A. I. MAL'TSEV'S PROBLEM ON OPERATIONS ON GROUPS UDC 512.543
A. Yu. Ol'shanskii
In response to A. I. Mal'tsev's problem (1948), associative operations are constructed on the class of all groups; these operations are distinct from the direct and free products and are hereditary for subgroups of the factors.
We say that an exact operation ~ is defined on the class of all groups if a group ~,, ~ J f ,
is assigned to each family of groups ~ = ~ ~
and monomorphisms ~ @ ,
are given
whose images (they are called the factors and are also denoted by ~,) generate the group ~, and all isomorphisms of the factors can be extended to isomorphisms of their ~ (i.e. the operation ~ is abstract). The classical examples of exact operations are the direct and free products of groups which also satisfy a number of additional postulates. These properties include: i.
Mal'tsev's Postulate.
inclusions ~$ ~
It says that for any subgroups ~'
can be extended to a monomorphism of the group
words, subgroups of the factors generate their ~ 2. the group
The Associativity Postulate.
unique isomorphism between ~
~
and
~o
~
into ~.
~,
the
In other
in
If a set Jr is partitioned into subsets J[~, ~.~,
~ is naturally isomorphic to the group
mented by the condition that if
of the factors
~o
~o ~.
then
This requirement is comple-
are trivial groups for ~ K ~ J K ,
then there exists a
~~
Main examples of assciative operations include the ~ -verbal multiplication which allows us to extend the free product inside an arbitrary variety of groups ~ to the class of all groups. Typical Mal'tsev operations are given by Grunberg--Shmel'kin multiplications. Axiomatic study of operations on groups has been initiated by O. N. Golovin in the 40s. The survey [i] deals with problems of classifying the operations. That article again raises, as one of its main open questions (Problem A), A. I. Mal'tsev's problem of 1948: do there exist associative Mal'tsev operations apart from the free and direct products? The authors of [i] view this question as "the central problem in the entire abstract theory of exact operations." It is also raised in the book [2] (p. 475). An approach to solving Mal'tsev's problem was found by S. I. Adyan [3] who noted that the classification of periodic words created in [4] and [5] can be naturally used here with modified rules of syntax (free product instead of a free group). The n-periodic multiplication introduced by S. I. Adyan (n is odd, n ~ 665) possesses a number of interesting and surprising properties; unfortunately, however, it can be defined only in the class of groups with no involutions [6] (and it is a Mal'tsev associative operation inside that class). The purpose of this article is constructing a Mal'tsev associative operation in the class of all groups, i.e., a solution to A. I. Mal'tsev's problem stated above. In the case of factors with no involutions and of a sufficiently large odd n, the constructed operation essentially coincides with S. I. Adyan's n-periodic product (however, the identification of these operations is a separate question because the scheme proposed in [7] substantially differs from the Novikov--Adyan approach even in its basic definitions). The operation defined
Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 14, pp. 225-249, 1989. Original article submitted May 20, 1987. 2468
0090-4104/90/5104-2468512.50 9 1990 Plenum Publishing Corporation
below for (even or odd) n _> i0 l~ and arbitrary factors 9~,- is denoted by ~n like in [3], although for an even n or in a presence of involutions in the factors the group (~ need not he periodic even if the subgroups ~/~. are periodic. Elements of ~ that can be written as products of pairs of involutions may have infinite orders. We turn to an exact definition of the group tained from the free product ~ 0 = * ~
@=~ngl~, n ~
I0I~
The group ~
will be ob-
by imposing on it additional relations of the form
C n = i, where C is a word in the alphabet ~, obtained as a disjoint union of the sets ~\{|). For i > 0 we shall define periods of rank i and groups ~$~. To this end, we consider the set of all words C of length i in the alphabet ~ subject to the conditions: !) C is not conjugate in the rank i - 1 (i.e. in with an element of any ~;
@i-~) with any word of length i, i.e.
2) C has infinite order in the rank i - i; 3) C is not equal to a product of any two involutions in @~-~ and choose in it an arbitrary maximal subset $F~ of pairwise nonconjugate, in the rank i - i, words Ci, K such that no Ci, K is conjugate to Ci,~ z~ in the rank i - i for ~ # v. Next, put ~ = { C "~ ~
U ~
~, C ~ } U ~ - ~ ,
where ~ 0 = ~ ,
we define @ =(@~0; R = I , R ~ >
f=l
and @~=(~0; R =l, R ~ ) .
and put
Finally, for
~r ~t6J~
THEOREM. For each n -> i0 ~~ the operation defined above is an exact associative Mal'tsev operation on the class of all groups. The group
~=~n
is infinite if at least two factors are not trivial.
Each element
g e ~ is either conjugate in ~ to an element of one of the factors ~, or is conjugate to a power of some Ci, K (and, thus gn = i), or has infinite order and is equal to a product of two involutions. Periods of all ranks have order n in ~. i.
The proof substantially relies on [7] whose lemmas are stated in a modified form in Sec. The values of all parameters introduced in [7] are preserved.
i.
Properties of the Groups (~ We
begin
with necessary changes and modifications
in terminology.
Each word in the alphabet ~ can be uniquely written in the form WI..oWk, where all letters appearing in Ws lie in one ~(0, and u(s # ~(~ + i) for s = i, .... k - I. Here, Ws is a syllable of the word W. Similarly, if in some map (p. 317 in [8]) a letter ~ (e) in ~, is assigned to each edge e in such a way that ~(e -l) =~(e) -I, then each path p in this map is divided into segments according to the division of the word ~(p) into syllables. The number of syllables of a word W (the number of segments of a path p) is called the IWI of the word W (the length IPl of the path p). The notation U ~ V is viewed in this article as the syllable-wise equality of the words U and V (instead of the graphical equality in [7]). An irreducible word is one in which each syllable al...a s possesses the property that ak...a ~ # 1 in the corresponding group ~ for 1 ~ k 5 s i s. If, in addition, the first and the last syllable of a word W lie in different ~, ~ (in the case of IWI > i), then W is said to be a cyclicaily irreducible word. A c_.vclic shift of a word W is defined as any word W' such that for some s we have W' ~ UW~+I...WkW~...Ws where VU = Ws in ~t). A subword in an irreducible word W is any irreducible word with a division into syllables of the form UWk...W~V, where k S s + I, and the syllables U and V lie in the same factors as the syllables Wk_ I and Ws respectively. A word C is said to be si__~le in the rank i ~ 0 if: i) C is not conjugate in the rank i to a power of any period of rank k 5 i; 2) C is not conjugate in the rank i to a power of any word of length less than ICI;
3) Icl e 2. 2469
The definition implies that a word simple in a rank i is simple in the ranks
Such a word T is contained in any subword of
In the definition of C-periodic words we add the condition ICI ~ 2, and in the definition of C-compatible decompositions of two C-periodic words X = XiX 2 and Y = YiY2 we add the requirement that each syllable of the word X (of the word Y) be entirely contained either in Xi(Yi) or in Xm(Y=). In the definition of compatibility of a region q and a cell H we continue to assume that the periods of the words ~(p) a n d ~ (q)-i are the same. To a smooth region of rank j we now assign not the period Cj • but, rather, one of the periods Cj, K-+i (any one if there are several such periods). Nevertheless, we often omit the second subscript < if it is clear from the context to what cell or to what smooth region Cj,~ belongs. The notions of smooth regions, connectives, incidence subdiagrams, incidence degree, and a regular rank diagram are taken from [7] verbatim. We will make some general connnents on changes in the proofs of Lemmas 2.1a-6.4a relative to 2.1-6.4 in [7]. First, the length ICj,
2470
main in force. There are no changes of this sort in Sec. 6 because the auxiliary vertices in Lemmas 6.1a-6.4a are points of division of regions of the contour into segments. The length of a cyclic region of the contour, or the perimeter of a cell is defined as the minimum of lengths of closed paths determining this cyclic region (the contour of the cell). In other words, the initial point in tracing is a primary vertex. Both the statements and (taking into account the remarks above) the proofs of Lem~nas 2.1a and 2.3a-4.5a remain unchanged. Lemma 2.2a is preserved with its original statement. In its proof an application of Lennna 6.4a (in a lower rank) leads to the same conclusion because Ck, < is not a product of two involutions in a rank j < k by the definition of periods of rank k. LEMMA 4.6a. i. Each word D is conjugate in a rank i to a letter in ~, or to a power of a period of a rank k ~ i, or to a power of a word D simple in the rank i, where IDI < IDI is not simple in the rank i; here, for a word D of finite order in the rank i either the first or the second possibility occurs. 2) No relation of the form Ck,
i. The periods of rank i + 1 are simple words in the ranks 5i. Proof. Statements i) and 2) obviously follow from the definition of periods of ranks i, 2, ... and the group ~, because in a dihedral group each element is a product of two involutions or is conjugate to one of the two generating involutions. In the case of 3), we choose a word D according to Lemma 1.5, IDI > i. Like in Le~ma 5.6 in [7], we deduce that D has infinite order in the rank i because [n/6] > 8. The word D is also not conjugate in the rank i to an element of ~, because considering a minimal ring diagram of such a conjugacy we arrive at a contradiction to Lemma 5.5a since [n/3] > 8. i if D = IJ is a product of two involutions in the rank i, then D and D -I are i conjugate in the rank i, D -I = IDI -~. Let A be a minimal ring diagram of rank i for conjugacy of the words D and D -I. By Lemma 5.5a, A has no cells because [n/3] > 8. Thus, D and D -I are conjugate in G0. But then the cyclic irreducibility of the word D would imply that D -l is a cyclic shift of the word D and, therefore, D has subwords S and S -l, where ISI 1/21D I, contrary to the choice of the word D. Finally,
Thus, if there are no periods of ranks i + 1 ..... ID[ - i, i.e.,~i = J ~ + l .... =~lOl--1, then, by definition, D is conjugate in the rank i to some period of the rank IDI > i.
