r p m ( D ) ;hence v,r = rpm on FG(D, E). Now if u = ( uo, ..., (Tk >,we calculate

which is the desired result. Our precise goal in the next two sections is now describable. If F, G E Cr , then one can check that we have F-I G,F = G,, whenever u, r E Gw and u F = r. If F E .$ and G E $ ! then, by Theorem 4.3, these equations can be written as relations which are true in $. From now on we write x * y for y-1x.y; and we now make a definition:

Definition 4.4. A function f~ w will be called .$-computable if there is a finite set of relations, cMf, which are valid in ,$I and from which the relations

466

K. Mctierizic. arid R.J. Thompson, Unsolvable word problems

can be derived. A set with these properties will be said to compute

.I:

Note that i i i this definition the relations qfare allowed to involve any finite subset of the letters L = {D,E,R,A,,A,, ...}; the relations assumed t o be derivable contain, on the other hand, only the three letters D , E and Af. Our immediate goal is t o prove the following theorem (whose converse is obvious): Theorem 4.5.Every (general) recursive function is $-computable.

35.The basic relations 9’3 We define a set of basic relations which will be useful in the proof of Theorem 4.5. Put M = E-lD,Q = D2RD-l, I? = M * R, A = M * R2.

B 1 : [ M , EDE-’ ] B2: [ M , E2DE-2] B3: R 3 B4: “ M , A l , X < l ) l

B7: [ R, X( 1 ,O,O,O) 1

J

This set of 1 1 relations we call 93. The words X( l ) , X( 1 ,O,O,O) were defined above (4.2). Lemma 5.1. The relutions 93 are valid in .().

R. McKenzie and R.J. Thompson, Unsolvable word problems

467

Proof. See Theorem 3.1 for B 1 and B2, and the definitions in § 2 for the others. Each of the commutator relations [W, X(u)] is true because W defines an element F E .$> such that u F = u. Lemma 5.2. Let W be an arbitrary word and let p , u , T E The two relations X(u) * W = X(T), X E { D, E }, imply f o r every Y E FG(D,E), the relation Y(u"p) * W = Y(T-p). Proof. By Definition 4.2, if A is any word in D and E, say A=X,nO. :.

X i k ( X o , ..., X k E { D , E } )

then A(u) = X,(u)"" A(T) = X,(T)~'

-

-

... Xk(u)nk;and

- ...

*

Xk(~)nk.

Hence obviously from X(u) * W = X(T) (X = D, E) one may derive A(u) * W = A(T). Let us take A = Y(p). Then A(u) = Y(u-p) and A(T) = Y(T^P),as follows from 4.2; hence we have the result. We will frequently use in the following derivations various immediate consequences of this lemma, such as for instance the fact that relations B4 imply, for any finite sequence u, [ [M, A ] , X(( 1 Y u ) ] = 1, or what is equivalent, X(( 1 Y o ) * MA = X(( 1 Y o ) * AM.

Theorem 5.3. Let X E { D, E } . For every triplet of nutiirul nuinbers i, j , k we have

* D = X(1,O); X(1,i) * M = X ( l , i +

(i)

I-cx3 X(1)

(ii)

I-q

1);

(iii)

X( 1, i, j , k ) * R = X( 1, j , k, i);

(iv)

X(1,i)

-

* Q = X(l,i, 0).

Proof. We write W, W , t o denote that W, = W , . By 3.1 amd 5.1, if W,, W , contain only the letters D and E then W, W , iff

-

468

R. McKenzie and R.J. Thompson, Unsolvable word problems

W, = W , holds in .Q. Hence (i) and (ii) follow immediately, since

M = E-'D. To prove (iii); by B4, B5, B6, for X E { D , E } we have (a) (1) X( 1)

(2) X(1) (3) X(1)

* MA - X( 1 ) * AM, * DA * Dr

- X ( l ) * MD, - X ( l ) * AD.

By 5.2 this remains true when we replace X( I ) by X(( 1 )nu). From this we obtain

-

* A X ( l , i , j + 1); (2)x(i,o,i,j) * r - x(i,o,i,j+i);

(b) (1) X ( l , i , j )

wtzenewr i, j E w and X E { D, E }. In fact, for i = 0 we have

* DA; by (i) and Lemma 5.2 X ( 1 , j ) * MD; by (a2) and 5.2 X( 1,0, j + 1); by (i), (ii) and 5.2.

X(l,O,j) * A - X(1,j)

-

-

If ( b l ) has been proved for i = i, then compute

X( 1, i,

+ 1, j ) * A - X( 1, i,,

- X ( l , i,

* MA; by (ii) and 5.2 + 1, j + 1); by ( a l ) , 5.2 and j)

the induction assumption. To get (b2), use (i), (a3) and ( b l ) : X(l,O, i , j ) *

r- X(1, i , j ) * D r

- X(1, i , j ) * AD

- X(l,O, i , j + 1). Now (b), together with (ii) and 5.2, yields

R . McKenzie and R.J. Thompson, Unsolvable word problems

X(l, i, j , k )

469

- X(l,O,O,O) * rkMZAi

-

If we conjugate both sides by R, taking note of B3 and B7, we have that I? * R A, M * R I', A * R M and

-

X( 1, i, j , k ) * R

-

- X( 1 ,0, 0,O) * Akr 'Mi

- X( 1, j , k, i);by (ii), (b) and 5.2.

To prove (iv); just apply (i) and (iii). 86. Proof of Theorem 4.5

The characterization of (general) recursive functions which we shall use is taken from J. Robinson [6] : We say that the function f E w is obtained by general recursion from g, h, u, u E * w if (i)jg = u, fh = ufand (ii) every natural number n belongs to the range of some one of the functions hkg ( k 2 0 ) . Theorem 6.1. [ 61 The class of (general) recursive functions is the smallest class o f numerical functions which is closed under composition und general recursion und contains the zero function e(O(n)= 0 ) und the successor function x(x(n) = 12 + 1).

The proof of Theorem 4.5 is contained in a series of lemmas, showing that the class of .()-computable functions contains x and 8 and is closed under composition and general recursion. We first extend Definition 4.4. Definition 6.2. Let f E w w and n E w . We say that f i s n-computable from a set of relations 92 iff the following are derivable from 92:

X(1, n, k ) * A f = X( 1 , n , k +f(n)) (X E { D, E } , k Lemma 6.3. x and 8 are .$-computable.

E

a).

470

R. McKenzie and R.J. Thompson, Unsolvable word problems

Proof. Referring back t o Definition 4.4 we see that 8 is computed by the set {A, = 1 j. Let 93, be obtained from the set CM ( 5 5) by adding the four relations:

Here, and in the future, we will not bother to show that the relations we write down are true in s j since these verifications are always routine. Moreover we always write to indicate equality provable from the set of relations being considered - in the present case CM,. The applications of Lemma 5.2 will not be explicitly mentioned. We now show by induction on 12 that IT is n-computable from ‘23, for every n. For n = 0:

-

X( 1,0, k ) * A,

- X( 1, k ) * DA,; by 5.3 (i)

- X ( l , k ) * MD; by ( 1 ) above -’

Assurning

IT

X(1,0, k + 1); by 5.3.

is n-computuhle jrom CM, ;

X(1, I I

+ 1, k ) * A,

- X(1, n, k ) * MA,; by 5.3 - X( l , n , k ) * AA,M; by (2) above - X(1,

I2

+ 1, k + N + 2);

by 5.3 (ii, iii), definition of A and the induction assumption. Thus IT is n-computable from “3, for all 1 2 ; hence it is Q-computable.

Definition 6.4. For any two words W,, and W , we put

471

R. McKenzie and R.J. Thompson, Unsolvable word problems

The two technical lemmas which follow considerably simplify the remaining discussion.

Lemma 6.5. Let W be any word and i, j , k, 1 E a.From the six relations (where X € { D, E } )

together with 9, one can derive (4)

X( 1 , i, j )

* W = X( 1 , k , 1) (X E { D, E }).

Lemma 6.6. Let W,, W, be any two words and i, j , k, I sider the five pairs of relations (where X € { D, E }):

E w.

Con-

In the presence of 9 one can derive I: (c) froin (a) and (b); II. (b) Jhom (a), (c), (d) and (el. Proof of 6.5. We calculate X( I , i, j )

* WDRrp

- X( 1, i, j ) * DRWFp; 6.5 (2) - X ( l , i, j , 0 ) * WFp; 5.3 - X ( l , i, 0 ) * rWP 6.5 (1) j,

l;

412

R. McKenzie and R.J. Thompson, Unsolvable word problems

- X ( l , i , j , 1)*WrpP1;5.3 anddef.ofI' - X(l,i,j,p) *W;

- X( 1, k, 1, p ) ; 6.5 (3) - X(1, k, I ) * DRI'P; 5.3; and then we conjugate by (DRI'P)-' to obtain 6.5 (4). Proof of 6.6. The derivation of (c) from (a), (b) and CM is a very interesting but perfectly straightforward computation (using 6.4 and 5.3) which we leave to the reader. To get (b) from (a), (c) - ( e ) and 33; first transform (c) - using (a) and 5.3 and conjugate both sides by QW,K to obtain the relations: X( 1 j , k, i ) * W , = X( 1, j , k + I, i). Then apply Lemma 6.5 taking W = W, . ~

Lemma 6.7. I f g and h are .$-computable then gh is .i>-computable. Proof. Let 93, and CM,, be finite sets of relations which compute g and h. Define CMsh to be CM U CM, U g h together with the two relations [(Ah 0 A,) Aih7X( l ) ] where X = D, E. Then apply Lemma 6.6.1.

-

Lemma 6.8. I f f is ohtuined by generul recursion ,from Q-computable ,functionsg , h , u and v, then f is .\:!-computable.

Proof. To 33 U 93, U 33, U 93, U qU we add the following relations to form a set CMf (where X E {D, E}):

R. McKenzie and R.J. Thompson, Unsolvable word problems

413

We are assuming that % ' ,I '%Ih, ... compute g, h , ..., repectively; hence '%If computes each of g, h, u, u. By the definition of general recursion, fg = u and j7z = uf. To show that W f computes f it will be enough to show, for each p 2 0 and for each m E rng h p g , that f i s rn-computable from Wf. This we do by induction on p . F o r p = O ; b y ( l ) w e h a v e , f o r n , k E w:

Hence by 6.6.11, (3) and (4) we obtain

which is the desired result since fg(n) = u(n). The induction step; assuming f is m-computable from qf for every m E rng h p g : consider m E rng h p + l g , say m = h ( n ) where n E rng h p g . By (2) we have, for any k E w :

by 6.6.1 and the induction assumption;

-

X( 1, n, m ) the since uf = F.Since we also have X( 1, n, 0) * A, desired result, X( 1, m, k ) * Af X( 1 , m, k + f ( m ) ) ,follows from ( 3 ) , (4) and 6.6.11.

-

97. The Novikov-Boone theorem

Theorem 7.1. ( [ 1 1 , [ 5 ] ) There exists a finitely presented group whose word problem is not recursively solvable.

4 14

R. McKenzie and R.J. Thompson, Unsolvable word problems

Proof. Let f b e a recursive function whose range is a non-recursive set. By Theorem 4.5 there exists a finite set of relations ‘;ofwhich computesf as per Definition 4.4. We can assume that 93 C ‘;of. Let lz be the function such that h ( n ) = 0 if n E rng f and h ( n ) = 1 if 11 @ rngf. Then the following relations are true in $:

in which X = D or E and Af 0 Ah was defined in 6.4. Let U be the set of relations ‘;of together with the above relations. Now it follows from Lemma 6.6.11 that

On the other hand, if n $Z rngf then the above relation is not true in (7, whence certainly

If

cll

E( 1, H , 0) * A, = E( 1, n, 0), if / I $ rngf.

Combining these two results, we get that the relations in the three letters D, E and A, which are derivable from U are non-recursive. Hence the finitely presented group ( L o ;U ), where L o consists of all the letters appearing in 2e, has an unsolvable word problem. 58. Concluding remarks

First, we should mention that by using Britton’s Lemma (see

[ 21 ) and suitably modifying the finite presentations given in this

paper the second author has succeeded in giving a proof for the existence of finitely presented groups with word problems of an arbitrary recursively enumerable truth-table degree. This proof will be published elsewhere by Thompson. Second, we remark that our definition of recursive function

R. McKenzie and R.J. Thompson, Unsolvable word problems

475

could have been taken from R.M. Robinson [ 71, instead of [ 6 ] . The proof, in that case, would have been simpler in some respects, since A f + gand A f A g have the same effect on the interval but it would have required us to prove a variant of [7; Theorem 3, p. 9401. Finally, we should like to recount a short history of our proof which will introduce some interesting related notions. We have dealt in this paper with a group G, whose members are homeomorphisms of the topological space o,provided with the product topology (so that it is homeomorphic to the space of irrational numbers). Let us consider now the Cantor space 2,, whose members are the infinite 01-sequences. We define @ 2 to be the set of all finite 01-sequei ces. Our Definition 0.1 is now transformed into the definition of a group Cr’ of homeomorphisms of 2. Theorem 0.2 is still true in this context but since 2 is compact every partition set C 8 2 is finite. Therefore 6‘ is a denumerable group. Actually Q’ is a finitely generated, algebraically simple group which has a finite presentation (as Thompson will show in a forthcoming paper). The group !$ which I plays a leading role in the present paper is isomorphic to the subgroup of 6’ composed of the functions which preserve the natural ordering of 2 (see below). Thompson discovered the groups 0’ and $’ in connection with his research in logic about 1965. He studied them thoroughly and obtained results which led eventually to our joint work on the word problem. Among his discoveries at that time were the armw notation, equivalents of Theorems 3.1 and 4.3, and a finite presentation for 6 ’ . The key observation which triggered our interest in the word problem was made by him: viz., that arbitrary numerical functionsf can be coded as permutations Bf of 2 (in various ways) so that f’is uniquely determined by the relations between Bf and the elements of 0‘. Working together the two of us completed, in December 1967, a proof rather similar to the one given in this paper, which referred however to the Cantor space. Later, McKenzie observed that a much neater argument is obtained by translating into o.

4 76

R. McKenzie and R.J. Thompson, Unsolvable word problems

The translation which was used is the following: Let A C w 2 be the set of 01-sequences which assume the value 0 infinitely many times. A is homeomorphic to w under the map q defined by

The set A is invariant under 6’ and only the identity element of 6 ’ acts trivially on A. Hence (one sees tha#) the map F - 77Fq-l embeds 6’ into 6. Under this map the two functions E ’ : ( 0 ) - ( O , O ) , ( 1 , O ) ( 0 , l ), ( 1 , l ) ( 1 ); and D’:( 0 ) ( O ) , ( l , O ) + ( l , O , O ) , ( l , l , O ) ~ l , O , l ~ , ~ l , l(,1l, ~l ) +a r e t a k e n o n t o the E and D defined in 3 2 which generate p. I n order t o see that E’ and D’ generate the subgroup of orderisomorphisms of 6‘ we shall now introduce Thompson’s original arrow notation. (Which, incidentally, should give an idea of the logical origin of the group 6’.) First we define an operation on finite subsets o f & ? . Given C, and C , , finite subsets of442, put +

--f

+

Clearly the set of all partition sets for “2 is generated by applying this operation repeatedly, beginning with the partition set { 8 } (@is the empty sequence). We now introduce a free groupoid generated by an infinite set of symbols X,, X ,, X , , .. . (The word “groupoid” is taken here t o mean an arbitrary algebraic system whose only fundamental operation is binary.) The elements of the groupoid we call exps, and we denote them by juxtaposition and the use of parentheses ( X u ,X,X,, X0(X1X2), etc.). With each exp, we correlate the partition set C, which it represents: (i) X,,X,,... all represent (0); (ii) the product of two exps s and t (in the groupoid) represents C, + C,. For two exps s and t we now have Cs = C, iff s can be obtained from t by replacing some of the generating symbols in t , at some of their occurrences, by other symbols. Thus we need consider only reduced exps, in which n o X , occurs more than once. For each reduced exp s there is a canonical one-one mapping

RMcKenzie and R.J. Thompson, Unsolvable word problems

477

gs from Cs o n t o the set of symbols in s: if C, = { u o , ..., un j where, lexicographically, uo < ul < ... < un , then s has length 12 + 1 ; upon Xi - we define gs(uk) = removing parentheses s looks like Xi,,... Xik. (For instance, if s = X0(X1X2)then C: = { ( O ) , ( 1 ,O ), ( 1 , l ) j and g s ( ( 0 ) ) = X,, g s ( (1,O)) = X I ,gs( ( 1 , l ) ) = X 2 .) Finally, take any expression s -, t where s and t are reduced

exps having the same symbols Xioccurring in them. We then consider the formal expression s -, t to represent the element of C’ defined by the partition sets Cs and C,, and the one-one map g;lgS from C, o n t o C,. With this convention, we see that the function E‘ E Q‘ is represented by the “associative law”:

and that the function D’is represented by the associative law “deferred to the right”:

The inverse of s + t is t -+ s, so we find that 15‘I-l D’ is represented by :

The representation defined above is a many-one map from the set of expressions {s t j o n t o Q’. It is very useful in the study of the group. The order-preserving members of 6’are obviously those which simply rearrange parentheses, and it is intuitively quite clear that they are generated by the t w o simple rearrangements E’ and D‘. -+

References [ I ] W.W. Boone, The word problem, Ann. of Math. (2) 70 (1959) 207-265 [ 2 ] J.L. Britton, The word problem, Ann. of Math. (2) 77 (1963) 16-32. [ 31 G. Higman, Subgroups of finitely presented groups, Proc. Roy. SOC.London, Ser. A, 262 (1961) 455-475.

478

R. McKenzie and R.J. Thompson, Unsolvable word problems

[ 4 ] W. Magnus, A. Karras and D. Solitar, Combinatorial Group Theory (John Wiley, New York, 1966). [ S ] P.S. Novikov, On the algorithmic unsolvability of the word problem in group theory, Trudy Mat. Inst. im. Steklov. No. 44, Izdat. Akad. Nauk SSSR, Moscow, 1955 [Russian]. 161 J. Robinson, Recursive functions of one variable, Proc. Amer. Math. Soc. 19 (1968) 815-820. [ 7 ] R. M. Robinson, Primitive recursive functions, Bull. Amer. Math. Soc. 5 3 (1947) 925-942.

A NON-ENUMERABILITY THEOREM FOR INFINITE CLASSES OF FINITE STRUCTURES* T.G. McLAUGHLIN University of Illinois, Urbana

Let $' 1 be a fixed, finitely axiomatized class of algebras with finitely many operations each of finite arity; e.g., let \)( = the class of all groups. (Axiomatizability is understood t o mean axiomatizability in the lower predicate calculus with equality.) Let "&, denote the set of all finite presentations, relative t o some fixed presentational alphabet A , for members of 91; let 3yl denote the class of all finite members of \)(; and let $'3; denote the class of all presentations in '$3,!, which determine elements of We assume a fixed Godel numbering g of the elements of !&, ;g is understood to possess all o f the usual effectivity and uniqueness properties demanded of such codings in recursion-theoretic contexts. The following easy lemma is used several times in our proof of Theorem 2.

z\Jl.

Lemma 1. The isomorphism problem is solvable unifortnly f o r ,finite members o f Yl, in the sense that there exists a unijbrm effective decision procedure (i.e., a partial recursive Junction) j b r determining f r o m any pair ( g ( P , ) ,g(P,)) such that P , and P, both belorig to p\,T whether or not P , and P, present isomorphic structures. We impose upon \)I the additional requirement that for all natural numbers k there exist members of Sr, having cardinality 2 k. * The present

paper suinmarizes the main features of sonic inaterial which will appear in detailed form elsewhere. Research supported by the U.S. National Science Foundation under contract No. GP-7421. 419

480

T.G. McLaughlin, A non-enurnerability theorem f o r infinite classes

Rogers’ book [ 4 ] is our source for notation and terminology in connection with the “arithmetical hierarchy” of Kleene and Mostowski (12, Chapter XI] ). Thus, we denote by E, (by n,) the class of all sets of natural numbers definable by a k-quantifier prenex arithmetical predicate having initial quantifier 3(having initial quantifier ’d); and by a coinplete set in C, (in II,) we mean a set which belongs t o C, (to II,) and has the property that any other set in C, (in II,) is recursively 1-1 reducible t o it. Let Inf ( ‘i?2 ) denote the family of all strongly infinite, recursively enumerable classes of elements of $gl * ; by a st%ngZy infinite class of presentations we mean one whose associated class of presented structures contains representatives of infinitely many different isomorphism types. Let Znf( !$?,*) denote the family of all classes 6 such that Q is the class of structures presented by (the elements of) some class I E I f ( p!,,*). By an index of a member of Znf( 5$? !,(*) we mean a Godel number of a class I E Znf (p,!,*);here our Godel numbering of classes of presentations is one which is induced (in any of a variety of natural ways) by the numberingg. Theorem 2. (a) The set a of all indices of members of Znf( $?$)is a complete set in n 2 . ( b ) I n f ( Q.l,*) is not “C2-enumerable”;i.e., there is no Z 2 set p, p c (Y, such that (YO ) [ 0 € Inf($31*l(==) ( 3 n ) In E P tk n is an index of 01 1 . Actually, we prove a theorem which is more “effective” than Theorem 2(b) as stated above, in that we establish a productivity property relative to C2-enumerable subfamilies of Znf ( !J3?;). (Investigations of productivity properties relating to classes and families of classes have previously been carried o u t in [ 13 and [31 .) The statement of the sharper form of the theorem, however, would involve technical terminology too cumbersome for the present note. We mention some consequences of Theorem 2. Corollary 3 . Let u finite presentation o f ’ u group be termed strongly finite in case the groiip presented by it is ,finite. Then: the

T.G. McLaughlin, A non-enumerability theorem f o r infinite classes

48 1

indices for the various strongly infinite r. e. classes of strongly finite group presentations form a complete 112 set, no C., subset of which sujfices to enumerate the family of all infinite r.e. classes offiriite groups. Corollary 4. The collection a0 o f all indices o f illfinite recursively enumerable sets o f natural numbers is complete in 112,and a. has no C 2 subset sufficient to enumerate the class of all such r.e. sets. (cf [ 1 , Theorems 3.3 and 6.41 ; using the sharper form o f Theorem 2 , we actually obtain productivity relative to C2-enumerable subclasses. ).

From Corollary 4 and an observation in [ 3 ] ,we can further show: Corollary 5 . The collection Po of all indices ofrecursiwly enumerable classes of infinite r.e. sets is complete in 112,and Po has no C 2 subset sufficient to enumerate t?ie family of all such classes.

We would like to have some information on the “levels of enumerability” of families of not necessarily strongly infinite classes of finitely presented structures in Sr,,for various %;and it can be demonstrated easily that Theorem 2 is not generally true if the strong infiniteness condition is dropped. References [ I ] J.C.E. Dekker and J. Myhill, Some theorems on classes of recursively enumerable sets, Trans. Amer. Math. Soc. 89 (1958) 25-59. [2] S.C. Kleene, Introduction to Metamathematics (D. Van Nostrand, Princeton, 1952). [ 3 ] T.G. McLaughlin,The family of all recursively enumerable classes of finite sets, to appear. (41 H. Rogers, Theory of recursive functions and effective computability (McCraw-Hill, New York, 1967).

SOME CONNECTIONS BETWEEN HILBERT’S 10th PROBLEM AND THE THEORY OF GROUPS Charles F. MILLER 111” The Institute for Advanced Study, Primeton

The tenth problem listed by Hilbert [9] in his famous address of 1900 is the following decision problem: to determine of an arbitrary polynomial equation P ( x l , _..,x n ) = 0 , with integer coefficients, whether or not it has a solution in integers. Whether or not this problem is recursively solvable is as yet unknown.** The purpose of this article is to examine some decision problems in the theory of groups which are equivalent*** to Hilbert‘s 10th problem, or which, if they were unsolvable, would imply the unsolvability of Hilbert’s 10th. Although there is essentially no literature on the connections between these group-theoretic decision problems and Hilbert’s 10th’ some of this material seems to have been known to several people. Consequently, this article might best be regarded as a survey of the possibilities of settling Hilbert’s problem via grouptheoretic problems. An effort has been made to supply sufficient detail for this work to be accessible to both logicians and group theorists.

* Work supported by U.S. National Science Foundation under contract No. GP-7421.

** This article was written prior to the publication of Matejasevich’s remarkable theorem [ 2 3 ] .

*** Equivalent

and reducible in this article generally mean Turing equivalent and Turing reducible as decision problems. 483

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C. T. Miller III, Hilbert's 10th problem and the theory of groups

8 1. The logical setting Interesting work on Hilbert's 10th is discussed in Davis [71 and J. Robinson [ 161. In contrast to their approach, we will pursue a development more oriented towards logic and model theory, and some familiarity with these subjects is assumed. First, we recall some definitions from universal algebra (see Cohn [5 I ): Let N = { 0, 1,2, ... } denote the natural numbers. An operator domairi is a set 52 together with a map a : 52 + N . If u E 52 and y1 = a(o), then u is called an n-ary operator. If A is a set, an 52algebra structure on A is an assignment to each u E 52 of a function from the Cartesian product A n to A where n = a(u). T h e set A with this structure is called an 52-algebra. Note that if u is a 0-ary operator, then u is just a distinguished element (constant) of A . For example, the ring of integers Z is an !&algebra where 52 consists of two 0-ary operators (for 0 and 1) and two binary operators (for + and Also, a group G is an 521algebra where 52' consists o f a 0-ary operator (for l ) , a unary operator (for inverse) and a binary operator (for group multiplication). The first order language associated with an operator domain a, written L(52), has as symbols the following: (1) a (countable) set of variables x, y , z , ... ; (2) an iz-ary function symbol for each iz-ary operator of 52 and the symbol =; (3) the usual logical symbols: 7 ,v , A , 3 , V , ( , ). A term of L(52) is defined inductively by (i) a variable or a 0-ary function symbol (i.e. a constant) is a term (ii) if t , , ._.,t,, are terms andf'is an n-ary function symbol, then f ( t l , ..., t n ) is a term. An a t o m i c f ~ i r m u l ais an expression of the form t , = t , where t , and t 2 are terms. Formulas are then defined inductively using atomic formulas and the logical symbols. An 52-algebra A is just an L(52)-structure in the usual terminology of logic. If B is a formula of L(52), then that B is valid (respectively satisfiable) in A is defined as usual. Intuitively, a formula B is valid in A if B is true for all possible ina).

C.F. Miller 111, Hilbert's 10th problem and the theory of groups

485

terpretations in A of the free variables of B. The formula B is satisfiable in A if B is true for at least one interpretation in A of the free variables of B. See Shoenfield [ 201 , Church [ 41 , or Stoll [ 2 1 ] for details. Let A be a class of a-algebras. By the elementary theory of A we mean the first order theory with language L(s2) whose nonlogical axioms are exactly those formulas of L(s1) which are valid in every A E A. The elementary theory of A will be denoted by Th, (A) or simply Th(A). If B is a formula, V B will denote the universal closure of B. Recall that, in Th(A), B is a theorem if and only if V B is a theorem. Also, B is a theorem if and only if B is not satisfiable in any A E A. Moreover, B is satisfiable in some A E A if and only if (3x ) B is satisfiable in some A E A. An open formula is a formula which contains no quantifiers. A formula of the form E , v E , v ... v E , where each Eiis either an atomic formula or the negation of an atomic formula will be called a basic formula. The disjunctive normal form theorem implies that if B is an open formula then B is logically equivalent to a disjunction D,v D, v ... v D, where each D iis a basic formula. The D ican be effectively found from B and will be called the disjunctive components of B. Clearly, B is satisfiable i n some A E A if and only if at least one of the disjunctive components of B is satisfiable in some A E A. We are particularly interested in the following three decision problems concerning Th(A): 7,

DPOF(A) - the decision problem for open formulas of Th(A), i.e., to determine of an arbitrary open formula B whether or not B is a theorem of Th(A). SPOF(A) - the satisfaction problem for open jtirmulus of Th(A), i.e. to determine of an arbitrary open formula B whether or not B is satisfiable in some A E A. SPBF(A) - the satisfaction problem for basic formulas of Th(A), i.e. to determine of an arbitrary basic formula B whether or not B is satisfiable in some A E A.

4 86

C.F. Miller, Hilbert’s 10th problem and the theory ofgroups

The following lemma is an immediate consequence of the above discussion:

Lemma 1 . DPOF(A) is equivalent to SPOF(A) und to SPBF(A). Moreover, the set of satisfiable open formulas is recursively enumerable if’and only if the set o.f Satisfiable basic formulas is recursively etiumerable.

82.Regarding Hilbert‘s 10th Recall that Z denotes the ring of integers. We now specialize the above discussion to A = { Z }, and consider Th(Z). An atomic formula of Th(Z) is simply an equation of the form P , = P, where PI and P, are polynomials with positive coefficients. Now P I = P , is (algebraically) equivalent to the polynomial equation ( P I - P z ) = 0. Conversely, a polynomial equation P = 0 is (algebraically) equivalent to an atomic formula obtained by separating the parts of P occurring with positive and negative coefficients. Indeed, we could suppose that Z has an additional binary operation “-”for minus. Consequently, for convenience it will be assumed that every atomic formula of Th(Z) has the form P = 0 where P is a polynomial (allowing minus signs); the negation of an atomic formula will be written as P # 0. Lemma 2 . Hilhert ’s 10th problem is equivulent to SPBF(Z) and hence to DPOF(Z) and to SPOF(Z). Proof. Since P = 0 is a basic formula, it is clear that Hilbert’s 10th problem is reducible to SPBF(Z). An arbitrary basic formula D of Th(Z) has the fonn

Let 1 1 , u, .Y, .v be variables not occurring in any of the Pi or Qi. In view of Lagrange’s theorem that every non-negative integer is a sum of four squares, it follows that D is satisfiable (in Z) if and only if the formula D, having form

C.F. Miller, Hilbert 's 10th problem and the theory of groups

PI A

=0 A

... A -

(1

A Q 2 f O A

Pk =

487

0

+ u2 + u2 + x 2 + y 2 ) = 0 ...

AQ,

#

O

is satisfiable (in Z). Continuing in this way, one obtains a basic formula D* having the form

such that D is satisfiable if and only if D* is satisfiable. Now D* is satisfiable if and only if the single equation E given by

is Satisfiable. Note that E can be effectively found from D. Hence SPBF(Z) is reducible t o Hilbert's 10th. This completes the proof.

Lemma 3. The set ofopen ,formulus o f Th(Z) which ure sutisfiuble in Z is recursively enumerable. Proof. Let B be any open formula. Then B is logically equivalent to D ,v ... v D, where the Di are basic formulas. As in the proof of Lemma 2, D jis satisfiable if and only if an associated polynomial equation Pi = 0 is satisfiable. But then the polynomial equqtion

-

P ( B ) = P I P,

*

_._ * P,, = 0

is satisfiable if and only if B is satisfiable. Further, P(B) can be effectively found from B. Now the set of polynomials Q such that Q = 0 has a solution in Z is easily seen to be recursively enumerable. Hence the set of satisfiable open formulas is recursively enumerable. Let R ( x l ,..., x n ) be a predicate of y1 variables concerning Z. Then R is called polynomial if there is a polynomial P(x,, ..., x l 1 ) such that

C.t;. Miller, Hilbert’s 10th problem and the theory of groups

488

R ( x , , ..., x??)* P ( x l , ..., x n ) = 0

R is called diophantinc (or existentially definable) if there is a po) that lynomial P(x l , . .. , x,,, .v l , ... , J ’ ~ such

Proofs of the following facts can be found in Davis [ 7 I : (1) If R , , R , are polynomial (resp. diophantine) then R A R , and R v R , are polynomial (resp. diophantine). (2) If R is diophantine, then ( 3 u ) R is diophantine. (3) Let rn,n be fixed integers and let x, y be variables. Then the following predicates are diophantine: (v) x = y mod n (i) x = J’ (vi) .Y = rn mod n (ii) x # y (iii) .x = I Z (vii) x is even (iv) x # n (viii) x is odd. (4) Every diophantine predicate R is recursively enumerable, i.e. the set of all n-tuples (xl, ..., x n ) such that R ( x l , ..., x n ) is recursively enumerable. (5) If there exists a diophantine predicate R which is not recursive, then Hilbert’s 10th problem is unsolvable.

$3.Diophantine matrix groups GL(n, Z ) will denote the group of n X n matrices with coefficients in Z having determinant 1. SL(n, Z) denotes the subgroup of GL(n, Z) consisting of those matrices with determinant + 1. (SL(n,Z ) has index 2 in GL(n, Z) - it is the kernel of the determinan t map.) Let G be an abstract group. G will be called a diophantine groiip if there exist an integer n > 0 and a faithful representation cp : G GL(n,Z ) and a diopahntine predicate R ( x l l , ..., x n n )of n2 variables such that:

*

+

C.F. Miller, Hilbert's 10th problem and the theoty groups

489

That is, the conditions which an arbitrary matrix with n2 variable entries must satisfy in order to belong to q( G ) are expressible as a diophantine predicate.

Example 1 (Sanov [ 171). Let F , be the free group of rank 2. Then F , is isomorphic to the subgroup F2 of SL(2, Z) generated by the matrices

Let

be an arbitrary 2 X 2 matrix. Then M E F2 if and only if the following conditions are satisfied: (1) uy - x u = I (2) u and y are congruent to 1 mod 4 ( 3 ) x and u are even. By our previous discussion (section 2), it follows that F , is a diophantine group.

Example 2. Let H be the free nilpotent group of class 2 and rank 2. Then H is isomorphic to the subgroup H" of SL(3, Z) cwsisting of all matrices of the form

490

C.F. Miller 111,Hilbert's 10th problem and the theory ofgroups

Again it is easy t o see that this characterization implies that H is a diophantine group. Example 3. GL(n, Z) and SL(n, 2) are diophantine groups. T h e conditions here are just the determinant conditions. Many arithmetic groups are diophantine (see Bore! and Harish-Chandra [3] ).

Remark. I t is easy t o show that the direct product of two diophantine groups is again a diophantine group. Let G be an arbitrary group and let A = { g , , ..., g, } be a finite (possibly empty) set of fixed elements of G. The pair ( G, A ) will be called a group with distinguished elements. ( G, A ) can be regarded as an Q'algebra where 52' is formed from the usual operator domain S2 for groups by adding new 0-ary operators in one-one correspondence with A. Formulas of L(52') then contain constants corresponding to elements of A and a formula B is valid in ( G, A ) if and only if U is valid in G when the g j are substituted for the corresponding constants (and similarly for satisfiable).

Theorem 1. L e t ( G, A ) be a group with distinguished elements and usmine that G is a diophantine group. Then SPBF(( G, A )) is redzlcihle t o Hilhert 's 10th problem. Hence, DPOF(( G, A )) and SPOF(( G, A )) are reducible to Hilbert 's 10th problem. Moreover, the set o f open formidas ofTh((G, A )) which are satisfiable in ( G, A ) is reciirsiwly enumerable. Rernurli. Theorem 2 will provide a converse t o this result. Namely, there is a diophantine group H with a single distinguished element c such that Hilbert's 10th problem is reducible to SPBF((H, { c } ) ) . Indeed, H is just the free nilpotent group of class 2 and rank 2. Proof. Theorem I is an immediate consequence of Lemmas 1 , 2 and 3 together with the following: Lemma 4.L e t ( G, A ) br u groiip with distinguished elernents and ussiinicl G is cliophantirw. Then there is un ejjective process which,

C.F. Miller III,Hilbert's 10th problem and the theory of groups

49 1

when applied to any basic formula D of Th(( C , A )), gives an open formula E D o f Th(Z) such that D is satisfiable in ( G , A ) ijand only if ED is satisfiable in Z. Proof. By hypothesis G is diophantine so for some fixed n there is a monomorphism cp : C GL(n, Z) and a diophantine predicate R ( x l l , ..., x,) of n2 variables such that +

if and only ifR(xl,, ..., x n n )or for simplicity R ( M ) . Now A = { g , , ..., gk } is finite so we can assume that R together with the matrices cp(gl), ..., cp(gk) are explicitly given ( R is specified by a polynomial). Let D be any basic formula of Th(( C, A )), and let 1 1 , , ..., u, be the variables occurring in D. Pick correspondingly m variables matrices M , , ..., M , with different variables (for integers) as entries - a total of men2 variables. Let D , be the formula concerning matrices formed from D as follows: replace u iby M i ; replace each constant corresponding to gi by the matrix cp(gi);replace 1 by the identity matrix; group multiplication is replaced by matrix multiplication ui-l is replaced by the formal inverse of M i which can be written down in terms of cofactors and the determinant of Mi. Now by simply carrying out all of the (matrix) multiplications in D , formally one obtains an open formula D , of Th(Z) of the following kind: D , is a conjunction of formulas Bi and 7 Bi where each Bi is a conjunction of n2 polynomial equations (corning from atomic parts of D , ). Let D , denote the formula

D,

A

R(M,)

A

... AR(M,)

492

C.F. Miller, 111,HilbertS 10th problem and the theory ofgroups

of Th(Z). Now D , is satisfiable in Z if and only if D is satisfiable in ( G, A ), since cp is monic and R characterizes cp(G). Let D, be the prenex form of D , . Then D4 has the form

where the Piare polynomials. Now D, is satisfiable if and only if D, is satisfiable. Finally, let ED be the open formula

ED is satisfiable if and only ifD, is satisfiable. Hence, ED is satisfiablc. Hence, E D is satisfiable in Z if and only if D is satisfiable in ( G, A ). Clearly, ED can be effectively found from D. This completes the proof of the lemma and thereby Theorem 1.

