Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen
1639
S rin er BPerlin g Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore
Tokyo
Karel Dekimpe
Almost-Bieberbach Groups: Affine and Polynomial Structures
Springer
Author Karel Dekimpe* Katholieke Universiteit Leuven Campus Kortrijk Universitaire Campus B-8500 Kortrijk, Belgium e-mail: Karel.Dekimpe @ kulak.ac.be * Postdoctoral Fellow of the Belgian National Fund for Scientific Research (N.EWO.)
Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Dekimpe, Karel: Almost Bieberbach groups: affine and polynomial structures / Karel Dekimpe. - Berlin ; Heidelberg ; New York ; Barcelona ; Budapest ; Hong Kong ; London ; Milan ; Paris ; Santa Clara ; S i n g a p o r e ; T o k y o 9 S p r i n g e r , 1996 (Lecture notes in mathematics ; 1639) ISBN 3-540-61899-6 NE: GT Mathematics Subject Classification (1991): Primary: 20H15, 57S30 Secondary: 20F18, 22E25 ISSN 0075-8434 ISBN 3-540-61899-6 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1996 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10479900 46/3142-543210 - Printed on acid-free paper
For Katleen, Charlotte and Sofie.
Preface The reader taking a first glance at this monograph nfight have the (wrong) impression that a lot of t o p o l o g y / g e o m e t r y is involved. Indeed, the objects we study in this book are a special kind of manifold, called the infra-nilmanifolds. This is a class of manifolds that can, and should, be viewed as a generalization of the flat Riemarmian manifolds. However, the reader fa,*niliar with the theory of the fiat Riemannian manifolds knows that such a manifold is completely determined by its fundamental group. Moreover, the groups that occur as such a fundamental group can be characterized in a purely algebraic way. More precisely, a group E is the fundamental group of a flat Riemannian manifold if and only if E is a finitely generated torsion free group containing a normal abelian subgroup of finite index. These groups are called Bieberbach groups. It follows that one can study the fiat Riemannian manifolds in a purely algebraic way. This group theoretical approach is also possible for the infra-nilmanifolds, which are obtained as a quotient space under the action of a group E on a simply connected nilpotent Lie group G, where E acts properly discontinuously and via isometries on G. (If G is abelian, then this quotient space is exactly a flat Riemannian manifold). The fundamental group of an infra-nilmanifold is referred to as an almost-Bieberbach group. It turns out that much of the theory of Bieberbach groups extends to the a l m o s t - B i e b e r b a c h groups. Thus for instance, a group E is the fro'idamental group of an infra-nilmanifold if and only if E is a finitely generated torsion free group containing a normal nilpotent subgroup of finite index. The aim of this book is twofold: 1. I wish to explain and describe (in full detail) some of the most imp o r t a n t group-theoretical properties of almost-Bieberbach groups.
VI
Preface I have the impression that the algebraic nature of almost-Bieberbach groups is far from well known, although many of their properties are just a straightforward generalization of the corresponding properties of the Bieberbach groups. On the other hand, I do not claim to be a specialist of Bieberbach (or more general crystallographic) groups and so a lot more of the theory o f Bieberbach (crystallographic) groups still has to be generalized. I hope therefore that this book might stimulate the reader to help in this generalization. . I also felt there is a need for a detailled classification of all a l m o s t Bieberbach groups in dimensions _< 4. We will see that an infranilmanifold is completely determined by its fundamental group. So m y classification of almost-Bieberbach groups can also be viewed as a classification of all infra-nilmanifolds of dimensions < 4. I myself use the tables of almost-Bieberbach groups not really as a classification but as an elaborated set of examples or "test cases" for new hypotheses. I hope that, one day, they can be of the same value to you too.