2471
Suppose now that some period C of the rank i + i is not simple in the rank i. By the definition of a period and by Lemma 4.6a, the word C is conjugate in the rank i to a power of some word C simple in the rank i, where ICI < ICI, i.e., j = ICI 5 i. But then C, like C, has infinite order both in the rank i and in the rank j - I. Thus, the definition of periods of rank j implies that C is a product of two involutions in @~_~. In this case, C is a product of two involutions in the rank i because a power of a product of two involutions is also a product of a pair of involutions. The obtained contradiction to the definition of a period of rank i + i shows that C is simple in the rank i and, thus, also in every rank k 5 i. The statement of Lemma 6.1a is as before. It appears that a new possibility arises in part 3) of the proof: D is conjugate in the rank i to an element C of ~ (the other two possibilities, i.e,, that D is conjugate in the rank i to a power of a word C simple in the rank i or to a power of some Ck, ~ have been considered in [7]). We will show that in fact it cannot arise. Indeed, like in part 6 of Lemma 6.1 in [7], in this case we obtain the equality i YZIX = Z2, w h e r e , ( Y ) is formed as a product of the labels of edges of a path p. Next, like in [7],
IZ~! + Iz~l< ~(p-'--l)lq~l +--~- !A[, and X is an A-periodic word,
IXI >- lq21 > k11Al"
Thus,
P-~ + --~- I A I >~(p-,--1 )lq21 +~-~ IAI+~>IZ~I+IYI+ IZ~l, p-~ iXl>--c-Iq~l which contradicts Lemma 4.2a applied to a minimal, i X = ZI-IY-IZ2 -I
in the rank i, diagram of the equality
Both the statement and the proof of Lemma 6.2a remain unchanged. LEMMA 6.3a. Let & be a minimal diagram of rank i with a contour PlqlP2q2, where ~(ql) a n d S ( q 2 ) are periodic words with a period A simple in the rank i and max(Ip~l , IP21) < ~IAI Then max(lqll , lq21) ! kslA I or A is a product of two involutions in
~{.
Proof.
If, say, lqll > kslAl, then using the notation of Lemma 6.3 in [7], we obtain i i the equality Z 2 = i as it was obtained there. Furthermore, the equality A = Z-IA-IZ implies i that (AZ) 2 = I (here, Z and AZ are not trivial in the rank i because, by Lemma 4.6a, a word A simple in the rank i has infinite order in ~{.) So A = (AZ)Z is a product of two involutions in ~{. 9 LEMMA 6.4a. Let A be a minimal diagram of rank i with a contour PlqlP2q2, where ~ (ql) and ~(q2) are periodic words with periods A and C simple in the rank i, lq21 ~ ~nlCl and max(let1,
IP21) < ~nb, where b = min(IAl,
ICl).
Then either
lqll < (i + e)IAl, or A is con-
jugate to C • in the rank i. In the latter case, if A m C • then either A m C -I and ql, q2 are compatible in A or A is a product of two involutions in ~ r The proof changes only at the moment when Lemma 6.3a is applied; a change in its statement has caused the corresponding modification in the statement of Lemma 6.4a relative to [7]. 2.
Mal'tsev's Postulate for n-Products
LEMMA i. In all groups ~i, and thus also in 6, each element of ~ is distinct from i, i.e., ~ is generated by isomorphic copies of the groups ~ , ~ J K . Proof.
The statement follows from Lemma 4.3a and the inequality on > I.
Henceforth, until the end of Sec. 2, ~='
are subgroups of the groups
~, ~'=U~'\{|}.
LEMMA 2. If all segments of a contour p of a minimal diagram A of rank i have labels in ~', then the segments of all cells of the diagram A also possess this property. Proof. Arguing "by contradiction," we choose a counter example A with a minimal number of cells. By Lemma i, IPl > i and the word ~ (p) may be assumed cyclically irreducible. By Lemma 5.5a, A has a cell ~ with a contour PlP2, where Pl is a subpath of p and Ipll > 1/318HI.
2472
Since n > 4, the labels of all segments H out of A and replacing the subpath Pl and a similar condition of the segments statement of the leamma, A does not have
of the cell ~ lie in ~'. Thus, excising the cell in p by p2 -I, we obtain a diagram A' with fewer cells of a contour. Since A' has no cells refuting the them either, a contradiction, a
LEMMA 3. Let q be a smooth contour of a minimal ring diagram A of rank i with contours q and t. Then if letters of ~ occur in ~(q), then they also occur in ~(t). Proof. The statement is obvious there exists a cell H whose incidence like in Lemma 5.5a, results in a cell and p is a cyclic subpath in t. Thus, possible.
for r(A) = 0. Otherwise, by Lemmas 2.3a and 3o6a, degree for t is greater than 8 - ~ > 1/3 + 400u which, ~ with a contour of the form pp, where IPl > 1/3 18~I like in the proof of Lemma 2, an induction on T(A) is
LEMMA 4. If words C -l and C are conjugate in rank i, then C is conjugate to an element of some factor ~, or C is conjugate to a power of some period of rank k 5 i, or C is a product of two involutions in U i Proof.
If neither of the first two possibilities occurs, then, by Lemma 4.6a~ the word
C is conjugate to a power of some word D simple in the rank i, i.e., D E ~ XD-~X -I, ~ > 0, for i some X. But then D s~ = XD-S~X-I. Applying Lemma 5.3a to the diagram of this conjugacy for a sufficiently large s, we excise out of it a subdiagram to which we already can apply Lemma 6.3a. This implies that D is a product of two involutions in ~i, because a power of a pair of involutions is again a product of two involutions. We now denote by
~i' the group obtained by imposing on C 0 " = , ~ /
relations of ranks
i, ..., i defined in the alphabet ~ in the same manner as relations of rank i have been defined in the alphabet ~. In order to avoid a confusion between relations of the two sorts, in the following lemma, while considering the group @/, we will talk on its relations of ranks i', 2', ..., i' in the alphabet ~. By Lem~a I, in the rank i' the group ~ ' is generated by subgroups isomorphic to the groups ~/. On the other hand, ~ / may also be viewed as subgroups of ~i, generating some subgroup @i in ~. LEMMA 5.
There exists a natural isomorphism between the groups
@/
and
~i
Proof. We perform an induction on i with an obvious base of i = 0. First, we will prove that each relation of the form Ci,K n = 1 of the group ~ / also holds in @~. Suppose first that a word Ci, ~ is conjugate in the rank i - 1 to an element of ~, and consider a minimal ring diagram A of rank i - 1 for this conjugacy with contours p and q, where ~(p)~---Ci,• ~(q)~. Applying Lemma 5.5a to A, we find a cell ~ in A with a contour PlP2, where Pl is a subpath of p and IPll > 1/313HI" Since n > 4, the labels of all segments of the contour of the cell H, like the labels of segment of the path p, lie in ~'. Since r(~) < i, by the induction hypothesis, the relation in ~ , corresponding to the cell H also holds in @'i-i, i.e., when the subpath Pl in p is replaced by p -i, we obtain a path p such that ~(p) m Ci,,<_and ~(p) are conjugate in the rank (i - i)'. Note that the cyclically irreducible form C for ~ (p) has the length >I, as follows from the definition of Ci, K. Continuing this process of excising cells out of A, we obtain a ring diagram with no cells, i.e. we deduce that in the free product ~ 0 a cyclically irreducible word of length >I is conjugate to an element of a free factor. The obtained contradiction means that Ci,,K is not conjugate in the rank i - 1 to a e ~. By definition, the word Ci,,~ has infinite order in the rank (i - i) ~, and thus, by Lemma 5 (in the rank i - i), Ci, < also has infinite order in ~_~. Suppose that Ci,,K can be written as a product and Ci,,~ -~ are conjugate in the rank i - I. Thus, as above, we deduce by means of Lemma 5.5a that the gate in the rank (i - i)' An application of Lemma tion of Ci,,<.