54.Applications to free groups h’or any cardinal number a , let Fa denote the free group o n Q generators and let F denote the class of all free groups (including the trivial and infinite cyclic groups). Consider an arbitrary open formula B of Th(F). Since B contains only finitely many variables, B is satisfiable in some I; E F if and only if B is satisfiable in FK,. Because subgroups of free groups are free and FK,,it follows that B is satisfiable in some F E F if and only if B is satisfiable in F , . Hence SPOF(F) is equivalent to SPOF(F,). According t o example 1 of section 2, F , is a diophantine group (with an empty set of distinguished elements). Applying Theorem 1 we conclude: Lemma 5. IIPOF(F), SI’BF(F), and SI’OF(F) are reducible to Hilbert ‘r 10th problem. Moreover, the set of open formulas o f Th(F) whicli are satisfiable in some F E F is recursively enumerable.

Any basic formula D of Th(F) is logically equivalent t o a basic formula of the form

C.F. Miller III, Hilbert's IOth problem and the theory of groups

R, = 1 A ... A R,

= 1 A w, # 1A

493

... A w k # 1

where the R iand wj are words in the free group on the variables, say x,, ..., x,, which occur in D. To give a useful interpretation to this formula, we introduce the finitely presented group G, = ( xl,..., x n ;R , = 1 , ...,R, = 1 ). Observe that G, does not depend on the inequalities which occur in D.Now D is satisfiable in some F E F if and only if there is a homomorphism $ from G, into some F E F such that $ ( w l ) # 1 A ... A $(wk)# 1. (This follows from the definition of satisfiable). However, since subgroups of free groups are free,it follows that D is satisfiable in some F E F if and only if G, has a quotient group F' which is free and such that the images of w l , ..., w k do not map to 1 in F' (take F' = $(G,) in the previous statement). For any group G, define

K,=

n K

{ K G I G / K E F )

(the intersection of all normal subgroups K of G such that G / K is free). Now G / K , is a residually free group. Moreover, if G, is any other homomorphic image of G which is residually free, then G, is a homomorphic image of G / K , . This property is in fact equivalent to the definition of K , .

Lemma 6. Let G be a finitely presented group. Then L , = G\ K , = { w I w is a word in the generators of G and w $ K , } is recursively enumerable. Moreover, the method of enumerating L , is uniform in G in the sense that given G we can effectively find the enurnerating process. Proof. Suppose G = ( X I , ..., x,; R , , ..., R, basic formula

R , = 1 A ... A R,

)

and let D, be the

= 1 A w # 1.

Then D, is satisfiable in some F E F if and only if w $ K,. The result now follows from Lemma 5.

4 94

C.F. Miller I l l , Hilbert’s 10th problem and the theory of groups

Lemma 7 . I f Hilbcrt ’s 10th problem is recursively solvable, then there is a iiritfi)rm algorithm A(G) which, when applied to any finitc.11. prcscnted group G , solves the word problem f o r GIK,. Lemma 7 ‘ .I f Hilbert’,~10th problem is recursively solvable, then tlierc is u iiriiform algorithm A‘(G) which, when applied to any finitell, Iwe.sentc.cl group G, entimerated the words o f K , . Proof. I n view of Lemma 6, Lemmas 7 and 7‘ are (logically) equivalent. Let C; and D ,be as in the proof of Lemma 6. Now D, is satisfiable in some F E F if and only if w = 1 in G / K , . The result now follows by Lemma 5. Problem 1 . Does there exist a finitely presented group G such that GIK,, is riot recursively presented? The following problem was posed several years ago by M.O. Rabin (unpublished) who knew of its connection with Hilbert’s 10th problem. Problem 2 (Rabin’s Problem). Does there exist a recursive class of finitely presented groups G , , G,, ... such that there is n o algorithm t o tell of an arbitrary Gi whether o r not Gi has a non-abelian free (1uo t ien t ? Clearly if the answer to Problem 1 is affirmative, then by Lemma 7’ Hilbert’s 10th problem would be unsolvable. Consider I’roblem 2. Let G I = (.Y, , ..., x , ~ ; R = 1, ..., R,, = 1 ). Now G, has a non-abelian free quotient if and only if one of the finitely many conimutatorc [ Y,, r,] # 1 in G/K,. But this is reducible to the word problem for G / K ( ; . Hence, if there were such a class of groups, by Lemma 7, Hilbert’s 10th problem would be unsolvable. Note that f G, I G, has a non-abelian free quotient } is recursively enunierable by Lemma 6. Problem 3 . Does there exist a recursive class of finitely presented groups G , , G,. ... with uniformly solvable word problem such that

F.C. Miller III, Hilbert’s 10th problem and the theory of groirps

495

{ Gi I Gi is residually free } is recursively enumerable, but not recursive? Here “uniformly solvable word problem” means there is a uniform algorithm which, when applied to a G j , solves the word problem for Gi.

If the answer to Problem 3 is affirmative, then Hilbert’s 10th problem would be unsolvable. To see this assume that Hilbert’s 10th were solvable and that G,,G, , ... is such a class. Now Gi is not residually free if and only if ( 3 w E G i ) (w f 1 in Gi A w E K G j ) . Since we are assuming Hilbert’s 10th is solvable, by Lemma 7’ it follows that S = { ( G i ,w)lw

E

Gi

A

w E Kci}

is recursively enumerable. Since the word problem for the G j is uniformly solvable the subset

T = { (Gi,w)IwE Gj A

W E

KGi A w f 1 in G i }

of S is also recursively enumerable. But Gi appears as the first component in some member of T if and only if G iis not residually free. Hence { Gi I Gi is not residually free } is recursively enumerable. But this contradicts o u r assumption o n the class G , , G,, ..., and the claim is established. Collins [ 61 has constructed a recursive class of finitely presented groups G , , G,, ... such that each Gi has solvable word problem (note: not uniformly) such that { G, I G j is residually free } is recursively enumerable, but not recursive. This does not settle Problem 3 since the word problem is not uniformly solvable. Indeed, his construction works exactly because the word problem for the G j is not uniformly solvable. $5. A free nilpotent group

In this section a construction of Malcev [ 131 will be used t o establish a “converse” t o Theorem 1 . H will denote the free nilpotent

496

C.F.Miller 111, Hilbert's 10th problem and the theory ofgroups

group of class 2 and rank 2. The commutator notation [ u , u ] = U U L I - ~ U - ' will be used frequently. Now H is generated by two elements, say a l and a 2 , which freely generate H (in the sense of varieties). H satisfies the law [ [ E l , HI , HI = 1 and the center of H , written C(H) is infinite cyclic with generator c = [ u 2 ,a l ] = a 2 a l a 2 - l a l - l . Recall that H has a faithful matrix representation H" in SL(3, Z) where H" consists of those matrices of the form

For convenience, H and H" will often be identified. In terms of matrices, one can take

LEI=

io J 1 0

0 1 0

0 1

i:: 1

'

a2=

1

0

The equation

,1

k,

171

2 c

n =

is easily verified hence every element of H is uniquely expressible in the form u , ku2mc" . Elements of H multiply according to the equation: ~

The general commutator is given by

C.F. Miller III, Hilbert's 10th problem and the theory ofgroups

497

[ ( a ,h a2icj 1, ( a l k a 2 m c n ) =] cikpmh. Finally, while the above equations are given for two particular free generators al , a2, if y 1 , y 2 are any other two free generators of H , then the same statements hold for y , ,y 2 in place of a l , a2 except that y l , y 2 correspond to different matrices. Lemma 8. Let z 1 = c k ,z 2 = c1 be arbitrary elements of C ( H ) . Then there exist elements x l , x 2 E H which satisfy the equations

xla2 =a2xl (*)

x2al = a l x 2

alxlal-lxl = z , a2x2a2-1x2-1= z 2

Moreover, if x l , x 2 are any two elements of H satisfying (*), then [ x 2 x, , ] = x2x1x2-1x1-1= C k . ' . Proof. To see that such x l , x 2 exist, put x 1 = a2-k and x 2 = all. Then clearly x l a 2 = a2xl and x2al = a l x 2 . Moreover, by our general rule for commutators:

as required. It also follows that [ x 2 x, l ] = [ a l l , a 2 - k ] = c - / . ( - =~ ) Ck.1. Now let x l , x 2 be arbitrary elements of H satisfying (*). Write x , = alha2mcnand consider the equation x l a 2 = a 2 x . Then 1 = [xl, a 2 ] = [ a l h a 2 m ~ na 2, ] = c - l e hand s o h = 0 and x1 = a 2 m ~ n . But now c k = z , = [ x , , a l l = [ a 2 m ~ na l, l = c - ~= .c ~- ~Hence . m = -k and x1 = a2-kcn.Similarly it follows that x 2 = a l l c s . But then [ x 2 ,x l ] = [ a l l c s ,a 2 - k ~ n=][ a1 ' a 2 - k ] = c k e las claimed. Lemma 9. Let z be any element of C ( H ) . Then there exist elements u, u E H such that z = [ u, u ] . Conversely, for any u, u E H , [ u, u ] E C ( H ) .

C.F. Miller III, Hilbert’s 10th problem and the theory of groups

498

Proof. That [id, u ] E C ( H ) is trivial since H satisfies the law [ [ H ,H I , HI = 1. Let z = ck be an arbitrary element of C(H). Then, taking u = a 2 , u = a l k , it follows that [ u , ul = [ a 2 ,a l k l = c k = z as required. Lemiiia 10. Let yl, y 2 be arbitrary elements o f H such that [ J ’ ~y, 1] = ~ ~ ~ y , y ~= -c. ~Then y , -y1~, y 2 freely generate H. In purticular, ull c i j the above results hold with y , ,y 2 in place o f UI,U2.

Proof. We first show that [ y 2 ,y 13 = c implies yl, y 2 generate H. Put = ulku2fTz c ” and y 2 = alha2‘cJ.From the general formula for commutators, it follows that [ y 2 ,y l ] = c ~ = c ,~and hence ~ ili - rizh = 1 . Let G be the subgroup of H generated by y , ,y 2 . Clearly C ( H ) C G. Now y I i = alika2micP for some p and .ir21fz = al’”’?a2micq for some 9 . Hence alikaZmiE G and o l f ~ z / ~ u 2E ’ nG i and u , ~ ~ ~ E~ G. u 2Thus ~ f alik-mh ~ f = al E G. Similarly a?- E G and so C; = EI as claimed. H is residually finite and therefore Hopfian, so according to H. Neumann [ I 5 1 (4 1.33) y 1 , y 2 freely generate H (in the sense of varieties). j

q

l

Next we consider Th(( H, { c } ) ) ,where ( H , { c } )is the group H with distinguished element c (the generator of C(H)). Let R ( z l , z 2 , x l , x 2 ,y , , y 2 ) be the conjunction of the following equations:

If the formula (3.13,)( 3 . v 2 ) R ( z 1 , z 2 , x l , x 2 , y 1 ,isy 2satisfied ) by particular assignment of the variables zl, z 2 , x l , x2 to elements of !I, by Lemmas 8, 9 and 10 it follows that zl, z 2 E C ( H ) and x 2 y- 1 x2 - l x 1 -1 = c k - 1 where z , = c k and z 2 = c l . Our purpose is to interpret Th(Z) in Th(( H, { c } ) )via the following scheme: :I

~

C.F. Miller III, Hilbert's 10th problem and the theory of groups

l+c, k

+1

-+

k. 1

z1 *

499

o-+ 1 Z,

= ck+l = ck"

-+

w h e r e z l = c k , z 2 = c ' a n d (3.~1) (3~,)R(zl,z2,xl,x2,.Yl,.Yz). Although it is possible to interpret any formula of Th(Z) in Th((H, { c})) by suitably adapting Malcev [ 131 , for our purposes it is only necessary to interpret polynomial equations.

Lemma 11. There is an effective process which ussociates with any polynomial equation P = 0 an open forinula B, of Th(( 11, { c} )) such that P = 0 is satisfiable in Z if and onl-v If' B, is satisfiable in (Zl,{c}). Proof. The equation P = 0 is, as previously noted, algebraically equivalent to an atomic formula P , = P, of Th(Z) (no minus signs allowed) where P , and P, are the positive and negative parts of P. First we show that any atomic formula P , = P, is logically equivalent to a conjunction of atomic formulas El A E2 A _ _ _ A El

which can be effectively found from P I = P, where each Eihas one of the following forms: tl = t,,

tl

+ t,

= t,,

t,

*

t , = t3

where each ti is 0, 1 o r a variable of Th(Z). A formula of the type

Eiwill be called elernentary. The proof is by induction o n the

number of occurrences of + and - in P , = P , (recall that P , , P, are both terms). If there are either zero or one occurrences of + and in P , = P, , the result is obvious from the definition of term. Assume at least two + o r occur in P , = P,. Then one of P I or P, has the form Q, + Q2 o r Q , Q2 where Q , , Q2 are terms, say P , = Q, + (2,. Let x, y , z be variables not occurring in P , or P, . Then P I = P, is logically equivalent t o

-

-

-

500

C.F. Miller 111,Hilhert’s 10th problem and the theory groups

( x = Q1)

A

( y = Q2)

A

( z = P 2 ) A (x + y

= z).

-

Each of the conjuncts contains fewer occurrences of + and than P I = P 2 . Hence by induction each conjunct is logically equivalent t o a conjunction of elementary formulas. Thus P, = P, is equivalent to a conjunction of elementary formulas. A similar argument applies in case P , has form Q , Q2 . Moreover, it is clear these instructions for obtaining the desired Ei’s are effective. This proves our claim. From P, = P2 one can effectively find a logically equivalent formula D of the form E , A E , A ... A E, where each Ei is elementary. Assume the variables of D are among u l , .._,u k . With each Ei we associate a formula Siof Th(( H , { c} )) with free variables among z l , ..., z k as follows: ( 1 ) If Ei has form u 1 = u2 take S j t o be

-

(2) If Ei has form zcl

+ u2 = u 3 take Si t o be

( 3 ) If E j has form

*

if1

i f 2 = 213

take Si t o be

where R is as defined above. (4) If Ei contains 0 or 1 instead of certain variables, find the corresponding Sias in ( 1 ) - (3) except that 0 or 1 becomes 1 or c respectively. Let T be the prenex form of S , A ... A S,. From Lemmas 8, 9, and 10 together with the remarks preceding the present lemma, it follows that P = 0 is satisfiable in Z if and only if

C.F. Miller III, HilbertS 10th problem and the theory of groups

501

T is satisfiable in ( H , { c } ) . But T has the form ( 3 u l ) ... ( 3um)B, where B, is an open formula. Hence P = 0 is satisfiable in Z if and only if B, is satisfiable in (H, { c } ). Moreover, B, can be effectively found from P = 0, as required. Since H is diophantine, Theorem 1 and Lemma 1 1 imply:

Theorem 2. L e t H be the free nilpotent group o f rank 2 and class 2 and let c E Hgenerate the center o f H. Then Hilbert’s 10th problem is Turing equivalent to SPBF(( H , { c j )) and hence also to SPOF((H, { c j )) and DPOF(( H, { c } )). Remarks. We have actually proved somewhat more than Theorem 2. In particular, since the Bp found in Lemma 11 contains no negation signs, it follows that Hilbert’s 10th problem and SPBF(( H, { c } )) are equivalent to the satisfaction problem for conjunctions of atomic formulas in ( H , { c j ). In a different direction, it would now be a simple matter t o prove most of the results in Malcev [ 131 . In particular, it now follows easily that Th(( H, { c } )) and Th(H) are undecidable. See [ 131 for more details. It would be desirable to replace SPBF((H, { c } )) by SPBF(H) in Theorem 2. In order to do this, we would need an open formula B ( x , y l , ..., y,) not involving c such that ( 3 y , ) ... ( 3 y , ) B ( x , y l , ..., y,) holds in H if and only if x = c. The author has not been able to do this. There are more complicated formulas which define c , but they do not help for Theorem 2. Over in Th(Z), the formula ( x x = x ) A (x f 0) holds in Z if and only if x = 1. However, the translation of this formula into Th(( H , { c } )) involves the formula R and hence

-

C.

Toh [ 221 has obtained some partial results concerning SPBF(H). Toh’s paper contains, among other things, some interesting examples which depend on substantial results from number theory. In view of Theorem 2, this dependence might well be expected. The following result is weaker than Theorem 2, and follows immediately from Theorems 1 and 2:

502

C.F. Miller I l l , Hilbert s 10th problem and the theory groups

Theorem 3. Hilbert’s 10th problem is unsolvable if and only if there exists a diophuntine group G with distinguished elements A szicli that SPBF(( G, A ) ) is unsolvable. $ 6 . Applications to the conjugacy problem

Recall that the conjugacy problem for a group G is the problem of deciding for an arbitrary pair u, u of words on the generators of G whether or not I ( and u are conjugate in G, i.e. whether o r not ( 311’ E G) (wlLIW = u in G). A group L is a diophantine group o f matrices, if for some integer n > 0, L C GL(n, Z ) and there is a diophantine predicate R of n2 variables such that for M any n X n matrix, R ( M ) * M E L . Lemma 12. Let L be a diophantine group of matrices, as above. Then the conjitgacy problem for L is reducibel to Hilbert s 10th pro Hem. Proof. Let M , , M 2 be arbitrary (but fixed) matrices belonging to L . Now M , and M , are conjugate in L if and only if the formula M , M = M M , concerning matrices is satisfiable by some M E L . Call this formula D , . As in the proof of Lemma 4,* there is an effective process which, when applied to D , , gives an open formula E of Th(Z) such that E is satisfiable in Z if and only if D , is satisfiable in L , i.e. if and only if M , and M , are conjugate in L . By Lemma 2 , the result follows. Theorem 4.Let G he a finitely generated diophantine group. Then c,~ f o r G is reducible to Hilbert’s 10th probthe ~ ~ o n j u g uproblem lem. Proof. Let g , , ..., g k denote a finite set of generators for G. Since G is diophantine, there is a faithful representation cp : G + GL(n, Z) for some r? > 0 so that p(G) is a diophantine group of matrices. Now put I , = p(g,), ..., 1, = cp(gk) and let L = cp(G). The matrices li generate L . Clearly, there is an effective process which, when ap* The starting

point in Lemma 4 was somewhat different. However, it is easy’ to see the proof of that Lemma also applies to these circumstances.

C.F. Miller I l l , Hilbert’s 10th problem and the theory ofgroups

503

plied to any word u ( g j ) on the gi in G, gives the matrix cp(u(gj)) - namely cp(u(gi)) = u(li). Since two words u, u on the gi are conjugate in G if and only if cp(u), p ( u ) are conjugate in L , the result follows from Lemma 12.

Remark. Consider a diophantine group with finitely many distinguished elements ( G, A ). If A generates G, it follows that the conjugacy problem for G is reducible to SPOF(( C, A >)and hence to Hilbert’s 10th problem. To see this, let u, u be arbitrary words in the generators A. Then w-l uw = u is satisfiable in ( G, A ) if and only if u and u are conjugate in G. But w-l U W = u is an open formula of Th(( G, A )), as claimed. These considerations led W.W. Boone (unpublished) to ask the following question? Problem 4 (Boone). Does there exist a finitely generated diophantine group with unsolvable conjugacy problem? In view of Theorem 4, an affirmative answer to Problem 4 would show Hilbert’s 10th problem unsolvable. In this connection, Miller [ 141 has given an example of a finitely generated subgroup of GL(4, Z ) with an unsolvable conjugacy problem. Whether or not this subgroup is diophantine is not known.

37.Concluding remarks In view of the connections established above between diophantine groups and Hilbert’s 10th problem, the following vague question seems appropriate: Problem 5 . Which groups are diophantine?

In trying to answer this question, one may as well consider groups of matrices and ask which are diophantine. Since a diophantine predicate is recursively enumerable (section 2), it follows that:

504

C.F. Miller 111,HilbertS l o t h problem and the theory of groups

(1) A diophantine subgroup of GL(n, Z) is a recursively enumerable set of matrices. Some of the work of Davis, Putnam, and J. Robinson [8] on Hilbert’s 10th problem is concerned with the following question which is as yet unanswered: Is every recursively enumerable predicate diophan tine? An affirmative answer would of course show Hilbert’slOth unsolvable (see section 2 and [ 7 ] ) and would provide an answer to Problem 5. As Boone and Davis have pointed out, an appropriate answer t o any of problems 1 through 4 could show Hilbert’s 10th unsolvable without answering the above question. Added in proof: Matijasevic“ [ 231 has now proved the remarkable theorem that every recursively enumerable predicate diophantine. This answers problems 4 and 5 . Problems 1 , 2 , and 3 still seem t o be open. Next we shall list several known results concerning groups of matrices which seem relevant to Problem 5 . The following notation will be used: cf, = arbitrary field of characteristic zero C = field of complex numbers Q = field of rational numbers R = field of real numbers GL(n, a)) = invertible n X n matrices with coefficients in a. Clearly, for any such a, one has

GL(n, Z)

C

GL(n, a).

It is known that GL(n, Z) is finitely presented (see [ 101 for instance). (2) A finitely generated subgroup of GL(n, a) is residually finite (Malcev [ 121 ). Since residual finiteness is heriditary, it follows that any diophantine group is residually finite. ( 3 ) Let G be a finitely generated subgroup of GL(n, a). Then G contains a torsion free normal subgroup of finite index (Selberg [IS]).

C.F. Miller I l l , Hilbert’s 10th problem and the theory of groups

505

(4) A solvable subgroup of GL(n, Z) is polycyclic (Malcev [ 1 1 I ). (5) The order of a finite subgroup of GL(n, Q ) is bounded by a function of n (see for instance Serre [ 191). An important class of groups related to diophantine groups is the class of arithmetic groups [ 1 ] . For example, the free nilpotent group of section 4 is an arithmetic group. We will briefly indicate connections between diophantine and arithmetic groups. A complex algebraic group is a subgroup G of GL(n, C) which consists of all invertible matrices M = ( m i j )whose coefficients annihilate some set of polynomials { P a ( x l l , ..., x n n ) }with complex coefficients. G is said to be defined over Q if the Pa may be chosen to have rational coefficients. In view of the Hilbert basis theorem one can assume the number of Pa is finite. Assume hereafter that G is an algebraic groups defined over Q by polynomials P,,..., P k . Let G, = G n GL(n, R) and G , = G n GL(n, Z). The group G, is called an arithmetic subgroup (or group of units) of G, . It is easy to show that G, consists of exactly those matrices in GL(n, Z) which satisfy a suitable polynomial equation with integer coefficients constructed from P,,..., P k . In particular, G, is a diophantine group. It is known that G, is finitely presented (see Borel [21). G, is a discrete subgroup of the real Lie group G, . Borel and HarishChandra [ 31 have given necessary and sufficient conditions under which G,/G, is compact, or of finite invariant measure. A theory of fundamental sets for G, has been developed. Of course the study of arithmetic groups relies on the study of algebraic groups. In particular, the methods of algebraic geometry and Lie theory are involved. To what extent similar techniques might be helpful in investigating diophantine groups is not clear to the author. References [ 11 A. Borel, Reduction theory for arithmetic groups, Proc. Sympos. Pure Math., vol.

9 (Amer. Math. SOC.,Providence, R.I., 1966) 20-25. [ 21 A. Borel, Arithmetic properties of linear algebraic groups, Proc. International Congress of Math. (Stockholm, 1962) 10-22.

5 06

C.F. Miller III, Hilbert's 10th problem and the theory of groups

[ 31 A. Bore1 and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962) 485-535. [ 4 ] A. Church, Introduction to mathematical logic, vol. I (Princeton University Press, Princeton, N.J., 1956). [S] P. M. Cohn, Universal Algebra (Harper and Row, New York, 1965). 161 D.J. Collins, t o be published. 171 M. Davis, Computability and unsolvability (McGraw-Hill, New York, 1958). [ti] M. Davis, 11. Putnam and J. Robinson, The decision problem for exponential diophantine equations, Ann. of Math. 74 (1961) 425-436. [ 9 ] D. Hilbert, Mathematical problems, Bull. of Amer. Math. Sac. 8 (1901-2) 437-479). [ 101 W. Magnus, A. Karras and D. Solitar, Combinatorial group theory (Wiley, New York, 1966). [ 1 1 J A.I. Malcev, On some classes of infinite soluble groups, Amer. Math. Soc. Translations, series 2, vol. 2 (1956) 1-22. [ 121 A.I. Malccv, On the faithful representation of infinite groups by matrices, Amer. Math. SOC.Translations, series 2, vol. 4 5 (1965) 1-18. [ 131 A.I. Malcev. On a correspondence between rings and groups, Amer. Math. Soc. Translations, series 2, vol. 4 5 (196.5) 221-232. [ 141 C.I.'. Miller 111, Unsolvable problems in direct products of free groups and in unimodular groups, unpublished. [ I S ] I f . Neumann, Varieties of Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 37 (Springer, Berlin, 1967). [ 161 J. Robinson, Diophantine decision problems, in: W.J. LeVeque, ed., MAA Studies In Math., Vol. 6 (Prentice-Hall, 1969). [ 17 I 1.N. Sanov, A property of a certain representation of a free group (Russian), Doklady Akad. Nauk SSSR. vol. 5 7 (1947) 657-659. [ 181 A. Selberg, On discontinuous groups in higher-dimensional symmetric spaces, Internat. Colloq. Function Theory, 147-164, Tata Inst. of Fund. Research, Bombay, 1960. [ 191 J.-P. Serre, Lie Algebras and Lie groups (Benjamin, New York, 1965). [ 201 J.R. Shoenfield, Mathematical logic (Addison-Wesley, Reading, Mass., 1967). 1211 R.R. Stoll, Set theory and logic (Freeman, San Francisco, 1963). (221 K.11. Toh, Problems concerning residual finiteness in nilpotent groups, to be published. [ 23 I Ju.V. Matijasevic", Enumerable sets are diophantine (Russian), Dokl. Akad. Nauk. SSSR, Tom 191 (1970), No. 2 - English translation: Soviet Math. Dokl., Val. 11 ( 1 970), No. 2.

DECISION PROBLEMS IN ALGEBRAIC CLASSES OF GROUPS ( A SURVEY) Ch.F.MILLER, III* The Institute for Advanced Study, Princeton

This is a survey of what is known concerning the world problem, conjugacy problem, and other decision problems in certain classes of finitely presented groups. Each of the decision problems is known to be unsolvable for finitely presented groups in general. Our aim is to investigate these decision problems in classes of finitely presented groups which are in some sense “elementary”. To methods of “measuring” the elementary nature of finitely presented groups will be considered: (1 ) algebraic constructions how a group is built from free groups by algebraic constructions such as free products with amalgamation; and (2) algebraic classes - groups defined by algebraic properties such as nilpotence and residual finiteness. Most of the results discussed are summarized in Tables 1 and 2, and the text might be regarded as an explanation of these tables. Section 1 consists mostly of terminology and has been included to make our survey accessible to a more diverse audience. Section 2 is concerned with Table 1, and Section 3 is concerned with Table 3. Section 4 contains an example relevant to both tables, and Section 5 consists of additional remarks and problems. All groups are assumed to be finitely presented unless otherwise stated. Regarding the tables: The notation “+WP” in a table means the word problem is known to be solvable for the corresponding class of groups; “,WP” means examples are known for which the word

* Work supported by U.S.National Science Foundation under contract No. GP-7421. 507

508

Ch.F.MiNer, Decision problems in algebraic classes of groups Table 1

\

Construction

---.--\_

Iterations

free products with amalgamation

Split Zxtensions

Direct Products

+WP ,CP 7GwP

+wP + CP -GWP

(Boone ,WP)

Level 3

Level 2

____

HNN-construc tion (= Britton ex tensions)

-WP

(Collins ,CP) 7WP

+WP 7CP -GwP

iWP 7CP -GWP

""\_

f.g. free groups +GWP

(all groups are assumed to be f.p.)

problem is unsolvable; and "?WP" means the status of the word problem is an open question. Similarly for the other problems considered. 1. Some terminology By a presentation of a group we mean an ordered pair CSP) where S is a set and D a collection of words on the elements of S and their inverses. By the group G presented by CSD)we mean the quotient group of the free group on S by the normal closure of the words in D.(For further details see [ 121 .) We usually write

Ch.F.Miller, Decision problems in algebraic classes of groups

5 09

w

Table 2 f.p. hopfian

,IsoP

+WP GWP -CPZISOP

f.p. residually finite

I ? \

I

j

solvable f.p. subgroups of (GL(n,Z)

,GWP ?IsoP

\

f.p. residually nilpo ten t

Polycyclic \

f.p. residually free

I

_

_

~

f.g. nilpotent I

f.g. abelian (all groups are assumed to be f.p.)

G = (SO> in this situation. Except when necessary, we wil, not distinguish between a group (as an abstract algebraic object) and its presentation (as a notation in some logical system)*. G = tS,D is said to be finitely generated ( f g ) if S is finite and finitely presented (f.p.) when S and D are both finite. Note that being generated by a finite set of elements or having a finite presentation is an algebraic property of groups (preserved under iso-

* There are other possible viewpoints towards presentations. can be viewed as logical systems

-

In particular, presentations see Boone [6] for this approach.

510

Ch.F.Miller, Decision problems in algebraic classes of groups

morphism). Finitely presented groups arise naturally in topology: they are exactly the fundamental groups of finite simplicia1 complexes o r alteriiatively tlie fundamental groups of closed differentiable ri-manifolds ( 0 4 ) . Viewing tlie group G = (SO)as the quotient of the free group F on S by the normal closure of D ,recall that a word W = 1 (the empty word) in G if and only if W is equal in F t o a product of conjugates of words in D.Moreover, W , = W 2 in G if and only if there is a word 2 = 1 in G such that. W , = ZW2 in F. In case G is finitely generated, the word probEem (WP) (or in some languages the identity problem) for G is the algorithmic problem c f deciding for arbitrary words W of G whether or not W = 1 in G. In case G = CSJ) is such that S is finite and D is a recursively enumerable (r.e.) set of words, G is said to be recursively prese!ztcd*. I n particular, f.p. groups are recursively presented. Observe that the word problem is r.e. for recursively presented gro u p s. ‘The algorithmic problem of deciding for an arbitrary pair of words U, V-E G whether or not U and V are conjugate in G (i.e. whether or not there exists W E G such that U = W-’ VW in G) is called the coiijugacy problem (CP) or transjbrmation problem for G. For recursively presented groups, the conjugacy problem is r.e. Observe that the word problem for G is one-one reducible to the conjugacy problem for G since U = 1 in G if and only if U is conjugate to 1 in G. Let 9 = { rTi,i>O} be a recursive class of finite presentations of groups (say on some fixed alphabet). The isomorphism problem (IsoP) for 9 is the algorithmic problem of deciding whether or not 1 ~ I 2i 1-1, for arbitrary i and j (i.e. whether or not and are presentations for the same abstract group). The class 9 is required t o be recursive so that the algorithmic problem is well-posed. In the context of topology, the word problem corresponds to the problem of deciding whether a closed path is contractible to a

ni

* By a device due lo Craig, 3 grcup is recursively presented (in our sense) if and only if

it has some presentation 6 ’ D ’ )where S’ is finite and D’is a recursive set of words. See Boone [ 6 ] in this connection.

Ch.F.Miller, Decision problems in algebraic classes of groups

511

point. The conjugacy problem corresponds to the problem of deciding whether two closed paths are free homotopic. Since homotopy equivalent spaces have isomorphic fundamental groups, the isomorphism problem is related to the problem of honiotopy equivalence. Finally consider a f.g. group G = CSP) and a finite set of words W , , ..., W , . Let H be the subgroup of G generated by W , ,..., W,. The generalized word problem (GWP) or membership problem for H in G is the algorithmic problem of deciding whether or not an arbitrary word U E G belongs t o the subgroup H (i.e. whether or not there is a product of the Wi and their inverses which is equal to U in G). The GWP is r.e. provided G is recursively presented. Clearly, the word problem for G is just the GWP for the trivial subgroup in G . The GWP as formulated here is often called the extended word problem, but we follow [ 121 in our choice of terminology. A property P of groups is called a poly-property if, whenever N and GIN have the property P, so has G. Here N is a normal subgroup of G. Proposition 1 . The following are poly-properties: ( 1) being finitely generated ( 2 ) having a finite presentation ( 3 ) satisfying the maximum condition f o r subgroups (4) being finitely generated and having a solvable word problem. That (1) - (3) are poly-properties is shown in P. Hall [ 91 , and that (4) is a poly-property is easily verified. A group G is poly-P if it can be obtained from the trivial group by a finite succession of extensions by groups in P,i.e. if there is a finite series of subgroups

G = Go

3

G,

3

.._3 G, = { l }

such that G j + l is normal in Gi and Gi/Gi+, E P. As an application of Proposition 1 , it follows that polycyclic groups are finitely presented and satisfy the maximum condition for subgroups. Observe that solvable groups are just the polyabelian groups.

512

Ch.F.Miller, Decision problems in algebraic classes of groups

Let P be a property of groups. A group G is said to be residually P if for every 1 # W E G there is a normal subgroup N, of G such that W & N , and GIN, has the property P. Equivalently, G is residually P if the intersection of the normal subgroups N of G such that GIN has P is the identity. For instance, free groups are residually nilpotent and residually finite (see [ 121 ). Baumslag [3] has shown that the automorphism group of a f.g. residually finite group is again residually finite. A group G is called hopfiun if N Q G and GIN 1 G imply that N = { 1 } . Equivalently, G is liopfian if every epic endomorphism of G is an automorphism. F.g. residually finite groups are hopfian (see 1211 41.44). 2. Algebraic constructions

This section concerns decision problems in f.p. groups which are “built” (allowing iterations) from free groups by the following constructions: ( 1 ) free products with amalgamated subgroup (2) the construction of Higman, Neumann and Neumann [ 101 (also called “Britton extensions”) ( 3 ) split extensions (4) direct products. From the viewpoint of decision problems, free groups are rather pleasant. Namely, the word problem, the conjugacy problem, and the generalized word problem for finitely generated subgroups are all recursively solvable (see [ 121 ). Each of the above constructions occurs quite frequently in infinite group theory. If a f.p. group is built from free groups by one application of these constructions, it is in a sense “elementary” say it is Level 1 above free groups (with respect to that construction). If the same construction is applied to Level 1 groups, obtaining a Level 2 group, the resulting group may be more complicated and less elementary. Next we inquire as to the status of the various decision problems at Level 1 , Level 2, etc. It turns out that, for these four con-

Ch.FMiller, Decision problems in algebraic classes of groups

513

structions, a rather complete picture is available. In general, the various decision problems are unsolvable at the lowest “reasonable” level. First we recall the Higman-Neumann-Neumann (HNN) construction in the case of f.p. groups. Let E be a f.p. group and let A , , ..., A , be f.g. subgroups of E. Assume that for each i there is an isomorphic mapping tpi from A i onto a subgroup B j of E. Then the group E* obtained from E by adding new generators p l , ..., p n and new relations pyl aipi = tpi(ai) for all ai E A ihas the following properties: (i) E is embedded naturally in E* and (ii) p l , ..., p , freely generate a free subgroup of E* . (For a proof see [ 151, [ 201, or [ 221 .) From the proof of this result one can easily deduce the following: Assume E is a free group and let Fn denote the free group of rank n . Then I!?= E* * Fn (ordinary free product) is the free product with amalgamation of two free groups. Clearly, the WP and CP for E* and E are equivalent. Moreover, if E* has unsolvable GWP, then so does J??. Note that E* has Level 1 in the HNN-construction and E h a s Level 1 using free products with amalgamation.