I tried to write this monograph both for topologists/geometers as for algebraists. Therefore, I made an effort to keep the prerequisites as low as possible. However, the reader should have at least an idea of what a Lie group is. Also, a little knowledge of the theory of covering spaces can be helpful now and then. From the algebraic point of view, I assume that the reader is fairly familiar with nllpotent groups and that he is acquainted with group extensions and its relation to cohomology of groups. Although this work is divided into eight chapters, there are really three parts to distinguish. . In the first part (Chapter 1 to Chapter 3), we define almost-crystallographic and almost-Bieberbach groups. We spend a lot of time in providing alternative definitions for them. Also we show how the three famous theorems of L. Bieberbach on crystallographic groups can be generalized to the case of almost-crystallographic groups. These first chapters could already suffice to let the reader start his own investigation of almost-crystallographic groups. . Chapter 4 forms a part on its own. It deals mainly with m y own field of interest, namely the canonical type representations. These are representations of a polycyclic-by-finite group (in our situation always virtually nilpotent), which respect in some sense a given
VII
Preface
filtration of that group. We discuss both affme and polynomial representations and present some nice existence and uniqueness results. The reason for considering polycyclic-by-finite groups is n a t u r a l in the light of Auslander's conjecture. . The last part of this monograph (Chapter 5 to Chapter 8) describes a way to classify almost-Bieberbach groups. We also give a complete list of all almost-Bieberbach groups in dimensions _< 4, which were obtained using the given method. Moreover, we show how it was possible to use these tables and find in a pure algebraic way some topological invariants (e.g. Betti numbers) of the corresponding infra-nilmanifolds. Finally, I would like to say a few words of thanks. To Professor Paul Igodt who introduced me to the world of infra-nilmanifolds and who proposed me to investigate the possibility of classifying the a l m o s t Bieberbach groups. I a m also grateful to Professor Kyung Bai Lee, since I owe much of m y knowledge on almost-Bieberbach groups to him. But most of all I must thank m y wife Katleen, for her encouragement when I was doing m a t h e m a t i c s in general and especially for her support and practical help when I was writing this book. \
Karel Dekimpe, Kortrijk, August 19, 1996
Contents Preliminaries and notational conventions 1.1 N i l p o t e n t groups . . . . . . . . . . . . . . . . . . . . . . . 1.2 N i l p o t e n t Lie groups . . . . . . . . . . . . . . . . . . . . .
1 1 7
Infra-nilmanifolds and Almost-Bieberbach groups 2.1 C r y s t a l l o g r a p h i c a n d B i e b e r b a c h groups . . . . . . . . . . 2.2 A l m o s t - c r y s t a l l o g r a p h i c groups . . . . . . . . . . . . . . . 2.3 How t o generalize the t h i r d B i e b e r b a c h t h e o r e m . . . . . 2.4 T h e first p r o o f of t h e generalized t h i r d B i e b e r b a c h t h e o r e m revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 A new p r o o f for the generalized t h i r d B i e b e r b a c h t h e o r e m Algebraic characterizations of almost-crystallographic group s 3.1 A l m o s t - c r y s t a l l o g r a p h i c groups a n d essential extensions . 3.2 T o r s i o n in the centralizer of a finite i n d e x s u b g r o u p . . . . 3.3 T o w a r d s a g e n e r a l i z a t i o n of A C - g r o u p s . . . . . . . . . . . 3.4 T h e closure of the F i t t i n g s u b g r o u p . . . . . . . . . . . . . 3.5 A l m o s t t o r s i o n free groups f r o m the t o p o l o g i c a l p o i n t of view
4
Canonical t y p e r e p r e s e n t a t i o n s 4.1 4.2 4.3 4.4
Introduction .......................... D e f i n i t i o n of canonical t y p e s t r u c t u r e s . . . . . . . . . . . A n algebraic d e s c r i p t i o n of the Seifert F i b e r Space struction . . . . . . . . . . . . . . . . . . . . . . . . . . . C a n o n i c a l t y p e affine r e p r e s e n t a t i o n s . . . . . . . . . . . . 4.4.1 I t e r a t i n g c a n o n i c a l t y p e affine r e p r e s e n t a t i o n s 4.4.2 Canonical type representations and matrices p o l y n o m i a l rings . . . . . . . . . . . . . . . . . . . 4.4.3 V i r t u a l l y 2-step n i l p o t e n t groups . . . . . . . . . . 4.4.4 V i r t u a l l y 3-step n i l p o t e n t groups . . . . . . . . . . 4.4.5 W h a t a b o u t the general case? . . . . . . . . . . . .