of two involutions in @~-~. Then Ci, K considering the diagram of this conjugacy words Ci, ~ and Ci,,< -1 are also conju4 produces a contradiction to the defini-
Thus, for the word Ci, ~ we have verified conditions 1)-3) in the definition of a period of rank i. Therefore, this word is conjugate to the • power of a period of rank i. i Hence Ci,,
2473
i Conversely, let W = I, where W is a cyclically irreducible word in the alphabet ~'. As follows from Lemma 2, this equality follows only from relations of the form Ck,~n = i, where k ~ i, and the words Ck, ~ are written in the alphabet ~. So it suffices to show that Ck,~n = 1 is a corollary to relations of ranks !i'. But, by the definition of periods of rank k, Ck, ~ is not conjugate in the rank k - 1 < i (nor in the rank (k - i)' by virtue of the isomorphism ~a-~--~-@'~-~) to elements of ~; Ck, < has infinite period in the rank k - 1 (and thus also in the rank (k - i)'); Ck, ~ is not equal to a product of two involutions in the rank k - 1 (nor, therefore, in the rank (k - i)'). So the word Ck, m is conjugate in the k' rank k - 1 to the il-st power of some period of rank k' and Ck,~n = i. The lemma is proved because we have established that each relation of the group ~ / holds in ~ and vice versa. 9 3.
Duplicates of Periods
To prove the associativity of n-products, we have to consider a portion o of the set ~ff of indices of the factors into a disjoint union of subsets J/{~, ~ . Let ~ = U ~{]}Then for each word W in the alphabet ~ one can uniquely define its decomposition into osyllables, i.e., maximal subwords each of which consists of letters of one of the subalphabets ~. The number of o-syllables s is called the norm of the word W and, distinguishing it from the length IWI in Sec. 3, 4, we denote it by llWll = s. For a path p, we put by definition ilpll = li~(p)ll. If for some period C we have IICll > i, where C is a period of rank Ji or a simple word in the rank i, then C is called a large period, otherwise it is a small period. As follows from the definition of periods, by replacing C by its cyclic shift, one can make the first and the last letters of a large period lie in distinct subalphabets. Accordingly, we define large and small cells in diagrams. If C is a period of rank i + 1 or C is any word simple in the rank i, ItCIl > i, then a duplicate of the period C of rank i + 1 (a duplicate of the word C in the rank i) is defined as some fixed word C' with the least norm among words conjugate to C in the rank i; in addition, we require the o-syllables in cyclic shifts of the word C' to be minimal in the rank i. If lICil = i, then, by definition, C' m C. As can be seen from Lemma 3, lJC'll > 1 if iICI] > I. Unlike the length ICj El, the norm lICj
usuall9
Let ~ be a cell of a diagram A with a large period C k < and let q be a region of the contour of the diagram A, where ~(q) Is a (Ck, K ) -perlodlc word, I~(q) l ~ ICk,K'[ 9 Suppose that vertices o I and 22 are selected on the contour of the cell ~ and on q, in which cuts define cyclic shifts C and C' of the words Ck, < and Ck,K'. Then the cell ~ is said to be compatible with the Ck,K'-region q if A has a path t = oi-o 2 with no self-intersections k--I
such that ~(t)C'~(t)-l=C-1.
Similarly, one can define C'-compatibility of two regions of the
contour whose labels are C and (C')-1-periodic words for a period C simple in the rank i. A (cyclic) region q of the contour of a diagram A with a period C k' is said to be o-smooth or Ck'-smooth if A has no cells compatible with q, and in the cyclic case ~(q) m (Ck')~. A (cyclic) region with a period C', where C is a simple word in the rank i, is also said to be o-smooth in a diagram of rank i. A minimal (ring) diagram of rank i is said to be o-regular if each incidence subdiagram of an arbitrary cell ~ to a o-smooth region of the contour is of rank
~(m) =X,, iiX~ll<eSvnilC~'ll,Z=I, 2474
~
The proof is obvious for O-connectives. Next, we perform induction on r(~), where ~ is a principal cell of the connective E. By definition, j = r(~) < r(H1). We will use the notation of Lemma 2.19 If both cells, ~ and H i are small, then, by the induction hypothesis, the labels of edges of the paths pl I, p2 I, pl 2, p22 lie in the same .~P~ where the labels of edges of the cell ~ reside, as well as the initial edges of the paths ql +-'I and q2 -+I. If H is a small cell and H l is a large one, then, like in the preceding paragraph, llplli = i, i.e., IJXIIJ < 23ynllCk'H. Let ~ be a large cell 9 By Lemma 4.1a, r(A) = j < i. So, by Lemma 6 in the rank j, for the paths p i and p2 ~ there exist words Yi and Y= equal to their labels in the rank j - i, j-~ with a norm <23yniJCj'lJ, and if ~ is a small cell, then we deduce from F~ that ~(q i) = V, where
JlVII < 46~nlICj'll 9
(Cj')+l-periodic word, lIWll
On the other hand, by Lemma 18, ~(q2 ~) j-1 = Z~YZm, where Y is a j-~ IIYII > (n~ - 4)IICj'II, and IIZ~II, JlZ211 < 51jCj'li. Hence Y = W, where
< (46yn + IO)IICj'II, contrary to Lemma 12 applied to the diagram of this equality because
9 (n[~-4) >46yn+lO. Thus, ~
is also a large cell9
By Lemma 2.2a, j < ~ / 2 k .
(1) By Lemma 19,
IICj'l] < 22u
and, by Lemma 18, the label of a subpath of the contour of the cell ~ is equal in the rank j - 1 to a word U, where
Hence
fl X~ [[ < 22y (n + 10)II C~ II + 23yn II Cj J] < (22y (n + 10) -~ 22y. 23yn) [j C~ J] < 23yn [I C~ [tLEMMA 7. Let A be a o-regular minimal (ring) diagram of rank i, F the subdiagram of incidence of a large cell ~ with a period C k to a (cyclic) o-smooth region with a period C', where C is simple in the rank i or C = - C s + i , ~ < i. Then, if the degree of F-incidence of the cell ~ is no less than e - ~, then IJF A qll < (! + e)IJC'II. Proof. Since A is o-regular, j = r(F) < k. By Lemma 6, one can apply Lenmla 21 in the rank j < i to r, and if the conclusion of Lemma 7 were false, then Ck -I and C' would be conjugate in the rank k - 1 and the cell ~ would be compatible with q, contrary to the c-smoothness of the region q. LEMMA 8. Let A be a o-regular minimal (ring) diagram of rank i, F the subdiagram of incidence of degree ~8 - ~ of some cell ~ of rank k in A to a c-smooth (cyclic) region q with a period C~'. Then k < I. Proof. q~ = F A q.
First, assume that ~ is a large cell. Let PlqlP2q2 be the contour for F, where Like in the proof of Lemma 6, we conclude that C~ is a large period [inequality Z-1 (i) is preserved when ~ is replaced by e - a). By Lemmas 7 and 18, ~(q2) = V, where IV! < IoJc~J. The last equality also holds in the rank k - 1 if s ~ k. By Lemmas 2.1a and 4.2a, this implies that p(e - a)nJCkJ < IOJCs + 4ynICkJ, and thus [CkJ < JC~I , i.e., k < s If ~ is a small cell, then, by Lemma 6 for Pl and P2 and by Lemma 12 in the rank j, ;lq2;l < p-llfplqlp=ll = p-l, i.e., llq2H = i. Assuming that Cs is a large period, we deduce that k < E as before. If Cs is a small period, then q is a smooth region, and k < E by Lemma 2.3a. i LEMMA 9. In a o-regular (ring) diagram of rank i the degree of incidence @ of each cell ~ to a c-smooth (cyclic) region q of the contour is less than ~. Proof. Let F be the correspondent subdiagram of incidence with a contour PiqlPzq2, where ql = F A ~, q2 = F A q. By the hypothesis, j = r(F) < k = r(~). Let C' be a period for q, where C is simple in the rank i or a period of rank E > k by Lemma 8. If ~ is a small cell, then, like in Lemma 8, Jlq2iJ = i. Assuming that ~ > ~, we arrive at a contradiction like in the proof of Lemma 2.3a (to the minimality of o-syllables of the word C' in the rank j < s Henceforth we assume that ~ is a large cell. 2475
By Lemma 6, the w o r d s ~ (p~) and ~(p=) are equal in the rank k - 1 to words x~ and x2 k-z such that llx~ll, fix211 < 23~nllCk'll and, by Lemma 18 in the rank k - i < i , ~ ( q ~ ) = Z~YZ=, where Y is a Ck-periodic word, [IYII > (na - 2)llCk'll, IIZ~II, llZ211 < 511Ck'll. Since C is simple in the rank k - i, one can apply Lemma 12 to a region with the label Y in a minimal diagram k-~ A0 of the equality Y = ( Z = X ~ ( q 2 ) X ~ Z z ) -~ and p ( a n - - 2 ) l I C / [ l <
i.e.,
(lO+46yn) I
'
' '
'
II~(q2)ll > n ( p ~ -- 47~)llCk'll.