Level 1. If a group G is the free product with amalgamation of two f.g. free groups and if G is f.p., then (i) (Baumslag [ 21 ) the amalgamated subgroup must be f.g. and so (ii) ([ 121 , p. 272) the WP for G is solvable. That is, at Level 1 using free products with amalgamation, the WP is solvable. Moreover, by the previous remarks, it follows that a Level 1 using the HNN-construction the WP is solvable. Since having a solvable WP is a poly-property, the split extension of one f.g. free group by another must have solvable WP. (Observe this also follows from the fact that such a split extension can be viewed as an HNN-construction over a free group.) Clearly, the direct product of two f.g. free groups has solvable WP and solvable CP. Miller [ 161 (see Section 4) has shown, there exists a split extension G of one f.g. free group by another having unsolvable CP and a f.g. subgroup H C G having unsolvable GWP. Moreover, G is obtained from a free group by one application of the HNN-construction. Hence, by the above remarks, the CP and GWP can be

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unsolvable at Level 1 using any of the first three kinds of construction. On the other hand, the direct product of t w o free groups has solvable CP. However, Mihailova [ 141 has shown there is a f.g. subgroup of the direct product of two free groups with unsolvable GWP. Higher Levels. I t is clear that if we take the direct product of groups at Level 1 in the direct product column, the result is still a direct product of (possibly many) free groups. Hence the status of the various decision problems at higher levels of direct products is the same as a t Level 1. Suppose G is the split extension of two Level 1 split extension groups. Then, since having solvable WP is a poly-property, G has solvable WP. Since the CP and GWP are already unsolvable at Level 1 , the status of the various decision problems a t higher levels of split extensions is the same as a t Level 1. Let E be a f.p. group at Level 1 in the HNN-construction column which has a f.g. subgroup A with unsolvable GWP in E (as noted before such A and E exist). Say A is generated by words w l , ..., w,. Now apply the HNN-construction to E as follows: E* is obtained from E by adding a new generator p and n new relations p-l wip = w,( i = l ,....n ) . Then E* is at Level 2 in the HNN-construction column. But E* has unsolvable WP since p-' upu-l = 1 in E* if and only if 11 E A where LL is a word on the generators of E and the GWP for A in E is unsolvable (we are using Britton's Lemma, see [ 221 ). Thus a t Level 2 in the HNN-construction column, the WP can be unsolvable. Similarly, let H be a f.p. group at Level 1 in the free product with rimalgamation column which has a f.g. subgroup A with unsolvable GWP in H . Let H , be another copy of H with corresponding subgroup A , . Now form the free product with amalgamation G = ( H * H , ;a=ul for all a E A ) . Now if h is an arbitrary word of H and h l the corresponding word in H , , then hh;' = 1 in G if and only if h E A . Since the GWP for A in H is unsolvable, this shows the WP for G is unsolvable. But G has Level 2 and can be f.p., so at Level 2 in the free product with amalgamation column, the WP can be unsolvable.

Ch.F.Miller, Decision problems in algebraic classes of groups

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Recall that the WP is reducible to the CP and the GWP. Thus at Levels 2 and higher in the HNN-construction column and the free products with amalgamation column, any of the WP, CP, and GWP can be unsolvable.

Historical remark. The groups studied by Boone and Britton (see [ 221 ) in showing the WP unsolvable for f.p. groups are located at Level 3 in the HNN-construction column. In [ 71 , Collins exhibits a f.p. group with solvable WP and unsolvable CP. Collins’ groups are located at Level 2 in the HNN-construction column. These are indicated in Table 1.

3.Algebraic cljlsses In Table 2 various classes of f.p. groups characterized by algebraic properties are listed, together with the known status of the various decision problems for groups in those classes. If two classes are connected by an upward line, then the “lower” class is contained in the “higher” class. Note that if one of the decision problems considered is solvable in a class of groups, then it is also solvable in any subclass. Accordingly, the solvability of WP has not been repeated for subclasses of the f.p. residually finite groups.

Hopfian groups. The largest class considered is the class of f.p. hopfian groups. Schupp and the author [ 181 have shown that any f.p. group G can be embedded in a f.p. hopfian group H in such a way that the WP for G is (Turing) equivalent to the WP for H . It follows that the WP and IsoP for f.p. hopfian groups can be unsolvable. Hence the CP and GWP can also be unsolvable. Residually finite groups. Any f.g. residually finite group is hopfian (for instance, see [ 2 1 1 41.44). Dyson [ 81 and Mostowski [ 191 have shown that f.p. residually finitcgroups have solvable WP. Briefly, the argument is as follows: Let G be f.p. residually finite, and let w be a word of G. Then w # 1 in G if and only if w # 1 in some finite quotient of G. Since G is f.p., one can con-

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struct a list of all finite quotients of G and check to see whether or not w f 1 in successive quotients. Hence, one can enumerate { w l w f l in G}. On the other hand, since G is f.p., { w(w=1 in G } can be enumerated. Since every word of G appears in exactly one of these lists, this solves the word problem for G. Miller [ 161 (see also Section 4 below) has given an example of a f.p. residually finite group with unsolvable CP and unsolvable GWP. Moreover, he shows that there exists a recursive class of f.p. residually finite groups for which IsoP is unsolvable. Observe that the property of being residually finite is hereditary, i.e. subgroups of residually finite groups are residually finite. Subgroups ofGL(n,Z). Let GL(n,Z) denote the group of n X n matrices with integer entries and determinant 5 1. Let A , denote the free abelian group of rank n . Then GL(n,Z) is isomorphic to the automorphism group of A , . Now A n is f.p. residually finite, so by Baumslag [ 31 it follows that GL(n,Z) is residually finite. Finite presentations for GL(n,Z) are known (see [ 121 , p. 168). Thus any f.p. subgroup of GL(n,Z) is residually finite and hence has solvable WP. Since GL(4,Z) contains a “nice” isomorphic copy of the direct product of two free groups, by Milailova [ 141 it follows that GL(4,Z) has unsolvable GWP. The status of the CP and IsoP for f.p. subgroups of GL(n,Z) seems to be unknown. Even the CP for GL(n,Z) itself seems to be unsettled. In this connection, Miller [ 171 has shown the CP is unsolvable for finitely generated subgroups of GL(n,Z) and that there is no algorithm to tell of two finite sets of matrices whether or not they generate the same subgroup of GL(n,Z) (or even isomorphic subgroups). Solvuble groups. For f.p. solvable groups in general, there seems to be little known concerning the various decision problems. It is remored that several Polish mathematicians have shown the WP is solvable for f.p. solvable groups, but the author knows of n o published proof.

Pdycyclic groups. Polycyclic groups are certainly solvable, in-

Ch.F.MiNer, Decision problems in algebraic classes of groups

517

deed a group is polycyclic if and only if it is solvable and satisfies the maximum condition for subgroups. As noted above, polycyclic groups are f.p. Malcev [ 131 has shown that a f.g. solvable subgroup of GL(n,Z) is polycyclic. Conversely, Auslander [ 11 has proved that any polycylic group can be embedded in GL(n,Z) for suitable n. It follows that polycyclic groups are residually finite, a result due to Hirsch which can be proved directly (see [ 21 ] 32.1 ). Toh [ 241 has shown that polycyclic groups are subgroup separable, i.e. if H is a subgroup of the polycyclic group G and w & H , then there exists an epimorphism cp: G + F where F is a finite group such that cp(w)& cp(H). Thus, by an argument similar to that showing the WP is solvable for f.p. residually finite groups, Toh has shown the GWP is solvable for polycyclic groups. The CP and IsoP are open for polycyclic groups. Regarding the CP, it does not seem to be known whether or not polycyclic groups are conjugacy separable. (G is conjugacy separable means that if u, u are not conjugate in G then their images in some finite quotient of G are not conjugate. If G is conjugacy separable and f.p., then G has solvable CP.) Added in proof: Remeslennikov [ 251 has now shown that polycyclic groups are conjugacy separable and hence have solvable conjugacy problem.

F.g. nilpotent groups. A f.g. nilpotent group is polycyclic and hence is f.p. (see [ 1 1 3 , vol. 11, p. 232, or [ 2 1 ] ). In particular, f.g. nilpotent groups are residually finite and have solvable WP. Blackburn [ 51 has shown that f.g. nilpotent groups are conjugacy separable and hence have solvable CP. Since polycylic groups are subgroup separable, so are f.g. nilpotent groups. Thus the GWP is solvable for f.g. nilpotent groups. (This was known before Toh’s work -.see Mostowski [ 191 for this and other decision problems.) The IsoP for nilpotent groups seems to be open and very interesting because of connections with Hilbert’s 10th problem. F.g. abelian groups. That the WP, CP, GWP, IsoP are solvable for f.g. abelian groups follows easily from the “fundamental theorem

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Ch.FMiller, Decision problems in algebraic classes of groups

for f.g. abelian groups” (see for instance [ 1 1 1 , vol. I, p. 145). Clearly, f.g. abelian groups are f.p. and nilpotent. Residually free groups. Since free groups are residually nilpotent (see [ 121, p. 3 1 I ) , it follows that residually free groups are residually nilpotent and hence are residually finite. In particular, f.p. residually free groups have solvable WP. However, the direct product of two free groups is residually free, so Mihailova [ 141 has shown the GWP can be unsolvable in f.p. residually free groups. The CP and IsoP for f.p. residually free groups seem to be open. IHowever, for f.g. recursively presented residually free groups, Miller [ 171 has shown the CP can be unsolvable. Residually nilpotent groups. A f.p. residually nilpotent group is residually finite because f.g. nilpotent groups are residually finite. I n particular, such groups have solvable WP. Beyond this, no more seems to be known other than the resulis cited above for the subclass of f.p. residually free groups. We remark that a polycyclic, residually nilpotent group need not be nilpotent. For example the infinite dihedral group D = k,)j2= ; 1,x-lyx=y-l) is residually a finite 2-group and polycylic, but not nilpotent.

4. An example

In this section, the author’s example [ 161 of a f.p. residually finite group G with unsolvable CP is presented. The group G is also the split extension of one free group by another and accounts for most of the unsolvability results for the CP discussed in the above classifications. Let H = 6,,s2,...,s, ;Rl,...,R,) be a f.p. group with unsolvable WP. Put F = (q,sl,...,s,), a free group of rank n+l on the listed generators. Finally define the f.p. group G as follows: Generators:

Ch.EMiller, Decision problems in algebraic classes of groups

5 19

Relations:

tT1qti

= 4Ri

t;ls,ti

= ,s

d;' qd, = sil qs,

l
,

l
where the Ri are the words in si which appear as the given defining relations for H .

Lemma 1 . G is obtained f r o m F b y the HNN-construction adding the letters { tl ,...,t,,dl ,...,d n > . Proof. Each of the subgroups in question is F itself. For example, for fixed i, the subgroups involved with ti are generated by ( 4 . ~ 1,...,s n } and {qRi,sl,...,s n ) . Since each of these sets consist of n+l elements of F which generate F which has rank n+ 1, they freely generate F. Consequently, the map 4 + 4Ri, s, + s, generates an automorphism of F as was to be verified. The same argument applies to the subgroups associated with each d,. Let T be the subgroup of G generated by { t l ,...,t, ,dl ,.._, d,}. Then, as noted above regarding the HNN-construction, T is a free group freely generated by { t l ,...,t, ,dl ,.. . ,dn } . Observe that every element of T normalizes F in view of the defining relations. Hence

Lemma 2. G is the split extension of the free group F by the free group T. In particular G has solvable WP. Lemma 3. G is residually finite. This lemma follows immediately from the following result which seems to have been known to several people. The proof is omitted (see [ 161 ):

Ch.F.Miller, Decision problems in algebraic classes of groups

520

Extension Lemma. Suppose that 1 + A +. E +. B +. 1 is an exact sequence ofgroups (i.e. E is an extension of A by B ) where A and B are residually finite and A is f g . Assume that any one o f the following conditions holds: (1) A has trivial center, ( 2 ) The sequence splits (i.e. E is a split extension or semi-direct product of A by B). ( 3 j B is free or A is non-abelian free. Then E is residually finite. Lemma 4. Let X be an arbitrary word on the .s, jugate in G to q if and only if X = 1 in H.

Then q X is con-

From Lemmas 3 and 4, since H has unsolvable WP, it follows that: Theorem. The j:p. residually finite group G has unsolvable CP. Proof of Lemma 4. Assume that q X is conjugate in G t o q . Define G H t o be the epimorphism obtained by mapping q +. 1, ti + 1 , di 1 , s, -+ .s, Then 1 = cp(q) and hence cp(qX) = 1 in H . But cp(qX) = cp(q).p(X) = cp(X) = X = 1 in H as claimed. Because of notational complexity, we only sketch the proof that if X = 1 in H then q is conjugate t o q X in H. First notice that the ti and di commute with the .s, Let Y , Z be words on the s, and consider YqZ. Conjugating YqZ by ti gives YqRiZ. Thus conjugating YqZ by ti corresponds to inserting Ri between Y and Z in the formal product Y Z , i.e. YZ becomes YRiZ. Now conjugating YqZ by d, gives Ys;' qs,Z. This corresponds to inserting si's, in the formal product Y Z t o get Ys;'s,Z. The q keeps track (in G ) of the position at which we are operating in the formal product. Finally, conjugating Ys,qZ by d, gives Ys,s;' ys,Z = Yqs,Z. Thus, by conjugations with d,, the position of the marking symbol 4 can be changed. Since X = 1 in H , X is freely equal to a product of conjugates of the Ri. Thus X can be built up from the empty word by inserting inverse pairs of generators and inserting Rf' , and finally keep freely reducing (after all insertions). But as argued above this corresponds cp:

-+

-+

Ch.F.Miller, Decision problems in algebraic classes of groups

521

to conjugating q by some sequence of ti and da t o give q X . This completes the sketch of the proof. Finally we remark that the GWP the subgroup of G generated by T and y is unsolvable. Moreover, using the fact that T acting on F by conjugation induces an automorphism of F , it can be shown that the GWP for Aut(F) is unsolvable.

5. Remarks and problems

‘The above discussion leaves as open questions the status of certain decision problems in several algebraic classes of groups. Here we consider some additional open problems and make some remarks related to the ibove. As noted in Proposition 1, “being finitely presented and having solvable WP” is a poly-property. The group G of Section 4 shows that “being f.p. and having solvable CP” is not a poly-property. Having solvable WP is hereditary in the sense that a f.p. subgroup of a group with solvable WP again has solvable WP. This is false for the CP. By modifying the example of Section 4, Miller [ 161 has constructed a f.p. residually finite group with solvable CP which contains a f.p. subgroup having unsolvable CP. Related to this is the following:

Problem 1. (B.H.Neumann, Schupp, Miller). Let G be a f.p. group and H a subgroup of finite index in G. Is the conjugacy problem for G (Turing) equivalent to the conjugacy problem for H? If not, is one always reducible to the other? It would suffice to prove the equivalence for H also normal. For suppose H has index n in G. Let K be the intersection of all subgroups of index n in G. Then K is contained in H. characteristic in G and normal in H , and has finite index in both.

Problem 2. Does there exist a f.p. group G such that (1 ) G has unsolvable WP and (2) G satisfies a non-trivial identity?

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Ch.F.Miller, Decision problems in algebraic classes of groups

Note that identities are iiot allowed in defining G - only finitely many relations. For f.g. recursively presented groups, the answer is affirmative. An example may be easily derived from a group constructed in P. Hall [ 91 . For almost all of the unsolvability results given in Tables 1 and 2, an “arbitrary r.e. degree result” has been proved, in the sense that for suitable choice of the groups involved, the decision problem has preassigned r.e. degree of unsolvability.

Problem 3. (Boone, Collins, Miller). Given two r.e. degrees D,, D 2 such that D, G D,, does there exist a f.p. group G with WP of degree D, and CP of degree D2? For f.g. recursively presented groups Miller [ 161 has shown the answer is affirmative. Collins has made some progress on the f.p. case, but the problem remains open. For D, = 0, the answer is affirmative and due to Collins [71. A d d d in p r o o f : Collins [ 26 J has now given an affirmative answer to Problem 3.

This survey has omitted several important topics - among them me tabelian groups, single relator groups, and groups satisfying a cancellation condition. Fortunately, the latter are discussed by Scliupp [ 231 in a survey in this volume. References [ 11 L.Auslander, On a problem of Philip Hall, Ann. of Math. 86 (1967) 112-116. [ 21 G.Baumslag, A remark on generalized free products, Proc. of Amer. Math. Soc. 13

(1962) 53-54.

[ 31 G.Baumslag, Automorphism groups of residually finite groups, Journal London

Math. Soc. 38 (1963) 117-118. 141 G.Baumslag, Finitely presented groups, Proc. Internat. Conf. Theory of Groups, Australian National University, Canberra, August, 1965, pp. 37-50, Gordon and Breach, New York, 1967. [ 5 ] N.Blackburn, Conjugacy in nilpotent goups, Proc. of Amer. Math. Soc. 16 (1965) 143-148.

Ch.RMiller, Decision problems in algebraic classes of groups

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(61 W.W.Boone, The word problem, Ann. of Math. 70 (1959) 207-265. [ 71 D.J.Collins, Recursively enumerable degrees and the conjugacy problem, Acta mathernatica 1.22 (1969) 115-160. [ 8 ] V.H.Dyson, The word problem and residually finite gorups, Notices of Amer. Math. SOC.11 (1964) 743. [ 9 J P.Hall, Finiteness conditions for soluble groups, Proc. London Math. SOC.(3) 4 (1954) 419-436. [ 101 G.Higman, B.H.Neumann, and H.Neumann, Embedding theorems for groups, Journal London Math. SOC.24 (1949) 247-254. [ 111 A.G.Kurosh, The theory of groups, Vols. I and I1 (Chelsea, New York, 1956). [ 121 W.Magnus, A.Karrass and D.Solitar, Combinatorial group theory (Wiley, New York, 1966). [ 13 J A.I.Malcev, On some classes of infinite solvable groups, A.M.S. Translations, series 2, vol. 2 (1956). [ 141 K.A.Mihailova, The occurrence problem for direct products of groups, Dokl. Akad. Nauk SSSR 119 (1958) 1103-1105 (Russian). [15] C.F.Miller, 111, On Britton’s theorem A, Proc. of Amer. Math. SOC.19 (1968) 1151-1154. [ 161 C.F.Miller, 111, On group-theoretic decision problems and their classification, Annals of Math. Studies, No. 68, Princeton, Princeton University Press, 197 1. [ 171 C.F.MiUer, 111, Unsolvable problems in direct products of free groups and in unimodular groups, unpublished. [ 181 C.F. Miller, 111 and P.E. Schupp, Embedding into hopfian groups, Journal of Algebra, 17 (1971), pp. 171-176. [ 191 A.W.Mostowski, On the decidability of some problems in special classes of groups, Fund. Math. LIX (1966) 123-135. [20] B.H.Neumann, An essay on free products of groups with amalgamations, Phil. Trans. Royal SOC.of London, No. 919, 246 (1954) 503-554. [21] H.Neumann, Varieties of groups, Ergebnisse der Mathematik und ihrer Grenzgebiete 37 (Springer, Berlin, 1967). [22] J.J.Rotman, The theory of groups (Allyn and Bacon, Boston, 1965). [23] P.E.Schupp, A survey of small cancellation theory, this volume. [24] K.H.Toh, Problems concerning residual finiteness in nilpotent groups, to be published. [ 251 V.N. Remeslennikov, Conjugacy in polycyclic groups, Algebrai i Logika, Seminar, Vol. 8, NO. 6 (1969) 712-725. [26] D.J. Collins, Representation of Turing reducibility by word and conjugacy problems in finitely presented groups, to appear in Acta Mathematica.

UNIFORM ALGORITHMS FOR DECIDING GROUP-THEORETIC PROBLEhlS A. Wlodzimierz MOSTOWSKI Mathematical Institute of the University of Gdahsk

5 1. Introduction This paper is devoted to an investigation of the concept of a uniform algorithm for deciding group-theoretic problems, for any group-theoretical predicate with a fixed number of variables, and invariant under isomorphism. Many problems such as the word problem, conjugacy problem and others are connected with such predicates, cf. Schupp [ 121. F o r the word problem (w.P.), uniform algorithms were investigated in Boone and Rogers [ 3 I . Collins [ 4 ] and also in Mostowski [ l o ] , but in the last paper the notion was not explicitly mentioned. Given a recursively enumerable (r.e.) sequence

of finite presentations, a uniform algorithm is a decision method applicable to any member of (1). Preliminaries are in Section 2. A precise definition of a uniform algorithm, in the language of recursive functions, is given in Section 3. Some reasons for limiting the notion to r.e. sets of presentations are explained. The main Theorem 5 concerning extensions of uniform algorithms to the closure of (1 ) under isomorphism is given in Section 4. The proof is based on Rabin [ 1 1 ] . Applications are given in Section 5 . The extension of the theory to predicates meaningful for any finite number of variables is given in Section 6. An example of a problem described by such a predicate is 525

5 26

A . W.Mostowski, Uniform algorithms

Magnus’ extended word problem. A method of treating the isomorphism problem is suggested in Theorem 9 of Section 7. The last section, viz. 8, is devoted t o a relativisation of the theory to the decidability problem with some oracle. Therein is t o be found Theorem 1 1, in which we prove that grouptheoretic problems for isomorphic presentations have the same degree of unsolvability.

$2. Preliminaries We shall deal with a fixed set S = (sl, s2, ...) of symbols and an effective enumeration of the free group F ( S ) . For this fixed enumeration, we shall denote by 7 a number of 7 E F ( S ) , and conversely by a 7 E F ( S ) such that 7 = k . A finite presentation of a group is a pair P = ( X ;R ) where the set X of generators is a finite subset of S , and the set R of relations is a finite subset of F ( X ) . We shall deal with a fixed Godel enumeration of the set P of all finite presentations. The number of P E P will be denoted by n(P). We shall use the notations P = Pm and n ( P ) = rn as well. An rz-variable predicate r ( x , , ..., x n ) where n = 0, 1 , 2 , ... is a grouptheoretical predicate iff: for any group G and g , , .. . , g, E G

r(g , , . .. , glI) is true or false and r is invariant under any isomorphism of G. The decision problem for r relative t o a class K c P is to decide for any P = ( X ;R ) E K and any T , , ..., 7 , E F ( X ) whether r ( g l , ..., 8,) is true or false,

where g , , ..., g,, are elements of the group G ( P ) presented by the words T ~...,, 7,. Note that because of the invariancy of under isomorphism the problem is well posed, i.e., is independent o n any isomorphic copy of a group G(P>presented by P. Now let us define a new predicate r P ( T 1 , ..., T ~ which ) is partial and connected with r. For P = ( X ; R ) E P, and T ~...,, 7, E F ( X )

527

A . W. Mostowski, Uniform algorithms

presenting elements g,, ..., g, of a group G ( P ) with a presentation P, we define

If for some i = 1 , 2 , ..., n , T~ 4 F ( X ) , i.e., 7 i contains a letter si or s-l not belonging to X , then r p ( T 1 , ..., 7,) is undefined. i Since we can effectively decide for any P = ( X ;R ) E P and any T,, ..., 7, E F ( S ) whether or not some of T ~..., , 7, contains a letter o u t of X , then the decision problem for I? is the same as the decision problem whether or not r p ( T 1 , ..., 7,) is true o r false. So the decidability of r is equivalent to the computability of the following functionf(rn, k , , ..., k,) on integers:

f ( n z , k l , ..., k,) =

for

rp ( E l , ..., k,) m

-

[

false r::efined.

$3. Decision functions and uniform algorithms In the literature, see [ 3 , 4 ] , a notion of uniform algorithm for deciding a problem for some recursively enumerable set C C P is used. The idea of the notion is, that having effectively listed the presentations of C as (Pa,, Pa , ...), we need one effective method Pa*,and so on. for deciding the problem for For some interesting classes of groups it is not known whether or not the sets of all finite presentations of these groups are r.e. sets. That is, nothing is known about the recursive enumerability of the following sets:

R , all finite presentations of residually finite groups S , all finite presentations of simple nontrivial groups. For both of these classes it is known that there is a uniform method of solving the word problem. For residually finite groups cf., e.g., [ 101 ; for simple nontrivial groups see the known method

A . W. Mostowski, Uniform algorithms

528

presented in Section 4. According to the supposition that both R and S are neither E1 nor n1 in Kleene-Mostowski hierarchy, both R and S are o u t of the notion of uniform algorithm for the w.p. despite the existence of uniform methods of solving the w.p. relative t o this classes. We shall look for a notion more extensive than the notion of uniform algorithm, t o cover both these cases. It is given in the following definition based on our preliminaries. Definition 1. A partial recursive function f ( m ,k , , ..., k,) is called a decision function for a grouptheoretical predicate r ( x l ,..., x n ) relative t o a class K C P iff for any integer rn such that P, E K, (a) the function f , ( k , , ._.,k,) =f(rn, k , , ..., k,) is total; (b) moreover,

for

rpm(E,, ..., E n )

defined and

false;

(El, ..., En) undefined, the value f m ( k , , ..., k,) can be (c) for rPm any integer different from 0 and 1. Note that the definition can be inessentially weakened since, according to our preliminaries, (a)-(c) can be replaced by the following: for any integer rn such that Pm E K , (d) for

rpnz(zl, ..., k,z) -

being

false

the value

Using the definition of a decision function, we can give a following precise definition of a uniform algorithm.

A . W.Mostowski, Uniform algorithms

5 29

Definition 2. A function f ( i ,m l , ..., m,) is a uniform algorithm for deciding r ( x l ,..., x,) relative to a r.e. set C C P iff there exists a decision function f ‘(k,m l ,..., m,) for r relative to C such that for some enumeration a(i) of C f ( i , m l , ..., m,) = f ‘(a(i),m l , ..., m,) for i = 1 , 2,...

From the definition it follows that f is a computable function. Thus, from the definition, we have the following.

Theorem 1 . I f a decision function f ‘(k,ml , ... , m,) for r relative to K C P exists, then for any r.e. subset I: of K there exists a uniform algorithm for deciding r relative to C . Proof. Let a ( i ) be an enumeration of C , the f ( i ,m l , ..., m,) = = f ‘(a(i),m l , ..., m,) is the desired uniform algorithm. Now we shall prove the following.

Theorem 2. Let f ( i ,m l , ..., m,) and h(s,m l , ..., m,) be two uniform algorithms for deciding r relative to a r.e. set, then there exists a computable function t(s)such that f ( i , m l ,..., m,) = h(t(i),m l , ..., m,). Proof. Let a(i)and p(s) be two enumerations of 2 and f ‘(k,m l , ..., m,) and h’(k,m l , ..., m,) be two decision functions relative to C such that f ( i ,m l , ..., m,) = f ’(a(i),m l , ..., m,)

for i = 1,2, ...

h(s,ml,..., m,) =h’(P(s),m l , ..., m,)

for s = I , 2, ... .

We have for k

E

n(C)

f ’ ( k , m l , ..., m,) = h‘(k,m l , ..., m,)

with both sides defined.

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A. W.Mostowski, Uniform algorithms

The function to ( k ) is defined as follows: to ( k ) = minimum i such that p(i) = k is a partial recursive function. The function t ( x ) = t o ( a ( x ) )is recursive and total, so computable. , m,) for Now h(to(I),m , , ..., m k ) = h‘(Z,m l , ..., m,) = f ’ ( l , m l ..., any I E n ( C ) , according t o P(to(l)= 1. Then h ( t ( i ) ,m l , ..., m,) = m l , _..,m,) =h’(a(i),m l , ..., m,) = f ‘ ( a ( i )m , , , ..., m,) = = h(to(a(i)), = f ( i , n z l ..., , m,).This proves that t ( x ) has the desired property. Now we shall prove:

Theorem 3. if there exists uniform algorithms f l and f 2 for deciding r relatively to r.e. subsets C, and C 2 o f P, then there exists a uniform algorithm fo for deciding relative to Co = C , U Z2. Proof. Let a , be enumeration of C1 and C, such that for suitable decision functions and fi

fi

The function:

S(i) =

a ( j ) for i = 2j B ( j ) for i = 2 j + 1

is an enumeration for 2 , = C , u C2. Note that for k such that k = S ( 2 j ) = 6(2j‘+ 1 ) for some integers j , j ‘ , .f;( k ,m l , ..., m,) = f . ( k , m , ,..., m,) so the function

f i ( k , m l , ..., m,) fi(k,m l , _ _m,) _, = fi ( k , m,,..., m,)

for k = S ( 2 j ) for k = 6 ( 2 j ’ + 1)

is a well-determined partial recursive function. Evidently, f d is a decision function relative t o C,. Now

is the desired uniform algorithm for 2,

A . W.Mostowski, Uniform algorithm

531

From Theorems 1 and 3 we obtain immediately.

Corollary. The family of subsets o f P with uniform algorithms for r forms an ideal of the lattice of all r.e. subsets of p. The following example due t o V. Dyson explains difficulties connected with the notion of a decision function, i.e., a decision process relative t o a class K which is not a recursively enumerable set of presentations. The example shows that decision processes for two classes K1 and K2 of presentations can give different results when used for a PE K1 u K2 even in case they both terminate. Let K1 be the class R of all finite presentations of residually finite groups. The decision process described in Mostowski [ 101 for deciding the word problem is as follows: R ) , list all elements For P = (X;

of the normal subgroup N R generated by R in F ( X ) . Similarly list all homomorphisms

of the group G(P) onto finite groups, and for a T E F ( X ) look for p1( r ) ,p2( r ) ,... step by step. Then either r is in a sequence ( 1 ) and then r = 1 in G(P) or for some i pi(r)# 1 , and then T # 1 in G(P). For details, see [ 101 . Let K 2 be the class S of nontrivial simple groups. The decision process goes as follows: For P = ( X ; R )list all elements (1) of the normal subgroup NR Similarly list all presentations

of trivial groups. For a r E F ( X ) , either T E NR and is in the sequence ( 1 ). That is

532

A . W.Mostowski, Uniform algorithms

1 in G(P). Or in the case T # 1 in G ( P ) , the presentation P' = ( X ;R , T ) is a presentation of a trivial group (since G is simple). Then P' is in the sequence (2). For a presentation P of a trivial group, the first process always gives the answer T = 1 for any T E F ( X ) . The second process can give the wrong answer T # 1, for some T . That is the case when we first assert P' as being in the sequence (2), before asserting that T is in the sequence (1). case

T =

54. Extensions of uniform algorithms

In Rabin's paper [ 1 1 ] it is proved that there exists a (total) recursive function e(n, k ) such that in the sequence

there are all finite presentations of groups isomorphic t o a group presented by P, E P.Denote b y C * a set obtained from C C P by adding all prescntations Q E P isomorphic t o some P E C. Then if C is a r.e. set and a ( i ) an enumeration for C , then for an enumeration c(i,j ) of pairs of integers, and projection functions I ( k ) and r ( k ) such that li = c(Z(k),r ( k ) ) the function

is an enumeration of Z * . Thus C * is a r.e. set when C is a r.e. set. Now let P' = ( S ' ;R ' ) and P" = ( S " ;R " ) be two finite presentations. A mapping $ of S' into F(S") such that the extension \Ilo of $ t o a homomorphism induces an isomorphism i of G ( P ' ) with G ( P " ) such that $ o ( ~ )= i(7) in G ( P " ) , for any T E F ( S ' ) , which we shall call an isomorphism of pi-esentations. Note that the definition is independent of the choice of G ( P ' ) and G ( P " ) . In fact, let i, : G ( P ' ) + G1 and i,: G ( P " ) + G, be t w o isomorphisms and I)~(T= ) i(T) where $ o ( ~ )is an element of G ( P " ) and T in i(7) is an . i 2 i i;' is an element of G ( P ' ) ,then i, $ o ( ~ )= i 2 i i i l ( i l ( T ) )Now isomorphism of G, + G,. And ~ , ( I ) ~ ( T )is) an element $ o ( ~ )in G,,

A. W.Mostowski, Uniform algorithms

533

and i, (7)is an element r in G I . So G0(r)= i,i i f l ( 7 )for $ o ( ~ )in G, and 7 in G,. Now we shall prove the following. Theorem 4. There exists a family $ n , k of isomorphisms, each isomorphism between Pn and Peen , k ) which is described eifectively; strictly speaking, there exists a (total)recursive function t(n, k, m ) such that

for r being a word in the letters of the presentation P, Proof. Let T be a pair ( s i , 0 )where si E S and w E F ( S ) . For any finite presentation Q = ( X ; R ) and any pair T = ( s j , w ), define a presentation Q ( T ) as

(

( X , si; R , siw-l)

QV)=

for si 4 X and w E F ( X ) for either sj E X or w fl!F ( X ) .

Let E be an equation, i.e. an expression of the form

*

where n > 0, ei = I , r , , ..., T,, w E F ( S ) and y , , ..., y , are some letters out of S . For a finite presentation Q = ( X ;R ) and an equation E of the form (1 ) we define a new presentation

Q ( E )=

( X ;R , w ) in case r , , _..,7, E F ( X ) and for some r , , ..., rn E R ~;'rp1T~ ... 7 i 1 r: T~~= w in F ( x ) ,

Q A sequence

in the opposite case.

A . W. Mostowski, Uniform algorithms

534

where A , , ._.,A,yare finite presentations and each B,, ..., B,-l is either a pair or an equation, is called a proofofisomorphy iff for i = 0, ..., s - 1, either

According t o Tietze's theorem, for P, Q E P, the groups G ( P ) and G(Q) are isomorphic iff there exists a proof of isomorphy r such that P = A and Q = A , . All sequences (2) can be effectively enumerated and for any sequence r it can be effectively checked whether or not r is a proof of isomorphy owing t o the finiteness of all presentations A , , ._.A , , in r. Thus all proofs of isomorphy can be effectively enumerated as

The above investigations are completely Rabin's. The function

d r z , k ) is defined by him as follows: For rrC = ( A , , B,, ..., A , ) , c(n, k ) =

(

; ( A , ) if n = r z ( ~ , ) if n # n ( A o )

Now we wish t o define for e(n, k ) a family of isomorphisms and a function t(n, k , i n ) . It does not matter whether we define a family of $ n , k or $ 0 , 2 , k . First for any proof of isomorphism I?, we define a homomorphism $,. For r = ( A , , B . A ) being a proof of isomorphy, we define a homomorphism $, and thereby an isomorphism $ of A , and A as follows: I f B is an equation, $, is the identity: $ o ( ~ )= T. If U is a pair ( &s,, c3 ) and F a proof of isomorphy, such that is the identity. I f A ( B ) = A , , two cases are t o be A,(B) = A d ist i ngu ished : ( 1 ) A , = A (Lz) = A then $, is an identity.

I),

A . W. Mostowski, Uniform algorithms

535

(2) A , = A ( B ) # A 1, then we delete the letter sj from A , containing A 1. In this case $, is defined as follows: for 7 = 7(x1, ..., x,, S j ) ,

= T(X1, . . . J n , d q ? - . , X , l ) ) .

$,(TI

Then for a proof of isomorphy r = ( A , , B,, ..., A , ) the homomorphism $o(!?) is a product of homomorphisms for proofs of isomorphy ( A o , B , , A l ),..., ( A s - l , B s - l A s ) . Then $,(r)induces an isomorphisms i of groups G ( A , ) and G ( A , ) such that $,(I')(T) = i(7).For rk = ( A , , B,, ..., A , ), $(n, k ) is defined as follows:

Then $(n, k ) is an isomorphism of Pn and Pe(, k ) . To determine the function t(n, k, r n ) such that

for rn such that iii E F ( X ) where Pn = ( X ;R ) we shall first notice that the function leading from a triple x,y,j of natural numbers t o a number n of a word obtained from X by the substitution of Y to a letter s j , is a total recursive function. Now let all pairs and equations be effectively enumerated as

We shall define a function

Z(X,

rn) as

for T , being an equation, for Tm being a pair ( s j , w and in X = 7(x1, ..., x,), all xl, ..., x, different from sj. 7(x1, ... , x,, o) for T , being a pair ( s j , w ) and 3- = T ( X 1 , .. . , X,*,Sj).