13 13 15 19 21 28
31 31 36 39 41 45 47 47 48
con. . . . over
53 57 58 61 70 75 78
X
Contents 4.5
Canonical type polynomial representations ......... 4.5.1 T h e first a p p r o a c h . . . . . . . . . . . . . . . . . . 4.5.2 T h e second a p p r o a c h . . . . . . . . . . . . . . . . . 4.5.3 E x i s t e n c e a n d u n i q u e n e s s of P o l y n o m i a l M a n i f o l d s 4.5.4 G r o u p s of affiaae defect one . . . . . . . . . . . . .
81 82 85 91 95
The Cohomology of virtually nilpotent groups 103 5.1 T h e n e e d of c o h o m o l o g y c o m p u t a t i o n s . . . . . . . . . . . 103 5.2 T h e c o h o m o l o g y for s o m e v i r t u a l l y n i l p o t e n t g r o u p s . . . 103 117 5.3 M o r e a b o u t t h e c o h o m o l o g y of v i r t u a l l y a b e l i a n g r o u p s 5.4 A p p l i c a t i o n to t h e c o n s t r u c t i o n of a l m o s t - c r y s t a l l o g r a p h i c groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Infra-nilmanifolds and their topological invariants 6.1 3 - d i m e n s i o n a l A l m o s t - B i e b e r b a c h g r o u p s . . . . . . . . . 6.2 C l a s s i f i c a t i o n o f r a n k 4 n i l p o t e n t g r o u p s . . . . . . . . . . 6.2.1 N is abe]Jan (class 1) . . . . . . . . . . . . . . . . 6.2.2 N is of class 2 . . . . . . . . . . . . . . . . . . . . 6.2.3 N is of class 3 . . . . . . . . . . . . . . . . . . . . 6.3 4 - d i m e n s i o n a l A l m o s t - B i e b e r b a c h g r o u p s . . . . . . . . . 6.4 O n t h e B e t t i n u m b e r s of I n f r a - n i l m a n i f o l d s . . . . . . . . 6.5 Seifert i n v a r i a n t s of 3 - d i m e n s i o n a l i n f r a - n i l m a n i f o l d s 6.6 I n v e s t i g a t i o n o f t o r s i o n . . . . . . . . . . . . . . . . . . . . 7
Classification survey 7.1 3- d i m e n s i o n a l A C - g r o u p s 7.2 4 - d i m e n s i o n a l A B - g r o u p s , subgroup . . . . . . . . . . 7.3 4 - d i m e n s i o n a l A B - g r o u p s , subgroup . . . . . . . . . .
The A.1 A.2 A.3 A.4 A.5
.................. with 2-step nilpotent . . . . . . . . . . . . . . with 3-step nilpotent . . . . . . . . . . . . . .
121 121 123 124 124 127 131 134 . . . 144 154 159 159
Fitting . . . Fitting . . .
168 219
use of Mathematica | 231 Choose a crystallographic group Q ............. 231 D e t e r m i n a t i o n of c o m p u t a t i o n a l c o n s i s t e n t p r e s e n t a t i o n s . 233 C o m p u t a t i o n of H2(Q, Z) . . . . . . . . . . . . . . . . . . 240 Investigation of the torsion ................. 243 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
Bibliography
2 51
Index
256
Chapter 1
P r e l i m i n a r i e s and notational conventions 1.1
N i l p o t e n t groups
In this first chapter we discuss the fundamental results needed to understand this book. Our primary objects of study are virtually nilpotent groups. R e m e m b e r that a group G is said to be virtually P , where :P is a property of groups, if and only if G contains a normal subgroup of finite index which is :P. Although we assume familiarity with the concept of a nilpotent group, we recall some special aspects of this theory in order to fix some notations. Let N be any group, then the upper central series of N Z,(N):
Zo(N) = 1C_ Z I ( N ) C_ ... C_ Z i ( N ) . . .
is defined inductively by the condition that
Z i + I ( N ) / Z i ( N ) = Z(N/Z~(N)) where Z(G) denotes the center of a group G. The group N is said to be nilpotent if the upper central series reaches N after a finite number of steps, i.e. there exists a positive integer c such that Zc(N) = N. If c is the smallest positive integer such that Zc(N) = N , we say that N is c-step nilpotent or N is nilpotent of class c. Another frequently used central series is the lower central series. This series uses the c o m m u t a t o r subgroups of a group N . We use the convention that the c o m m u t a t o r [a, b] = a - l b - l a b for all a, b E N. Conjugation
2
C h a p t e r 1: Preliminaries and n o t a t i o n a l conventions
in N with a is indicated by It(a).