On the other hand, by Lemma 18 for a path complementing q~ to a contour of the cell ~, k we conclude that ~(q2) = W, where IIWIJ < (46~n + i0 + (i - a)n)llCk'll < (i - a + 47~)nJICk'JJ. Now, if lJq2Jl 5 IIC'II, then we have a contradiction like in Lemma 2.3a because
0a-47?> 1-~+47y.
(2)
If Jlq211 > llC'll, then, again like in the proof of Lemma 2.3a, we get a contradiction (by means of Lemma 7 instead of Lemma 2.2a) because
(p~-47y) (l-e)> Lemmas 10-14 a r e p r o v e d by a c o l l e c t i v e
(I-a+47?) (I+e)+2n-L induction
on r ( A ) .
LEMMA 10. L e t F be t h e s u b d i a g r a m o f i n c i d e n c e o f a l a r g e c e i l ~ w i t h a p e r i o d Ck t o a ( c y c l i c ) r e g i o n q o f t h e c o n t o u r o f a m i n i m a l ( r i n g ) d i a g r a m A w i t h an i n c i d e n c e d e g r e e ~, l e t PxqlP2q2 be i t s c o n t o u r , w h e r e q l = F A ~, q2 = F A q, and q c o m p l e m e n t s q l t o c o n t o u r o f the cell ~.
Then
I[ q~ II~ (p~--47y) n l[ C'~ II, a ~
-
1
k--I
(p~q- P2) = W ,
r~e Jl W ]] <
<(1--~2 + 47y) nil C2 I1. Proof. All necessary arguments have been produced in the proof of Lemma 9 [40 need not have rank k - i, but ~(A o) < ~(4), so Lemma 12 is applicable]. LEMMA II. Proof. the contour.
A minimal (ring) diagram of rank i is o-regular.
Consider the subdiagram of incidence F of a cell ~ to a o-smooth region q of By Lermma 6, we may apply Lemma 14 to F, (~(F) < ~(4)), and obtain r(r) < r(~).
For a cyclic word W the number of the word W.
lIWll is defined as the minimum of norms of cyclic shifts
LEMMA 12. Suppose A is a minimal diagram of rank i with a contour qt or A is a minimal ring diagram of rank i with contours q and t. If q is a (cyclic) o-smooth region, then pJlqlJ ! lltl[. Proof. Considering the former case, we choose a 0-cell ~ and use the notation of Lemma 4.2a. If ~ is a small cell, then, by Lemma 6, a replacement of the subpath t I by sz-luq2ws2 -I would not increase Jftll but would decrease T(&). So we assume that ~ is a large cell. By Lemma 9, ~z < ~, and Lemma I0 allows us also to assume that the degree of incidence of each cell to t is less than ~ by (2). Thus, first, e - ~ < ~z, ~2 < ~, and, second, we may apply Lemma 14 to F 2 (~(F 2) < ~(A)), i.e., r(r 2) < r(~) = j. Next, r(r I) < j also by Lemma 14, and j < E by Lemma 8 if C&' is a period for q. Thus, in the rank j = r(A'), C is a simple word if C is a period for q. By Lemmas 6 and 18, the label of the path pl-Zws2-1t1-1s1-1up2 -1 is equal in the rank j - i to some word U, where
IlUll < ( ( 1 - - O ) n +20)[IC} II + 9 2 ? n i l e ) ! l + IIt, II < <(1 - - o +93?) n lJ Cj [I + II tl tl. Together with the inequality
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IIqlll < (i + e)lJC'il, Lemma 7 implies,
like in Lemma 4.2a, that
(t --~) (I + ~)-x [I qa II < ( i --0 + 93y) n It C}II § II t~ II. (3) By Lemma 10, we a l s o have f o r Fi IIq~ll > (~(8 - ~) - 47y)nllCj' II and, a p p l y i n g (3), we o b t a i n 9 ql [l~i]~a]l+ [ i - - 0 § t
1-e \ I+e
9) [9 (0--~)--47~]} n I[ C} II-
(4)
Next, we consider separately diagrams Al and A= with contours q'pl-lws2-Zt ' and q"t"-isl-lup2 -I. By means of Lemmas 6 and 18, we replace in them the subpaths p1-!ws2 -l and sl-lup2 -I by paths with labels V I and V=, where IIVIII + IIV211 < (i - 8 + 93y)nIICj'II. The modified diagrams A l and A 2 have type lower than ~(A) (they do not contain the cell ~ of rank j). So, by the induction hypothesis,
p (llq'll+llq"ll) < llt'll §
+ (1--0+93V) nil C/i].
Adding it to (4), we obtain pllql[ <- lltllbecause the additional term is less than -2. The ring case is studied in a similar manner (cf. 4.2a). LEMMA 13. Let C k be a period of a large cell v of a minimal diagram A of rank i with a contour t. Then on[ICk'll <- Iltll. Proof. Excising the cell ~ from A, we obtain a ring diagram A I with contours t and p. One can attach to p the diagram A 2 of rank k - 1 of conjugacy of the words (Ck')n and Ckn. We obtain a ring diagram A 3 [with T(A 3) < T(A)]; as a result of reducing it to a minimal one in the rank i only cells of ranks <_k - 1 may be cancelled because of the minimality of A. For the same reason the cyclic region with the label (Ck') +n in the obtained ring diagram Aq is o-smooth. Therefore, by Lemma 12 for A4 we obtain the required inequality. LEMMA 14. Let A be a minimal diagrm~, of rank i with a contour PlqlP2q2, where ql is a o-smooth region or a region with an irreducible label for which A has no cells with degree of incidence to qz no less than ~; q2 is a smooth region with a period C, where IICII <- k, [PII, ]P21 < 2ynk. Then r(A) < 22yk. Proof. By Lemmas 9 and II, we may assume that the latter condition of the lemma holds ql- Like in Lemma 4.4a, we choose a O-cell ~ and consider a number of cases using in each of them the notation of Lemma 4.4a. I.
By Lemma 4.2a for F, It2[ < p-1(2ynk + 4ynICj;), whence
n,Cil = ,On, < ~ - ' , / ~ [ < (1 1_2_p.4-4V)-' 9-' (2vnk *. 4vn[ C/i ). i.e., [Cjl < 5yk. Furthermore, Is2{2sll < p/2n]Cj[ < 3ynk, i.e., the perimeter for A' is less than 8ynk and thus, by Lemma 4.3a, r(A') < p-18~k < 9yk. Next, like in part I) of the proof of Lemma 4.4a, we perform the induction step [on T(A)]. 2.
Like in part 2) of Lemma 4.4a, j = ICjl < 27nlC I.
The rest is like in 4.4ao
3. Suppose the sum of incidence degrees by means of Fp and Fq is greater than 8 - 26. Like in Lemma 4.4a,
IOn! <(9(O--~--2g)--4y)-~lf~l<2yn(p(O--~--2~)--4y)-~f Hence r(~) < 5yk.
C i<5ynrCJ.
The perimeter of the subdiagram Fp (or upon adding ~ to Fp) does not ex-
ceed 2~kn + (4y + i/2)IOv I < 5~nk and, by Lemma 4.3a, r(Fp) < 6yk. Next, r(rq) < r(~) by Lemma ii or 4.1a. One should also note that Is2w-lsl I < 2yn[C] which is derived like in part 3 of Lemma 4.4a for [s2w-ls1[. So the rank of the subdiagram with the contour UlUI(Sl)-IWS2 -I is less than 22yk because its type is less than T(A) (in its contour a smooth region has zero length). Thus, one can turn to A w by induction as in part 3) of Lemma 4.4a. 4. Suppose the degree of F1-incidence is greater than 1/2(0 - ~ - $), and the degree of F2-incidence is greater than ~ (see the notation of Fig. 12 in [7]). Let r(~) = j. Then, by Lemmas 2.1a and 4.2a for Fl,
2477
[t~l
+ 4 v n t C / I ) < 9-'(2gnk 5 4vn]).
Comparing this inequality to the condition It=t1 > 1/2(8 - ~ - ~)nj, we conclude that j < 107k. Similarly, considering F 2 [replacing 1/2(8 - ~ -- ~) by ~], we have Its21 > Is~21. Thus, I~l-ls21, l~2(s~a)-ml < 2u and one may apply Lemma 14 to the subdiagram A with the contour ~ = s i = w ~ ( s = ~ ) - i ~ q m
because ~(~) < ~(~),
i.e., r(A) < 227k.