X X

~~~

According t o the definition z(x, rn) is total recursive.

A. W.Mostowski,Uniform algorithms

536

From the definition of $, for a proof of isomorphy I? = ( A,, T,, A ) it follows exactly that

for x such that

X E F ( X ) where A ,

rk= ( A,,13,,...,As)

wedefine

=(

X ;R ) . Now for

The function t(n, k , x) is total recursive and

where A , = ( X ;R ) . The proof of the theorem is thereby finished Now we return to uniform algorithms. Suppose that for a predicate I'(x,, ..., x n ) there exists an uniform algorithni f(i, m l , ..., m,) relative t o a r.e. set 2. Suppose a ( i ) is an enumeration for C. Then the set C * is r.e. and suppose a*(s) is s its enumeration. O u r aim is to define an uniform algorithm f * ( s , I , , _ _ill) _ , relative to C". When definingj"k(s,1,, _..,I n ) , two cases must be distinguished. Suppose Pa*(s) = ( S ' ;R ' ) . The first case: Some TI, _ _ _ contains , a letter sp 4 S ' . We define

Cl

f'ys, I,,

_

_

_

In>

)

=

2

(or any fixed integer different from 0 and 1 ). The second case: All letters i n ..., 7, are in S ' . Then we find i, s,t n l ,..., m, such that

4,

d(S)

(3)

= e(a(i),.x)

t(a(i),x, nz,) = li for j = I , ..., ti f'(i,i n l ,

., m t l )is

either 0 or 1

A . W.Mostowski, Uniform algorithms

537

and define

To show the correctness of the definition it is to be proven that m;,..., mk such i, x, m l , ..., m, exist, and that for any other i', such that ( 3 ) , the equality XI,

holds. E C", there exists Py E C isomorphic t o P a t ( s ) so for Since Pa*(sj some i, j , e ( y , 1 ) = a*(s) and a( i ) = y . Now we consider the ]somorphism $o o l ( i ) , x .When the second case of the definition takes place, then I,, ..., I, are in letters of Pa*($)and some words r l , ..., r, in letters of P a ( i )exist such that

$o a ( i ) , x ( r=? j)

for j = 1, ..., n.

-

For m l =7,,..., m, = r,, we have

t ( a ( i ) ,x, mi)= Zi. Since rP ( r l ,..., 7), is determined, the values of f ( i , m l , ..., m,) 40 are zeros and ones. Now suppose that for some i', m i , ..., 7n; the equalities (3) are satisfied. Then according to e ( a ( i ) ,x) = a * ( s ) = e ( a ( i ' ) ,x') we have isomorphisms $, = $o a ( i ) , x , $2 = $o a ( i ) , x onto Pa*(!)such that $o a ( i ) , x (mi) and $o a ( i ' ) , x (mi)= Zi. Then $5' , ) I is an iso( H i ) = Ei for morphism of Pa(i) and Pa(i,, and moreover $5, j = 1, ..., m. According t o the invariance of r we have XI,

=q

This proves the equality (4). The function f'*(s, 1, , ..., 1,) is well determined and by definition is a computable function.

A . W. Mostowski, Uniform algorithms

538

Now according t o the definitions of uniform algorithm and decision function, the following holds:

f ( i , m i ,..., m,)

0 =

but due t o the invariance of

, false I'~,aci,(El ,..., m,) true -

1

iff

rp . (ml, ..., %,) 4 1 )

false undefined;

r under the isomorphism Q0

undefined

iff

rp

ff*(S)

false (11, ..., l n ) true undefined.

This proves that f * ( s , I , ,..., I,) is a uniform algorithm for deciding r relatively t o C*. So we have proved the following main theorem in this section.

Theorem 5. I f for a predicate I? there exists a uniform algorithm f relative to a r.e. set I: then there exists a uniform algorithm f * for r relative to a necessarily r.e. closure I:* of Z under isomorph-

ism.

Note: When we deal with a decision function relative to a set K C P which is not r.e., the question of whether o r not the decision function relative t o K * exists is open. In the proof of the latter theorem the recursive enumerability of C was used 'very strongly. $5. Applications of the main theorem to the word problem and similar problems

In the sequel we shall give some applications of Theorem 5 . First we consider applications t o the word problem. We obtain the word problem when we set the predicate I' as

r(x,)iff

x l = 1.

A. W. Mostowski, Uniform algorithms

539

For the word problem, the set D of all finite presentations with decidable word problem is neither a r.e. set nor the complement of a r.e. set. See Boone and Rogers [ 3 ] .It is also proved there that there exists a r.e. set C C D such that n o uniform algorithm for the w.p. relative t o C exists. From this it follows that n o decision function for the w.p. relative to D exists. For some interesting classes of presentations such as R (residually finite groups) and S (nontrivial simple groups), decision functions for the w.p. exist; cf. the investigations of Section 3. Note that for both R and S it is not known whether or not they are r.e. or complements of r.e. sets. Now we shall give some examples of r.e. sets with uniform algorithms for the word problew Let C C D be a r.e. set with uniform algorithm. Then according t o Theorem 5 the uniform algorithm for C * exists. The first interesting case is when C = ( P ) , i.e., C consists of one given presentation with decidable w.p. Then C * is the class of all finite presentations of the group G ( P ) . The result can be stated that when P has decidable word problem, then there is one uniform method t o decide the word problem for any presentation of G ( P ) . The second interesting case is the class M of all finite presentations with a single defining relation. It is known (see Magnus [61) that there exists a uniform algorithm for deciding the word problem relative to the (evidently r.e.) set M. The class M * is the class of all finite presentations of groups which can be defined by a single defining relation. Then according t o Theorem 5, there exists a uniform algorithm for deciding the w.p. relative t o M*. The third case is the class F of all finite presentations of finite groups. It is known that there exists an enumeration a ( i ) such that in the sequence

there is exactly one presentation of each finite group (meaning different groups as nonisomorphic). In Mostowski [ 101 Eq. (1 ) is given as the set of group tables. Then Z * = F is an r.e. set. From Theorem 5 follows the well-known result that the uniform algorithm relative t o F* exists.

540

A . W.Mostowski, Uniform algorithm

The fourth application is to nilpotent groups. Let N, be the class of all finite presentations of nilpotent groups of nilpotency < c, and N = U:=l N, be the class of all finite presentations of nilpotent groups. We shall prove the following.

Lemma. The sets N and N, f o r c = 1 , 2 , ... are r.e. sets. Proof. A group G = {gl,..., g n } is nilpotent of nilpotency E (1, ..., n ) for any a l , ...,

< c , iff

w h e r e ( x l , X 2 ) = x ~ ' x ~ 1 x 1(xx2l , ..., X ~ , X ~ + ~ ) = (,..., ( X X~ ~ ) , X See, e.g., Mostowski [ 9 ] , Lemma 3 . Now for a presentation P = (sl, ..., sn;R ) of a group we. form a new presentation P(') by adding new relations

~ + ~ ) .

Then P(') is a presentation of a nilpotent group of nilpotency < c. The function P P(') maps P into N,. Moreover, P E N, iff P and P(,)are presentations of isomorphic groups. I t is easy t o see that the two argument function -+

g ( k , c): ik

-+

n(Pf))

is a total recursive function. This proves that P (') and U:=l r.e. sets. So N, = P(')* and N = (U:=l P ('))* are r.e. sets.

P(')

are

The proof of the lemma is finished. Since N , N, C R c = 1, 2, ... according to the lemma just proved and t o Theorem I , there exist uniform algorithms for deciding the w.p. relative t o N and N, for c = 1 , 2 , ... . The next example is an application of Theorem 5 t o the conjugacy problem, i.e., a problem for a predicate

I'(xl, x 2 ) iff there exists z such that zxl = x 2 z .

A . W. Mostowski, Uniform algorithm

54 1

It is known that for the conjugacy problem there exists a decision function relative the class C of all finite presentations of conjugacy separable groups. It is known (see Blackburn [ 21 ) that N C C. Owing to the recursive enumerability of N, there exists a uniform algorithm for the conjugacy problem relative t o N. All that we can obtain as a corollary t o Theorem 5 is the following: If there exists a uniform algorithm for the conjugacy problem relative t o an r.e. set set C , then there exists a unfiorm algorithm relative to the set C*. There is a long list of other applications of Theorem 5. We shall describe a general idea. Let A be an algebraic property, i.e., a property of groups shared by isomorphisms. The first case: Define a zero argument predicate I'as r true for a group having A , r false for a group having non A . The interesting examples of A are: triviality, finiteness, abelianity, nilpotency, solvability, having decidable word problem, and so on. A very long list of such properties is described in [ 1 1 . The other case is when we define an n-argument predicate r as follows

r ( g , , ..., g n ) iff an n-generator subgroup { g l ,..., gn} of the group G has a property A . For both cases, Theorem 5 can be restated as the following.

Theorem 6 . I f there exists a uniform algorithm f o r deciding whether the group has an algebraic property A , relative t o a r.e. set C o f finite presentations, then there exists a uniform algorithm relative to C *. I f there exists a uniform algorithm f o r deciding whether an n-generator subgroup o f the group has a property A relative t o a r. e. set E, then there exists a uniform algorithm relative t o C*. But note that all except the last of the examples of the properties A listed above, are strong Markov properties, hence there exists a r.e. set C (depending of A ) such that there is n o uniform algorithm for deciding whether the group has the property A relative t o 2 . For details see Collins [ 4 ] .

542

A . W. Mostowski, Uniform algorithm

Other possible applications of the second part of Theorem 6 arise when we take as A the following subgroup properties: being of finite index, being normal, being isomorphic with a given group, being a finitely related group, and so on; cf. also [ 1 1 . At the end of this section we shall mention the inclusion problem, sometimes called Magnus’ extended word problem. For a single presentation P = ( X ;R ) , there are in fact three inclusion problems; some of them surely different; c.f. Michailowa [71 and [81: ( 1 ) Weak inclirsion problem: For a given T ~...,, 7, E F ( X ) decide for any T , + ~E F ( X ) whether o r not

(2) n-generutor inclzision problem. The n is fixed. The problem is t o decide for any T ~...,, T , , T , + ~E F ( X ) whether or not

( 3 ) The inclztsion problem (the strong inclusion problem). This is the union of all n-generator inclusion problems for all n. The weak inclusion problem does not make sense for a class of presentations. The n-generators inclusion problem allows a treatment by our theory, since the n + 1 variable predicate r

r(.xl,..., x , , + ~iff )

E

{xl, _ _x_n,} in G ( P )

is algebraic. So frorn Theorem 5 we obtain a corollary.

Corollary. I f there exists u unijbrm algorithm .for solving the n-gencrutor inclzision problem relative to a r. e. set C then the unijorm ulgorithm relative to C * exists.

For a discussion of the (strong) inclusion problem see the next section.

A . W. Mostowski, Uniform algorithm

543

56. Families of predicates

In the previous section, we were dealing with decision problems such as: the n-generator subgroup has algebraic property A or the ( n - 1 )-generator inclusion problem. But these problems can be treated in another sense as follows: a finitely generated subgroup has an algebraic property A or a strong inclusion problem. In both cases we have a family r n ,n = 1 , 2 , 3 , ... of algebraic predicates rn = rn(xl, ..., x n ) such that

rn( X I , ..., x n )

iff

{ x l , ..., x n } has a property A

or

and we wish t o decide the problems for all predicates from the family rnrelative t o the same class K of finite presentations. This can be translated into the language of recursive functions as follows: Let us denote by II, ( X ) a set of n-tuples

u = ( T ~...,, T , ) where

T

~..., , T, E

F(X)

for X C S put m

n(x)= u

n=l

rI,JX).

We fix some Godel enumeration of the set n(S)and denote by U a number of z i E TI(S), and for a natural number k , we denote by K an n(k)-tuple ti E n(S) such that i 7 = k. For a family rn = rn(xl, ..., x n ) of algebraic predicates, define a new partial predicate r p ( u )where u E II(S)and P = ( X ; R ) is a presentation P c P as follows. For u = ( -rl, ... , 7, ) E II, ( X ) ,

rP(u)iff

r , l ( T 1 , ..., T , ~ )

in the group G ( P )

A. W.Mostowski, Uniform algorithm

544

and for u E n ( S ) \ n ( X ) , the predicate r P ( u )i:i undefined. Then a notion of a decision function and a.uniform algorithm can be extended t o the family rn as follows: Definition 1'. A partial recursive function f ( m ,k ) is called a decin = I , 2, ..., of predicates relatively t o sion function for a family ril: a class K c P iff for any integer rn such that Pm E K

( d ' ) for

rpm(E) being

false

the value f ( m ,k ) =

(A.

Definition 2'. A function f ( i , m ) is a uniform algorithm for deciding a family r',?;n = I , 2, ..., of algebraic predicates relative t o a r.e. set C C P iff there exists a decision function f ' ( k ,rn) for the family such that for some enumeration a ( i ) of C

Theorems 1 , 2 , and 3 can be easily proved for a family rn of algebraic predicates (instead of for an algebraic predicate). The analog of the main Theorem 5 can be proved also. Theorem 5'. I f for a f a m i l y rn,n = 1 , 2 , ..., oj'ulgebraic predicates there exists u m i f o r m algorithm frelutivelv to a r.e. set C c P, then there exists a irnifbrm algorithm f * f o r the jamily, relative to the necessurilv r.e. closure C * o f C under isomorphism. The proof needs only small technical changes in the proof of Theorern 5 , given in Section 4. As an application of Theorem 5' or families ( 1 ) we obtain immediately the following corollaries. Corollary. If there e.uists u iitiiform algorithm for deciding whether a finitely generated siibgrozip has at1 algebraic property A , relatively t o u r.e. set C C P, then there exists a uniform algorithm relative to C * .

A. W.Mostowski,Uniform algorithm

545

Corollary. I f there exists a uniform algorithm for deciding the (strong) inclusion problem relative to a r. e. set C C P , then there exists a uniform algorithm relative t o C *,

8 7. Isomorphism problem Let G be a fixed group. Define a zero argument predicate follows. For a P E P defining a group isomorphic t o G

r as

r is true and for a P E P defining a group nonisomorphic t o G

r is false. Then evidently J? is algebraic. The problem of deciding r relative t o a class K C P is simply a weak isomorphism problem, i.e. the problem of isomorphism with a fixed group. The decidability of a weak isomorphism means simply the existence of a decision function for r (or a uniform algorithm in the case when K is an r.e. set) relative t o K . So the following theorem holds.

Theorem 7. I f a weak isomorphism problem is decidable relatively to a r.e. set C, then it is decidable relative to the set C.*. The isomorphism problem is the most interesting ; i.e., the problem of whether for presentations P and Q the groups G ( P ) and G ( Q ) are isomorphic. The theory developed above does not fit exactly in this case. We now develop some modifications. First, the decidability of the problem must be stated in terms of recursive functions. Definition. The isomorphism problem is decidable relative t o a class K of presentations if there exists a recursive function f ( n ,k ) such that

546

(1 1 f’(wk ) =

A . W. Mostowski, Uniform algorithms

0 for P,,

Pk E

K and G(P,l)and c ( p k ) non isomorphic

1 for P,,

Pk E

K and G(P,) and

The value of f ( n ,k ) , in the case Pn 4 K or P k fined. Now we shall prove the following

c(pk)

isomorphic.

4 K , need not be de-

Theorem 8. If the isomorphism problem is decidable relative to a r.e, set C o f presentations, then the problem is decidable relative

t o the set C*. For the proof, we need to define a recursive function f ’(n,k ) satisfying ( 1 ) relative t o C*. Let g ( i ) be an enumeration far C, i.e., C = (Pgcl,,Pg(2),...). For P,, Pk E C *, there exists some s, t , i, j such that

Set f ’ ( n , k ) = f ( g ( i ) ,g ( j ) )in case there exist,s, t, i, j such that (2) and undefined in the other case. The function f’(n, k)is well defined. Suppose there exist s’, t‘, i’, j ‘ such that

Then groups G(P,), G(Pgci,)and C(Pg(i,))are isomorphic since G(P,) is isomorphic t o GU‘e(x,y$.Similarly the groups c ( p k ) , G ( P g ( j )and ) G(Pg(j$ are isomorphic. This proves that f(di)> g ( i ) ) = f ( g ( i ’ ) g, ( i ’ ) > . The function f ‘ ( n ,k ) is recursive by definition, and evidently satisfies ( I ). From the theorem just proven immediately follows

Theorem 9. I f there exists a recursive set C o,f finite presentations such that different presentations give nonisomorphic groups, then

.for C * the isomorphism problem is decidable.

A . W.Mostowski, Uniform algorithm

547

Proof. In the case of the theorem the function

satisfies (1 ) relatively to C. Now we shall give some well-known facts which are corollaries from this theorem. Let A , be a set of presentations of the following form

P = (sl,

..., s r ; s y 1 , ..., s f l k , (si,si) for all 1

< i <j < r)

where k < r , and n1 < ... < 17% are all primes or powers of primes. Evidently A , is recursively enumerable; different presentations from A , present nonisomorphic groups. Moreover, since any finitely generated abelian group is a direct sum of cyclic groups, A: is the set of all finite presentations of abelian groups. According t o our theorem, the isomorphism problem for abelian groups is decidable (cf. also Rabiii [ 1 1 ] ). In Mostowski [ 101 it is proved that there exists a recursive sequence C of presentations of finite groups (given by tables) such that different tables present nonisomorphic groups and any finite group has a presentation by tables. From this follows the wellknown result: the isomorphism problem is decidable relative t o finite groups.

5 8. Relativisation and degrees of unsolvability Let F be a set of total functions from integers to integers and T ( F ) be the smallest class of functions containing 0, x + 1, I , (x,, xz) = xl, Z2 (x,, x2) = x2, and the set F , and closed under substitution, recursion scheme and minimization. Note that F , C F2 implies T ( F , ) C .I ( F 2 ) ,and T T ( F ) = T ( F ) . If F is empty, T ( F ) = Tis the class of all (total) recursive functions. If F consists of a single function g , we shall set T ( g ) = T ( F ) . The equality T ( g ) = Tmeans g is total and recursive.

548

A. W. Mostowski, Uniform algorithms

Call T(g) the degree of unsolvability of the function g . The degree of g will be denoted as deg(g) and deg(g) < deg(h) will mean that T(g) C T(h).All total recursive functions have the lowest de= gree of unsolvability. For f being a characteristic fiinction of a set A of integers, d e g ( f ) is simply the degree of unsolvability of A (strictly speaking the degree of unsolvability of the problem x E A ) ; cf. [ 5 ] . Let r(.xl,.._,x n )be an algebraic predicate and E C P a r.e. set with an enumeration a ( i ) . Define a total function f(i, m l , ..., m,) as follows: (a) For Pa(i)= (X; R ) and fil, ..., Ezn E F ( X ) (1) f ( i ,m l , ._.,m,) =

0

r(yM1,..., E n ) false in G(Pacj,)

1

r ( M , , ..., Hi)true in G(Pa($

(b) For some of E l , ..., Win not belonging t o F ( X ) , set

f ( i , m l ,..., in,) = 2. Note that since the set U of ( n + 1)-tuples ( i, m l , ..., m, ) such that some of f i , , ..., %, does not belong t o F ( X ) is a computable set (cf. Section 3), then for any total function g(i, m l , ..., m,) satisfying (a) we have T ( f ) C T(g). From the proof of Theorem 2, it follows that if, for any other enumeration p(i) of the set C we have a total function g satisfying (a), (b), then there exists a function t ( x ) E T such that

This proves that T ( f ) C T(g) and interchanging the roles o f f and g, T(g) C T(f), and therefore T ( f ) = T(g). Then we shall call deg(f), the degree of unsolvability of C for the predicate I' and we shall write

(2)

deg, C = deg f:

A. W. Mostowski, Uniform algorithms

549

For a fixed r, we shall write deg C = deg, C. Now let r = r(xl,..., x n ) be fixed. If f l is a function relative a r.e. set 2 ,

satisfying (a), (b) and C is a r.e. subset of C,. Then there exists a function f satisfying (a), (b) relative to Z such that (cf. proof of Theorem 1 )

For fl and f 2 being functions satisfying (a), (b) relative to r.e. sets Z1 and Z2, there exists a functionfsatisfying (a), (b) relative to the set Z = C, U C2, and such that (cf. proof of Theorem 3)

So we have the following. Theorem 10. The function deg(C) is defined on an ideal of r. e. subsets of P, and moreover C C C1 implies deg C

< deg X I ,

and

(5)

deg(C,) U deg(C2) = deg(Zl u C2).

Proof. From (3) we have T ( f ) G T(fl) and thereby deg(C) <

G deg(Zl). Now from (4)

Now, by relstivisation of the proof of Theorem 5 to T ( f ) (instead of 7 )we obtain the following: If there exists a function f

550

A . W.Mostowski, Uniform algorithms

satisfying (a), (b) relative t o C , then there exists a function f * satisfying (a), (b) relative to a (necessarily r.e.) closure C * of C under isomorphism, and such that f * E T ( f ) . So the theorem.

Theorem 1 1 . For an algebraic predicate r' and any r. e. subset C c P and its closure C * under isomorphism deg, C = deg, C * Proof. Since Z: C Z *, deg, C < deg, C * and since f * E T ( f ) , then T ( f * ) C T ( f ) and thereby deg, Z * < deg, C. The most interesting application of Theorem 11 is a case when C = ( P ) is a class containing a single presentation. Then deg, C is simply the degree of unsolmbility of r for P. Now when Q is a presentation of a group isomorphic with that for P, then (Q)* = Z*, and deg, ( P ) = deg, C * = deg, (Q).

So we have the following Corollary. For an algebraic predicate r all presentations of the same group have the same degree o f unsolvability. As a n interesting special cases of this corollary we obtain the following result:

All presentations of the same group have the same degree of unsolvability of the word problem, the conjugacy problem, and the n-generator inclusion problem. References [ 1 ] G . Baumslag, W.W. Boone and B.H. Neumann, Some unsolvable problems about ele-

ments and subgroups ofgroups, Math. Scand. I ( 1 9 5 9 ) 191-201. [ 2 ] N. Blackburn, Conjugacy in nilpotent groups, Proc. Amer. Math. Soc. (1965) 143-148.

A . W.Mostowski, Uniform algorithms

551

[ 31 W.W. Boone and H. Rogers, On a problem of J.H.C. Whitehead and a problem of Alonzo Church, Math. Scand. 19 (1966) 185-192. [4] D.J. Collins, On recognising properties of groups which have solvable word problem, Archiv d Math. 25 (1970) 31-39. [5] S.C. Kleene and E.L. Post, The uppersemilattice of degrees of unsolvability, Ann. Math. (2) 59 (1954) 379-407. [6] W. Magnus, Die Identhts problem fur Gruppen mit einer definirenden Relationen, Math. Ann. 106 (1932) 295-307. [7] K.A. Mihailova, Inclusion problem for a direct product of groups, D.A.N. 119 NO. 6 (1958) 1103-1105. [8] K.A.Mihailova, Inclusion problem for a free product of groups DAN 127 No. 4 (1959) 746-748. [9] A.W. Mostowski, Computational algorithms for deciding some problems for nilpotent groups, F.M. 59 (1966) 137-152. [ 101 A.W. Mostowski, On decidability of some problems in special classes of groups, F.M. 59 (1966) 123-135. [ 111 M.O. Rabin, Recursive unsolvability in group theoretic problems, Ann. Math. 67 (1958) 172-194. [ 121 P.E. Schupp, A note on recursively enumerable predicates in groups, F.M. LXVI (1969) 61-63.

THE ISOMORPHISM PROBLEM FOR ALGEBRAICALLY CLOSED GROUPS B.H.NEUMANN Vanderbilt University, The University of Cambridge, and The Australian National University

1. Known facts and a conjecture

Let @ be a class of algebraic systems (briefly: algebras), all of the same species; for example the class of groups; and let A be an algebra in @.Consider a set C of sentences made from elements of A , element variables xl, x2, ..., y , z, ..., the algebraic operations of the species of 6 , equality, and the operations of the lower predicate calculus. We call Z consistent over A if A can be embedded in an algebra A’ in 6 such that Z is satisfied, that is to say, all sentences of Z: are valid in A ‘ . Next let G be a class of such sets C of sentences. Then A is called algebraically closed (or, if necessary, 6 -Galgebraically closed) if every set Z E E that is consistent over A is satisfied in A . Take, for example, to be the class of (commutative) fields, and G the class of singletons C = { u }, where each sentence u is a polynomial equation in a single variable prefaced by an existential quantifier binding the variable: then we get the usual notion of an algebraically closed field. Again, let (5 consist of singletons C = { u } , where now each sentence u is a finite conjunction of equations

and negations of such equations, 553

554

B.H.Neurnann, The isomorphism problem for algebraically closed groups

prefaced by existential quantifiers binding all the variables; here L f',g, g' are words formed with the relevant algebraic operations from variables and constants, that is elements of the algebra A under consideration. When 6 is the class of all groups, the 6-Ealgebraically closed algebras are the algebraically closed groups introduced by Scott [71. Other classes of algebraic systems will also serve instead of groups, and in particular the facts that will be stated and the conjecture t o be discussed about algebraically closed groups apply also, mutatis mutandis, t o algebraically closed semigroups [ 61. The following theorem and corollary were proved by Scott [ 71 : Theorem 1 . l . Every group can be embedded in an algebraically closed group. Corollary 1.2. Every countable group can be embedded in a countable algebraically closed group. This latter fact is a corollary of the proof method rather than of the theorem; and the proof applies equally t o other classes of algebras, provided that they admit direct limits and their species is defined by finitely many finitary operations: it is in essence the proof devised by Steinitz [ 81 to prove the corresponding statemen t for fields. We deduce an easy consequence: Theorem 1.3. There are 2 * O mutually non-isomorphic countable algebraically closed groups.

To see this we need only remember that there are 2*0 mutually non-isomorphic 2-generator groups [ 41 , and each can, by Corollary 1.2, be embedded in a countable algebraically closed group; but each countable group contains only countably many pairs of elements and thus only countably many 2-generator groups. Hence countable algebraically closed groups are needed to accom-

B.H.Neumann, The isomorphism problem for algebraically closed groups

555

modate all the 2-generator groups. On the other hand there can not be more than 2N0 isomorphism classes of countable algebraically closed groups, because there are no more than 2 N o isomorphism classes of countable groups altogether. We note in passing that the same argument will show the same fact for semigroups instead of groups; but not for fields: there are only countably many countable algebraically closed fields, one for each combination of characteristic (a positive prime number or zero) and transcendence degree (a cardinal number, in this case

< No).

However, no algebraically closed group is explicitly known, the existence proof being highly non-constructive. This stems in part from the fact that there is no useful criterion known that tells one what sentences are or are not consistent over a given group. But assume now that we are given the knowledge that two countable groups, say A and B , are algebraically closed. We then ask how one can tell whether or not they are isomorphic. Now I conjecture that this problem, the isomorphism problem for countable algebraically closed groups, is algorithmically unsolvable. The aim of this note is to give some reasons for entertaining this conjecture. 2. Absolute presentations of groups

A proof of the conjecture would first of all require a more precise formulation of it; in particular we would need to know just how the groups A and B are given, and what algorithms are admitted. But as I am not proving the conjecture, I shall allow myself much imprecision. We may think of A and B as given, for example, in some presentation, that is by defining relations in some set of generators; and we note a fact that is not really relevant:

Theorem 2.1. A n algebraically closed group can not be finitely generated, nor finitely related. Proof. If gl,g z , ..., gn are finitely many elements in an arbitrary group G, then the sentence

556

B.H.Neumann, The isomorphism problem for algebraically closed groups

32 : g1z = zg1 .&. g 2 z = zg2 .&. ... .&.gnz = zg, .&. z # 1

is consistent over G, as it is satisfied in any direct product of G and a non-trivial group. It follows that in an algebraically closed group every finitely generated subgroup has a non-trivial centralizer; but it is known [ 51 that every algebraically closed group is simple, and thus has trivial centre: hence it can not be finitely generated. Now in a finitely related group in infinitely many generators infinitely many of these generators occur in none of the defining relations and thus generate freely a free group that is a free factor of the whole group: take any two of these free generators, and observe that the centralizer of the subgroup they generate is trivial: but in an algebraically closed group this can not happen, as we have seen, and the theorem follows. How could we hope to prove that B is not isomorphic to A? We know that B can not contain (isomorphic copies of) all possible 2-generator groups as subgroups; so we might try to spot one that is not in B , but is in A . Thus we pick two elements, say a l and a 2 , of A and try to show that the subgroup H, say, they generate can not be matched in B. To this end we need to know what this subgroup His. A set of defining relations of H in terms of a1 and a 2 , that is a presentation of H , suffices if we are given that it is a presentation, that is to say, if we know that the only relations between a l and a2 are the consequences of the given relations. We observe that to be able to decide whether an arbitrary word in a l and a2 equals the unit element in consequence of the given defining relations, we should be able to solve the word problem for this presentation of H ; and to be able t o decide whether H is isomorphic t o the subgroup K , say, of B generated by a pair b,, b2 of elements picked from B , we should be able to solve the isomorphism problem for our presentation of H . If H is finitely presented, say by

B.H.Neumann, The isomorphism problem for algebraically closed groups

551

then the sentence

3 x 1 . 3 x 2 : r 1 ( x 1 , x 2=) 1 .&. r 2 ( x 1 , x 2=) 1 .&. ... .&. rm ( x 1 , x 2 )= 1 .&. x1 # 1 is consistent over every group G, as it is valid in G X H , where we have assumed, without loss of generality as we shall presently see (Lemma 2.5), that H is not cyclic; hence this sentence is satisfied in B. This does not quite mean that H is matched in B and thus useless for telling B from A : if b, , b, in B satisfy this sentence, the group K they generate might still satisfy further relations that are not satisfied by a l and a, in H. This leads us to think of H as specified more precisely by what we shall call an absolute presentation, that is a set of relations

(2.2)

r l ( a l , a 2 ) = 1 , r 2 ( a l , a 2 ) =1 , ...

together with a set of “irrelations”

that jointly ensure that if b, , b , are two group elements that satisfy them in place of a,, a,, then mapping al on b , and a2 on b , necessarily generates an isomorphism of H onto the group K generated by b , and b,. If a presentation by generators and defining relations only is to be distinguished from an absolute presentation, it will be called a relative presentution. We note in passing that all the classical problems on relative group presentations, such as the word problem, the conjugacy problem, the isomorphism problem, also make sense for absolute presentations, and are still in need of solving, or unsolving. It is easy t o see that if H is finitely absolutely presented, then it is matched in B and thus useless for distinguishing B from A :

Lemma 2.4. Every algebraically closed group contains un isomorphic copy of every finitely absolutely presented group.

558

B.H.Neumann, The isomorphism problem for algebraically closed groups

Wc need only observe that the sentence obtained by conjunction of the defining relations and irrelations, written in variables xl,x2 that are then bound by existential quantifiers, is consistent over an arbitrary group, hence satisfied in every algebraically closed group. We have already, in claiming that n o generality is lost by assuming u1 # 1 , made use of a fact that is almost a corollary of this lemma:

Lemma 2.5. Every algebraically closed group contains cyclic groups oj' every order. Cyclic groups of finite order are, like all finite groups, finitely absolutely presented, hence contained in every algebraically closed group by Lemma 2.4. For an infinite cyclic group we use the sentence 3x, .3x,.3x3.3x4 : x;2 = x; .&. x;3

= x22 .&. x';4 = x; .&.

whicli is satisfied in a group introduced by Graham Higman [ 1 ] , and which has the property that in every non-trivial epimorph the generators have infinite orders: this sentence must be satisfied in every algebraically closed group, and the lemma follows.

Corollary 2.6. A n ulgebraicully closed group cun not be periodic. 3. Infinite presentations

It follows from what has been said that our hope of picking two elements u l , u2 from the algebraically closed group A such that the subgroup H they generate is not matched in the algebraically closed group B , so as to distinguish B from A , now rests with an H that can not be finitely absolutely presented, and is not infinite cyclic either. However, as we seek an effective procedure that distinguishes our groups, it seems to me that we need to assume a t least

B.H.Neumann, The isomorphism problem for algebraical1.v closed groups

559

that H is recursively absolutely presented, in the sense that in some effective enumeration of the elements of the free group on u l , a2 the relators r , (a1,a2 ), r2 (al ,a2 ), r3 (al ,a2 ), ... are recursively enumerated, and so are the irrelators s1 (a1,a2 ), s2 (ul ,u2 ), s3 (al ,u2 ), ..., where the absolute presentation is given by (2.2), (2.3). Note that one and the same group can have absolute presentations that are recursive and others that are not: for example the infinite cyclic group generated by an element a can be given by an arbitrary infinite sequence of irrelations of the form

and the recursiveness or otherwise of this absolute presentation depends on the sequence of numbers n ( i ) chosen. We need a procedure to reduce a countable set of irrelations to a single irrelation; the method is adapted from [ 51 . For notational convenience we describe it here only for a 2-generator group: it is clear that it applies in the same way to groups without this restriction. Lemma 3.1. Let the group H be generated b y a l , u 2 with the defining relations

(3.1.1)

r i ( a l , a 2 ) = l , i = 1 , 2,...,

and assume that H also satisjies the irrelutiotzs (3.1.2)

al f 1 ,

(3.1.3)

s j ( a l , a 2 ) # 1 , j = 1 , 2, ... ;

here i and j range over some or all positive integers. Then H can be embedded in the group H , generated b y H and three elemeuts c, c’, d with the defining relations of H und (3.1.4)

c2 = 1, ( c c ’ ) ~= 1, d 2 = u l ,

560

B.H.Neumann. The isomorphism problem for algebraically closed groups

(3.1.5)

( d - - ~ c d ~ . ~ ~ ( u=, 1 , u. j~=) )1,~2, ...,

with j runging over the sume integers us in (3.1.3). Corollary 3.2. With the same notation, i j H is absolutely presented by (3.1.1 ), (3.1.2), (3.1.3), then every homomorphism o f H , under which the image oj'u, is not 1 is u monomorphism on H. Corollary 3.3. ? f ' H is recursively absolutely presented by (3.1. l), (3.1.21, (3.1.31, then H , is recursively relutively presented b y (3.1.1), (3.1.4), (3.1.5).

Proof of the lemma. Denote the order of a, by a ,the order of s,(ul .u2 1 by au). Form the group C generated by c, c' and al with the defining relations c2 =

1,(cc')2 = 1,c'2 = a , , u y = 1 ;

and for each relevant j the group Cj generated by cj and sj(al , a 2 ) with the defining relations

It is known that the order of ul in C i s a, and the order of

sj(ul ,u2 ) in Cj is (YO): hence we can form the generalized free product of H and C, C,, C,, ..., amalgamating u1 in H and C, then s1( u l .u2 ) in the product so obtained and C, , and so on. In the re-

sulting group the elements c, c1 , c 2 , ... all have order 2 and are otherwise not related among one another: that is to say, they generate the free product of their cyclic groups of order 2. The partial automorphism of this free product generated by mapping c on e l , c1 on c 2 , ... can be obtained by conjugation [3] by an element d that we now adjoin. the resulting group is the H , of the lemma. Note that now c, = d--JcdJ, so that the relations c2 = 1 follow from J c2 = 1 and may be omitted. This completes the proof of the lemma. The corollaries are obvious.

B.H.Neumann, The isomorphism problem for algebraically closed groups

56 1

Note that if H is embedded in a simple group, this would serve equally well for Corollary 3.2; but we need an embedding that is constructive, so as t o retain recursiveness; and it is convenient to have H, finitely generated We now use Graham Higman’s famous embedding theorem [ 2 1 t o make our final embedding.

Theorem 3.4.If H is recursively absolutely presented by (3.1.1 ), (3.1.2), (3.1.3), then H can be so embedded in a finite1.y relatively presented group H * that every homomorphism of H* under which the image of al is not 1 is a monomorphism on H. We need only embed H I , which is recursively relatively presented, in a finitely relatively presented group: the rest follows from the lemma and its corollaries.

Corollary 3.5. Every algebraically closed group contains isomorphic copies of every recursively absolutely presented 2-generator group. This is hardly scrprising, as there are only a countable infinity of such groups; but it seems t o me t o leave no hope of distinguishing two algebraically closed groups by picking out pairs of elements and looking a t the subgroups they generate. References [ 11 G . Nigman, A finitely generated infinite simple group, J. London Math. SOC.26 (1951) 61-64. [2] GHigman, Subgroups of finitely presented groups, Proc. Roy. Soc. London (A) 262 (1961) 455-475. [ 31 G. Higman, B.H. Neumann and H. Neumann, Embedding theorems for groups, J . London Math. Soc. 24 (1949) 247-254. [4] B.H.Neumann, Some remarks on infinite groups, J . London Math. Soc. 12 (1937) 122-127. (51 B.H.Neumann, A note on algebraically closed groups, J. London Math. Soc. 27 (1952) 247-249. 161 B.11. Neumann, Algebraically closed semigroups, Studies in Pure Mathematics (ed. L. Mirsky) (Academic Press, New York, London, 1971) 185-194.

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B.H.Neumann, The isomorphism problem f o r algebraically closed groups

[7] W.R.Scott, Algebraically closed groups, Proc. Amer. Math. SOC. 2 (1951) 118-121. [ 8 ] E. Steinitz, Alcebraische Theorie der KBrper, J . reine angew. Math. 137 (1910) 167-309, especially 6 2 1 .

EQUATIONS OVER GROUPS Helmut SCHIEK University of Bonn Dedicated to W.Krull on his 70th birthday

Introduction