Sometimes we use ba -- a - l b a =
,(a-1)(b). The lower central series of N is the central series N : 71(N) _~ 72(N) _~ -'" _~ 7i(N) _~ " " where t h e / - f o l d c o m m u t a t o r subgroups 7i(N) are defined inductively by the formula 7i+1(N) = [N, Ti(N)]. The groups we are interested in are the finitely generated torsion free nilpotent groups for which it makes sence to consider central series
iv,: I = N o C N 1 C N ~ C . . . C N c = N with torsion free quotients N i / N i _ I for 1 < i < c. We will refer to such a central series N . as a torsion free central series. Given such a torsion free central series, there exists integers ki E such t h a t N i / N i - 1 ~ ~k~. We write Ki = ~ kj. We also write K for K1, which is the rank or Hirsch number of N. A set of generators al,1,
a1,2,
9 9 9 ~ al ,kl ~ a2,1 ~ 9 9 9 as,k~,
a3,1,
99 9
ac,kc}
of N will be called compatible with N. iff Vi E { 1 , 2 , . . . , c } : a l , l , a l , 2 , . . . , a i , k ~ generate Ni. It is clear at once t h a t any torsion free central series of N admits a compatible set of generators. Such a compatible set of generators m a y be obtained in the following way: First we choose kl generators of N1, say a1,1, al,2,. 9 al,kl. Then we complete this set to a set of generators for N2. So we have to choose elements a2,1,...a2,k~. We continue this way and finally we find the last kc generators a~,l,. 9 ac,k~. Any element n E N can now be written uniquely in the form Xc,1
~c,2
~Cl,kl
n = ac, 1 a~,2 . . . al,k~
C N , for some mi,j E ~ .
This shows t h a t we m a y identify n with its coordinate vector (~1,1, xl,2, 9 9 xl,k~, ~2,1, 9 9 xi,j, 9 9 Xc,k~) E E K. For all torsion free finitely generated nilpotent groups N, the upper central series determines a torsion free central series, while in general the lower central series fails to have torsion free factors. However, we can alter the lower central series slightly in order to get a torsion free central series. To explain this we need the concept of the isolator:
Section 1.1 Nilpotent groups
3
D e f i n i t i o n 1.1.1 (see also [56], [59]) Let G be a group. For H a subgroup of G, the i s o l a t o r of H in G (sometimes called the r o o t s e t ) is defined by
= {g E G II gk ~ H for svme k > 1}. In general, the isolator of a subgroup H in G doesn't have to be a subgroup itself. E.g. if H = 1 then ~ is exactly the set of torsion elements of G, which needn't be a group in general. We will only need the isolator of a c o m m u t a t o r subgroup. L e m m a 1.1.2 Let G be any group. Then,
1. Yk E No: ~ 2. Vk C No: G/ ~
is a characteristic subgroup of G. is torsion free.
3. Vk, l e N0: [
c
For the proof of this lemma we refer the reader to [56, page 473]. It follows that for any finitely generated, torsion free c-step nilpotent group N the series
is a torsion free central series. We will refer to this series as t h e a d a p t e d lower central series. For any group G, the groups ~/~-(G) can be determined by means of a universal property. Write T~(G) = G/~/7~+1(G) and denote the canonical projection of G onto T~(G) by p. The group T~(G) is the biggest possible torsion free quotient of G, which is nilpotent of class _< i. Formally L e m m a 1.1.3 U n i v e r s a l p r o p e r t y o f ri(G). Let G be any group and suppose that N is a torsion free nilpotent group of class < i. Given a group homomorphism ~ : G ~ N , there exists a unique morphism r : Ti(G) -+ N such that ~ = r o p. I.e. the following diagram commutes: G
P
. ~(C)
N
4
Chapter 1: Preliminaries and notational conventions
Proof: As N is nilpotent of class _< i, all i + 1-fold commutators are m a p p e d trivially. So the morphism ~ factors through G/Ti+I(G). Also, as N is torsion free, the characteristic subgroup v(G/Ti+l(G)) consisting of all torsion elements of G/Ti(G) is m a p p e d trivially. Therefore, there is a factorization :a
~ N.