It remains to note that
the perimeter of the subdiagram with the contour p m t ~ s = ~ w ~ - m s ~ 2 t i = p ~ q = the cell v from it) is less than
(or, upon excising
by Lemma 4.2a and its rank is less than 22yk by Lemma 4.3a. The assertion of the lemma follows from the cases that we have considered. 4.
Auxiliary Cuts on Diagrams
LEMMA 15. a. Suppose A is a minimal diagram of rank i with a contour tq or ~ is a minimal ring diagram of rank i with contours t and q; let q be a (cyclic) o-smooth region. i Then for each subpath q of the path q there exists a word U such that ~(~) = U, IUI < 4/31t I + 1/2. b. Let A be a minimal diagram of rank i with a contour PlqlP2q2, where ql is a o-smooth region and q2 is a smooth region with a period C [where C is a word simple in the rank i or a period of rank s = r(q=)], [Pll, IP21 < 27nlCIThen each vertex o of the path ql can be i joined in A to some vertex of the path q2 by a path s such that for some word S ~(s) = S and
Isl < 2m nlcl. c,
Suppose that
A is a minimal
ring
diagram
of rank
i 0 < 22~k 5 i w i t h
contours
ql and
q2, where ql is a o-smooth region and q2 is a smooth region, or A is a minimal diagram of rank i 0 with a contour PlqlP2q2, where ql is a o-smooth and q2 a smooth region, IPII, IP21 < 22ynk. Then each vertex o of the path ql can be joined in A to some vertex of the path q= i0 by a path s such that ~(s) = S, where ISI < 105~nk. Proof.
We will prove the lemma by induction on ~(A).
a. The assertion is obvious if A has no cells. In the general case Lemma 3.6a enables us to select a e-cell ~ with incidence diagrams r i and F 2 to q and t, respectively (see the notation of Lemma 4.2a). Like in Lemma 4.2a, we may assume that ~2 < ~, i.e., F I indeed exists, and @i < ~ by Lemma 9. Thus, @2 > 8 - ~. Statement b) is applicable to F i because 9 (FI) < ~(A), and IPll, IP21 < 2u by Lemma 2.1a. So for each subpath q of the path ql i there exists a word U I such t h a t ~ ( q l ) = UI, IUll < (42y + i/2)18~ I. For the subpath q' in q', by the induction hypothesis for a subdiagram of a lower type with the contour q'p1-1ws2-1t ' , i we have a word U', where ~(q') = U', IU'I < 4/3(It' I + lwl + 4718~I). The inequality for q" i is similar. As a result, for the subpath q there exists a word U, where ~(q) = U and
IUI<
(427-t---~- I) la l.
f!t'l ,'--lt"l ~-(1--O§
By Lemmas 2.1a and 4.2a for F2, we have
lt~
I U I < ~4--Itl--[-~a ( 1 --t3 4- 8!e) § 42y
> (p(s - =) - 4y)la~ I, whence
1 2
4 (p(O__.GC)__4~)] ]0.~ ] < _ ~ ]tl " 3
The ring case is studied similarly. b. The assertion is obvious if A has no cells. consider four cases as in Lemma 4.4a. i.
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It is complete analogous
Otherwise,
to case i) of Lemma 4.4a.
we choose a e-cell ~ and
2. In this case, like in 2) of Lemma 4.4a, IPI[, Ip21 < 2ynIcl and, in the light of inductive reasoning, it suffices to consider the case of o e {q11}. Then, by Lemma 15, for i F~ there exists a path s' joining o to a point o' of the path q2', where ~(s') = S', IS'I < 21u I . Lengthening this path by a subpath of the contour of the cell ~ and by the path i p~a [or (p2=)-~], we obtain a path s, ~(s) = S, and
9 +-7-(1--1~)+ 1 iSl < (91v 2v) [oal<- Fl[oal But, like in part 2 of Lemma 4.4a,
13~I < (p8 - 4y)-l(l + E)IC I. I
IsI<-?-(~--4~)-~(~ 3.
Thus,
+~)lCl<~nlCi.
We consider two possibilities separately.
3'. Fq is the subdiagram of incidence to ql. Like in Lemma 4.4a, IS2W-lSltlull < 2[pll < 4ynIC I. Therefore, by statement a), a vertex o of the path ult I may be joined to i the end of the path Pl by a path 2', w h e r e ~ ( s ' ) = S', Is'I < 4/3"4u I + 1/2. Putting s = s'pl -I, we obtain IS < 8ynlC I. If o does not lie on ult l, then one has to use the induction hypothesis for 4" because Pll < IPll [cf. part 3) of Lemma 4.4a]. 3". Fq is the subdiagram of incidence to q=. cause Ipll < IpiI, T(A") < x(A). 4.
Then we are immediately lead to &" be-
Like in part 4) of Lemma 4.4a, [q21
(1+8y)
-~ (O--~--~)--4y
IC]
+ 4 '
T h e r e f o r e , by s t a t e m e n t a o f Lemma 15, e a c h s u b p a t h s ' i n qz h a s a l a b e l e q u a l i n t h e r a n k i to a word S', where IS'I < 4/3(i0~n + 4~n)[C[. Extending s' by means of Pa, we obtain a path i s w i t h ~ ( s ) = S, ISI < 21ynlC I.
c. The assertion is obvious for a diagram with no cells. Considering, henceforth, the second version of the statement, we will study four cases like in part b. Case i is completely analogous. In case 2, IPII, IP21 < 227nk, because 18~I < 22ynk by hypothesis. Hence IsI < 1/218~ I < llynk. In case 3 a substitution for IPll (22ynk for 2ynlC I) results in the inequality [SI < 82ynk. In part 4,
[q~]
iS]< (@62 +22) vnk<105 ynk.
The ring case is studied in a similar manner. LEMMA 16. a. Suppose that A is a minimal diagram of rank i with a contour tq or A is a minimal ring diagram of rank i with contours t and q; let q be a smooth region with a period simple in the rank i . Then for each subpath q of the path q there exists a word U i such that ~(q) = U, HUfJ < 4/31t I + 1/2. b. Let A be a minimal diagram of rank i with a contour qlqlqmq=, where ql is a smooth region with a period C simple in the rank i, q2 is o-smooth, llplil, IIp2;I < 237nlIC'Jl (C' is some duplicate for C in the rank i). Then each vertex o of the path ql may be joined in A to some vertex of the path q2 by a path s for which there exists a word S such that ~ (s) = S, [ISII < 162ynHC'II. Proof.
We will prove the lermma by induction on ~(A).
2479
a. The assertion is obvious if A has no cells. In the general case we select a e-cell of rank j (cf. the notation in the proof of part a of Lemma 15). If ~ is a small cell, then, by Lemma 6, we may replace t I by s1-1uq2ws2 -I without increasing iltl] but decreasing T(A). Henceforth we assume that ~ is a large cell. Lemma i0 together with inequality (2) allows us to assume that ~2 < ~, i.e., F I indeed exists, and ~z < ~ by Lemma 2.3a. Therefore, ~2 > e. By Lemma i0,
Ilhil> (p(e-a)-47v)nllC/ll.
(5)
By Lemma 6, one can r e p l a c e t h e p a t h s p~ and P2 in t h e subdiagram rz by pz, P2, where IIps
< 23~nllCj'll,
and
(pg)
j - =~
~(p~),
s = i, 2.
For the obtained diagram F z' we have
9 (Fz') < ~(A) since the cell ~ of rank j has been excluded from A. So, by statement b) of Lemma 16 (and Lemma 18 in the rank j - i), for each subpath ql of the path qz there exists a i word U~ such that ~(q~) = U~ and
II G 11< (2. 162yn + 10 + ~ -1
n) II C}]l-
(6)
In the diagrams A I and A 2 with the contours q'pz-Zws2-1t ' and q"t"sl-Zup2 -I, using Lemmas 6 and 18, one can replace the subpaths of the contours p1-Zws2 -i and sz-lup2 -l by u' and u", respectively, preserving the labels in the rank j - i, and
II u ' Ii + !1 u" II < (92u + 20 + (1 - - O) n) JJC} lJ. Applying Lemma 16(a) to the obtained diagrams
At' and A 2' [their types are less than t(A)], i we obtain for any subpaths ql and q2 of q' and q" words Vz and V 2 such that ~(q~) = V~, ~ = i, 2, and
!1V, II + li v, II < + I(93~, + l--e) n I! C} II + IIt' II + II t" I1]By (5), (6), and (7), for the subpath q there exists a word U such t h a t , ( q )
IIUll<--3-[llt'll+ll
1t+(93W§
(7) i
= U and
n[IC}I]<
324y-r-
4 nllC;.li<-f-iltil.