Let G be any group and X = OC) an infinite cyclic group, let G [ x ]be the free product G * X and R = R ( x ) = agxilal...a,-lxi~a, with a,,, ..., a, E G be any element of G [ x ] . We say, that the equation R ( x ) = 1 has a solution over G if and only if R ( h ) = 1 for some element h of a group H containing G. A solution exists if and only if the normal closure N of R in G [ x ]has trivial intersection with G. The study of equations over groups was initiated by B.H. Neumann [ 4 ] . Levin [ 3 ] showed that a solution always exists provided n 2 1 and i, , ..., in > 0. Gerstenhaber and Rothaus [ 11 showed that a solution always exists provided the exponent sum e ( R )= i , +...+in does not vanish and that G can be embedded in a compact connected Lie group. If e ( R ) = 0 we don't have always a solution. For example, the equation ux-l bx = 1 can have a solution only if u and b have the same order, while it is a special case of a result of Higman, Neumann and Neumann [ 21 that if a and b d o have the same order a solution always exists. The case e ( R )= 0 has been studied by the author in several papers [ 6,7]. Here we use results from these earlier papers t o obtain the following new result. Theorem. Let R ( x )= aE-lbE where a and b are elements o f G of the same order and where E has the form E = x-'lalxS1 ...x-SnanxSnwith n 2 1 , all non-trivial u l , .._,a, E G, E # 1 and all o f s , , ..., s, # 0. Then the equation R ( x )= 1 has a solution. 563

564

H. Schiek, Equations over groups

Proof of the theorem. We first state a result from Schiek [ 6 ] . Let H be any group. Assume we have isomorphic copies Hi s H for every integer i E Z, If, n Hi = 1 for i # j . Let u i be the isomorphism with o i ( H ) = Hiand let H* be the free product H* = * Hi ( i E Z ) ; in special cases we can have Hi= Hx' for a subgroup H of a group G and .Y an element of G. We define l ( W ) for an element W of H* t o be the least s such that for some i, W E Hi+, *...*Hi+s. We define R E H* t o be iiicomprcwible with respect to the factorization of H* into the factors Hi if for every non-trivial element S in the nornial closure of R in H* we have l ( S ) 2 Z(R);if Hi = Hx' we also say: K is incompressible with respect to H and x. In H* we have an automorphism u , defined by u [ u i ( u ) ]= uj+l( a ) (for every u E N,every j E Z ) and the homomorphy property. Let Ri = u i ( R )for every integer i ; then for every non-trivial T i n the normal closure of the set of all Ri ( i E Z ) we have: Z(7') 2 l(R);for Hi= Hx' every non-trivial T i n the normal closure of all Rx' has this property. Define a series of subgroups of G [ x ] as follows. Let KO = G [ x ] and, for all i Z 0, let Ki+l be the normal closure of G in Ki.This series was studied in Schiek [ 51 . Evidently K , consists of all W such that e ( W ) = 0, and K , = * Gi where Gi = Gx'. Also K , = * G E , where E runs over all elements of K l with normal form E = E , ...E n , n Z 0, where every Ei( 1d idn) has the form X - ~ ~ U ~ X ~ ~ ui E G , ui # I , s i # 0, si # si+l for 1 < i d 17- 1 (for n = 0 we have E = 1, G E = G). Under the hypothesis of the theorem it is clear that R E G * G", whence l ( R ) < 2. Imposing the relation R = 1 o n G * GE amounts to passing from G * GE t o a free product with amalgamation of the cyclic group generated by u-l in G and that generated by bE in GE . 'This extends t o a homomorphism from H = *GES with kernel the normal closure of R in H . It follows that n o non-trivial element in the normal closure of R in H lies in any G E S ,hence that R is incompressible with respect t o G and E (of course we can assume I(R)= 2, otherwise a, b and R would be trivial). From the result cited above it follows that the normal closure M of the set of R E S in N has trivial intersection with G. Now K , = H * P, where P is the free product of all factors GF of K , with F # Es (for every integer s). The normal closure N , in K , of the set of R F ,one for

,

H.Schiek, Equations over groups

565

each factor GF of K 2 , has intersection M with H , hence trivial intersection with G. Because N , is the normal closure of R in K , , we have shown that the normal closure of R in K , has trivial intersection with G; we want to show the same is true of the normal closure N of R in KO = G[x] . We treat first the special case that E = x-jcxJ with 1 # c E G and j # 0. For each i let pi be the natural projection of K , = * Gi onto Gi. If i # 0, we evidently have p,(R) = 1, whence p i ( N , ) = 1 and it follows that N , has no non-trivial element in Gi. We have seen above that N , has no non-trivial element in Go = G. Thus N , has no non-trivial element of length less than I(R)= 2, and R is incompressible with respect to G and x. It now follows from the result cited earlier that the normal closure in K , of the set of Ri has no non-trivial element of length 1. This normal closure is N and we have shown that N has trivial intersection with G. For the general case we use induction. Let E = El ...En , Ei = 1 # ai E G, si # si+, (l is the set of all i (1< i < n ) for which si # s1 . Surely we have t 2 1 (S2fS1). R* = aE*-l bE* has a normal form of type t < n. According to the induction hypothesis R* = 1 has a solution. Let N* be the normal closure of R* in G [ x ], H = G [ x ]/ N * , then we have H 2 G. Now we get R* from R if we introduce the commutator relations [ G,, ,KT ] = 1 in K , ; that means: R-l R* is in the normal closure of [ G,, ,K; ] in K , . We will prove that the normal closure Q of the set theoretical union of [ G,, ,K; ] and R* in G [ x ] has trivial intersection with G ; then we have N C_ Q, N n G = 1 and R ( x )= 1 has a solution. Let y & H , Y = ( y ) an infinite cyclic group, and let Hi = HY', L , =*Hi, LT = * H i (i#s,). We can introduce the relations

566

H.Schiek, Equations over groups

[H,, ,LT ] = 1 without changing H 2 G; we take the direct product L* = Hsl X LT, L* 2 H 2 G. Let z & L * , Z = ( z ) an infinite cyclic group, and L [ z ] = L * Z . In L [ z 1 we can introduce relations: (1) (2)

= 1

for all a E G

(aZ)-lax = 1 for all a E G

.

The projection of (az)-,& = 2-la-l zy-l ay in to G is trivial, also the projection of (az)-'ax into G. Let fi be the normal closure of the union of all (a")-laJ' and all (aZ)-lax with a E G in L* [ z ] , then we have: 8 n G = 1. In this group we have the relations: ax = az = aY, G,, = Hsl , KT = LT2 R-l R* = 1 , R * = 1 and therefore R = 1. The group L* [ z l / N contains G. Therefore we have proved that R ( x ) = 1 has a solution.

Remarks. R ( x )is an element of K , - K , . We can ask if we can prove the existence of a solution of an equation Ri(x) = 1 , Ri(x) being an element of K , of a similar structure. In [ 51 the structure of K , is given by induction on i. For Ki we have a system of generators: S, = Tiu G. The set T , is defined by T , = {axil l#a€G,i#O}. If Tiis given, we define A i as the set of all products E = El ...E, (for some t 2 1 ), all El ,...,E, E Ti. We define: Ti+l = { a E I l#a€G,EEAi}. then we can ask if Ri(x) = 1 has a Let R , ( x ) = aF, F = bE E Ti; solution. R , ( x ) is an element of Ki - K For every R E G [ x ] - G there is an integer i such that R E K , - K,,, (in [ 51 it is proved that n K , = G). It might be possible that one can use induction on i in order t o get more general results; this would need quite a lot of technique.

H.Schiek, Equations over groups

56 7

References [ 11 MGerstenhaber and O.S.Rothaus, The solutions of sets of equations in groups, Proc. Nat. Acad. Sci. U.S. 48 (1962) 1951-1953. [ 2 ] G . Higman, B.H. Neumann and Hanna Neumann, Embedding theorems for groups, Journal London Math. SOC.24 (1949) 247-254. [ 3 ] F.Levin, Solutions of equations over groups, Bulletin American Math. Soc. 68 (1962) 603-604. [4] B.H.Neumann, Adjunction of elements to groups, Journal London Math. SOC.18 (1943) 4-11. [ 5 ] H.Schiek, b e r eine spezielle Reihe von Normalteilern, Archiv der Mathematik 9 (1958) 236-240. [ 6 ] H.Schiek, Adjunktionsproblem und inkompressible Relationen, Math. Annalen 146 (1962) 314-320. [ 71 H.Schiek, Adjunktionsproblem und inkompressible Relationen 11, Math. Annalen 161 (1965) 163-170.

A SURVEY OF SMALL CANCELLATION THEORY Paul E.SCHUPP* university of Illinois, Urbana

1 . Introduction

1.1. A preview

In 191 1 M.Dehn [ 51 posed the word and conjugacy problems for groups in general and provided algorithms which solved these problems for the fundamental groups of closed orientable twodimensional manifolds. A crucial feature of these groups is that (with trivial exceptions) they are defined by a single relator r with the property that if s is any cyclic conjugate of r or r l s,# r-l , there is very little cancellation in forming the product rs. Dehn’s algorithms have been extended to large classes of groups possessing presentations in which the defining relations have a similar “small cancellation” property. At first, investigations were concerned with the solution of the word problem for groups G presented as a “small cancellation” quotient of a free group F. The theory was subsequently extended to the case where F is a free product or a free product with amalgamation. Moreover, strong results were obtained about algebraic properties; for example, one can classify torsion elements and commuting elements in “small cancellation” quotients. Dehn’s methods were geometric, making use of regular tessellations of the hyperbolic plane. The first extensions of Dehn’s re-

* This manuscript was prepared while the author was a visiting member at the Courant

Institute of Mathematical Sciences, New York University, and was supported in part by the New York State Science and Technology Foundation, Grant SSF-(8>8. The author thanks Bruce Chandler and Roger Lyndon for making many valuable suggestions. 569

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sults to larger classes of groups were obtained using cancellation arguments of combinatorial group theory, independent of any geometric considerations. More recently, the geometric character of Dehn’s argument has been restored in the form of elementary combinatorial geometry. “Small cancellation” theory is now emerging as a unified and powerful theory. In what follows, we shall outline the central ideas of this theory and present some important and typical results. 1.2. The hypotheses We now turn to tlie formulation of the conditions which, for suitable values of the parameters, allow one to “do” small cancellation theory. In order to fix our notation and terminology, let F be a free group on a set X of generators. A letter is an element of the set Y of generators and inverses of generators. A word w is a finite string of letters, w = y l . . . y m . We shall not distinguish between w and the element of F that it denotes. We denote the identity of F by 1. Each element of F other than the identity has a unique representation as a reduced word IY = y1 ...yn in which n o two successive letters yiyi+l form an inverse pair xix;’ o r x;lxi. The integer ti is the length of w , which we denote by IwI. A reduced word w is called cyclically reduced if y,., is not the inverse o f j ’ , . If there is n o cancellation in forming the product z = z i , ... 21, we write z = u 1... u , ~ . A subset R of F is called symmetrized if all elements of R are cycliciilly reduced and. for each r in R , all cyclically reduced conjugates of both r and r1also belong t o R . Suppose that r1 and r2 are distinct elements of R with r1 = bcl and r 2 = bc,. Then b is called a piece relative to the set R . (Since we only work with one symmetrized set a t a time, we will omit the phrase “relative to R” and simply say that b is a piece.) Since b is cancelled i n the product r ; l r 2 , and R is symmetrized, a piece is simply a subword of an element of R which can be cancelled by tlie multiplication of two non-inverse elements of R . The hypotheses of “small cancellation” assert that pieces are relatively small parts of elements of R . The most usual condition takes a metric form, C‘(h),where h is a positive real number.

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Condition C'(X): If r E R,r

571

= bc where b is a piece, then Ibl < hlrl.

A closely related, non-metric, condition is C@) where p is a natural number. Condition C@): No element of R is a product of fewer than p pieces. Observe that C'(h) implies C ( p ) for X < 1/ ( p - 1). As illustration, the fundamental group of a closed, orientable 2-manifold of genus g has a presentation

In this case R consists of all cyclic permutations of r and r-l where r is al blai' b i I... ugbgaglbg' . Clearly, pieces are single letters and R satisfies C'( 1/4g- 1 ) and C(4g). Groups which have a presentation G = (X,R>where R satisfies C'( 1/6) are sometimes called sixthgroups. Analogously, we have eight-groups, etc. We shall sometimes need a condition T(y), for q a natural number, whose intuitive meaning will be clarified shortly. Condition T(q):Let 3 < h

< q . Suppose r l , ..., r h

are elements of

R with no successive elements r i , ri+l an inverse pair. Then at least one of the pairs r l r z , ..., r h - l r h ,rhrl is reduced without cancellation.

The cancellation conditions just introduced can be extended naturally to the case where F is a free product or a free product with amalgamation by using the appropriate normal forms and associated length functions. The definitions are then essentially the same as for F a free group. One must be careful on a few points, however, so we defer precise definitions of the cancellation conditions in the more general cases to Section 5. As an illustration of the power of small cancellation theory over fre6 products we note that the theory applies to "most" Fuchsian groups

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We shall consistently use the notation introduced above. F will be a free group on generators X , o r a free product, or a free product with amalgamation. R will be a symmetrized subset of F with N the normal closure of R in F. G will be the quotient group FIN. If F is a free group, G has a presentation tX,R). The natural map will be denoted by v : F + FIN.

1.3. A brief historical sketch Dehn’s methods were geometric [ 5 , 6 ] . He used the fact that with the fundamental group G of an orientable closed 2-manifold there is an associated regular tessellation of the hyperbolic plane which is composed of transforms of a fundamental region for G. Using the hyperbolic metric, Dehn inferred that a non-trivial word w equal t o 1 in G contained more than half of an element of R . Reidemeister [ 371 pointed o u t that Dehn’s conclusion followed from the combinatorial properties of the tessellation, without metric considerations. In 1949, V.A.Tartakovskii [ 47,48,49] initiated the algebraic study of small cancellation theory. Tartakovskii solved the word problem for finitely presented quotients of free products of cyclic groups by symmetrized R satisfying C(7). J.Britton [ 3 ] ,in 1957, independently investigated quotient groups of arbitrary free products by R satisfying C’( 1/6). The triangle condition, Condition T(4), was introduced in 1956 by Schiek 1381, who solved the word problem for R satisfying C’( 1 /4) and T(4). Greendlinger [ 1 1,121 , in 1960, solved the conjugacy problem for C’(1 /8), gave a new proof of the solvability of the word problem for C’(1/6), and obtained several other important results. Greendlinger 1 15 1 subsequently also investigated the C’(1 /4) and T(4) hypothesis. Very few group presentations have Cayley diagrams which are embeddable in the plane. However, it turns o u t that for any group G = (X$\, if kv is in N,there exists a finite planar diagram M , each

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edge of which is labelled by an element of F , and such that each region (face) D of M has as label on its boundary an element of R , while the label on the boundary of the entire diagram M is the reduced word w . The existence of such a diagram M was observed by Van Kampen [50] in 1933. Van Kampen’s paper seems to have been totally ignored until Weinbaum [ 5 1 1 , 1966, used the ideas to prove some of the results of Greendlinger. The above ideas were rediscovered independently by R.C.Lyndon [ 291 in his 1966 paper “On Dehn’s Algorithm”, which provided a unification, simplification, and generalization of many previous results. Lyndon observed that the Condition C(p) asserts that every interior region of the diagram M borders on at least p other regions. Condition T(y) expresses the dual condition that each interior vertex of M (excluding vertices of degree two) has at least 4 incident edges. Lyndon solved the word problem for finite R satisfying one of the hypotheses C ( p )and T ( q )where @ , q )is one of the pairs (6,3), (4,4), or (3,6). (Condition T(3) is vacuous.) These hypotheses correspond naturally t o the three regalar tessellations of the Euclidean plane. For example, the hypothesis C(4) and T ( 4 )corresponds t o the regular tessellation of the plane by squares. In the regular tesselation, all vertices and all regions have degree four. In the diagrams considered under the hypothesis C(4) and T ( 4 ) ,all interior regions and interior vertices have degree greater than or equal to four. 2. Decision problems

2.1. Dehn’s algorithm and Greendlinger’s lemma In his study of the word problem for fundamental groups of orientable 2-manifolds, Dehn concluded that if a freely reduced non-trivial word u is equal to 1 in the fundamental group, then u contains more than half of some cyclic permutation of the defining relator or its inverse. This conclusion gives Dehn’s algorithm for the word problem. Suppose a group G has a presentation G = (xl ,...,x n 4 )where R is

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a finite symmetrized set of defining relators and it has been estab!ished that freely reduced non-trivial words which are equal to 1 in G contain more than half of some element of R . Let w be a nontrivial word of G. If w = 1 in G, then w has some factorization w = bcd where, for some r in R , r = ct with It1 < Icl. In G then, w = bt-l d , a word of shorter length. A finite number of such reductions either leads to 1, giving a “proof” that w = 1 in G, or to a word w* which cannot be so shortened, establishing w # 1 in G. The most fundamental result of small cancellation theory is that Dehn’s algorithm is valid for R satisfying one of the metric hypotheses C’( 1 /6), or C’( 1/4) and T(4).Actually, as first discovered by Greendlinger [ 1 I ] , considerably more is true. In order to state the sharper results we need to single out certain “large” subwords of elements of R . Definition. A word s is called a j-remnant (with respect to R ) if some r E R has the form r = sb bi where b , ,...,bj are pieces.

,...

Theorem 1 (Greendlinger’s lemma). Let F’be a free group, free product, or free product with amalgamation. Let R be a symmetrized subset of F with N the normal closure of R. Assume that R satisfies the hypothesis C @ ) and T(q)where (p,q)is one of the pairs (6,3), (4,4), or (3,6). I f w E N , w # 1 , then w E R or some cyclically reduced conjugate w* o j w has the form w* = u l s l...umsm where each sk isan i(sk )-remnant. The number m of the sk and the numbers i(sk)satisf y the relation

”+Z-i(s,)

k= 1 4

1

2p

I f the hypothesis is C‘(6)we have

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For this inequality to hold w must contain a j-remnant with j < 3. If the hypothesis is strengthened to C’( 1/6) we conclude that w contains more than half of an element of R . The conclusion that an element w of N contains more than half of an element of R was obtained under varying hypotheses by Britton [ 3 ] and Schiek [381. Theorem 1 was proved by Greendlinger [ 1 11 for F free and R satisfying C’( 1/6). He subsequently [ 151 proved the theorem for R satisfying C‘( 1/4) and T(4). Weinbaum [ 5 1 ] gave a geometric proof for the C’( 1 /6) case. The general formula m

is due to Lyndon [ 291. In the geometric approach, the nature of this formula is revealed as a combinatorial “curvature formula”. The relationship of the curvature formula to Greendlinger’s Lemma is discussed by Schupp [421.

2.2. The word problem Greendlinger’s Lemma is independent of any cardinality or effectiveness assumptions on either F o r R. The conclusion that a non-trivial element of N contains more than half of an element of R allows one to use Dehn’s algorithm to solve the word problem in many cases where F is a free product and R is infinite. We consider the following effectiveness condition on F and R . Condition E ( k ) : Assume that F is a finitely generated free group or free product. Suppose that R is a symmetrized subset of F satisfying C’(1 /6) o r C’( I /4) and T(4).Assume that there is an algorithm which decides, given w E F , whether or not there exist r E R such that irI < klwl and r = ws,and, if so, produces all such r. (In particular, there are only finitely many such r . In most cases of interest, if such an r exists it will be unique by the cancellation condition.) The following corollary is then immediate from Theorem 1.

Corollary 1 . Let F a n d R satisfy Condition E ( 2 ) . Then the word

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problem for G = FIN is reducible to the word problem f o r F and conversely. (In the language of recursive function theory, the word problems for Fund G have the same Turing degree.) Under the non-metric hypothesis C@) and T(y) a rather different argument is needed to solve the word problem. The following theorem is due to Lyndon [ 291.

Theorem 2. Let F be a free group, and let R be a finite symmetrized subset o f F satisjying thh hypothesis C@) and T ( 4 )for @ , q ) one o f (6,3), (4,4), or (3,6). Let N be the normal closure of R in F. Then FIN has solvable word problem. The proof of Theorem 2 involves a combinatorial “area formula” which shows that if w E N , one can calculate from w a bound on the number of conjugates of elements of R necessary to form a product ( u l r l u;’ )...(urnr, ti-,‘) = w. The lengths of the conjugating elements u i are also bounded. One could then try the finite number of possible products to see if w can be so formed.

2.3. The conjugacy problem Our discussion of conjugacy begins with Dehn’s algorithm for the conjugacy problem which Dehn [ 61 discovered for fundamental groups G of 2-manifolds, Let w and u be two words of G. We can effectively replace w and u by words w‘ and u’ which are conjugate in G t o w and u respectively, which are cyclically reduced, and which are “cyclically R-reduced” in the sense that n o cyclic permutation of w’ or u‘ contains more than half of an element of R . If one of w‘,u‘ is the identity 1 and the other is not, then w and u are not conjugate in G. If both w’ and u‘ are 1 they are certainly conjugate in G. Assuming that both w ’and u’ are non-trivial, then w’ and u’ are conjugate in G if and only if for some cyclic permutations w * ,u* of w‘ and u‘ respectively, the equation w * = uu*u-l holds in G for u a subword of some r E R . There are only finitely many such u , and, in view of the solvability of the word problem for G, one can decide if the above equation holds. Dehn also obtained a bound on IuI in terms of Irl, namely 1241 < Irl/8.

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Greendlinger [ 121 showed that Dehn’s algorithm solves the conjugacy problem if F is a free group and R is finite satisfying C’( 1/8). Greendlinger subsequently showed that generalizations of Dehn’s algorithm solve the conjugacy problem for R satisfying C’(1/6) [ 171, and C‘(1/4) and T ( 4 ) [ 151. A geometric proof extending these results essentially unchanged to free products is given by Schupp [ 4 1 I . To investigate conjugacy when F is a free product we need a mild additional assumption which is automatically true if F is a free €TOUP.

Condition J : If r E R , then r is not conjugate to r-l in F . Theorem 3. Let F be a free group or free product, and let R be a symmetrized subset of F satisfying C’( 1/6)7 or C’(1/4) and T(4). Assume that R satisfies Condition J and that R and F satisfy Condition E(2q) ( q is 3 and 4 for C’( 1/6)7and C’(1/4)and T ( 4 ) respectively). Then the conjugacy problem for G = FIN and F have the same Turing degree. As with the word problem, a somewhat different argument is needed to solve the conjugacy problem under the non-metric hypotheses. This was done by Schupp 1401. Theorem 4. Let F be a free group, and let R be a finite symmetrized subset of F satisfying the hypothesis C @ )and T ( q )for (p,q) one o f (6,3), (4,4), or (3,6). Then G = FIN has solvable conjugacy problem. 3. Algebraic applications

3.1. Small cancellation products In this section we discuss some of the algebraic applications of small cancellation theory. Definition. A group G is a product of the grotips X i , i E I , if G

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F / P where F is tlie free product of tlie Xi,and P is a normal subgroup of F such that the natural map v : F -+ F / P embeds each of the groups Xi.(Viewing each Xias a subgroup of F , we require V(Xi)

= Xi.)

We see immediately from Greendlinger’s Lemma that if F is a free product of groups X I and R satisfies one of the cancellation hypotheses C‘( 1 /6), or C’(1/4) and T(4), then N contains n o elements of length one. Hence, FIN is a product of the Xi. Thus we have some new products of groups at our disposal, which we will call sniull curicellatiotl products. We shall see that these products have several “nice” properties and are powerful tools in dealing with embedding problems and adjunctions of solutions to equations over groups. The theorem of Higman-Neumann-Neumann [ 191 that a countable group can be embedded in a two-generator group is now wellknown. As an illustration of tlie use of small cancellation products we will prove a much stronger result: The idea of using small cancellation theory t o prove embedding theorems is originally due to J.L.Britton (unpublished) and has been used by McCool [ 3 3 ] , Levin [ 21 1 , and Miller and Schupp [ 351. The following proof is Brit ton’s. Definition. A countable group K is called S-Q iiniversal if every countable group can be embedded in a quotient group of K . Theorem 5. Let F be uriy lion-trivial j i e e prodiict, F = X 4: Y . with the single ercqrtiori of the j k e prodtict of t w o copies of C, , the cyclic group of order two. Theti F is S-Q urziversal. Proof. Since we have excluded the case of C, * C,, we can pick distinct elements x 1, x 2 , neither tlie identity. in one group, X say, and an element I‘# 1 i n Y.Let H be a countable group with presentation 11 = ( l z l ,...;51. S is a set of defining relations among the 11,.

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L e t F ’ = H * X * Y . Let

and, in general, let

ri = h , ( ~ , y ) * O ( ~ - ~ ).+. .~(x~~ ~y y ) ~ O . ~x~y Let R be the symmetrized set generated by the ri. It is not difficult t o see that R satisfies the cancellation condition C’( 1 / 10). This follows from the fact that no piece can contain a subword of the form [ ~ ~ y ( x ~ y ~ ~ x ~ y ( x ”~ .yLet ) ~N+be ~ xthe ~ ynormal ] closure of R in F‘. G = F’/N is a sma!l cancellation product and hence embeds all the factors of F ’ , in particular H. Now G is certainly generated by the images of X and Y , since each ri = 1 in G and each hi is thus equal to a word on xl, x2,and y . lndeed, since each ri contains precisely one h i , we can eliminate the relations R by Tietze transformations, rewriting the relations S in terms of x l , x 2 , and y . Hence G is actually a quotient group of X * Y which embeds H . This completes the proof.

3.2. Torsion elements If R satisfies the metric hypothesis C‘( 1/6) one can completely classify the elements of finite order in FIN.

Theorem 6. Let F be a free group or a free product, uiid let R be a symmetrized subset of Fsutisjying C‘( 1/6). Let v ; F -+ FIN be the natural map. Ij’w has finite order in G = FIN then either (i) w = v ( w ’ )where w ’is uri element of firiitc order iri F,or (ii) there is an r E R whicli is u proper power it? F, suy r = Vn, n > 1 , und w is conjiigute in G to a power oj v. Theorem 6 says that the only elements of finite order in G are the “obvious” ones. In particular, if R contains no proper powers and F is torsion free (certainly the case if F is free) then FIN is torsion free.

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Theorem 6 was proved for F free by Greendlinger [ 121 , and for F a free product by McCool 13 1 1 . The same result was proved for F free and C'( 1 /4) and T by Soldatova [ 45 1 .

3 . 3 . Commuting elements Like elements of finite order, commuting elements can be successfully cliaracterized in suitable small cancellation quotients. Theorem 7. Let F be u f r e e group or u free product. Let R be a syinirietrized subset o f F sutisjj!irig Condition J , and which ulso sutisjies o m of C ' ( 1/6), or C ' ( 1 /4) arid T(4). Let v 1 F + FIN be the riutiiml niup. Then two eleineiita u arid u of G' = FIN cornmute i j atid only if u = v ( u ' )urid v = v ( v ' )f o r elements ii', U' which commute in F. Greendlinger [ 121 and Lipschutz [ 221 first investigated commuting elements for the case F free and R satisfying C'( 1 /6). Greendlinger [ 141 then showed that for F free and R satisfying C'( 1 /S), two elements of G commute if and only if they are powers of a comnion~element.Greendlinger [ 171 then extended the result t o C'( 1/6). Schupp [44] gave a geometric proof which extends the results to free products and t o the C'( 1/4) and T(4) hypothesis.

3.4. Endomorphisms and hopficity I t turns out that endomorphisms of certain small cancellation products may be severely limited. We need t o recall some definitions. A group K is 1iopfiuii if every endomorphism of K o n t o K is an automorphism. Dually, K is co-hopfiuri if all 1-1 endomorphisms of K are onto. K is complete if all automorpliisms o f K are inner and K has trivial center. Let F ' = H * Cm * C,, where N is an arbitrary countable group, and Cm and C, are cyclic of orders nz 2 3 and ri 2 2. Miller and Schupp [ 3 5 ] investigate the endomorphisms of G = F ' / N where N is the normal closure of the set R defined in the proof of Theorem 5. They prove that G is always complete and liopfian. I n addition, if I1 has no elements of order In, or no elements of order t i , then G is co-hopfian. I n particular, we have

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Theorem 8. Any countable group H can be embedded in a complete, hopfian quotient G H o f the modular group C, * C,. I f H is finitely presented then so is G H .

3.5.A uniqueness theorem Theorem 9. Let F be a free group or a free product. Let R and R’ be symmetrized sets of F which satisfy C’( 1 /6) or C’( 1 / 4 )and-T(4). I f R and R’ have the same normal closure, then R = R’. Theorem 9 was discovered by Greendlinger [ 131 for F free and C’(1/6). The theorem is essentially a direct consequence of Greendlinger’s Lemma. Schupp [42] observed that the above generalization holds.

3.6. Small cancellation theory over free products with amalgamation It turns out that small cancellation theory can be done over free products with amalgamation. For precise definitions of how the hypotheses are t o be interpreted see Section 5 . Greendlinger’s Lemma, when properly interpreted, and the uniqueness theorem, Theorem 9, continue to be true for F a free product with amalgamation and R satisfying C’( 1/6). One can use the existence of small cancellation theory over free products with amalgamation to prove that many free products with amalgamation are S-Q universal. The idea is to attempt to imitate the proof of Theorem 5. In order to do so, we need a defini tion. Definition. Let H be a group with subgroup A . Let { x l ,x2} be a pair of distinct elements of H , neither of which are in A . The pair { x1,x2} is called a blocking pair for A in H if the following two conditions hold: (i) xex! & A , 1 < i, j < 2 , E = f 1,6 = f 1 unless the product I ! XfX! = 1. I ! (ii) I f a E A , a # 1,thenx;ux; & A , l < i , j < 2 , ~ = + 1 , 6 = + 1 . The existence of a blocking pair for a subgroup is not an un-

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rcasonable condition in groups which have a fair amount of “freeness”. For instance, if H is free and A is a finitely generated subgroup of infinite index in H , then there does exist a blocking pair for A in H . (See Burns [ 4 ] .) Schupp [43] proves Theorem 10. Let F = (H*J;A=B)be a free product with amalgamutiori, where A and B ure proper subgroups o f H and J respectively I / there exists u blocking pair f o r A in H then F is S-Q universal. Perhaps the most surprising group to which Theorem 10 applies is the group G = (u,b,c,d;b-lub=a2,c-lbc=b2,d-l cd=c2,a-lda=d2)

which G.Higman [ 201 used to prove tlie existence of a finitely generated infinite simple group.

4.Some open questions There are some interesting open questions in small cancellation theory. For this section, a “suitable small cancellation group” will be a finitely presented group G = ( X $ ) where R satisfies C’(h)for some “sufficiently small” X and R also satisfies “suitable” additional hypotheses. What additional “suitable” hypotheses are necessary to make progress on the questions listed is not known, but it is assumed that they eliminate tlie obvious counterexamples not thought of by tlie author. Unfortunately, all the questions are probably quite difficult, with the possible exception of question 4. 1 . Does a suitable small cancellation group have solvable generalized word problem?

The generalized word problem asks for an algorithm which, when given a word w of G and a finitely generated subgroup H of G , decides whether o r not w E H . Asking for a solvable generalized word problem is a very strong condition. It is known (Mikliailova

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[ 341 ) that even the direct product, F2 X F 2 , of two free groups of rank two has unsolvable generalized word problem! D.J .Collins has observed that the obvious presentation of F2 X F2 satisfies C(4) and T(4). On the positive side, Lipschutz [ 25,281 has shown that for R satisfying either C’( 1 /6), o r C’( 1/4) and T, tlie generalized word problem restricted t o cyclic subgroups is solvable. A tempting case in which t o hope for a positive answer is that of tlie Fuchsian groups.

2. (C.F.Miller). What can we say about the algebraic properties of subgroups of suitable small cancellation groups? For example, is the intersection of two finitely generated subgroups again finitely generated? (This latter property is sometimes called the Howson property.)

3. What can one say about the endomorphisms of suitable small cancellation groups? I t is reasonable to expect that endomorphisms should be quite restricted. We remarked in Section 3.4 that certain very special small cancellation groups are hopfian. Are suitable small cancellation groups always hopfian‘? Perhaps suitable small cancellation groups are even residually finite. This is true, of course, in the Fuchsian case. What about the automorphisms of small cancellation groups? A possible conjecture is that all automorpliisms of G are induced by automorphisms of the free group F which fix the set R . This was proved by Nielsen [ 361 for the fundamental groups of 2-manifolds. (The conjecture is not quite true for free products of cyclic groups, so among the “suitable” additional hypotheses should be one saying that R has elements involving all the generators of G.)

4. (Lipschutz). If G is a suitable small cancellation group, is the centralizer of every element cyclic?

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We already know that, assuming only C’( 1 / 6 ) , t v o elements commute if and only if they are powers of a colnmon element. What is desired here is t o establish that if un and urn commute then u and u are powers of a common element. This should be true and perhaps even approachable.

5 . What “is” a small cancellation group? What is desired here is a geometric characterization of small cancellation groups. For example, the Fuchsian groups are essentially the groups with planar Cayley diagrams. The geometric approach to small cancellation theory suggests that there should be a characterization of small cancellation groups by means of “natural” geometric properties of their Cayley diagrams, or in terms of their possible action on other complexes. Such a characterization would bring us full circle back to Dehn. 5. The hypothesis revisited

I n this section we want to clarify exactly what the small cancellation hypotheses are in case F is a free product o r a free product with amalgamation. If F is the free product of non-trivial groups Xi then each non-identity element w of F has a unique representation in normal jorm as w = y ,...yn where each of the fetters y i is a non-trivial element of one of the factors Xi,and where n o adjacent yi,yi+, come from the same factor. The integer I E is the length of w,written IwI. If u = y 1..._y k c1_..ct and u = c’; ... cT1d , ...d , in normal form where dl # . v i l , we say that the letters cl, ..., ct are cancelled in forming the product uu. If y k and d , are in different factors of F , then w = u u has normal form y,...ykdl...ds. It is possible that d , and y k are in the same factor of F with d , # yil . Let a = y k d , . Then 1.1, = uu has normal form y1...yk-,ad2...dt. We say that y k and dl have been consolidated t o give the single letter a in the normal form of uu. We say that a word w has semi-reduced forrn uu if there is n o

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cancellation in forming the product uu, and write w = uu. Consolidation is expressly allowed. (Our notation w = uu in this context is somewhat a departure from usual usage since = often means “identically equal”.) More generally, we write w =- ul...u, if there is no-cancellation in bringing the product u1 ...u, into normal form. There are two reasonable definitions of “cyclically reduced” for elements of free products. The most common is to mean that (wI= 1 or if w = yl...y, in normal form, theny, a n d y l are in different factors of F. This is equivalent to asserting that w is not conjugate in F t o an element of shorter length. We shall say that w is strictly cyclically reduced in this situation. We shall say that w is cyclicully reduced if IwI = 1 or if w = yl...y, in normal form and y , # yi’. Thus there is no cancellation between y , and y1 although consolidation is allowed. A subset R of F is called symmetrized if every r E R is cyclically reduced and every cyclically reduced conjugate of r and r-l is also in R . A word b is called a piece if there are distinct elements r1 and r2 with semi-reduced forms r1 = bel and r2 = b c 2 . Note that the last letter of b does not have to be a letter of the normal form of r1 or r2.

Condition C’(h):If r E R , r = bc in semi-reduced form where b is a piece, then Ibl < hlrl. To avoid pathological cases, we further require that if r E R then Irl > l/h. Condition C ( k ) :N o element of R is a product in semi-reduced form of fewer than k pieces. If r E R then IrI 2 k . The triangle condition, Condition T(4), splits into two parts: (i) If r, s, t are strictly cyclically reduced elements of R then at least one of the products rs, st, tr is reduced as stands without cancellation or consolidation. (ii) If each of y l , y 2 , y 3 is a letter occurring in the normal forms of strictly cyclically reduced elements r, s, t of R , then ~ 1 ~ 2 f. ~1- 3

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I].,?. Schupp, A survey of small cancellation theory