(1.1)
But ~7X/-~+I(G) consists exactly of those elements which are m a p p e d into the set of torsion elements r(c/7~+~(a))under the canonical projection of G onto C / ~ i + l ( a ) . So, =
from which it follows that the factorization (1.1) mentioned above can be written as:
~ : a P,r~(C) r This establishes the existence of the map r
The uniqueness is obvious.
The above proposition determines the subgroup ~ (G) of G completely. For suppose there exists another normal subgroup A of G (together with a canonical projection q : G ~ G/A), such that any morphism qo : G --* N as above can be written in the form ~ = r o q. In this case let N be equal to G~ ~ ( G ) and let ~ = p. It is obvious that the map r in tiffs case maps the coset g A onto g ~/Ti+l(G), for all g C G. By reversing the roles, we obtain a morphism r G~ ~ ( e ) ~ A : g ~/Ti+1(G) g A, which is the inverse of r Therefore, the groups A and coincide. As an application of the above universal property we find: L e m m a 1.1.4 Let G be any group. For all j >_ i, there is a canonical isomorphism ~
-
It follows that for G =
G
.
C)
Proofi By the universal property of ri(G) there is a canonical morphism el: ,~(a)-~ ~(~j(c)) which maps the coset of an element g of G onto the coset of g G in r & j ( G ) ) . Conversely, we have the following commutative diagram
Section 1.1 Nilpotent groups
5
G
where 1. the non labeled arrows are canonical projections onto a quotient group. 2. c is induced by the fact that ~ / ~ + I ( G ) i s contained in ~ ( G )
(j > i). 3. c2 is obtained by the universal property of T~(Tj(G)). It is clear now that cl and c2 are each others inverse. The last claim of the lamina, concerning the equahty of the two subgroups of G / G , follows from the comments preceding this lemma. []
The following technical laminas will be needed at special occasions during our t r e a t m e n t of almost-crystMtographic groups. L e m m a 1.1.5 Let H be a torsion free, normal subgroup of finite index in a group G. A s s u m e z E Z ( H ) and x 9 G such that [~,z] ~ 1. Then any commutator of the f o r m [~, [... [x, [~, z]]...] is not trivial. P r o o f : Consider the sequence (ci)ie N in Z ( H ) defined by Co = z and ci+l = [~, ci]. We proceed by induction. Assume ci r 1 and ci+l = 1 for i _> 1. If [G : H] = m, x T M E H and hence it commutes with ci-1. A trivial computation shows that 1 = [xm, c _l] =
1] x j=l
-j
z
m m I I E~ j----1
3
m ~---Ci .
Since H is torsion free ci = 1, which is a contradiction.
L e m m a 1.1.6 If 0 --* H ~ G ~ K ~ I determines G as a central extension of an abelian group H by a group K which is nilpotent of class ~_ c, then G is nilpotent of class <_ ( c + 1). The proof is straightforward and left to the reader.
6
Chapter 1: Preliminaries and notational conventions
L e m m a 1.1.7 If 1 --~ H --~ G ~ ~k ~ 1 determines G as a nilpotent extension of a torsion-free, c-step nilpotent group H by a finite cyclic group, then G itself is c-step nilpotent. P r o o f : The p r o o f goes by induction on the nilpotency class of H . So, assume H is abelian. Take g E G and h E H . If [g,h] ~ 1, lemma 1.1.5 implies that G is not nilpotent. Consequently, it follows that the extension is a central one. Having chosen a section s : Zk ~ G, elements g E G can be written as hs(x l) (h E H, x is a generator of Zk and 0 _< l _~ k - 1). Now, it is clear that if gl and g2 are in G, [gl,g2] = [s(xl'),s(zl2)]. Since s(x) l and s(x l) belong to the same coset of H , it follows that [gl,g2] = [s(x) 5, s(x) t~] -- 1, so G is abelian. Now assume H is of class c. Lemma 1.1.5 implies that Z ( H ) C Z ( G ) when G is nilpotent. Consider the short exact sequence 1 ~ H / Z ( H ) G / Z ( H ) ~ Zk ~ 1. Here, H / Z ( H ) is torsion-free ( c - 1)-step nilpotent. By induction G / Z ( H ) itself is ( c - 1)-step nilpotent. Apply lemma 1.1.6 to the extension 1 ~ Z ( H ) ~ G ~ G / Z ( H ) ~ 1 and deduce that G is nilpotent of class < c.