4 4 1 < 43 (11t,II+llt, ii+ !It II)+ (,-~-+--~-(l--O)----9(O--cz)+450y ' 3 , '
b. The assertion is obvious if A has no cells. Otherwise, we choose a 8-cell ~ of rank j and consider a number of cases like in the proofs of Lemmas 4.4a and 15, using the same notation. i. This case is completely analogous to the case I of Lemma 15, however, Lemma I0 is used to replace Pl by Pl if ~ is a large cell. If ~ is a small cell, then llpzil <- ilplll by Lemma 6. 2. Let ~ be a large cell. In this case, like in Lemma 6, ~(p~) j-1 = X~, ~ = i, 2, where liX~il < 23u and by inductive reasoning it suffices to consider the case of o e {qll}. Then, by Lemma 16 for rz, (more exactly, for Fl' , like in part a), there exists a path s', j-1 where (s') = S', IIS'II < 16211Cj'll~n. Extending this path by a subpath of the contour of i the cell ~ and by the path pl 2, we obtain a path s for which ~(s) = S and, by Lemma 18,
IISll<
(1 6 2 7 n + 2 3 ? n + y +n
)
I0 "llC}l[
But, by Lemma 19 [in the rank r(F1) < r(~) by virtue of Lemma 4.1a], iiCj'li < 22~ilC'iI, whence 11Sil < 22ynllC'll. If ~ is a small cell, then, in the light of Lemma 6 (and Lemma 16 for FI) , the estimate is only strengthened.
2480
3. 3'.
i.e.,
Assuming
that ~ is a large cell, we consider two possibilities.
Fq is the subdiagram of incidence to q~.
By Lemma i0,
> (p (0--~--~)--47y)n ][ C} II, (8) By Lemma 16(a), each subpath fi of the path u ~ has a l a b e l equal
IWi < 1/201pzl - 1/2.
in the rank i to a word U, where
llU]] < +
(1 + ~-~--)I{P~/i < 33yn it C'/-
So the statement of the lemma is true for o e {u l} (33 + 23 < 162). Let o ~ {t} I.
Then,
by Lemma 16 for Fq (more exactly, for Fq'), o is joined to a point j-z on t 2 by a path s, w h e r e , ( s ) = S', IS'I < 162ynlCj'l. Extending the path s' by a subpath of the contour of the cell v, the path sl, and a subpath of the path pl -I, we obtain, using i Lemmas 6 and 18, a path s,~ (s) = S,
[IS II < ( + +24y § 162y)n [] Ciil +23yn I] C' '[, which is l e s s than 52ynlC'l by (8). If o does not lie on u~t I, then one should use the induction hypothesis for A" (more exactly, one should replace Pl in the contour of 8A" by Pl by means of Lemmas 6 and 18 and inequality (8), where l~(pl)l < Ipll, because 1 - O + $ +47~' < p(8 - ~ - B) -47u 3".
Fq is the subdiagram of incidence to qi.
Then we i ~ e d i a t e l y
turn to A '~ because
llplli < IPll. If ~ is a small cell, then, due to Lemma 6, the estimates are only improved~ 4.
If ~ is a small cell, then, by Lemma 16(a) for A and Lem~na 6, for each subpath i of the path q there exists a word U, where (4) = U, IUI < 4/3 46v.nllC'il + 1/2 < 62~nlC'I, whence ISI < (62 + 23)~nlC'I < 162~nlC'I.
I,.,II>(~ ~2(sli)-Zw1(si~)-~1)
in the rank j - I), we conclude by Lemma 16(a) that for each subpath i of the path ql there exists a word U such that ~(q) = U and
Hence
ISI < (23 + 139)ynlC'l
= 162~nlC'I.
LEMMA 17. Let A be a minimal ring diagram of rank i with contours t and q, where ~(q) C is a word simple in the rank i and ~(t) m C'. Then: a) for each (cyclic) subpath ~ of the path ql there exists a word U such that ~(q) =i ilUl < 4/3 IC'I + 1/2. There exists a path s with no self-intersections in A connecting i points on q and t such that ~(s) = S, ISI < max(0.021tl; i);
U,
b) for each subpath t of the path t there exists a word U such that q (~) =i u, juJ < ICI + 1/2. There exists a path s with no self-intersections connecting t and q such that i ~(s) = S, ISI < 0.021q I. 4/3
Proof. The first half of statement a follows from Lemma 16a. To prove its second half, we select a e-cell ~ with incidence degrees ~1 and ~2 like in the proof of Lemma 16(a) (using the same notation). If ~ is a small cell, then ISI 5 1 by Lemma 6. If ~ is large, then, by
2481
Lemmas 6 and 18, one of the paths s2w-lpl,
s1-1up2 -I has a label equal in the rank i to a
word S, where ilSU < (467 + l--8/2)nliCj'll. [cf. (5)]. Thus, IISli < 0.02iltli.
On the other hand,
lltll ~ (p(8 - ~) - 477)nllCj'il
b. The first half of the statement follows from Lemma 15 (a). Its second half is obtained like in part a (using Lemmas 2.1a and 4.2a instead of Lemma I0). LEMMA 18.
a.
Let X be a periodic word with a large period C simple in the rank i, i ~IC1 ~ txi < (s + I)IC I for some s There exist words Z l, Y, Z 2 such that X = ZIYZ 2, Y is a C'-periodic word, ilZ111, UZ2U < 511C'll, (s - 2)ilC'll ~ liYii ~ s b.
Let X be a C'-periodic word, where C is a large period simple in the rank i, s 5 i llXil < (s + l)llC'li. Then there exist words ZI, Y, Z 2 such that X = ZIYZ2, Y is a C-periodic word, (s - 2)Ic I 5 IYi ~ Pica, and IZ~[, Iz=i < 41c i . Proof.
Let X ~ xIckx2, where s - 2 < k 5 s and XI and X= are the end and the bei ginning of the word C. By Lemma 17, C = ZC'Z -I, where llZll < 1/2 + 4/3 ilC'li + max (ilC'li/50; i I), and X d = Ud, d = I, 2, where ilUdll < 1/2 + 4/3 liC'll + i. This implies that it suffices to put ZI m UIZ, Z2 ~ Z-IU2. b.
a.
This is proved like part a, except that we use Lemma 17(b).
LEMMA 19. Let A be a minimal diagram of rank i with a contour PlqlP=q2, where ~ (ql) and ~ (q2) are periodic words with periods C and D, simple in the rank i, such that C is a i large period. Furthermore, let IP~I, IP=I < 27nICl and for some words X~, ~ = i, 2, ~ (ps = Xs and lIXs < 237niiC'II. Then, if lq~l ~ ~niCl, then either IIC'II < 227llD'II, or C and D • are conjugate in the rank i. In the latter case, if C m D • then either C m D -~ and q~, q= are C-compatible in A or C is a product of two involutions in ~ i. Proof. It is easily seen that relaxing the estimate for IPs in half in the proof of Lemma 6.4a results in a similar statement with e replaced by 2e. Therefore, we may assume i that lq=i < (i + 2e)ID I. It follows from Lemma 18(a) that ~(q=) = W, where ilWil < lliID'il. i Taking into account the hypotheses of the lemma, we have ~ (q~) = V, liVil < llllD'il + 467nliC'II. i = Z~YZ=, where lIZ~li, iIZ2il < 511C'll, and Y is a i C'-periodic word with ilYii > (n~ - 2)ilC'll. Since ~(q~) = V, V is equal in the rank i to some word U, where iIUil < llllD'li + 477nlIC'll. We apply Lemma 12 to a minimal, in the rank i, diagram of this equality: On the other hand, by Lemma 18, ~ ( q ~ )
p(n~--2)llC'll
ilC'11 < ll(p~ - 2 - 477)-in-liID'tl < 227iiD'11.
LEMMA 20. Let A be a minimal diagram of rank i with a contour PlqlP2q2, where ql is a smooth region with a period C simple in the rank i, q2 a o-smooth region, IPdl < 27nlCI, and i ~(Pd) = Ud, where ilUdll < 237n11C'II, d = i, 2. Then one can choose a number of vertices o0, ..., o s on the path ql (where o0 and o s are the beginning and the end of the path ql) in such a way that each point os may be joined to some vertex vs on q2 by a path ts such that i its < 27nlC I and @(ts = Ts where Hrs < 237nilC'Ii, also Ios - os < 167nlCI, v 0 and u s are the beginning and the end of the path q2 -I, and the paths ts do not intersect one another. Proof. Here we call a cell v a G-cell if its incidence degrees to ql and q2 by means of some subdiagrams of incidence are no less than 6. If lqli < 167nlCi, then we may take s = i, to = pl -I, tl = P2. Assuming furthermore that lql > 4(ipll + 1P21); IPll, IP21 < 27nICl, we will prove the existence of a common edge in the paths ql and q2 -~ or the existence of a G-cell in A. Performing induction on T(A) (the statement is obvious if A has no cells), we choose a 8-cell ~ and consider four cases like in the proofs of Legends 4.4a and 15.