We now turn to the case of F a free product with amalgamation. i E I , be a collection of groups with proper subgroups Let Xi, A iC Xi,each isomorphic to a fixed group A . Let I) : A + Ai, i E f, be isomorphisms. Let F =(*Xi;$i(A)=I)i(A)). I n discussing a normal form for elements of F it is usual t o choose coset representatives for each A i in X i . We specifically d o not want to d o this. We thus depart from the usual usage of “normal form”. Definition. An element \v # 1 if F is said to be in norrnul f o r m if is ~ ‘ ~ . . . . ywhere , the successive y i come from different factors of F and no y i is in the amalgamated part A unless n = 1 . NJ

Under this definition, an element may have infinitely many normal forms. It is well known, however, that if it’ also has a normal , then t?? = n. The integer n is the length of w and we form j,;...J~;,~ again write Iwi = 1 2 . Suppose that zi and u are elements of F with normal forms I [ = j ’ l. . . j i l 1 and u = x 1...x , respectively. If y n x l is in the amalgamated part A we say that there is cancellution between ci and u in forming tlic product iv = m . If y n and x1 are in the same factor of F but y n . x l & A we say that y n and x1 are consolidated in forming a normal form of ~ i u . A word w is said t o have semi-reduced form ul...ii, if there is no cancellation i n bringing the product i i l ... 11, into a normal form. Again, we write w = [ l 1 ... u, . As for free products, if w = y ,...y, in normal form, we say that w is strictlj, cj-clicallv redziced if y n and y are in different factors of F. I V is c : , ~ c ~ l i c rediiced dl~~ i f y n y l is not in the amalgamated part. The definition of “symmetrized” is as before. A word b is said t o be a piece if there exist distinct elements r l , r2 of R (i.c.. r1# r 2 in F ) such that r , = bcl and r 2 = bc2 in semi-reduced form. The definition of the cancellation conditions is now exactly as for F a free product. The cancellation conditions are quite strong in the case of free products with amalgamation since uZZ normal forms of r 1 a n d r2 must be used t o determine whether or not “sub-

P. E.Schupp, A survey of small cancellation theory

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words" b are pieces. A great deal thus depends on the amalgamated subgroup. As an illustration of what can happen, let us consider the following example. In the case of an ordinary free product, no amalgamation, if x and y are elements from different factors, then the symmetrized set generated by ( ~ ywill, ) ~for sufficiently large H , satisfy whatever cancellation condition C ' ( h )we desire. Now let F = (oC)+(y);x2=y2), the free product of two infinite cyclic groups with the squares of the generators identified. Consider the symmetrized set R generated by (xy)". Now ( x ~ #) ~ (x-ly-l)" in F , but

Thus (xy)"-lx is a piece relative to R and any cancellation condition fails badly. References [ 11 C.Blanc, Une interprdtation elementaire des th6orSms fondamentaux d e M.Nevanlinna, Comm. Math. Helv. 12 (1940) 153-163. [ 2 ] C.Blanc, Les rcseaux Riemanniens, Comm. Math. Helv. 13 (1941) 54-67. [ 3 ] J.L.Britton, Solution of the word problem for certain types of groups, I, 11, Proc. Glasgow Math. Assoc. 3 (1956) 45-54 (1958) 68-90. [ 4 ] R.G.Burns, A note on free groups, Proc. Amer. Math. SOC.23 (1969) 14-17. [ 5 ] M.Dehn, h e r unendliche diskontinuierliche Gruppen, Math. Ann. 71 (1911) 116-144. (61 M.Dehn, Transformation der Kurven auf zweiseitigen Flichen, Math. Ann. 72 (1912) 413-421. [ 7 ] F.Fiala, Sur les polykdres faces triangulaires, Comm. Math. Helv. 19 (1946) 83-90. [8] A.V.Gladkii, On the nilpotency classes of groups with 6-bases, Doklady Akad. Nauk SSSR 125 (1959) 963-965. [ 9 ] A.V.Gladkii, On simple Dyck words, Sib. Mat. Zh. 2 (1961) 36-45. [ l o ] A.V.Gladkii, On groups with k-reducible bases, Sib. Mat. Zh. 2 (1961) 366-383. These results were announed in Doklady Akad. Nauk SSSR 134, 16-18. [ 111 MCreendlinger, O n Dehn's algorithm for the word problem, Comm. Pure Appl. Math. 13 (1960) 67-83. [ 121 MGeendlinger, On Dehn's algorithms for the word and conjugacy problems with applications, Comm. Pure Appl. Math. 13 (1960) 641-677.

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[ 131 M.Greendlinger, An analogue of a theorem of Magnus, Archiv d. Math. 12 (1961)

94-96. [ 141 MGrcendlinger, A class of groups all of whose elements have trivial centralizers, Math. Z. 78 (1962) 91-96. 1151 M.Greendlinger, On the word problem and the conjugacy problem, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965) 245-268. These results were announced in Doklady Akad. Nauk SSSR 154 (1964) 507-509. [ 161 MGrecndlinger, Strengthened forms of two theorems for one class of groups, Sib. Mat. Zh. 6 (1965) 972-985. [ 171 MCrccndlinger, Problem of conjugacy and coincidence with the anticenter in group theory, Sib. Mat. Zh. 7 (1966) 785-803. Enghsh translation, Siberian Math. J. 7 (1966) 626-640. These results were announced in Doklady Akad. Nauk SSSR 158 (1964) 1254-1257. [ 181 M.IIa11, Jr., Generators and relations in g o u p s - the Burnside Problem, in: Lectures in Modern Mathematics, Vol. 11, ed. T.Saaty (New York, 1964). [ 191 G.Higman, B.H.Neumann and H.Neumann, Embedding theorems for groups, J. London Math. SOC.24 (1949) 247-254. (201 G.Higman, A finitely generated infinite simple group, J. London Math. SOC.26 (1951) 61-64. [21] F.Levin, Factor groups of the modular group, J. London Math. SOC.43 (1968) 195-203. [ 221 S.Lipschutz, Elements in S-groups with trivial centralizers, Comm. Pure Appl. Math. 13 (1960) 679-683. [23] S.Lipschutz, On powers of elements in S-groups, Proc. Amer. SOC.13 (1962) 181-186. [24] S.Lipschutz, On square roots in eight-groups, Comm. Pure Appl. Math. 15 (1962) 39-43. [25] S.Lipschutz, An extension of Greendlinger’s results on the word problem, Proc. Amer. Math. SOC.15 (1964) 37-43. [ 261 S.Lipschutz, Powers in eight-groups, Proc. Amer. Math. SOC.16 (1965) 1105-1106. [ 271 S.Lipschutz, On thc conjugacy problem and Greendlinger’s eight groups, Proc. Amer. Math. SOC.23 (1969) 101-106. [ 281 S. Lipschutz, On the word problem and T-fourth-groups, this volume. [29] R.C.Lyndon, On Dehn’s algorithm, Math. Ann. 166 (1966) 208-228. [30] R.C.Lyndon, A maximum principle for graphs, J. Combinatorial Theory 3 (1967) 34-37. [31] J.McCoo1, Elements of finite order in free product sixth-groups, Glasgow Math. J. 9 (1968) 128-145. [ 321 J.McCoo1, The order problem and the power problem for free product sixth-groups, Glasgow Math. J. 10 (1969) 1-9. [ 3 3 ] J.McCoo1, Embedding theorems for countable groups, to appear. [ 341 K.A.Mikhailova, The occurrence problem for direct products of groups, Doklady Akad. Nauk SSSR 119 (1958) 1103-1105. [ 351 C.F. Miller and P.E. Schupp, Embeddings into hopfian groups, J. Algebra, 17 (1971) 171-176. [ 361 J .Nicken, Untersuchungen zur Topologie der geschlossenen zweiseitigen Flachen, I, 11, 111, Acta Math. 50 (1927) 189-358; 53 (1929) 1-76; 58 (1931) 87-167.

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[ 371 K.Reidemeister, Einfiihrung in die kombinatorische Topologie (Braunschweig, 1932). [38] H.Schiek, Ahnlichkeitsanalyse von Gruppenrelationen, Acta Math. 96 (1956) 157-251. [39] H.Schiek, Das Adjunktionsproblem der Gruppentheorie, Math. Ann. 147 (1962) 159-165. [40] P.E.Schupp, On Dehn’s algorithm and the conjugacy problem, Math. Ann. 178 (1968) 119-130. [41] P.E.Schupp, On the conjugacy problem in certain quotient groups of free products, Math. Ann. 186 (1970) 123-129. [42] P.E.Schupp, On Greendlinger’s Lemma, Comm. Pure Appl. Math. 23 (1970) 233-240. [43] P.E.Schupp, Small cancellation theory over free products with amalgamation, to Math. Ann., to appear. [44] P.E.Schupp, Commuting elements in small cancellation groups, to appear. 1 [45] V.V.Soldatova, On groups with a &basis, for 6 6 , Izv. Akad. Nauk SSSR, Ser. Mat. 13 (1949) 483-494. English translation of [47,48,49], Amer. Math. SOC.Translations 6 0 (1952), reprint 1 (1962). [50] E.R.Van Kampen, On some lemmas in the theory of groups, Amer. J. Math. 55 (1933) 268-273. [5 11 C.M.Weinbaum, Visualizing the word problem, with an application to sixth groups, Pacific J. Math. 16 (1966) 557-578. [52] C.M. Weinbaum, The word and conjugacy problems for the knot group of any tame, prime, alternating knot, Proc. Amer. Math. SOC.,to appear.

I would like to mention t w o interesting results obtained since this survey article was written. C.M. Weinbaum [ 521 has shown that if G is the group of a prime alternating knot, then G * ( x ) (the free product of G and an infinite cyclic group) has a presentation satisfying C(4) and T(4). Thus both the word and conjugacy problems for G are solvable. We have not touched on small cancellation theory for semigroups in this article. John Remmers (Thesis, Univ. of Michigan, 197 1) has recently developed a geometric version of small cancellation theory for semigroups and extended the previous results.

THE ASSOCIATIVITY PROBLEM FOR MONOIDS AND THE WORD PROBLEM FOR SEMIGROUPS AND GROUPS Dov TAMARI Department ofMathematics, State University of New York at Buffalo

5 1. Associativity of monoids Definition of associativity A word with formally correct binary bracketing is a monomial. Parentheses are a priori indispensable for giving words of length > 2 an interpretation by computation in terms of the given, possibly partial, binary operation. Associativity of such a multiplicative system ( M , - ) is usually intended to mean the independence of the “value” or “sense” of words (if there is one!) from the distribution of the parentheses. The complexity and, in spite of it, insufficiency of such a concept of general associativity are well hidden for complete, i.e. everywhere defined, operations : M X M + M . We call these systems monads *, although they are more often referred t o as groupoids *; associative monads are semigroups. General systems with a partial operation : PM + M , where PM c M X M is an arbitrary binary relation over M , will be called

-

-

* Terminological remark: The legitimate use of the term “groupoid”

refers t o Brandt groupoids or relatively slight generalizations, synonymous to “partial groups”, “group germs” or “local groups” and “categories of morphisms”. “Partial groupoids” is a pleonasm and compounded abuse of language. T o correct the misused “groupoid” to “Ore groupoid” is unwanted; why should one honor the unlucky choice of terminology? It was unfortunate that Ore ignored the legitimate use of “groupoid”; but it was fortunate for category theory (or should one say for category theorists?) that, as a consequence, the limited rights of Brandt could be ignored in good conscience. Confusion is compounded by the use of “monoid” for “semigroup with 1” under the influence of Bourbaki (who probably intended t o abolish the term “semigroup” for reasons of linguistic ta\te). 59 1

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D. Tamari, The associativity problem for monoids

nzonoids", although they are more often referred t o as partial groupoids". One is inevitably led t o a problem of definition: What does or should associativity of monoids mean? This question is reduced t o that of monads as follows: M is associative if it can be completed to, i.e. embedded in, a semigroup; in other words, if there is an injective homomorphism (rnonomorphisrn)M S, Sa sernigroiip. One can choose S = S,, the semigroup generated by M (universal object in the category of morphisms of M into semigroups). M is therefore associative iff its distinct elements, considered as the generators or one-letter words of S , represent mutually distinct elements of S : s1 # s2 in M * s1 # s2 in S . The canonical homomorphism 4: M S maps each s E M into the equivalence class s@containing the one-letter word s. M is associative, or what is the same, 4 is a monomorphism, iff s is the only one-letter word in s@. -+

-+

Immediate consequences M associative card M = card M v< card S ; M associative & card M > 1 card S > 1, i.e. S is not trivial; M non-associative 3 card M > card M @* 3 a couple s1 # s2 in M but s1 = s2 in S. M @is the greatest associative homomorphic imuge o f M ( u p to isomorphism). Why has this simple conceptual definition of associativity not been used before? Its meaning in explicit, constructive terms, while simple and definite for monads, is combinatorially complicated for general monoids and cannot be written down in ordinary finite terms. For symmetrical monoids, or partial inverse property loops 19, 331 the definition in constructive terms is still infinite, b u t simplified by the "one-mountain theorem" which can be traced back t o Newman [ 2 5 ] . =j

=j

Construction of S , FM/EM Fin is the free semigroup over the set (support) IMI of M , EM the congruence generated over F, by the relations ab = c in M , acting as elementary substitutions: ab + c (products or contrac;F

See Terminological remark (preceding page).

D. Tamari, The associativity problem for monoids

593

tions) and c --z ab (factorizations or expansions). For A , B E F M , A = B in S , * A B (mod E M )* 3 a finite chain of substitutions B . A3 W o + W , +. ... +. Wi-l + Wi + ... +. W n B , CA.

-

where either Wi-l = Uiaibi 6 and Wi = UiciVi, or vice versa. “M associative” means that a chain contains at most one one-letter word, although possibly several times. Call chains starting and ending with one-letter words a, b special chains Cab; then: “M associative” * “C,b 3 a = b”. An inner piece of a special chain is a general chain. Conversely, every general chain can be extended to, or embedded in, a special chain over a suitable extension monoid.

Non-validity of the one-moun tain theorem for general chains Arranging words in levels by length, chains consist of alternating sequences of expansions or “ascents” and contractions or “descents”. The one-mountain theorem, valid for symmetrical monoids, permits limitation to chains with a single ascent followed by a descent. The monoid

M = {a, b, c, d, f,f ‘ ; a b= ac

= bc = cb = d , dc

= J d b = f ‘>

shows that the one-mountain theorem is not valid in general: f

+.

dc + abc +. ad

+.

acb + d b +. f’;

i.e., f = f ‘ in S, but not in M ( M non-associative). Yet, there is no word contracting over M t o f as well as t o f ‘. *

Universal and partial algebras A class A of algebras is equationally defined by basic identities (laws, axioms); what is the natural generalization t o partial or incomplete A-algebras? A partial algebra P with the “same” operations as A is an incomplete A-algebra if it can be completed t o , i.e. * In this example it is essential that M is not cancellative as (left) cancellation would

imply b = c and thus f = f ’ ;however, one can easily construct somewhat larger cancellation monoids similarly exhibiting non-validity of the one-mountain theorem.

D. Tamari, The associativity problem for monoids

5 94

tvnbedded in, an A-algebra A. As before one can choose the universal object in the category of morphisms of P into A-algebras, i.e. A = A , , the A-algebra generated by P. Unconditional equations, i.e. identities

I : f(xl , .. . , x , ~ =) g ( x l ,..., x n ) , f; g polynomials, are replaced, or “translated”, by implications from chains

where C; is a chain of elementary substitution relations o i ( i i 1 , ..., ~ 4 ~ =; ui, ) the oibasic operations, the ui and zii intermediary variables, c o m p u t i n g f ( x , , ..., x,,) = y and g(xl, ..., x n ) = z . Denote: C the axiom of complete operations; I. the set of basic laws of A; I = Up=, Ii the r.e. set of all valid identities of A in some convenient graduation I i ; I;, resp. I* = U,o IIF, the corresponding “translation” to implicational chains. In A

but in the corresponding class of incomplete A-algebras n o finite suhset I&) = U Z , I: needs imply all oJ’I*. Therefore the original, finite set of laws I , of A must be replaced by the infinity of implications I”, or, at least, by an infinite subset D * C I* such that D.4 1.4. j

Associativity chains For A being the class of semigroups, one single binary operation and one single identity ( x y ) z = x(yz) suffice as a set of basic operations and laws. For the class of incomplete semigroups, o r associative monoids, one needs an infinity of axioms”, a r.e. system of conditional identities in the form of implications from finite chains of ever-increasing length and complexity. These associativity chains * This insight had

been obtained independently and indirectly by Evans [ 121. From the fact that the word problem for semigroups is unsolvable he concludes that “no finite set of axioms for semigroups will admit an embedding theorem”.

D. Tamari, The associativity problem for monoids

595

are similar to the Malcev conditions for the embeddability of semigroups into groups. They are most conveniently pictured by literally the same graphs which are geometrical models of Malcev chains as introduced by Tamari [ 321. Their deduction and algebraic interpretation are even simpler and more general. To some degree this more basic meaning is already present in the older work of the early 50’s. The interesting theory of these associativity chains will appear in detail elsewhere.*

5 2. Presentations The proof of the equivalence of the associativity and the word problem is based on conceptual work, in particular on wider interpretation of the concept of presentations as partial algebras.

Preliminaries A word is a shorthand for all its monomials and the associativity relations equating them. These relations are a kind of a priori implied “trivial” defining relations. With the postulated existence of a monomial, the existence of all its components or szibrnonomials is implied; with the existence of a word, also that of the associativity relations in its subwords.”” The concept of presentation of semigroups ( G ,R ) , G an alphabet or set of generators, R a set of defining relations, is extended t o presentation of monoids (Q,R ) , Q a set of existing monomials (letters included) and R a set of defining relations among members of Q. Monoids themselves are presentations of themselves, as well * For a first informal account see “Le probleme de l’associativite des monoides e t le probleme des mots pour les demi-groupes: Algebres partielles et chaines elementaires”, Sem. Dubreil-Pisot 24 (1970/71) 8.01-15.

** These preliminary

conventions are not equivalent to consideration of semigroups in the category of monads. While abelian groups are singled out in the category of groups by the “trivial” commutativity relations of couples of generators, g&. = gg.. I I’ and groups in the category of semigroups by adjoining an identity, formal inverses, and the “trivial” defining relationsgig;’ = g;’gi = 1 , the associativity relations and new gencrators of the standard presentation below will not make the generated monad a semigroup. Thus associativity is a deeper property.

596

D. Tamari, The associativity problem for monoids

as of other structures, in particular of the semigroups generated by them. Conversely, every presentation of a semigroup can b e standardized t o a monoid generating the same semigroup (up t o isomorphism); finite presentations standardize t o finite monoids. Extensions A presentation of a semigroup, standardized or not, can be extended by adjunction t o Q of monomials, in particular of words. The extended standard presentations are extension monoids of the original standard presentation: The alphabet and the set of defining relations are extended by new letters for each new subword and the implied associativity relations. Construction of standard presentation Let M be a monoid, A = silsi2 ... sin an incontractible word over M. Construct first an extension M ( A ) of M with a chain sA E M ( A ) , SMcA,= S,: Screen and name the distinct two-letter subwords in A among the n - 1 neighbor couples sii sji+ sii ii+ ; the distinct three-letter subwords among its n - 2 neighbor triples introducing new symbols and couples of elementary relations

eA,

s i . s i . + l i j + 2 = Sii ii+ i i + 2 I

1

-

- Sii

ii+l Sii+ ;

the distinct k-letter words, k < n, among the n - k + 1 neighbor k-tuples, introducing new symbols and sets of k - 1 new elementary relations; finally,

with one new symbol and n - 1 new relations. Thus M is extended bya finite number of new letters and relations t o M ( A ) in which the word A is contractible t o one letter s A . Similarly, one can adjoin to M a set of words ( A , B, ...}. Assume all words incontractible over M , or replace them by incontractible contractions.* Screen * A contractible word may admit several incontractible contractions; any one will do.

D. Tamari, The associativity problem for monoids

597

and name their distinct neighbor couples, triples, ..., k-tuples, until the exhaustion of the longest words, introducing new symbols and sets of new elementary relations. One obtains an extension monoid M ( A , B , ...) generating the same S, because its elements are words over M and its relations anyhow valid in S,. Any C: can be embedded in a special chain CsB = C A C: C F . Prescribed relations, say A = B, are taken in accouns%y pifting sA sB and considering them one and the same symbol. Given any ordinary presentation of a semigroup ( G , R ) ,start with its set of generators and no relations; i.e., consider the abstract set G as a monoid with empty multiplication table. Adjoin the set of all words appearing in R and “take account” of all defining relations of R . This standardizes the given presentation t o a monoid. The result of the construction is unique up t o isomorphism. As each (monoidal) presentation can be extended by adjunction of any word, one can thus associate with each finite semigroup presentation r.e. classes of finite monoids, all presenting the same semigroup, for any r.e. set of words, in particular the r.e. set ojall words. Non-associativity Non-associativity of finite o r countably infinite symmetrical monoids is a readily assertable property. It suffices t o enumerate the words and t o subject each t o a finite number of finite computations. A word is non-associative if it can be contracted t o at least two distinct elements of M . I f M is non-associative, one must come across a first non-associative word, indeed across any non-associative word, in a finite number of steps. Therefore the set of nonassociative words over M - if not empty - is effectively constructible. For general, finite or countably infinite monoids for which the one-mountain theorem is not valid, non-associativity is still readily assertable. For finite monoids, the number of chains of all types of given length is finite because a word admits only a finite number of elementary contractions and expansions. This is still so for infinite monoids with the property that each element appears only

598

D. Tarnari, The assoeiativity problem for monoids

;I finite number of times in the multiplication table, i.e., admits only a finite number of binary factorizations. Even without such a property, one can enumerate all chains by rank r = I + i, I the length of the chain, i the highest index of letters in any of its words. A chain is non-associative if it connects two distinct letters. Thus, given an enumeration of all chains in order of increasing rank, if a monoid is non-associative one can find the first, and ultimately, a n y non-associative chain. Therefore the set of non-associative chains is either empty or effectively constructible. It follows that u.s long as one can decide i f ‘ afinite monoid is associative or not, one can ef,y?jctiwlyconstri ict its greatest associa t ive homomorp hic image M o M / E , by constructing the congruence generated by its finitely many “rion-associative” couples of elements connected by non-associative chains.

Homomorphic images A presenting monoid may o r may not be associative. I f it is nonassociative all its extensions are so, too. One can make sure that associative monoids are among the standard presentations of a semigroup by including liomomorphic images, i.e., closing the class for homomorphisms. Among them must be associative homomorphic images. The greatest associative homomorphic image still presents S and is identifiable with a generating subset of S and an extract of its multiplication table. A finite monoid admits only a finite number of congruences and corresponding homomorphic images. Thus a r.e. class of finite monoids remains r.e. by homomorphic closure. On the other hand, homomorphic closure introduces pre sentations of new semigroups which are proper homomorphic images of the originally presented semigroup. If one has an associativity test for the class of finite monoids under consideration, this can be avoided by limiting oneself t o congruences induced by non-associative couples only. Iterating these quotient constructions a finite number of times, one is stopped at the first, i.e. greatest associative homomorphic image. For monoids presenting S with a word comparison test one admits only congruences generated by ii f b in M , but u =s b in S . The meaning of “homomorphic image” is thus restricted from now on.

599

D. Tamari, The associativity problem for monoids

93. Reduction of the word problem of semigroups to the associativity problem for monoids Theorem. To any given finite presentation o f a semigroup one can construct a r.e. class ojfinite monoids such that an algorithmic solution of the associativity problem f b r this class implies a solirtion of the word problem jbr the semigroup, and vice wrsa. Corollary. The known existence of finitely presented semigroups with unsolvable word problem [ 27,34, 101 implies iinsoIvability o f the associztiuity problem f o r the corresponding classes o ffinite monoids and for the class o f finite monoids in general.

Remark. Checking associativity of finite monads is a finite routine task; passing t o monoids makes holes in the multiplication table; therefore, the “devil of unsolvability” slips in through these holes at the passage t o partial systems. Proof. The hypothetical solution of the associativity problem by an assocativity test AsT for a r.e. class Ms of finite monoids associated with S leads t o a solution of the word problem for S. AsT can be imagined as an oracle, black box or machine putting out the right answer t o the question “associative or not” for every monoid of the class fed into its input. Narrowing down the class apparently strengthens the result; but it is convenient to start with the whole class M of finite monoids; in fact, considerably smaller classes will do.* A priori the class is not r.e., even not a set. One means the finite monoids in standard notation with elements sl, s2, ..., sm from an infinite standard set of symbols sl, s 2 ,... . For each m exist ( m + I ) m 2 partial multiplication tables; therefore the standard class of finite monoids is r.e.; the isomorphism type of each finite monoid is represented in it.:::’:’

* The more restricted classes actually used are implicit in the construction of the proof.

Explicit definition of r.e. classes, good for both reductions, seems difficult and is itself a problem (see discussion of unsolvability degrees).

* * One can constructively assert the first appearance o f an isomorphism cursive enumeration of finite multiplication tables.

type in a re-

600

D. Tamari, The associativity problem for monoids

Let M be a standard presentation of S. AsT either asserts M as associative, or it enables one t o pick out its greatest associative homorphic image. Without loss of generality one can assume M = 114, associative. One constructs an ascending chain of finite associative monoids exhausting S and solving its word problem: 00

M 0 C M , C M , C . . - C S - J ~ ~ M , = U M,. n=O

Assume M , with elements s1 ,...,, ,s and its partial multiplication table already constructed. Arrange the incontractible words over M , in an order 8, progressing first by length, then by highest indexed letter in a word, and last by alphabet; e.g.:

nci;ab, ha, bh;ac,bc, ca, cb, cc;ad, ...;aaa;aab,aha, abb,baa, bab, hba, bhb; aac, ... . I n this way the last couples contain the last generator and are of form s ~ s , , , ~,,,s si;similarly for triples, etc. Denote a, h, = c',,, the first in contractible two-letter word over M,. Consider c, a new letter and u, b, = c,, a new relation yielding an extension MA = M , (c,). (c,, is of course the lettersm,+,.) Apply AsT toMA and distinguish two cases ( a ) MI, is ussociutive: Put MA = M n + l . Words containing u,b, which were incontractible become contractible: a, b, is replaced everywhere by c,. One gets a new enumeration en+, of incontractible words over M r z + : a, b, disappears from the head of the twoletter word sequence, while new incontractible words of form sicn, c,si appear at its end.* d (b) MA non-associative: 3 ! d, E M , I CCnn in M I , , i.e. d, c, = M:>. E , is a trivial congruence; its equivalence (mod E M ,). Mn class are $ngletons except for {d,, c,}. As convened, standardize by replacing c,, by d,, 3 s k (h- < m,). The supports of M , and M,+l are the same, but M,,, has an additional relation, anbn = d,. In the new order O n + , of incontractible words over M,+, again the first two-letter word of 0, disappears, but no mu' two-letter words

-

Possibly also C,C, if b,a, is incontractible over M,. Words heading the three- and more letter word sequences disappear and words with c, appear a t the ends.

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are added: formerly incontractible three- or more letter words containing anbn shrink by one or more letters. (c) From M , + l proceed as from M,. If the non-associative case b) repeats itself a sufficient number of times, one will be stopped at M n + p , with IM,+p, 1 = IM, I and completely filled multiplication has empty places. M n + p , = S is a finite semigroup; its table, if word problem is solved." (d) Otherwise one must encounter the associative case (a) sufficiently often, replenishing the supply of new incontractible twoletter words. One keeps going ad infiniturn. At each step a new relation is added; words shrink in length t o become replaced by existing or new incontractibles; from time to time a new generator is added. One exhausts all original two-letter incontractibles over M , and reaches M n + p , . All incontractibles over M , have disappeared and are replaced by shorter words consisting mostly of new generators. One continues filling the 4, S p , new lines and columns corresponding to the 4, new l e t t e r s ~ , , + ~ , s , , + ~ , ..., s,,+~,, etc. Any given, originally incontractible word of arbitrary length will eventually be reached, shrunk to one letter at some stage of sufficiently high index M,. Each word contracts finally t o one and only one letter representing an element of S v lim M,; its word problem is solved. One can express the result as follows: The unsolvability degree of the associativity problem for finite monoids, in particular also for the r.e. classes constructively associated t o a given semigroup, is greater or equal the unsolvability degree of the word problem of any finitely presented semigroup.

h,,

$4. The converse reduction

In order t o prove the equivalence of the word and the associativity problems one needs a converse reducing the associativity problem for sufficiently large r.e. classes M of finite monoids to the word problem of finitely presented semigroups SM associated This solves the finiteness problem as a special case of the associativity problem.

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with this class. Assumption of an algorithmic solution for the word problem of Sbd leads to the solution of the associativity problem for M. ‘The algorithmic solution of the word problem can be imagined ;is a word comparison test WcT (oracle, black box or machine) admitting as input couples of words in the generators and yielding as output correct answers ‘‘=s’’or “f,” (equal or not equal in S ) . Each monoid known t o be a presentation of S has elements expressible as words over the original presentation. Apply WcT to the finite number of couples of distinct elements written as such words. Getting each time the same answer “f,“ means “M is associative”, getting once “non-associative”. Doing this for the standard class of all finite monoids presenting S would solve the associativity problem for this class. The trouble is that this class is not r.e.; there is no decision procedure for the isomorphism problem of finitely presented semigroups. One has therefore t o be satisfied with a smaller, less elegantly defined, r.e. class of finite monoids, e.g., that defined by the construction of the former reduction. There the argument can evidently be inverse.d. This r.e. class M, starts with the finite monoid M and includes all its modifications by adjunction of words in a suitable enumeration and b y restricted homomorphic image. M generates the semigroup S, for which the WcT is assumed. A finite number of applications of WcT decides for each monoid of the class if it has distinct elements u, b with u =s b or not, i.e., if it is non-associative or not, if it admits a homomorphic image or not in the class, solving its associativity problcm. One has thus proven that the degree of unsolvability of the word problem for S = S, is at least as great as that of the associativity problem for the implicitly and vaguely defined r.e. class M M of finite monoids. As the classes M M used in both reductions are the same, one has thus proven that the word problem and the associativity problem are equivalent in this sense.

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55. Unsolvability degrees, open questions Contrary to what was said for the first reduction, the converse theorem gets stronger if the r.e. class of finite monoids, for which the associativity problem is reduced t o the word problem for a certain ad hoc constructed finitely presented semigroup, becomes wider. Clearly, classes which are too small can not do. Consider the extreme case of classes reduced to a single monoid; there is no genuine, i.e., algorithmic associativity problem. For an individual monoid the problem is decidable: it is either associative or not. One may say that its unsolvability degree is 0. True, the solution of the word problem for the semigroup generated by M solves the “associativity problem” for M as its generator or one-letter word problem; but to use an algorithmic solution for the word problem of S, for its solution seems an abuse.* Denote (YM the unsolvability degree of the associativity problem for a r.e. class M of finite monoids, PS that of the word problem of a finitely presented semigroup S. One has:

I (first reduction): For all finitely presented semigroups S = S, there exist r.e. classes Ms of finite monoids such that aMs 2 PS,. I1 (second reduction): Any finite monoid M generates a r.e. class of finite monoids M , and a semigroup S = S, such that P S M 2 aMM. I becomes stronger if M s gets smaller, I1 becomes stronger if M , gets larger. It is therefore desirable that Ms be as small and MM as large as possible, MM 3 Ms, and S = S M .This, certainly, would justify the conclusion crM = PS, i.e., a strong equivalence of both problems; but M, = Ms suffices. As a matter of fact the definition of MM = Ms is not very satisfactory. One can doubt that they are well defined r.e. classes, or even suspect circular definition. More t o the point is the recognition of a “grey” area (instead of a clear borderline) containing r.e. M M , Ms, which needs further re-

* “Like driving a small nail with a sledge hammer” (Boone), shooting birds with cannons, or, quite usual today, using universal computers for a few bills.

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D. Tarnari, The associativity problem for monoids

search. Does an ideal borderline exist’! If so, d o there exist r.e. M 01: it? Is there a “principle of continuity” assuring existence of r.e. classes on it? Can Mnf be replaced by larger and simpler r.e. classes? More generally, can one construct t o any r.e. class of finite monoids a fznitelv generated semigroup such that the solution of its word problem solves the associativity problem for the class? If so, this would in particular apply t o the standard class of all finite monoids and imply the existence of a “universal” finitely presented semigroup with word problem of unsolvability degree equal t o that of the associativity problem for the class of all finite monoids. Presently one can only assert that the unsolvability degree of this associativity problem is at least equal that of the word problem for fi 11i tely presented semigroups.

Another open question A direct proof of the unsolvability of the associativity problem for finite monoids (for certain r.e. classes), say by a diagonal method, is desirable. I t would immediately yield a more significant proof for the unsolvability of the word problem for semigroups and groups. Originally the search for such a method delayed publication of this work in the early ~ O ’ S until , it was lost.” At that time the author believed himself near his aim; the question is still open. $ 6 . Symmetrical monoids and groups

S,i’rnnzerricuZmoizoids (= partial inverse property loops; for definition see [ 9 , 3 3 1 ) generate by themselves semigroups which are groups. Symmetrical presentations standardize by themselves t o symmetrical monoids. For arbitrary monoids and presentations one needs a theory of symmetrization and symmetrizability. In this way methods and results transfer from semigroups t o groups. One has the advantage of the one-mountain theorem simplifying associativity, but the disadvantage of the more difficult word problem for groups. This method with emphasis on groups was pre* A handbag containing manuscripts and documents of the author disappeared in the har-

bor of Haifa, Israel, September 9, 1962. In spite of rewards offered it was never recovered

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valent in the earlier work of the author in the beginnings of the 60’s. An informal, partial account of some of its results is given in [ 3 3 b ] . Among others, it is proven there that a cancellation semigroup is embeddable in a group if and on& if its symmetrization is associative. Acknowledgements

As already mentioned the first ideas of this paper go back t o the beginnings of the 50’s [ 32a, b ] . T h e author feels deeply obliged to Professors H. Cartan, A. Chitelet and P. Dubreil, who made this earlier work in France possible. He further owes a great intellectual debt to Malcev; also a moral one for encouragement when he met him personally at the occasion of the I.C.M. Stockholm 1962. This paper owes its origin t o an attempt. t o freely reconstruct and to update, as far as feasible, work lost in 1962.” It is part of this attempt of reconstruction, which has perhaps partially succeeded. The author takes this occasion to thank those French, Dutch and American Mathematicians who helped him in his need by offering shelter, interest and generosity of mind; in particular in direct connection with this work thanks are due t o Professor Godel who invited him a second time to the Institute for Advanced Study at Princeton 1967/68; and t o the Organizers and Participants of these Proceedings, in particular to Professor Boone, for hospitality and critical discussions. Updating t o o is only partial, due t o the limited capacity and difficult circumstances of the author. During the last decade progress in the fields of Algebra, Universal and Quasi-Universal Algebra, and of Decision Problems has been great, generating a vast volume of literature. New, powerful tools like Category, Automata and Language Theories have become readily available. There are some near approaches in the direction of our subject, but the most relevant one, Evans’ papers [ 121, were already available in the early 50’s. Yet, t o the best of the author’s knowledge, aiid fortunately for this paper, and others to follow, the point of view of this work and its results have not been covered elsewhere. * See historical