L e m m a 1.1.8 Let G be any group and suppose that T is a torsion free normal subgroup of G, while F is a finite normal subgroup of G. Then
[T,F] = 1. Proof: Let t C T and f E F , then
[t, f] = t - l f - l t f = t -1 f - l t f EF
CT AF = 1
ET
L e m m a 1.1.9 Let ~ be any automorphism of ~k. I f there exists a subgroup A of finite index in Z k on which ~ is the identity, then ~ is the identity automorphism. Proof." ~ can be represented by an invertible matrix M with integral entries. Seen as an element of Gl(n, [~), this matrix represents a linear mapping leaving fixed a generating set (i.e. A) of the real vector space R '~. This implies that M is the n • n-identity matrix.
S e c t i o n 1.2 N i l p o t e n t L i e g r o u p s
1.2
7
N i l p o t e n t Lie groups
Although we intend to keep the topological/geometrical background needed to understand this book as small as possible, we need at least some knowledge concerning nilpotent Lie groups. In fact, most of the stuff we will use can be found in the magnificent paper of A.I. Mal'cev [51]. We refer to this paper for all the proofs of the claims we make here. Throughout this section G will denote a connected and simply connected nilpotent Lie group. We use 9 to indicate the Lie algebra of G. This Lie algebra 9 has the same dimension and nilpotency class as G. Moreover, in the case of connected and simply .connected nilpotent Lie groups it is known that the exponential map exp : 9 ~ G is bijective. We denote its inverse by log. The exponential map earns its name because of the fact that for matrix groups/algebras the exponential map is indeed oo A n given by the exponentiation of matrices. I.e. exp(A) = ~ n = 0 ~ - . If H is another connected and simply connected nilpotent Lie group, with Lie algebra D, then we have the following properties: 9 For any morphism ~ : G ~ H of Lie groups, there exists a unique morphism dg~ : g ~ [~ (differential of ~) of Lie algebras, making the following diagram commutative: r
G
)
H exp
(1.2)
dqo
9 Conversely, for any morphism d~ : 9 --* I~) of Lie algebras, there exists a unique morphism ~ : G ~ H of Lie groups, making the above diagram commutative. In G, it makes sense to speak of a x where a (== G and z E R. (E.g. consider the one-parameter subgroup of G passing through a). A formal definition m a y look as follows: D e f i n i t i o n 1.2.1 a~ =exp(xloga),
V a E G, Vx E •.
The definition satisfies all the expected conditions: 1. a . a -1 = a - l . a =
1;
a ~ = 1,
8
Chapter 1: Preliminaries and notational conventions 2. a'~ = ~
,
i f n E N,
n times 3.
4.
a;
=a, ifnE
N,
= a xy,
5. aX.a y = a az+y, 6. If ~o : G ---+ H is a morphism between connected and simply connected nilpotent Lie groups, then ~(a x) = (~(a)) x. We give a p r o o f of this last property, using the commutative diagram (1.2): ~o(ax)
log a)
=
~o(exp z
=
exp(d~(ae log a))
=
exp(xd~(log a))
= =
exp(x log(cp(a))) (~(a)) x .
We also mention the famous C a m p b e l l - B a k e r - H a u s d o r f f formula:
VA, B E 9 :
e x p ( A ) . e x p ( B ) = e x p ( A , B),
where
(1.3)
o0
A*B=A+B+~
Cm(A,B).