2482
1. Replacing pz by p~, where like in part i of Lemma 4.4a.
Ip~I < [p~[, we turn to a subdiagram
2.
In this case ~ is a required cell by definition.
3.
Like in the proof of Lemma 15b, we consider two variants.
A" of
lower type,
3'. Fq is the subdiagram of incidence to q~. Then, like in part 2 of the proof of Lemma 5.2a, [q~l - lq~[ < 2(IPll - IP~[), which allows us to apply the induction hypothesis to a diagram of lower type with the contour p~q~p=q= (the inequalities with the parameters and 8 remain true in part 2 of Lemma 5.2a when 8 is replaced by 8 - ~ and D by 2). 3".
Fq is the subdiagram of incidence to q= (see part 3 of Lemma 15).
4. Like in part 4 of Lemma 4.4a, 18~I < [i/2~(8 - ~ - ~) - 4~]-Zlp~I, which enables us (since ~ is incident to both p~ and p=) to join the beginning and the end of the path q~ in A by a path of length less than 7 § !Pz])(1 § ( + § 4Y)(+P(O--~--~)--4y)-l)<-~-(Ip~, + IP~I,.
IP~[ § ,pz,§247 But then, by Lemma 4.2a, sible.
[ql[ < 7/2p-I(IPI[
+ IP2[) < 4(IPII + [P2[), i.e., case 4 is impos-
We will now prove the statement of Lemma 20 by induction on lqiI. If ql and q2 -l have a common edge, then the choice of the beginning and the end of this edge as division points o~ and os allows us to perform the induction step. Otherwise, let ~ be a S-cell (see the notation in part 2 of the proof of Lemma 4.4a). Like there, IPI[, [P21 < 2~nICl and, like i in Lemma 6, (pd) = Xd, d = !, 2, where l]Xdll < 23ynUC'll. For the role of the paths t~ and ts I we choose pl -z and P2. It remains to estimate Ios - os = q~ But, by Lemma 2.2a, lqz~l < (i + s)[Cl < 16ynlCl, as required. LEMMA 21. Let A be a minimal diagram of rank i with a contour P~qlP2q=, where ~ (q~) is a periodic word with a large period C simple in the rank i; lq~l ~ (8 - ~)nIcI, f0
Ipkl<2ynlCl, e~(pk)-~X~, !!Xkl[<23ynI[C'l,
k---l, 2, io~[m3/2!Ct];
q2 is an A'-smooth region, where A is a word simple in the rank i or a period of rank i I 5 i. Then either ;lq2Jl < (i + s)lJA'tJ or C and A • are conjugate in the rank i, and, in the latter case, if A ~ C • then either A ~ C -I and the regions ql and q2 are A'-eompatib!e is a product of two involutions in ~i.
in A or C
Proof. By Lemma 14, j = r(A) < 22u I . Let A 0 be a word of minimal length conjugate to A' in the rank k = min(il - i, j), such that A0 m A if i I S j and A is a period of rank i I. Since, by definition, the word A is not conjugate in the rank k to powers of periods of ranks ~k, the word A 0 conjugate to it also possesses this property~ Furthermore, by the definition of the word A, IAoI ~ IAI ~ 2. By virtue of Lemmas 4.6a and 4.5a, the word A, and thus also A0, are not true powers in @k. Thus, conditions 1-3 of the definition of a word simple in the rank k hold for A 0. Clearly, A' is a duplicate of the word A 0 in the rank k. By means of part c of Lemma 15 for the diagram of conjugacy of the words A f and A0 in k the rank k < 22~ICI, we derive the equality ~(q2 ) = ZIYZ2, where Y is an A0-periodic word, IZII, IZ21 < 105ynICl, and in the diagram A' of this equality with the contour zlqlz2q2 -l [where Y ~ ( q ' ) ] the regions q=-i and q' and A'-compatible. We attach A' to A and, performing cancellations of cells if necessary, we obtain a minimal diagram A0 of rank j with the contour z2pzqlp~zlq' , where Iz2piI, Ip2zll < 105~nlC I +
2ynlCl.
Assume that i I S j.
Then
IAI <
22~IC1 and
in A 0 ~(ql) ~ ~(Pl-lz2 -l) • W (Zl-lp2-1),
where W =j Y and IW[ < nIA I < 22ynlC I because A n =j i. Thus, ~(ql) is equal in the rank i < IC[ to a word of length <236ynICl, which contradicts Lemma 4.2a because p(@ - ~) > 236yo So k = j and A 0 is a simple word in the rank j. Similarly, one can prove that i I > i 0"
2483
One can repeat the estimates of Lemma 6.4a, replacing A by A0, e by e/2, lq=[ > Sn[cl by [q=[ > (8 - ~)n[C[, and the conditions [pz[, [p~[ < 27n[C[ by [z~pz[, Ip~z~[ < 107yn[C[. Like there, we deduce that either [q'[ < (i + e/2)[A0[, or C • and rank j. In the latter case we may choose A0 m C +z-. Then, like in a product of two involutions in ~, or A 0 m C -m and the paths ql -~ in A 0. In the latter case A' and C -~ are conjugate in the rank i, C -~ and q~ and q= are A'-compatible in A.
A 0 are conjugate in the Lemma 6.4a, either C is and q' are C-compatible and if A m C • then A
It remains to consider the case of [q'[ < (i + e/2)[A0[. Suppose that llq21l~ (l+e)llA'll. We choose a vertex u on the path qz in such a way that it would divide the path q= = q= 1 q= 2 , 0
where ~ ( q i) m ~, is a cyclic shift of the word A' or, perhaps, ~(q=~) = A'a' for some letter a' e ~ (if q= has no secondary vertex determining a cyclic shift of the word A'). According to Lemma 20, we choose vertices os os z on the path qz and vs vs on the path q= in such a way that the point u lies on the subpath t" = vs - vs For the path p consisting of the edges of the paths t" and q22 we have [[pll ~ ~(i + g-l)llqa]I. Thus, if W ~ ~(p), where p = o0 - os then, by Lemma 12 for the subdiagram with the contour p~pts
II W II > ~ (1 + ~)-x II q~ ll--46vn II C' IIThis inequality also holds if C' is replaced by C, a duplicate for C in the rank j, because []~I[ ~ llC'll. Let [q~[ = mlC [.
By Lemma 18, ~(ql) j= Z~XZ2, where X is a C-periodic word,
HXII > (m - 2)IICll; IIZIII, llZ2ll < 5U~U. (pm - 47xn)lldll. Thus,
Comparing X with ~(q2),
IIWIl>p2e(I § If tradicts
by Lemma 12, we h a v e
IIq2ll >
9
0.8 mlcI, then, b y Lemma 18, ~ ( p ) ~ W, w h e r e the estimate (9) since m ~ (8 - ~)n. Therefore,
(9)
tIWll < ( 0 . 8 e m + 10)IICII w h i c h c o n -
Ipl > 0.8eml C I. We d e n o t e
t'
= os - os
and consider
(10)
a subdiagram
in A with
the
contour
t~-~t'ts
''.
By Lemma 20, [t~-It'ts < 20xn[C[. Since the region t" is A'-smooth, by Lemma 15(a) for the subpath s = vs - u there exists a word S such that
S =ir Let u'
be a vertex
of the
I S I < T4 .20ynICI path
q' which divides
+}!-<27ynlCl. 2 the
path
q'
(11) = ql'q2',
where ~(qz')
~
0
is
a cyclic
shift
o f t h e w o r d A0 o r ,
perhaps,
~ ( q l 1) = Aa f o r
some l e t t e r
a e
~.
Since
q2 -1
and q' are compatible in A', A' =j ZIAZI -l, whence (a')-I(A')-IZIAa =j (a')-iZla, i.e., for the label of the path (q2')-Izlql ' joining the vertices u and u' there exists a word T equal to it in the rank j, of length <2 + 105yn[C[. Since os I is joined to u by the path ts the last estimate, (ii) and Lemma 20 imply that in A 0 the vertices os I and u' are joined by a path r, where ~(r) ~ R,
[R[ < 135ynlC[.
Hence, by Lemma 4.2a for the diagram of rank j with
t h e c o n t o u r z 2 p l p r q 2 ' , w h e r e ~ ( r ) ~ R, [ q 2 ' [ > PIP[ - [ z 2 P l [ - [R[ > ( 0 . 8 p e m - 2 4 2 y n ) [ C [ the hypotheses o f t h e lemma, t h e d e f i n i t i o n o f Z2, a n d i n e q u a l i t y (10). So
IAol > (e/2)-11 qg I > (l.6pm--60 000yn) I C IOn the other hand, by Lemma 4.2 in the rank j for A 0
IAo 1<~ [q' 1< P- 1 Iz~Plq,P2Zx ]< P - ' ( Iql I § 214yn IC I) < p-1 (m + 214yn) IC I. The l a s t
two i n e q u a l i t i e s
are
contradictory
because
m ~ (8 - ~ ) n a n d
( 1 . 6 p - p -1) (0--~) > 6 0 250y. Thus, the inequality llq211 ~ (i + ~)lIA'll is false. 2484
The lemma is proved.
by
5.