remark in footnote preceding page.

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Bibliography and References [ 1] S.I. Adjan, Algorithmic undecidability of certain group properties, Doklady Akad. Nauk. SSSR 103 (1955) 533-535 (Russian). [ 2 ] S.I. Adjan, Defining relations and algorithmic problems for groups and semigroups, Trud Mat. Inst. Steklov 85 (1966). Translated A.M.S. 1967, IV + 152. [ 31 R. Baer, Free sums of groups and their generalizations. An analysis of the associative law. 1-111. Am. J. Math. 71 (1949) 706-742; 72 (1950) 625-646; 647-670. [4] G.E. Bates, Free loops and nets and their generalizations, Am. J. Math. 69 (1947) 499-550. [ S ] W.W. Boone, The word problem, Ann. Math. 70 (1959) 207-265. (61 W.W. Boone, Word problems and recursively enumerable degrees of unsolvability, A sequel on finitely presented groups, Ann. Math. 84 (1966) 49-84. [7] J.L. Britton, The word problem for groups; Proc. London Math. SOC.8 (1958) 493-506. [8] C.C. Bush, The embedding theorems of Malcev and Lambek; Canad. J. Math. 15 (1963) 49-58. [ 91 Carvalho and Tamari, Sur l’associativitk partielle des symbtrisations de semigroupes; Portug. Mat. 21 (1962) 157 169. [ 10) G.S. Cejtin, An associative calculus with an unsolvable problem of equivalence; Trudi Mat. Inst. Steklov 52 (1958) 172-189 (Russian). [ 1 1 ] R. Doss, Sur I’immcrsion d’un semi-groupe dans un group. Bull. Sci. Math. 72 (1948) 139 150. [I21 T. Evans, (a) The word problem for abstract algebras, J . London Math. SOC.26 (1951) 64- 71; (b) Embeddability and the word problem, ibid. 28 (1953) 76-80. [ 1 3 ) Freedman and Tamari, P r o b l h e s d’associativitd: Une structure de treillis finis induite par tine loi demi-associative; J . Comb. Th. 2 (1967) 215-242. [ 14 I A . A . Fridman, Degrees of Unsolvability of Word Problems for Groups (Nauka, 190 pp. (Russian). M o ~ c o w 1967) , [ 15 I A. Ginzburg and D. Tamari, Representation of binary systems and generalized groups by families of binary relations, Israel J . Math. 7 (1969) 21-45. [ 161 G. Higman, Subgroups of finitely presented groups, Proc. Roy. SOC.Ser. A 262 (1961) 455-475. [ 17 1 J . Lambek, The immersibility of a semigroup into a group, Canad. J. Math. 3 (1951) 34-43. [IS] A. Malcev, On the immersion of an algebraic ring into a field, Math. Ann. 113 (1937) 686-691. [ 191 A. Malcev, On embedding associative systems into groups I, Mat. Sbornik 6 (1939) 331-336; 11, ibid. 8 (1940) 251-263. [20] A . Malcev, Sur les groupes topologiques locaux et complets; Comptes Rendus (Dokl.) 32 (1941) 606-608. 121 ] A. Malcev, Quasiprimitive classes of abstract algebras (Russian); Doklady Akad. Nauk SSSR 108 (1956) 187-189. 1221 A. Malcev, Some questions bordering o n algebra and mathematical logic; AMS t ransl. [23] K.A. Mihailova, The occurrence problems for direct and free products of groups; Doklady. Akad. Nauk SSSR 119 (1958) 1103-05; ibid. 127 (1959) 746-748.

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[24] H. Neumann, Generalized free products with amalgamated subgroups, Am. J. Math. 70 (1948) 590-625. [25] M.H.A. Newman, On theories with a combinatorial definition of “equivalence”; Ann. Math. 4 3 (1942) 223-243. [26] N.S. Novikov, On the algorithmic unsolvability of the word problem in groups; Trudi Mat. Inst. Steklov, 44 (1955) 1-143. [27] E.L. Post, Recursive unsolvability of a problem of THUE, J. Symbol. Logic 12 (1947) 1-11. [28] V. Ptak, Immersibility of semigroups; Acta Fac. Nat. Un. Gar. Prague 192 (1949) 16. [29] M. Rabin, Recursive unsolvability of group theoretic problems, Ann. Math. 67 (1958) 172-194. [30] S. Swierczkowski, Some examples of non-associative monoids (symmetric or not) of low order; refers t o Malcev [20] ; a private letter communicated by J. Mycielski. [ 3 1 ] D. Tamari, Les images homomorphes des groupoydes de Brandt,Comptes Rendus (Paris) 229 (1949) 1291-1293; Representations isomorphes par des systemes de relations; ibid. 232 (1951) 1332-1334. [32] D. Tamari, (a) Malcev chains and generalized Malcev conditions, Commun. Int. Congr. Math. Cambridge 1950; (b) Mono‘ides pr6ordonnCs et chaines de Malcev; Thkse, Paris 1951; (c) part of it in Bull. SOC.Math. France 82 (1954) 53-96. [33] D. Tamari, (a) Imbeddings of partial (incomplete) multisystems (monoids), associativity and word problem, Abstract, Amer. Math. SOC.Notices, 7 (1960) 760. (b) Problkmes d’associativith des monoides et problkmes dcs mots pour les groupes; Sem. Dubreil-Pisot 16 (1962/63) 7.01 -30. [34] A.M. Turing, The word problem in semigroups with cancellation; Ann. Math. 5 2 (1950) 491-505.

MAXIMAL MODELS AND REFUTATION COMPLETENESS: SEMIDECISION PROCEDURES IN AUTOMATIC THEOREM PROVING* Lawrence WOS

George ROBINSON

Argonne National Laboratory, Argonne, Illinois

Stanford Linear Accelerator, Stanford, California

3 1. Introduction In recent years the idea of using electronic computers to search for proofs of theorems of quantification theory has drawn considerable attention. One of the more successful methods of attack on the problem has stemmed from the work of Quine [ 121 , Gilmore [ 5 ] , Davis and Putnam [4] and J . Alan Robinson [ 161. This paper is concerned with a portion of the theory underlying an extension of this line of development to systems - first-order theories with equality - in which there is a distinguished relation symbol for equality. The field of mathematics upon which we have concentrated our computer experiments in order to study various properties of our procedure is first-order group theory. To say that a particular property is decidable for some class of statements means that there is a single uniform procedure which will correctly determine whether the property holds when presented with any given statement from the class. Church showed [2] that, for any fixed procedure, there exists a statement of firstorder predicate calculus for which the procedure will not be able to answer correctly the question: is this statement a theorem? For group theory, the question of theoremhood is also known to be undecidable as proved by Tarski [ 171 . The situation is, however, far from hopeless for first-order theories at least. There do * Work performed in part under the auspices of the United States Atomic Energy Commission. 609

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exist procedures which will, if presented with a set of first-order axioms for a theory and a first-order statement that happens to be a theorem thereof, correctly identify the statement as a theorem. Such a procedure is called a semidecision procedure for theoremhood. The basis for such a procedure is often a set of inference rules (rules for reasoning from the axioms of the theory to the statement whose theoremhood is in question), in which case it is natural to call the procedure a proof procedure since the procedure generates a proof of the chosen statement if it is a theorem. In order for a procedure t o be of interest from the computational viewpoint, it must also be reasonably efficient. The indications are that the inference rules given in this paper may provide the basis for an efficient semidecision procedure for theoremhood not only for first-order group theory but for other first-order theories with equality as well. The reasons for expecting efficiency are: equality is treated as a special logical symbol distinguished from the relation symbols of the mathematical theory, the inferences deduced with the rules have a certain important property of generality that causes us t o call the rules conservative, and finally a number of inferences which are ordinarily obtained immediately in more classical systems are not deducible with these rules. This last property of nondeducibility is referred t o as the lack of deduction completencss. For example, not all theorems of the familiar first-order predicate calculus can be deduced from the proposed set of rules. Contrary to intuition, this is often an advantage computationally. The set, Il, of inference rules to be studied in this paper consists of resolution, factoring, and paramodulation. The first two are generalizations of well known infe-rence rules for the propositional calculus. Paramodulation is a generalization of a substitution rule €or equality. The formulation of resolution and factoring is essentially that of J. Robinson [ 161 while paramodulation originated with the present authors [ 13,151. It is desirable that the underlying set of inference rules have certain logical properties related to semidecidability. The key property is that of R-refutation completeness. A refutation is a proof of contradiction. It is proved herein that in the presence of functional reflexivity II is R-refutation complete, i.e., a refutation employing

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just the rules from FI exists for any set of statements (each of which is in the appropriate logical form) which possesses no equality model. In other words, if one starts with a statement which should lead to a contradiction with the usual meaning of equality, n is strcng enough to yield a contradiction in the presence of functional reflexivity. The general idea is as follows. To prove that a statement is a theorem of some first-order theory with equality, one proceeds by first denying the statement. Then this denial and the axioms of the theory in question are converted t o a particular logical form which may be said t o be, loosely speaking, a conjunction of statements each of which is a disjunction. The existential quantifiers have been replaced by Skolem functions and the remaining variables are considered t o be universally quantified precisely over the entire conjunct in which they occur. The resulting statements are called clauses. A contradiction is deducible (using the rules of II) from the set of clauses if and only if the original Statement was true in all equality models of the theory. The sufficiency requirement leads t o a definition of R-refutation completeness. The necessity leads to a corresponding soundness concept. Several examples from first-order group theory are discussed in Appendix I, and refutations (within II) for two of them are given. It is not within the scope of this paper to discuss the strategies which lead t o a promising semidecision procedure. Intuitively, “strategy” may refer t o the order in which the inference rules are applied or t o certain constraints placed on their application. In order t o have an efficient semidecision procedure, one needs, in addition t o good inference rules, various strategies. The procedure used in Corollary 2 is not one which is recommended. One of the most successful approaches is that which allows only those inferences which are in part traceable t o the special hypothesis of the theorem or t o the denial of its conclusion. This strategy is a special case of the set of support strategy. The set of support strategy is known to be R-refutation complete when coupled with n in the presence of functional reflexivity [ 2 1 1 . To illustrate the general approach, consider the theorem: if x 2 = e for all x in a group G, then G is commutative. In the nota-

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tion of first-order predicate calculus the axiom for the existence of an identity element (two-sided) is, ( 3 ~ ’()x ) [ Py x x A P x y x ] ,where Y denotes product. Replacing the existential quantifier by a Skolem function and then putting the results into the desired form (a sei of clauses), one has the clauses Pexx and Pxex. Since the computer attempts to find contradiction from assuming the theorem false, one has, in addition to the clauses corresponding to the other axioms, the clauses Pxxe and Pubc and Pbac. The last two clauses correspond t o the assumption that the group is n o t commutative, i x . , that there exists a pair of elements which d o not commute. The system n uses n o logical axioms nor does it contain any of the more familiar classical rules of inference such as universal instantiation. In the example above, the procedure would consist of applying the various rules of n, resolution, factoring, and paramodulation, in some order until a contradiction was found. Thus we are vitally interested in the property of R-refutation completeness. The use of Herbrand models throughout the paper rather than the more familiar concrete models provides a convenient tool for proving the theorems underlying the theory of the approach discussed herein. The theorem of logic which permits use of Herbrand models states in effect that a set has an Herbrand model if and only if it has a concrete model. In Section 2 we give definitions of a number of concepts, such as iritcrpretution and rnodel, which are necessary for our study. In Section 3 we prove the Muxirriul Model Theorem which states that, given a particular interpretation for a given satisfiab!e set of disjunctions, there is a model for that set of disjunctions, whose settheoretic intersection with the particular interpretation is as small as it can be without losing the property of modelhood. In Sections 4 and 5 we give the inference rules. In Section 6 we introduce the terms, rejiitatioii complete, deduction complete, and conservative and prove that the inference system (set of inference rules) n under study has the desired logical property of R-refutation completeness for firnctioizullv refZexivc sets. The usual axiom set for basic group theory [ 131 and also that for basic ring theory, for example, are functionally reflexive relative t o n.

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Although many classical inference systems have the property of refutation completeness, they are also deduction complete and not conservative. The system, I'I, on the other hand, is conservative but not deduction complete. In the light of experience with existing computer programs, this seems to be advantageous in using computers to search for proofs of theorems.

5 2.Definitions We shall deal with a language having as primitive symbols individual variables x l , x,, ..., x,, ... ; individual constants a l , a,, ..., a,, ...;p redicate constants (relation symbols) P: , P:, ..., Pi,...; and function letters f ; , f ; , ..., f i , ... (where superscripts indicate the number of arguments). (The equality predicate P: will frequently be abbreviated to R.) The remaining primitive symbol is a bar for negation. Individual constants and variables are terms; a function letter applied to n terms t , , .. . , t,, is also a term. A predicate letter PF applied t o n terms is an atounic,forruiulu or a t o m A literal is an atom q or the negation 4 thereof. If q is an atom, the negation of 4 is just taken to be q . The ubsolute value I12 I of a literal h is the atom y such that either 12 is q or h is 4 . A clause is a finite set of literals. Intuitively, one may view a clause as the disjunction of its literals, universally quantified (over the entire disjunction) on its individual variables. Also intuitively, existential quantification is (in effect) accomplished through the use of (Skolem) function letters. A ground clause (literal, term) is one that has no variables occurring in it. The result C8 of a substitution 8 = [ t , lit,, ..., t n / u nI (where the u i are all distinct) on a clause C (literal, term) is the clause (literal, term) obtained by replacing uniformly and simultaneously each occurrence (if any) of each variable t i i (i = 1 , ..., n ) by the corresponding term ti. Here the clause (literal, term) C8 is called an instance of C. Suppose that a substitution 8 transforms S , some set of literals, into a set SO. Then if for all literals h , and h , in SO, Ih, I = I h , I , we shall call 8 a unifier (or match) of S . The set

fi"

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S is said to be unifzuble if such a 8 exists.* If 0 transforms all members of a set T of terms into a single term, 8 is likewise called a iirzifi'eror match of T. The process of finding and applying a unifier or match is called uniji'cation or matchiuig. If a substitution 0 can be obtained by composing two substitutions 0 and 8, in that order, then 8 is an instance of 0 . If one clause (literal, term, substitution) is an instance of a second and the second is also an instance of the first, then each is a variant of the other. A finite, non-empty set of clauses (literals, terms, substitutions) that have an instance in common have a most general commo~zinstance (not necessarily unique), i.e., one that itself has as instances all common instances of the originals. Similarly, if a finitc, non-empty set S of literals (terms) is unifiable, then it has a most general ziiziji'er (match), i.e., a substitution 6' that unifies S and such that every unifier of S is an instance of 8. If a tern1 t occurs as the i,-th argument of the ik-1 -st argument of ... of the i, -st argument of a literal h , then the ordered k-tuple (i, , ..., i k ) is called the position vector of that occurrence o f t in I?. Two terms are in the same position in their respective literals if they have the same position vectors. The Newbrand ziniijerse H,y of a set S of clauses (literals, terms) is the set of all ground terms that can be constructed from the symbols occurring in S. Here if no individual constant occurs in S , a l is to be added to the vocabulary. An Herbrand atom for a set S of clauses is a predicate letter Py from S applied to n terms from H,y. An interpretaticn I of S is a set of ground literals whose absolute values are a11 the Herbrand atoms for S , such that for each Herbrand atom j exactly one of j and 7is in I . An interpretation I sutisfks the ground clause C' if at least one of the literals of C' is in I , i.e., if I n C' f 0. The interpretation I satisfies the (possibly non-ground) clause C if it satisfies every ground instance of C (over the Herbrand universe under consideration); it satisfies the set S ojclauses if it satisfies each clause in S ; it condeinns the groiind c.la2r.w C' if it contains the negation of every literal of C'; * Whcn a finite set S of

literals is to be unified, we d o not usually find it particularly inrtructivc to view t h a t sct as a clause, although it is not set-theoretically distinguishable from a clause.

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61 5

it condernns a clause C if it condemns some ground instance of C (over the Herbrand universe under consideration); and it condeinns the set S of clauses if it condemns some clause in S . When the empty set of literals is viewed as a clause jalse, it can be satisfied by no interpretation and is vacuously condemned by every interpretation. Note however that the same set-theoretic object @, when viewed as a set of clauses, is vacuously satisfied by all interpretations. A model M o j t h e set S of clauses is an interpretation of S that satisfies S ; it is an R-model of S if, in addition, the set { ( t l ,t2)lRt,t2 E M ) 6%the set { ( t l ,t , ) l P ~ t , t ,E M } )is an equality relation for the terms occurring in M , i.e., if each of the following is true for all terms s and t in H, and all function letters

fk":

(i) (ii')

Rtt E M . If the literal k E M and Rst E M and if h is obtained by replacing in k some one occurrence of s by an occurrence oft, then h E M . It is often more convenient to replace (ii') by the condition If an atom k E M and Rst E M and if h is obtained, etc. (ii) since (i) and (ii) are equivalent to (i) and (ii'). The other familiar properties follows easily from (i) and (ii'): If Rst E M , then Rts E M. (iii) If Rst E M and Rtu E M , then Rsu E M . (iv) For all t o , t l , ..., t,, in H,, if Rtito E M and foccurs in (v) some literal in M , then R f ( t 1 , ..., tj-i. tj, ..., t n ) J ' ( t l ,..., t j p l , t o , ..., t n ) E M .

53.The Maximal Model Theorem One can associate with each interpretation T of a set S of clauses a partial ordering relative to T of the set of all interpretations of S , and hence of the set of models of S , given by: M I G T M , if and only if M , n T 3 M , n T. The maximal model theorem states that, given a set S of clauses

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possessing a model and an interpretation T of S , among the models of S there exists one, say M , such that n o other model of S has a strictly smaller set-theoretic intersection with T.

Maximal model theorem. I j a set S o f clauses has a model, then, g i w n any interpretation T cif S, S has a maximal model M, relative t o T. Proof. Let T' = { X I IX E T } ,o r equivalently the set of the negations of the literals in T. Then T' is an interpretation of S , and T' n T =

8.

I f we show that every simply ordered (relative to T ) set o f models of S has an upper bound which is itself a model, then an application of Zorn's lemma will suffice. Let the set { H , } a t A be a simply ordered (relative to T ) set of models of S . Let M=U,,,(H,n T ' ) u ( n a E A ( H , nT ) ) . M w i l l b e s h o w n t o b e both an upper bound for { H , } and a model of S. That 11.9 is an upper bound (if it is an interpretation) follows from considering IIo for an arbitrary /3 in A and noting that H , n T'c U, ( H , n T ' ) = M n T ' , since T n T' is empty. Thus H , n T ' c M n T', hence H , n T 3 M n T and H , G T M . To see that M is an interpretation, consider an arbitrary literal h in T since every Herbrand atom is represented in T either by itself or its negation. If b is in IZ, for all a in A, then b is in n, ( H , n T ) so h is in M . If for some (Y, the literal b is not in H a , then for that a , 5 is in Ha since each Ha is an interpretation. But b E T ' , so 6E U, (Ha n T') C M . This shows that at least one of b and 6 is in M . To see that only one of b and 6is in M assume that b is in M . Then since b is not in T' it cannot be in U, ( H , n T') and so must be in n, ( H , n T) and hence in ll, H,. But with b in every II,, 6 cannot be in any H , and hence not in U, ( H , n T ' ) and hence not in 11.2. To see that M is a model of S , consider some arbitrarily chosen ground instance D of some clause in S.

L. Wos and G. Robinson, Maximal models and refutation completeness

Case I .

D

6 11

C TI. Let H , be arbitrarily chosen. Since H , is a model

of S, H , n D is not empty and so contains some literal, say c. Since D is contained in T' by assumption, c is in T ' , so c E T' n H , c U, (Ha n T ' ) c M . Therefore, c E M n D.

Case 2. D $ TI. Then D n T = { cl, c2, ..., ck } for some finite k > 0, since D has only a finite number of literals and since all literals of all ground instances of all clauses of S are in T u T ' . If some ci, say c, is in all H a , then c is in fl, ( H , n T ) c M , and M n D is not empty. Otherwise assume that for 1 < i < k no ci is in all Ha. Therefore, for each ci, there is an H,, say Hai, with ci 9Hai. But the H , are models and hence interpretations, so Ci E Hai for 1 < i < k . Since c , , ..., ck are in T, C1, ..., ck are in T ' . Since the family { H , }cy,,is simply ordered relative to T , there is among H a , , ..., H a largest (relative to T ) H,,,, say H . From the crk definition of < T , if a literal is in some Hai n T ( 1 < i < k ) then it is in H n T ' . Therefore, { C1, ..., ck } C H n T ' . Since H is a model and hence an interpretation of S, no ci for 1 < i < k is in H . But H n D f 8 since H is a model of S , so H n D contains some element a. Since no ci is in H , a # ci for each i. Therefore, a @ T since T n D = { e l , c2, ..., ck }. So a is in T ' , and therefore a E H n T ' . But H n T' c U , ( H , n T ' ) c M , so again M n D # 8. Since the cases for D are exhausted, and since D was arbitrarily chosen, for each ground instance D of a clause in S, M n D # 8. M is thus a model of S. The conditions for Zorn's Lemma being satisfied, there exists a maximal (relative to the ordering associated with T ) model M , of S. Q.E.D.

In a similar fashion one can, by applying the maximal model theorem to T ' , show that there exists a minimal model of S with respect to T . In a later section we apply the theorem to the case where T is the set of Herbrand atoms for S. In this case, the key point is that if M , is a maximal model relative to T , for each atom k in M , we can find a ground instance D of a clause of S with D n M , = { k } .

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This in turn allows us certain applications of paramodulation, which we shall define in a later section. Corollary A. If'sis u set of clauses, T a n interpretation of S, and M a tna.wirna1 (relative to T ) model of's, then, .for each literal b in lcl n T, there exists a clause in S having a grotind instance (over /Is) D with LI n M = { b } . Proof. Let S be a set of clauses, T be an arbitrary but fixed interpretation, and M be a maximal (relative to T ) model of S . Assume by way of contradiction that there exists a literal b in M n T falsifying the theorem. Since M is a model of S we can conclude, therefore, that, for any ground instance (over H,) C of any clause in S, ( M n C ) - { b } is not empty. Let M * be obtained from M by replacing b by 6. M * , therefore, has a non-empty intersection with all ground instances (over H s ) of all clauses in S. For each Herbrand atom u, M" will still contain exactly one of a and Z. M" is, therefore, an interpretation of S and, hence, a model of S. Since h is in T by assumption and T is an interpretation, 6 is not in T. Therefore, M n T contains M* n T as a proper subset. This contradicts the maximality of M , and the proof is complete. Corollary A and the maximal model theorem establish that there are models (the maximal models) which possess the intersection property (given in the conclusion of corollary A) upon which the proof of R-refutation completeness rests. That this intersection property, however, does not characterize maximality can be seen from the following example. L c t S = { n , r ~ } , w h e r c A = { P a , Q aa}n d B = { P u , Q a } . L e t T = { Pa, QN} . Let /lI* = { Pa, Qu } . lcl* is not maximal relative t o T , for the model M = { pa, O u } is such that M* n T = { Pa, Q a } contains M n T = 8 as a proper subset. M is itself a maximal model relative t o T . M * , however, has the property that, for every literal h in /21* n T , there exists a clause in S one of whose ground instances intersects ill'#i n exactly b. This example shows that the property of rnaxiinality for models is not equivalent t o the intersection pro pert y .

L. Wos and G. Robinson, Maximal models and refutation completeness

6 19

Depending on S and T , one cannot be assured of the existence of a unique maximal model. In the example just given, M is the only maximal model. But, if one replaces S by S* = { A *, B * } , where A * = { Pa, Qa } and B* = {pa,o a }, there are two maximal models. M I = { Pa, Qa } and M , = { P a , Qu } are both maximal models relative to T. By definition, a model M is maximal relative t o T if and only if, whenever M n T contains M* n T for a model M * , M r'l T = M* n T. Using the definition of interpretation one can easily show that a model M is maximal relative t o T if and only if, whenever M n T contains M* n T for a model M * , M is equal t o M * settheoretically. 34. Resolution and Factoring

In this section we give a brief account of an inference rule called resolution. This rule may be viewed as a generalization of modus ponens and hypothetical syllogism.

Definition (Resolution). If A and B are clauses (with n o variables in common) with literals k and h respectively such that k and h are opposite in sign (i.e., exactly one of them is an atom) but I kl and I h I have a most general common instance rn, and if u and T are most general substitutions with m = I k I u = I h I T , then infer from (any variants A * and B* o f ) A and B the clause c' = ( A - { k } ) u u ( B - { h } ) T . C is called a reSoh~t7tof A'%and B* and is inferred by resolzition I 14,161.* Example 1. Premisses { Pubc} and { pxyz, P y x z } yield by resolution { P b a c } . In this example resolution has the effect of first instantiating the second premiss (which might be thought of as the commutative axiom in a group theory problem) so that one has two premisses t o which r n o c l ~ i s[ionem can be applied, arid then ap* Note that the prernissesA * and B* t o which the rule of inference is applicd. arc not

themselves subject to the re3triction of having no variables in common. Given an arbitrary A * and B * , in practice, one need merely re-letter B* t o obtain a variant B having n o variable in common with A * before constructing the resolvent C' from A * and B .

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plying rnodzcs ponens. The clauses which serve as premises for the modzis ponens applications are most general instances o f the given premises, most general with respect to permitting application of modus ponens. Example 2. Premisscs {Pax, Qx} and {Pyb, Qyi yield by resolution { Qa, Q b } . Syllogistic inference is not directly applicable to the given pair of clauses. The clause { Qa, Q b } ,however, can be inferred by instantiation followed by a syllogistic inference. By definition, resolution, in effect, seeks most general instances of the clauses which will permit syllogistic inference. Example 1 shows that resolution yields in the spirit of modus ponens an inference from pairs of premises even though rnodus ponens does not apply directly to those premises. Example 2 illustrates the corresponding relationship to syllogistic inference. By placing n o restriction o n the (finite) number of literals in either premiss, resolution extends both rnodzis ponens and syllogistic inference in yet another way. Example 3. From { Nx,Px} and { P y , Qyl infer by resolution { Ry, Q.y }. This is the familiar categorical syllogism, all F are G, all G are H , so all F are 19. Definition (Factoring). If A is a clause with literals k and h such that k and h have a most general common instance m , and if a is a most general substitution with ka = ha = m , then infer the clause A' = ( A - { k } ) o from A . A' is called an immediate factor of A . The factors of A are given by: A is a factor of A , and a immediate factor of a factor of A is a factor of A . Example 4. From { Qax, Qyb, Qziz} infer (among other clauses) as an immediate factor { Qab, Q u z } ,which in turn has an immediate factor { Qub }.

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$5. Paramodulation The concept of equality is ordinarily handled in first-order theories in one of two ways: 1) a set of explicit first-order axioms (or axiom schemata) is given for the equality predicate; 2) a substitution rule is supplied together with the axiom of reflexivity. It is in the context of the second approach that the inference rule paramodulation is to be understood. With the appropriate constraints on the variables in the terms s and t , one version of the substitution rule for equality permits inference of the formula p ( t ) from the formulae Rst (i.e., s = t ) and p(s). Paramodulation [ 13,15 ] extends this substitution rule by permitting inference from the formulae Rst and p ( u ) when the terms s and u , though not identical, have a common instance. (Since the only formulae of interest throughout this paper are clauses, paramodulation is defined only for clauses.) Definition of Paramodulation. Let A and B be clauses (with no variables in common) such that a literal Rst (or R t s ) occurs in A and a term u occurs in (a particular position in) B . Further assume that s and u have a most general common instance s’ = su = ur where u and r are most general substitutions such that su = ur. Where B is obtained by replacing by to the occurrence of ur in the position in Br corresponding to the particular position of the occurrence of u in B , infer from any variants A * and B* of A and B respectively the clause C = B U ( A - { Rst } ) u (or C = B U (A- { R t s } ) ~ C ) .i s called aparamodulant o f A * and B* (and also of B* and A *) and is said to be inferred by paramodulation from A * on the variant of Rst (or Rts) into B* on (the occurrence in the particular position in B* of)the variant of u . The variant of the literal Rst (or Rts) is called the literal oj’paramodulation and the occurrence of the variant of ZI is called the term ofparamodulation. * Before giving examples to aid in one’s understanding of paramodulation, we remark that examination of the definition shows * Sce footnote on page 000 for a comment o n how separation o f variables works out in practice.

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that the property of symmetry is inherent in the inference rule. It is also to be noted that, although in much of what follows we assume the presence of the reflexivity axiom, the definition of paramodulation could have been easily extended t o obviate the need for this axiom. The extension does not, however, seem t o be of practical value. Example 1. Premisses { R u b } and{ Q a } yield { Q b } by paramodulation o r by the usual substitution rule for equality. Example 2. Premisses { R u b } and { Qx} yield { Q b } by paramodulation. They also yield by paramodulation { Q u } . The substitution rule does not apply i n its usual form since neither term, u nor b , occurs in the premiss {Ox}.I n many systems from { Qx} one could infer { Q u } , which could then have been used as one premiss together with { R u b } and the substitution rule to yield { Q b } . Example 3. Premisses { R u b } and { Qx.fx} yield among others the clause { Qb, Pu} by paramodulation. Again in many systems { Qu, P u } could have been inferred from { Qx, Px}, providing a premiss t o which the substitution rule for equality would apply. Example 4. l’reniisses { Rxlz ( x ) } and { Qg(j>)}yield by paraniodulation i Qh(g( I.))}. Note that the presence among the constants occurring in the set of clauses under consideration of the individual constants a and h does not lead by paramodulation t o either the inference { Q h ( g ( a ) ) } or { Q h ( g ( h ) ) } .Paramodulation (in effect) first finds most general instances of the two premisses which permit straigh tfonvard application of the substitution rule for equality and then makes the equality substitution.

} {fyf(g(y)z)z}. Example 5 . Consider the premisses { R f ’ ( x g ( x ) ) e and For intuitive purposes think of P as product, f as product, and g as inverse. The functions f and g are frequently Skolem functions introduced in place of existential quantifiers. For this application of paramodulation let s be f (.xg(x)) and zl be j ( g ( y ) z ) . A most general common instance of Z I and s is f ( g ( i,)g(g(1%))).The infer-

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623

ence thus made is { P y e g ( g ( y ) ) } Note . that both premisses required non-trivial instantiation in order to apply the substitution rule. Example 6. Premisses { R , f ( x g ( x ) ) e Q , x } and { P y j ’ ( g ( y ) z ) zQ, z } yield by paramodulation with s and u as in Example 5 { P y e g ( g ( y ) ) ,Qg( y ) , Q g ( g ( y ) ) }. This example illustrates another way in which paramodulation extends the substitution rule, as both premisses in this example contain more than one literal. The substitution rule applies to pairs of formulae one of which is of the form { R s t } . The extension in the direction illustrated by Example 6 is needed for R-refutation completeness as can be seen by examining the following set of clauses: { { Rab, Qc }, { Rg(a)g(b),Qc 1, { R e d , & c L { R g ( c ) g ( d ) ,a c 1 , { R x x ) } . A most crucial property of paramodulation was illustrated by Example 4. From a theorem-proving viewpoint it is important that the inference rules have the property of being conservative (see Section 6). 36. Refutation Completeness

A set S of clauses implies (R-implies) a clause C if no model (R-model) of S condemns C, and S implies (R-implies) a set T of clauses if it implies (R-implies) each clause in T . We write S k C, S C , S t= T, S k R T respectively to express these relationships. If for clauses C and D, { C} I= D ,we also write C I= D and say that C implies D (and similarly for k R ) .If A implies (R-implies) B , then B is a consequence (R-consequence) of A . If S has no model (R-model), then S is unsatisjiable (R-unsatisfiable). A deduction D o j a clause C,, f r o m S in an injerence system C2 is a finite sequence of pairs (Ci,J i ) i = 1,2, ..., n where Ci is a clause and Ji is a “justification” of Cj in terms of one of the rules of inference of S2 and previous steps of D or in terms of membership of C j in S. (By inference system we mean here nothing more

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than a set of rules of inference.) If such a deduction exists we write S I k i 2 C,, . Except when the justifications J i are of particular interest [ 2 11, D will be identified with the sequence C , , C 2 , ..., C,. A rejiitutioti of S in 52 is a deduction of false from S in s2. Henceforth n will be the inference system whose rules of inference are paramodulation, resolution, and factoring and C will be that whose rules are just resolution and factoring. A system s2 is dediiction complete (R-dedtiction-complete) if C whenever S I= C for any set S of clauses and any clause C, S ( S I-n C if S kR C). I t is (R-) rejutution complete if for any (R-) unsatisfiable S, S I-s2 julse. The system C is well known to be refutation-complete [ 161. (An adaptation to the present formalism of the simple but rather elegant proof in [ 161 is given in Appendix 1 .) If C were deduction complete as well, one could easily prove (see next paragraph) that n is R-refutation complete (in the presence of the usual reflexivity axiom). (More generally, any deduction complete system s2, when augmented by paramodulation, becomes R-refutation complete i n the presence of reflexivity.) Both 2: and II would then, however, be virtually worthless for automatic theorem proving algorithms of the type in use today! Let f2 be any deduction complete system including the inference rule paramodulation and let S be any R-unsatisfiable set of clauses including { R x s } . Consider the tautology { R x y ,R x y } , a trivial logical consequence of S. Deduction completeness of s2 would give S { Rxy, Rxy }. By paramodulating from this tautology into { R x x 1, one could show that S I-n { R x y , R y x } . By paramodulating from this clause into itself, one can deduce { Rxy, Rzjl, Rxz } . Since { { Rxy, R-vx}, {&y, a z y , Rxz }} I= {Ex),, Rj*z,Kxz }. it follows by deduction completeness of s2 that S {fixj..Rj.3, R.Y:}, completing the equivalence relation properties for R . For the predicate substitution property, consider f i x l ... xk....xi, for arbitrary choice of i, j . and k . From the dexi,P,!xl ... xk...xi}. duction completeness of n,S I-n {
I,. Wos and G. Robinson, Maxinial models and refutation cornpleteness

625