I[A,B] + ~ m~3
Here Cm(A, B) stands for a rational linear combination of m - f o l d Lie brackets in A and B. Since our Lie algebras are nilpotent, the sum involved in A 9 B is always finite. As an immediate consequence of tiffs formula, one sees that
Va, b E G : log(a.b)= l o g a , logb. Of m a j o r importance to us, is the concept of
a
uniform lattice of G.
D e f i n i t i o n 1.2.2 Let G be a connected and simply connected nilpotent Lie group. A uniform lattice of G is a uniform discrete subgroup, i.e. a discrete subgroup with compact quotient, N of G.
Section 1.2 Nilpotent Lie groups
9
We remark that not all connected and simply connected nilpotent groups admit lattices. One of the nicests results of Mai'cev is the "urdque isomorphism extension property" T h e o r e m 1.2.3 Let G and H be two connected and simply connected nilpotent Lie groups. Suppose moreover that N and M are uniform lattices of G and H respectively. Then any isomorphism ~ : N --+ M extends uniquely to an isomorphism of Lie groups of G onto H . In case we use this property for M = N , we also say "the unique autom o r p h i s m extension property " V.V. GorbacevK ([33]) generalized this theorem as follows: Let N be a uniform lattice of a simply connected, connected nilpotent Lie group G and let H be an arbitrary simply connected, connected nilpotent Lie group. Then any morphism : N ~+ H extends uniquely to a morpkism G --+ H . Mal'cev also describes all possibilities of uniform lattices in a connected and simply connected nilpotent Lie group. T h e o r e m 1.2.4 A n y lattice N of a connected and simply connected nilpotent Lie group G is a finitely generated torsion free nilpotent group. Conversely, for any torsion free finitely generated nilpotent group N there exists (up to isomorphism) exactly one connected and simply connected nilpotent Lie group G containing N as a uniform lattice. We refer to this G as the M a l ~ c e v c o m p l e t i o n of N . Let N be a torsion free and finitely generated nilpotent group with a torsion free central series N . as in the previous section. Suppose moreover that {al,l~
al,2~
9 9 9 ~ al,ki~
a 2 , 1 ~ . 9 9 ~ ac,kc }
is a set of generators compatible with N . . Then the elements {Ai,i = log(al,i), Ai,2 = log(ai,2),...,Ac,k~ = log(ac,kc)} form a basis for the Lie algebra 9 of the Mal'cev completion G of N. It follows that arty element g of G can be written uniquely in the form ~r . . . al,kl zl,~l for some Xi,j E ]~. g = ac,~r1 ac,2 Any element can thus be identified with a coordinate vector (X1,l,~l,2,''',$1,kl,~2,1,
''',xi,j,''',~c,kr
E ~K.
Chapter 1: Preliminaries and notational conventions
10
Moreover, such an element belongs to the lattice N if and oi~ly if its coordinate vector belongs to Z g . Using the Campbell-Baker-Hausdorff formula one proves that multiplication in G (and so in N ) is given by a polynomial function of these coordinates. Finally we want to mention the following properties: L a m i n a 1.2.5 Let N be a uniform lattice of a connected and simply connected nilpotent Lie group G with Lie algebra g. Then,
1. ffi(N) and N V @ ~ ) are uniform lattices of ffi(G), 2. Z i ( N ) is a uniform lattice of Zi(G),
4. exp(Z (g))= Here the upper and lower central series of a Lie algebra g are defined analogously to the upper and lower central series of a group. We also find an alternative description of the isolator subgroup. L e m m ~ 1.2.6 Let N be a finitely generated, torsion free nilpotent group with Mal'cev completion G, then Vi : ~ = N • ?i(G). Proof: Using the universal property of the groups Ti(N) (see also the comments after leinma 1.1.3) it is enough to proof that for any morphism ~a : N ~ M where M is a torsion free nilpotent group of class < i there exists a unique morphism r : N / ( N n ffi(G)) making the following diagram commutative.
N
(i M
It is obvious that r if it exists, is unique. To prove the existence of r consider the finitely generated, torsion free nilpotent group M ~ of class < i given by M ~ = ~a(M). Denote the Mal'cev completion of M ~ by H. By the result of V.V. GorbaceviS, mentioned above, we know that the m o r p h i s m ~o extends (uniquely) to a homomorphism ~ : G ~ H .