The Proof of the Theorem
The exactness of the n-product has been established in Lemma 1 and Mal'tsev's postulate in Lemma 5. Adding trivial factors does not change the group ~, as follows from its definition. To prove the associativity,
we compare
@ with
@~=
Rn
@~, where
G~
~
Rn
~.
By the
above, both groups are generated by the subgroups ~, and we have to prove that the sets of relations between elements of distinct subgroups 9~g in G and in @o coincide. An irreducible word in the alphabet
9~ can be rewritten
in the alphabet
~=
U ~n \ {I}, g
when an element of some ~ is assigned to each o-syllable (as can be seen from Mal'tsev's postulate and the postulate on trivial factors, the subgroups ~,, ~ J f ~ , generate in ~ a subgroup naturally isomorphic to the subgroup ~ in ~o ). Turning from a word X as an element of the group @ to the same word as an element of @~, we mean rewriting of this kind; then IIXII = IX[o, the length relative to H. The reverse passage from ~ to ~[ is also possible, of course. As follows from the definition of duplicates of periods, the system of all relations of the form (Ci,K') n = 1 is defining for ~. Therefore, it suffices to prove, by collective I,K 'II and [Cj, K Io, that each defining relation (Ci,~a)n = i of the group induction on IIC" holds in Go, and all relations Cj,v n = 1 for @= are true in G. Let C i' be a duplicate of a period of rank i and suppose that (Ci')n # 1 in Goo Then it follows from the definition of periods and from Lemma 4.6a that the word C i' is conjugate in ~ to an element of one of the groups @n or is a product of two involutions. In the former case C i' = XaX -l in ~o, a ~ ~, and in the minimal diagram (over ~ ) of this conjugacy the lengths of perimeters of all cells are, by Lemmas 5.1a and 4.3a, less than Go
(i+2~) P-' ([I C/ll +1)
Cj n = 1 is false in
e.
Indeed,
otherwise we would have as a corollary that Ci f = XaX -l in ~, whence a n = 1 in ~, i.e., a n = 1 in ~ , and, therefore (Ci') n = (XaX-I) n = 1 in @o, contrary to the hypotheses. In the latter case C i' = X(CE')-IX -I in Go , again by virtue of relations with periods Cj, where [Cjl o < llCi'll. If all Cj n = 1 were true in G, then C i' = X(Ci')-IX -~ in @. Then, by Lemmas 5.1a and 4.3a, C i and Ci -I would be conjugate Lemma 4.
in Gi-1, which is impossible by
Thus, there exists a period Cj (for the group ~o ) such that in
~
[Cjl o < lICi'll and Cj n # 1
To complete the proof, we will establish an analogue of Lemma 5.1a.
LEMMA 22. Let A be a maximal ring diagram of rank i with contours p and q. Then some vertices of the paths p and q can be joined in A by a path t such that for some word T i
~(t)
= T,
IITII < Ilpll + Ilqll.
Proof. The statement is obvious if A has no cells or all of its cells are small. We now introduce m'(A) in a way similar to ~(A) but taking into account only large cells and perform induction on m'(A) and, for a fixed mi(A), on llp[l + l[q[l. Let A' be a ring diagram minimal in the rank i with contours p' and q' obtained by replacing the labels of segments of the paths p and q by words minimal in their respective factors. By Lemma 2, T'(A') S ~(A). Furthermore, A' has no small cells whose degree of incidence to one of the segments of the paths q' , p ' is no less than ~. So, if ~ is incident by means of a subdiagram F to qf (or to p') with degree >a, then IIF A q'll ~ 3 by Lemma 6. Thus, excising ~ together with F out of A', we lower llpll + llqll by at least two units. By the induction hypothesis, one can join the contours in the obtained diagram A 0 by a path t o , i where ~(t0) = To, liT011 < llpll + l[qll - 2. Using Lemma 6, we can extend the path t o to q' while liT011 is increased no more than by i. This implies the assertion of the lemma.
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Suppose, further, that ~ is a large cell with a period Ck whose degree of F-incidence to q' is >8. If t = ~ A F, p~tp2s is the contour for F, and t is an arc of the cell ~ comi plementary to t, then replacing s by a path t, where 9 (s) = ~(t), according to Lemma I0, results in a diagram A, T'(A) < T'(A) (because the cell ~ has been excised), for whose contour q we have 11qil < ilqU + (i - 8 - pe + 937)niICk'li. Applying Lemma 22 to A, we find a path i _ joining the contours, where ~(t) = T, iITII < liqi; + ilpli + (i - 6 - p8 + 937)niICk'll. Adding to it a subpath of the path q with norm 5(i - 8 + 477)nlICk'[l (see Lemma i0) along with a subpath of norm !l (in the passage from A' to A by means of Lemma 2), we obtain the required inequality for UTII. Finally, let ~ be a large 8-cell incident to both p' and q'. and 12,
Then, by Lemmas 6, 18,
IIp'll+llq'[[ > (p0+93u
(12)
i and f o r t h e p a t h t ' = o l -- 02 ( s e e t h e n o t a t i o n i n p a r t 2 o f Lemma 5 . 1 a ) ~ ( t ' ) = T ' , where IIT'II < (1 - 0 + 47~)ntlCk'll by Lemmas 6 and 18, w h i c h t o g e t h e r w i t h ( 1 2 ) i m p i i e s t h e r e q u i r e d i n e q u a l i t y f o r t h e word T w h i c h i s o b t a i n e d by e x t e n d i n g t h e word T' by s u b w o r d s o f norm ~1 i n t h e p a s s a g e t o A. The 1emma i s p r o v e d ; t o c o m p l e t e t h e p r o o f o f t h e t h e o r e m , we assume t h a t where Ci i s a p e r i o d o f r a n k i f o r ~o. A g a i n , two p o s s i b i l i t i e s may a r i s e .
Ci n ~ 1 i n
~,
1. Ci i s c o n j u g a t e i n ~ t o an e l e m e n t a ~ ~,. By Lemma 22, Ci = XaX - z i n @, w h e r e fIX < 2(llCill + i). By Lemma 13, the last equality holds by virtue of the conditions with periods Cj, where [iCjii < 8p-~n-lilCili = 8p-~n-~ICil o. If all (Cj') n = 1 in ~, then C i = XaX -i in ~, whence a n = 1 in ~ (since Ci n = 1 in ~ ), i.e., an = 1 in ~ (since a ~ ~,~ @ ). Hence c i n = (XaX-i) n = 1 in @, contrary to the assumption above.
2. C i = XCi-IX -I in ~. By Lemma 22, we may assume that llXi[ < 411Ciil and, by Lemma 13, the given equality holds by virtue of the conditions with periods Cj, where [iCj'il < 10p -~ x n-lllCiil = 10p-ln-llCilo. If all (Cj') n = 1 in ~o, then C i = XCi X-~ in ~o, which leads to a contradiction (as a similar equality in ~ above). Thus, there exists a period Cj for the group @, such that ilCj'li < aCiDo and (Cj') n # 1 in ~o. Together with the conclusion drawn before Lemma 22, this means that the assumptions are false, and the reserves of relations for ~ and @o are identical. The remaining assertions of the theorem follow from Lemmas 4.6a and 5.6a. LITERATURE CITED i. 2. 3. 4. 5. 6. 7. 8. 9.
O. N. Golovin and M. A. Bronshtein, "Axiomatic Classification of Exact Operations," in: Algebra and Logic [in Russian], Nauka, Novosibirsk (1973), pp. 40-96. A. G. Kurosh, Group Theory [in Russian], Nauka, Moscow (1967). S. I. Adyan, "Periodic products of groups," Trudy MIAN, 142, 3-21 (1976). P. S. Novikov, "On periodic groups," Dokl. Akad. Nauk SSSR, 127, No. 4, 749-752 (1959). P. S. Novikov and S. I. Adyan, "On infinite periodic groups," Izv. Akad. Nauk SSSR, Ser. Mat., 32, Nos. i, 2, 3, 212-244, 251-254, 709-731 (1968). S. I. Adyan, "On simplicity of periodic products of groups," Dokl. Akad. Nauk SSSR, 241, No. 4, 745-748 (1978). A. Yu. Ol'shanskii, "On the Novikov-Adyan theorem," Mat. Sb., 118 (160), No. 2, 203-235 (1982). R. Lyndon and P. Schupp, Combinational Group Theory [Russian translation], Mir, Moscow (1980). S. I. Adyan, Burnside's Problem and Identities in Groups [in Russian], Nauka, Moscow
(1975).
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