property, since { ~ x x } { R f : ( x x k . . . x i ) , t ; i ( x l. _ x_k_. . . x i ) } , deduction completeness of &? gives { R x x } I-R { R f y ( s ... , xk ... xi) f ; ( x l ... x k ... x i ) . Paramodulating from symmetry into this clause on x k would give { R x x l I k R { R X ~ + ~ R.f;'(x, X ~ ,... X ~ + ~ . . . XX ~ ) fi((xl... x k ... x i ) } . Hence S I-n S U t: where E is a set of equality axioms strong enough t o guarantee that any model of S U E would also be an R-model of S. Suppose then, that S is R-unsatisfiable and includes { R x x } .S u E could then have no model; hence S u E k j a l s e (vacuously). Again applying the hypothesis of deduction completeness of 52, one could show S LJ E I-R false and hence S I-n false. Since S was assumed only t o be R-unsatisfiable and t o include { R x x l , this shows that &? is R-refutation complete in the presence of { R x x ! . To assess the problems involved in proving refutation completeness of systems that, unlike the hypothetical &? above, are useful for (present techniques of) automatic theorem proving, one must take care to note that one of the principal rmsons that the systems under study here are wejtil is that they are devoid oj' the familiar deduction-completeness properties of many of the customary formulations of quantification theory, hypothesized for the system &? above. (So that, for example, the Godel completeness theorem fails to hold for C and II.) The simplest difference is that universal instantiation is not usually possible in C or n. For example, { R x a } { R a a } , but { R a a } is not deducible from { R x a } in either Z , or n. The property that makes systems such as C and II valuable for automatic theorem proving is that they are conservative in the following sense: Loosely speaking, an inference rule is called conservative if it does not allow a proper instance c" of a clause C to be inferred from, say A and B , when C itself could have been inferred [ 131. (A proper instance C' of C is an instance for which C is not an instance of C'.) Resolution is a conservative inference rule. From { P f ' ( x ) ,Q u } and { p y , Q y } one infers by resolution { Q f ( x ) , Q a } and not, for example, { Qf(a), Q a } . The latter, of course, can be inferred by instantiation followed by a syllogistic inference. The instantiation involved therein is, however, not most general among those permitting syllogistic inference. Factoring is also a conservative infer-

+

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ence rule, permitting, in effect, only the most general instantiations that allow application of the equivalence of p v p v q with P V

4.

Paramodulation, in effect, seeks instantiations that are most general among those that permit equality substitutions. In addition to extending substitution in the two ways already discussed, it is important that certain inferences are avoided. (See Example 4 in Section 5 . ) From a pair of premisses paramodulation yields an inference which could be made after some number (possibly zero) of applications of instantiation to either or both premisses, followed by an application of an equality substitution rule. I t combines instantiation with equality substitution in a way such that the property of being conservative is present. Among the possible instantiations which permit application of equality substitution, paramodulation. i n effect. allows only the most general instantiations. The intuitive comparison above of resolution and paramodulation with syllogism (or rnodus ponens) and substitution, respectively, is intended as motivation for, not as a precise characterization of, resolution and paramodulation. For the latter one must refer t o the definitions. The intuitive discussion might, for example lead one t o take C = ( A o - { k o } )U (BT - { h ~ }as) the resolvent instead of C = ( A { k ) o U ( B - { h) ) T as we intend in the present paper. If A = {I'll, P.u 1 and B = {Pa, Qh } , we intend the resolvent obtained from choosing P.x and Pa as focal literals to be c={ P U , Q h } , not c={ Q b } . A set S of clauses is said t o be functional1.v rrjlexive if { R x x } is in S , and for each function letterf'i occurring in S , ~ R . f ~ (.... r.u,)fk(x1 l ... xi)I is in S . S is said t o be firrzctioizallv reflexive relative to a n inference system R if S {Rx.xi and for eachf; occurring i n S, S I k R i Kf/(.rl ... xi)f'(.xl ... xi)}. T h e principal result of this section is that the inference system I1 is K-refutation complete for sets S that are functionally reflexive (or equivalently, for S functionally reflexive relative t o n).For deduction-complete systems R , the condition that S be functionally reflexive relative t o R reduces to the condition that S F { R x x l . Fortunately for efficiency in automatic theorem proving, TI is not deduction complete: consequently, the additional functional rcflcxivity properties are not so easily eliminated. (When paramodulation was first introduced a ~~

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few years ago, we conjectured that S I-n { R x x } was sufficient for R refutation completeness of II. N o counterexample has yet been discovered to this conjecture, but neither has a proof been given that the completeness theorem proved in this section can be replaced by the simpler and somewhat more attrative conjecture. An earlier, weaker version of the completeness theorem was proved in [ 151. There it was required that S bl, { R t t } for all terms t E H,, a much stronger restriction since there are infinitely many such clauses R t t , but there are only n + 1 functional-reflexivity clauses where n is the number of distinct function symbols occurring in S. Several systems, such as basic group theory [ 131, satisfy the hypothesis of the present completeness theorem (Corollary 1) but fail t o satisfy the hypothesis of the completeness theorem in [ 151 .)

Lemma 1 . Let A E Shave a ground iristance (over H,) A' = AT such that in A and A' respectively the terms t i * in the literal m* and u' in the literal m' = m*r are in the correspondirig positiorzs. Then there exists a factor E of A and a substitiition p such that (1) A' = E p is an instance of E, ( 2 ) E and A' have the same number o f literals, and ( 3 ) there exists a literal m in E uiid a term ~i in in with m p = m' and with u in m in the corresponding position to 10 in m'. Proof. The substitution 7 induces a partition of the clause A given by j equivalent t o k if and only if jr = k r , where j and k are literals in A . The sets A i of literals of the partition are each unifiable. There exists, therefore, a most general single substitution 8 which simultaneously unifies each of the A i . Since r unifies each set A j , there exists a substitution p with r = Op. Let E = AB. Let m = m*O, and let u = t i * B . The clause E has the desired properties. Lemma 2. Let A' und B' be respectively groirnd instarices (over Cf,) o j A und U in S, and let C' be a paramodulant o f A' utid B'. Fiirther assume that the literal, Rs't', ofpararnodiilation is in A' and that the term, i l ' , of'puramodulation is in the literal In' of' B'. Let the position ojri' irz m' be given by the vector ( y , , y z , ..., q H )Let . the substitution p and the literal m* in B be such that B p = B'

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and rn*p = m'. The11i f therc exists a term zi* in In* stich that u* and ii' are iii tho corresponding position respectiveli~in m * and m ' , theri tliere exists N clairse C huvirig c" as an iristance siich that C is a paramodtiluiit of some factors E mid I; b f A and B respectively

Proof. From Lemma 1 we can conclude the existence of factors E and F o f A and R respectively such that E and F have n o variables in common, E has A' as an instance, E and A ' have the same number of literals, F has B' as an instance, I.' and B' have the same number of literals, and F contains a literal I I I containing a term ZL with i i in rri and 11' in 111' in corresponding positions. Furthermore, we can conclude the existence of substitutions r and 8 with A' = Er, 13' = F O , 111' = me, and ii' = 118. E contains a literal K,t with Rstr = Rs't', where Rs't' is the literal of paramodulation in A ' . Assume without loss of generality that Y' is the argument involved in the inference by paramodulation of C ' . Then .\ and ZI have a common instance. Let h and p be most general substitutions such that sh = zip is a most general common instance of s and if. Let b be obtained from Fp by replacing irp in the position ( y I , y 2 , , q r i )in m p by t h . Let C = ( E { R s t } ) A u F c i s a paramodulant of E and F having C' as an instance, and the proof is complete. ~

Leninia 3. Let S be ci fiinctioliullv rellijuive Jet o f clairses closed z i d w ho th purai~zocliilutioriarid factoriiig / f A' and B' are respectiveh groiirid iiistuiicos ( o w r H,) o f ' A and B iti S a n d i f C' is a paramodiilant of A' u i i t l B ' , theii there exi.rts a C iiz S having C' as a11 itistuiice

Proof. Assume without loss of generality that the paramodulation is from -4'into B'. Let Rs't' in A' be the literal of paramodulation, and let the literal m' i n B' contain the term 11' of paramodulation in the position ( y l , q 2 . . ., q , l ) .Let p be such that B' = B p .

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629

Case 1. There exists a literal m* in B with m*p = m' and with m* containing a term u* in the position ( 4 1 , 4 2 , ..., 4 n ) . Lemma 2 applies to A , B, A ' , and B' to yield a clause C having C' as an instance. C is in S since S is closed under both paramodulation and factoring. Cast. 2. No such m* exists in B. Let m in B be such that mp = m'. There exists, therefore, a term (a variable) x in m in the position ( q l , 42, ..., q k ) with k < n. Thus there exist p = n - k functionsf) such that p contains the element f i (... f 2 (...(... f, (... u' ... ))))/x. Let G,, G,, ..., G, be the functional reflexivity axioms corresponding to f l , j , , ..., f , . G iare in S since S is functionally reflexive. A set containing G,, G,, ..., G, and B and closed under paramodulation contains a clause B, with the following properties: Bp8 = B' for some substitution 8, B, contains a literal m, with m,O = m ' , and m, contains a term up with up in m, and u' in m' in corresponding position. By applying the argument of Case 1 t o A , B,, A ' , and B ' , we can complete the proof. Theorem 1 . I f a functionally reflexive set S is closed under paramodulation and factoring, and ijrS is R-unsatisfiable, then S is unsatisfiable. Proof. Assume by way of contradiction that S is satisfiable, and hence, has a model. Let P be the set of atoms over H s . By the maximal model theorem S has a maximal model M relative to P. We shall show that M is an R-model of S thus contradicting the R-unsatisfiability of S. Since S is functionally reflexive, Rxx E S. For arbitrary t E H,, therefore, R t t E M since M is a model of S. Consider arbitrary s and t in H s , and assume Rst in M . Let k and h be atoms (in P ) such that h is obtained from k by replacing some one occurrence of s by t. Assume k E M . By Corollary A there exist ground clauses A' and B' such that A' n M = { Rst} and B' n M = { k } . Let A and B in S be such that A' and B' are respectively instances of A and B.

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Let C‘ = (A’ - {Rst}) u (B’ - { k } ) u { h } .C‘ is a paramodulant of A ’ and B’. By Lemma 3 we can conclude that there exists a C E S having C’ as an instance. Since M is a model of S , the intersection of M and C‘ is n o t empty. This intersection, therefore, equals { h } ,and so h E M . Thus ill has been shown t o be an R-model, and a contradiction is reached. The proof is complete. The desired proof, in the presence of functional reflexivity, of the R-refutation completeness of II follows as a corollary from Theorem 1 . Corollary 1 . For sets (consisting of clartses) functionally reflexive rclutivc to II, Il is R-refiitution complete. Proof. Let S be a functionally reflexive relative to II, R-unsatisfiable set of clauses and let S* be its closure under paramodulation, resolution, and factoring. S* is therefore functionally reflexive. By Theorem 1 , S* is unsatisfiable. By the Resolution Theorem (see Appendix l ) , fulse must then be in S*. But every clause C in S* is the last line of a deduction D , , ..., Dn of C from S in II with D iE S“ for i = 1 , ..., i i . Hence S* contains a refutation of S in II. Corollary 2, khr ji’iiite or eJfi7ctivel.y erizirnerable sets S that are fiiiic~tioiiallj~ rc~flesiverelative to II, there is u seiiii-decision procedirre f i )r K-urisati,sf~uhilit~,. The proof of this corollary is a form of a well known argument, whose details are supplied for the reader who may be unfamiliar with the formalism of paramodulation and resolution. The semidecision procedure exhibited in the proof is chosen t o make the argument transparent. It is tiot recommended as an efficient theorem-proving procedure; efficient procedures generally require a number of “strategies” for determining the sequence in which inferences are t o be made and for suppressing unwise applications of the rules of inference.

L. Wos and G. Robinson, Maximal models and refutation completeness

631

Proof. Let S be a finite or effectively enumerable R-unsatisfiable set of clauses functionally reflexive relative to II. Consider an effective enumeration C,, C,, .._of S. Let W 0 = 8. For i > 0, let Wi = W i - l u { Ci } U Qi, where Qi is the set of all clauses C such that C is a paramodulant, resolvent, or factor of clauses in Wiand no lexically earlier variant of C has that property.* Each Wi is finite and can be effectively generated from W i - l . For each clause C in the set S* defined in the proof of Corollary 1 , Ui Wi contains a variant of C. Hence false E U i Wi and thus is in some Wn.

To complete the proof, note that for any clause D,S k n D only D ,so that S knfalse only if S is R-unsatisfiable.

if S

By examining the proofs of Theorem 1 and Corollaries 1 and 2, we can easily obtain parallel results about an inference system in which paramodulation is subject to the following constraint: allow paramodulation only when the term of paramodulation is contained in a positive literal. Thus, in the clause {Pb, Q b } ,only the first occurrence of b would be admitted as a term of paramodulation. One can then prove, for example: if a functionally reflexive set S is closed under paramodulation (restricted by the constraint above) and factoring, and if S is R-unsatisfiable, then S is unsatisfiable. The proof is identical to that given in Theorem 1. The parallel to Corollary 1 also holds when paramodulation is so constrained. If the semidecision procedure of Corollary 2 is modified by constraining paramodulation as above, the property of semidecidability is not lost.

* From any pair of clauses that have at least one non-ground resolvent,

one can obtain infinitely many resolvents since any non-ground clause has infinitely many variants and any variant of a resolvent of A and B is also a resolvent o f A and B . Hence the need to select a single representative in order that Q' and W' remain finite. A similar situation exists for paramodulation and factoring.

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Appendix 1

Refutation Completeness Theorem for C . [ f ' S is un 1' unsutisjiuble set o j c~luuses,tllerl s txtcllse. Proof. (Adapted from [ 161 .) Let Y be the closure of S under resolution and factoring. Since for each clause c' in Y , Y contains (all the steps of) a deduction of C from S in C, we need only show that fhlse E Y.Consider an ordering C I , , u2,... of the Herbrand atoms for S. Let Ic>= 0, and for j > 0, let Ii = IiplU {ai } if IiU { Zi } condemns some clause in Y , otherwise let Ii = IiplU {Zi }. Let I = U i I j . I is an interpretation of Y , but cannot be a model of Y, since it would then be a model of S. but S is unsatisfiable. Hence I condemns some clause C in Y . that is, for some ground instance C" of C, the negation of each literal of C' is in I . Since there are only finitely many literals in C ' , C' must be condemned by I j for sonic finite j . Let J be the smallest non-negative integer such that Ii condemns (some ground instance c" o f ) some clause C in Y . The following establishes that j = 0: Suppose that J > 0. Then Ii..l condemns no clause i n Y while Ii condemns some claiise C'. I t must be that Ii = I i P 1 u {ai} since Ii = I i - , u {ai> only if Ii-., u iui} condemns no clause of Y,and by hypothesis Ii condemns C'. But I,. u tui t must also condemn some (groiuid instance D' of some) clause D in Y , since otherwise /i would be Ii L 1 u {cli 1. Ii by hypothesis condemns neither C' nor D'. The literal ifi must therefore occur in C' and ai must occur in D'. Hence there must be a set C" of literals contained in C with and ( C ~ - C * ) o= C ' - { L l j } a substitution u such that C*u = and a set D* C I1 with a substitution T such that D*r = { a i } and ( D - D * ) T = D' ~-{ uj 1 . Let u* and r" be most general unifiers for C" and D" respectively. Then C', = C'u" and D , =Or* are factors of C and D respectively and must thus be in Y . The factors C , and D , then resolve on the literals C*u* and D*r* to give a clause E also in Y . The clause E ' = ( C ' - ~{ Z i } ) u ( D ' - { u i } ) is a ground instance of E . Since Ii u { a i } condemns C', Iipl must condemn C'- {Zi};and similarly since Ii u { Z i } condemns D', must condemn D' { a i } . Hence Ii must condemn E , contrary t o hypothesis. Hence j 0, so j must be zero.

,

~

{q}

~

~

~

>

,

~

L. Wos and G. Robinson,Maximal models and refutation completeness

633

Thus 1, = @ condemns some clause Co in Y . But this is possible only if Co =false. Hence false E Y , completing the proof. To see that factoring is needed for refutation completeness one need only consider the example where S is composed of the clauses { Qa } , { Px, Py } , and { Pw,Pz } . Then the closure under resolution alone includes only variants of clauses in this unsatisfiable set S and of {Pu,P t } , but does not include false, so no refutation can be obtained. Resolution is sometimes defined in such a way as to subsume factoring. While this appears to simplify a few definitions and theorems, it leads, we feel, to undesirable consequences when applied to automatic theorem proving. Hence we prefer the older formulation in terms of separate rules for resolution and factoring.

Alternate Proof o f the Maximal Model Theorem: Since the proof of the maximal model theorem appearing in Section 3 was first given [ 191, a number of other proofs have been found. The simplest we have been able to work out to date is obtained by adapting an idea found in the proof of Lindenbaum's Lemma that is given in [ 111. Briefly, the alternative proof thus obtained runs as follows: Consider an enumeration a , , a,, ... of the interpretation T. Let bi = Zi if S u { { b l } , ..., { b i p l}, { Z j } } is satisfiable, otherwise let bj = aj. Let Q, = 8 and f o r j > 0 let Qi = { { b ,}, ..., { b j } } Let . M = { b , , b,, ...}. Then M is an interpretation, and furthermore if M is a model from the manner of construction, it must be a maximal one relative to T. To see that M is a model, suppose that it is not. Then, since M is an interpretation, it must be that M condemns some clause in S. Hence the set W = S U { { b , }, { b,}, ...} must be unsatisfiable. Some finite subset W' of W must then be unsatisfiable. Without loss of generality let W ' be such that no proper subset of W ' is unsatisfiable. Then let W ' n { { b l } , { b , ,... } } = { { b i l ,... } {bik}}andletnbethe largest subscript on b appearing therein. Now f o r j > 0, S U Qi must be satisfiable if S U Q j - l is, since otherwise both u {aj} would be unsatisfiable. S u Qj-l U {ai} and S u Hence, since S is satisfiable, S U Qn must by induction be satis-

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I,. Wos arid G. R o h i ~ s o nMaximal , models and refutation completeness

fiable, which contradicts the unsatisfiability of its subset W ’ . Thiis ill condemns no clause in S and must be a model of S, hence a maximal one relative to T. The construction given in the proof of the Refutation Completeness theorem for z1 can also be made to yield a maximal model. Failure to recognize the importance of closure of Y under resolution and factoring to the modelhood of I was, however, led to a number of spurious “constructive” proofs of the maximal model theorem. Indeed, if S is finite and the closure restriction is satisfied by S itself, I is a model, and an effective means of enxmerating 1 is provided by the technique given in the refutation completeness proof for C. Luckham has successfully applied a similar technique [ 9 ] to a useful special case involving a finite set of ground clauses. It appears [ 101 that attempts to give a proof of the general Maximal Model theorem embodying an effective enumeration of the maximal model may be foredoomed by the existence in first-order logic of sentences for which there exist n o effectively enumerable models, hence n o such maximal ones.

Appendix 2

I n this section we list a number of theorems from group theory and some from ring theory whose proofs have been obtained by one of our theorem-proving programs. One should not draw conclusions from comparing the times given to obtain proof, since the times obtained were affected materially by the fact that a number of different programs were employed. We shall also include the proofs of two theorems from group theory t o illustrate the inference rule, paramodulation. 1 . In a group, if xy = e , then y x = e. T h e time required to obtain a proof was 8 15 milliseconds. 2. If xy = e and ZJ’ = e, x = z . 5.3 sec. 3. The right inverse of the right inverse of x = x. 326 milliseconds. 4. (.rl )-l = .Y. 998 milliseconds.

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Example 3 differs from example 4, obviously, by not treating inverse as known t o be t w o sided.

5. A non-empty subset H of a group is a subgroup if and only if, for every x and y in H. xy-l is in H . The necessity was proved in 1.4 sec., assuming that identity and inverse of the subgroup were that of the group as a whole. The sufficiency was proved as a series of lemmas. That H contains e required 89 milliseconds t o prove; for every x in H , H contains x-l , 222 milliseconds; H is closed under multiplication, 3 2 sec. 6. Exponent 2 implies commutativity. 8 sec. 7. The axioms of right identity and right inverse are dependent axioms. The proofs were obtained respectively in 2 seconds and 3+ seconds. 8. Exponent 2 implies commutativity, but using only those axioms sufficient to obtain a proof. 460 milliseconds. The addition of various lemmas, just as the presence of a full axiom set, can help o r hurt the proof-search. In problems of some depth. lemmas will almost certainly be needed. 9. Subgroups of index 2 are normal. The theorem was proved by dividing it into two cases, as is often done in the standard proof. The proof for invariance with respect to elements of the subgroup was obtained in 4.9 sec., for elements outside the subgroup in 27.5 sec. 10. Boolean rings have characteristic 2. 4 1.7 sec 1 1. Boolean rings are commutative, assuming the lemma of characteristic 2. 12+ min. 12. If a set is closed under an associative operation, contains an element e with e2 = e , and every element has (with respect to e) at least one left inverse and at most one right inverse, then the set is a group. A proof was obtained in 5.8 sec. Of the examples given, example 12 is clearly the most difficult, mathematically speaking. The theorem is not easy t o prove for the average student of group theory. On the other hand, 18.2.8 on page 322 of [ 7 ] has not yet been provable by a computer. The lemma states that exponent 3 implies ((u, b ) , h ) = e for all u and b in the group. The proof is shortened

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I.. Wos and C. Robinson, Maximal models and refutation completeness

considerably (see appendix of [ 131 ) by the addition of paramodulation t o the most successful of the previous theorem-proving systems. That previous system was based on resolution. We now give a proof, employing paramodulation, of the theorem that the axiom of right inverse is dependent o n the set consisting of left identity, left inverse, and associativity. In the following, R is read as equals, f a s product, g as inverse, and e as the left identity. Assume by way of contradiction that there exists an element u lacking a right inverse. The resulting clause is { R f ( a y ) e } . Braces and commas will be omitted in the following. Proof. 1. Rf'(e.u)x. Left identity Left inverse 2. Rf'(g ( . x ) x ) e . 3 . R.f(-rf(.vz> ) . f ( f ( x).z~) . Associativity 4. R . f ( a . ~ ~ ) e . 5 . R.f(f(ea)js)e, paramodulate 1 on x into 4 on u. 6. Rf(f'(f'(s(w>w)u).1.')e, 2 into 5 on first occurrence of e. 7. R.f'(.f'(g(i.v>f'(wa))l~)e,3 into 6 on f ( f ( g ( w ) w ) u ) . 8. Kf'(.f'(g(g(u))e).v)e, 2 into 7 on J'(wa). 9. K f ' ( g ( g ( r r ) ) . l ' ( e ~ ) ) 3c , into 8 on f ( f ( g ( g ( u ) ) e ) v ) . 10. R f ( g ( g ( u ) ) j , ) e ,1 into 9 on f ( e y ) . 1 1. ,fulst., resolve 10 and 2.

For the 12th example, the following clauses were input t o the theorem-proving program, PG5 [ 201 , and the proof which follows the input (in essence) was obtained in 5.8 sec. 1. P X l , j ' ( X l ~ ) . Closure 2. P e w . 3 . Pg(.x).xe. Left inverse 4. pxJ.11pw11 P l l Z W PXVW. Associativity 5 . PXJ'Ll p v z v PVLJW Puzw. 6. ?%j.e p ~ z Rj?z. e 7 . RXY. Reflexivity 8. ??VJ'll PYJ'l'Rl4V. Well-definedness 9. R.uy RJ)ZRxz. Transitivity

L . Wos and G. Robinson, Maximal models and refutation completeness

10. Ruv Puxy Pvxy. 1 1. Ruv Pxuy Pxvy. 12. Ruv Pxyu Pxyv. b 13. Ruv Rf(ux)J’(vx). 14. Ruv Rf(xu)f(xv). 15. Ruv Rg(u)g(v). I 16. peaa.

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Substitutivity of equals

e is not a left identity

Proof. 1. Peee. 2. Pxyu Pyzv Fuzw Pxvw. 3. Pxye Fxze Ryz. 4. R U V Pxyu Pxyv. 5 . Peaa. 6.Pg (x)xe.

7.PXYfbY). 8.Rf(xy)v Pxyv,resolve clause 4 on second literal against clause 7

9. Rf(ea)a, 82 (i.e., second literal of clause 8) vs. 5. 10. Pyzv Ff(xy)zw Pxvw,2, vs. 7 . I 1 . Pf(xe)ew Pxew, 10, vs. 1. 12. Pxef(f(xe)e). 11 vs. 7. 13. Pg(z)ye Ryz, 3, vs. 6. 14. Fg(a)f(ea)e, 132 vs. 9. 15. pyzv pezw Pg(y)vw, 2, vs. 6. 16. pezw Pg(y).f(yz)w, 15, vs. 7. 17.Pg(y)f(ye)e, 16, vs. 1. 18. Rf(xe)x, 13, vs. 17. 19. Pezv Pf(f(xe)e)zw Pxvw,2, vs. 12. pezv Pxzw Pxvw, demodulate [Reference 201 second literal of 19 with 18. 20. Fxzw Pxf’(ez)w, 19; vs. 7. 21. Pg(x)f(ex)e, 20, vs. 6. 22. false, 21 vs. 14.

,

PG5 gave as output only 1 thru 5 , 9 , 12, 14, and 21. The following proof is also of example 12 but employing paramodulation as an additional inference rule.

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I,. W o s atid G Rohinson. Maxinial rnodels and reftitation completeness

Proof. 1 . R f ' ( c e e. ) 2. R.f'(.uj'(.vz)).f( f ( x y ) z ) . 3. R j ' ( x y )e R.f'(xz)e ~ y z . 4. K.f'('g(-u).r)c. 5 . R j ( CLl) a. 6. E j ' ( g ( z ) j . ) cK J ~ Z .resolve 3 , vs. 4. 7. R j ' ( g ( o ) j ' ( d ) eresolve , O7 and 5. 8. R j ' ( ~ ( ~ ) , ~ ' ( . i paramodulate , ~ ) ) / ( ~ ~ ~4) into , 2 on j'(.uy). 9. R . j ' ( ' g ( ~ , ) , j ' (paramodulate ~ , ~ ~ ) ) ~ ~ , 1 into 8 on j ' ( e z ) . 10. Rj(,t'e)!,,resolve 6 , and 9. 1 1 . K,f'(.~.f'(c,))j'(.uz), paramodulate 10 into 2 on J ' ( x y ) . 12. R.l'(g(x)f'(c.u))e, paramodulate 4 into 1 1 on .f'(xz). 13. . ~ ~ ~ I . Sresolve P, 7 and 12. Acknowledgements

The authors are deeply indebted to W.F. Miller for his support and cncouragement of the automatic theorem-proving effort over the years; to 11.L. Luckham and C.C. Green; and most particularly to W.W. Boone for his extensive and invaluable assistance in the preparntion and criticism of this paper. An important portion of the fundamental research underlying the results in this paper was wpported b y the Computer Science Department of the University o f Wisconsin and the University of Wisconsin Computing Center during 1966-67. The current work is supported by the U.S. Atomic Energy Commission. References [ I I A . ('liurcli, Introduction to ~llnthcmuticalLogic 1. Princeton (1956). 121 A. ('hurcli. A note on t l i c Fntsclicidungsproblern, J . Syinb. Logic 1, (1936) 40-41. 101-102. 13) .I. Darlington, Automatic theorem proving with equality substitutions and mathci 1 i : i t i c . d induction, in: I). Zlichie (ed.), Machine Intelligence (Edinburgh Univ. Press and Oliver and 13oyd, 1968).

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(41 M. Davis and H. Putnam, A computing procedure fo; quantification theory, J. Assn. Comput. Mach. 7 (1960) 201-215. [ 5 ] P.C. Gilmore, A proof method for quantification theory: its justification and realization, IBM Journal (Jan. 1960) 28-35. [ 6 ] C. Green, Theorem-proving by Resolution as a Basis for Question-answering Systems, in: B. Meltzer and D. Michie (eds.), Machine Intelligence 4 (Edinburgh Univ. Press and American Elsevier, N.Y., 1969). [ 7 ] M. Hall, The Theory of Groups (Macmillan, New York, 1959), p. 322. [ 8 ] R. Kowalski and P. Hayes, Semantic trees in automatic theorem proving, in: B. Meltzer and D. Michie (eds.), Machine Intelligence 4 (Edinburgh Univ. Press and American Elsevier, N.Y., 1969). [ 9 ] D. Luckham, Some tree-paring strategies for theorem-proving, in: D. Michie (ed.), Machine Intelligence 3 (Edinburgh Univ. Press, 1968). [ 101 D. Luckham, Personal communication (1969). [ 1 I ] E. Mendelson, Introduction to Mathematical Logic (Van Nostrand, 1964). [12] W . Quine, A proof procedure for quantification theory, J . Symb. Logic 20 (1955) I41 -149. [ 131 G. Robinson and L. Wos, Paramodulation and theorem-proving in first-order theories with equality, in: B. Meltzer and D. Michie (eds.), Machine Intelligence 4 (Edinburgh Univ. Press and American Elsevier, N.Y., 1969). [ 141 G. Robinson, L. Wos and D. Carson, Some theorem-proving strategies and their i n plementation. AMD Tech. Memo No. 72, Argonne National Laboratory (1964). [ 151 G. Robinson and L. Wos, Completeness of pardmodulation, J. Symb. Logic 34 (1969) 160 (abstract). [16] J.A. Robinson, A machine-oriented logic based on the resolution principle, J . Assn. Comput. Mach. 12 (1965) 23-41. [I71 A. Tarski, A. Mostowski and R. Robinson, Undecidable theories (North Holland, Amsterdam, 1953). [ 181 L. Wos, D. Carson and G. Robinson, The unit-preference strategy in theorem proving, AFIPS Conference Proceedings 26 (Spartan Books, Washington, D.C., 1964) 6 15-62]. [ 191 L. Wos and G. Robinson, The maximal model theorem, J . Symb. Logic 34 (1969) 159-160 (abstract). (201 L. Wos, G. Robinson, D. Carson and L. S h a h , The concept of demodulation in theorem proving, J. Assn. Comput. Mach. 14 (1967) 698-709. [ 211 L. Wos and G. Robinson, Paramodulation and Set of Support, Symposium on Automatic Demonstration, Lecture Notes in Mathematics 145 (Springer, Berlin, 1970).

Problems

1. Let II be the inference system consisting of the inference rules paramodulation, resolution, and factoring. For sets of clauses which contain a reflexivity axiom for equality but are not necessarily functionally reflexive, is II R-refutation complete? In other words, if the set S of clauses is R-unsatisfiable (has no equality model) and contains a reflexivity axiom, is II strong enough to guarantee the existence of a refutation of S solely employing the inference rules in II? Roughly, the question amounts to asking whether II is sufficiently strong as to provide the basis for a semidecision procedure for theoremhood for first-order theories with equality. (The natural approach to this problem would be to apply the so-called “lifting lemma”. Contrary to intuition, the standard type of “lifting lemma” does not hold, and the modified forms of the lemma that are known to hold have not yielded a proof of the desired completeness theorem.)

2. Find a “direct” proof of the unsolvability of the associativity problem for finite monoids by a diagonal procedure. 3. Solve the same question for the class of finite associative symmetrical monoids.

4. Or, as next best, consider proofs reducing directly to Turing machines as recursive functions, without the intervention of groups (or semi-groups). 5. Determine the existence of (Turing) degrees of unsolvability of associativity problems of monoids, symmetrical monoids, etc. (Conjecture: there are associativity problems of any given degree of unsolvability.)

6. Reduce Miller’s Tableau on special group classes with solvable and unsolvable problems to symmetrical monoids, in particular, for the corresponding embedding problems. 64 1

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Pro blerns

7. The non-associative finite tnonoids, in particular the symmetrical ones, are effectively enumerable (in principle). Establish a reasonable practical method of enumeration giving those of lowest orders without gap, and in particular, establish an enumeration for strongly non-associative monoids, i.e., having n o non-trivial associative homomorphic image.

8. Extend the method of group and semi-group diagrams (Dehn’s “Gruppenbilder”) t o monoids. 9. Is the word problem for Thue systems with one relation solvable? I n problems 10- 14 below let G be a Greendlinger group, (or a small cancellation group in general).

10. Does G have unique roots? T h a t is, does xn = y n imply x = y ?

1 1 . Does G contain a free nonabelian subgroup? 13. Does G satisfy the ascending chain condition for n-generator su bgro 11ps?

13. Is G conjiigacy separable? 14. Does C have a solvable commutator problem, i.e., given an arbitrary word it’ E G can we decide if w is a commutator?

IS. Is the conjugacy problem solvable for free products of Greendlinger groups with a cyclic amalgamation‘? 16. Are small cancellation groups Hopfian (residually finite)’? The gerierulized word problenz for the group I1 relative to the Jiiiitelt,geiicrntrd siibgroup K is the problem to determine for an arbitrary word i v E H whether o r not I.V E K. (Here, regard K as given by a finite set of words which generate K . ) Then the gcticruliied word problem for H is the union of the generalized word problems for H relative t o each K, as K ranges over all finitely generated subgroups of H.

17. Do small cancellation groups have solvable generalized word problems?

Problems

643

18. Are the conjugacy problem and the generalized word problem solvable for one-relator groups? 19. Is the isomorphism problem solvable for one-relator groups? 20. Is there a one-relator group whose word problem is not solvable by a primitive recursive function? 21. When are one-relator groups Hopfian (residually finite), (SQuniversal), (orderable)? 22. Is the conjugacy problem solvable for GL(rz, Z)? 23. Does the automorphism group of a finitely generated free group have solvable conjugacy problem? 24. Is the conjugacy problem solvable for finitely presented residually nilpotent o r residually free groups?

25. Is the word problem, conjugacy problem, and generalized word problem for finitely presented solvable groups solvable? (One probably wants t o consider “almost finitely presented” groups, i.e., finitely generated groups defined by a solvability law and a finite number of additional relations.) 26. Does there exist a finitely presented group which satisfies a non-trivial identity and has unsolvable word problem?

27. Does the free group of a finitely based variety of groups have solvable word problem? Or, to frame the question in a more combinatorial fashion: for any finite set L of laws defining a variety % ( L )of groups, the set Q ( L ) of all consequences of L , ix., the set of all laws holding in % ( L ) ,is obviously recursively enumerable. But is there a choice of L for which Q ( L ) is not recursive? 28. Does there exist a finitely presented infinite periodic group? Does there exist one of bounded exponent? 29. If G is a sixth group, urn# I , un # 1 in G and the commutator [ u r nu, n ] = 1 in G, must u and u be powers of a common element?

30. (Michael Rabin) Is there a decision procedure t o decide if a finitely presented group has a non-abelian free quotient? (Negative

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Pro blerns

answer gives an independent proof of the unsolvability of Hilbert’s Tenth Problem.)

3 1. Same as above replacing “free” by “finite”. 32. Let G be a finitely presented group. Let K be the intersection of all normal subgroups of finite index in G. Can G / K have unsolvable word problem? (A positive answer would show the open sentence problem for finite groups undecidable.) 33. Same as above with “finite” replaced by “free”. (A positive answer would show the open sentence problem for free groups, and hence Hilbert’s Tenth Problem, undecidable.)

34. Does there exist a recursive class i2 of groups with uniformly solvable word problem such that the set of groups in i2 which are residually free is recursively enumerable but not recursive. (A positive answer would give an independent proof of the unsolvability of Hilbert’s Tenth Problem.) 35. Same as above with “free” replaced by “finite”. 36. Do there exist finitely presented groups G , , G , with solvable power problem such that G , X G, has unsolvable power problem? 37. Let F , , F , be two free groups of the same finite rank with 6 : F , + F, an isomorphism. Let H , be a finitely generated subgroup of F , . Does the free product with amalgamation G = F , * B , H ,F , have solvable conjugacy problem?

38. Is the conjugacy problem solvable for knot groups? 39. Can one solve the word problem, conjugacy problem, and isomorphism problem for fundamental groups of 3-manifolds?

40. Does the automorphism group of a fundamental group of a 2-manifold have solvable conjugacy problem‘? 41. Call a finite presentation of a group bulunced if it has the same number of generators as defining relators. Is there an algorithm to decide if a balanced presentation is trivial?

42. Consider the class W of groups which can be built u p from free groups by taking Britton extensions and free products with

Problems

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amalgamation. Can those groups in W with solvable word problem be characterized algebraically? Similarly, can those groups in W which are fundamental groups of free manifolds be so characterized? 43. Can a finitely presented group with solvable word problem be embedded in a finitely presented group with solvable conjugacy problem?

44. Does a finite extension of a finitely presented group with solvable conjugacy problem have solvable conjugacy problem?

45. Does there exist a finitely generated (but not finitely related) residually finite group with unsolvable word problem? 46. What is the smallest number of relators a presentation with unsolvable word problem can have? Same for the conjugacy problem.

47. Does there exist a finitely presented residually finite group whose word problem is recursively but not primitive recursively solvable?

A finite simplicia1 complex, K , is called collapsible if there exists a sequence of subsets of K, K = K O ,K,, ..., K, such that K , consists of a single vertex and for i = 1 , 2, ..., y1 K j p l = K j u { A j ,Bj} where A is a proper face of no simplex of Kj- and Bjis a proper face of A j but Bjis not a proper face of any other simplex of Kj-l. A polyhedron P is called collapsible if there exists a finite simplicia1 complex K such that K is collapsible and the underlying polyhedron of K is piecewise linearly homeomorphic t o P, i.e., some triangulation of P is collapsible. 48. Let C be a collection of contractible polyhedra. Can one decide which members of C are collapsible? 49. Let C, be a collection of collapsible polyhedra. Is there an algorithm for finding some way of collapsing each element of C,?

(A collection of polyhedra that is of particular interest in connection with the 3-dimensional Poincare conjecture is the collection C = { Q X II Q is a 2-dimensional polyhedron and I = [ 0, 1 I}. )

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Problems

50. Let M be a collection of compact n-manifolds (respectively, piecewise linear ti-manifolds or differentiable /?-manifolds) such that for M , M ' E A[. M is homotopy equivalent to A4'. Is the homeomorphism (respectively, piecewise linear homeomorphism or diffeomorphism) problem solvable in M ?