Section 1.2 Nilpotent Lie groups
11
Moreover, as H is nilpotent of class _< i, ~5 factors through G/'yi+I(G). The map r is now obtained as a restriction of @ to the subgroup
N.~+I(G)/~+I(C) N/(N n~+l(C)). :
This shows that ~/Ti+l(N) = N N 7{+~(G).
Chapter 2
I n f r a - n i l m a n i f o l d s and Almost-Bieberbach groups 2.1
Crystallographic and Bieberbach groups
Let G be any connected and simply connected Lie group. We denote by Aut (G) the group of automorphisms of G (as a Lie group). The semidirect product G:~Aut (G) acts on G in a canonical way by Vg, x E G, Va C A u t ( G ) :
(g'~)x = g a ( x ) .
We denote the group G>~Aut (G) by Aft(G) and refer to it as the group of affme diffeomorphisms of G. If G = R n tlhe abelian Lie group, then Aft(G) = R n ~ G1 (n, R) the usual affine group (with the usual action on
Rn).
In this book we will be concerned with discrete subgroups H of Aft(G) which act properly discontinuously on G. If, moreover, the action of H on G is free (i.e. II is torsion free), then the quotient space I I \ G is a manifold. Let us look at a concrete case and suppose G is a connected and simply connected abelian Lie group. So G equals ]~n for some n. It is well known that the orthogonal group O(n) is a maximal compact subgroup of G l ( n , • ) . A uniform discrete subgroup II of Rn~O(n) C_Aff(R '~) is called a c r y s t a l l o g r a p h i c g r o u p of dimension n. If II is torsion free, II is said to be a B i e b e r b a c h g r o u p . These groups are well known by the work of L. Bieberbach. Standard references are [60] and [13]. We mention three famous theorems on Bieberbach groups (in fact on crystallographic groups) which were published around 1911 and 1912, by L. Bieberbach
14
C h a p t e r 2: I n f r a - n i l m a n i f o l d s and A B - g r o u p s
and G. FrSbenius ([6], [7], [32]). The order in which the theorems are listed is not the historical order in which they appeared. We will shortly sketch the history of these theorems after there statements. T h e o r e m 2.1.1 F i r s t B i e b e r b a c h t h e o r e m Let II be an n - d i m e n s i o n a l crystallographic group, then F = II • 2 '~ is a lattice of R n and I I / F is finite. This means that for an n-dimensional crystallographic group H, the translational part F = II N ]R'~ of II is a free abelian group isomorphic with Z ~, such that the vector space spanned by F is the whole space ]R~. If H = F, then H is torsion free and the manifold I I \ R '~ is an n-dimensional torus. For more general torsion free II we see that the manifold M = H\IR n inherits the flat Riemannian structure of IR'~ (as H is a group of distance preserving transformations) and so M is a compact flat Riemannian manifold. Moreover, all compact flat Riemannian manifolds are obtained in this way. Thus the Bieberbach groups are exactly the fundamental groups of the compact fiat Riemannian manifolds. T h e o r e m 2.1.2 S e c o n d B i e b e r b a c h t h e o r e m Let H and H ~ be two n - d i m e n s i o n a l crystallographic groups. 1 ] r : H ~ H' is an i s o m o r p h i s m , then there exists an element a E Aff(IRn) such that V"y ~ II : ~3(~/) = OtGOt-1. From this, one deduces that two fiat Riemannian manifolds with isomorphic fundamental groups are "affinely" diffeomorphic and so, a flat Riemannian manifold is up to a well understood diffeomorphism completely determined by its fundamental group. T h e o r e m 2.1.3 T h i r d B i e b e r b a c h t h e o r e m or F r 6 b e n i u s ' T h e orem Up to conjugation in Aft(IRa), there are only finitely m a n y n - d i m e n s i o n a l crystallographic groups. This implies that, up to affine diffeomorphism, there are only finitely many n-dimensional fiat Riemannian manifolds. These theorems constitute an answer to one of the famous Hilbert Problems, number 18 to be precise. The first Bieberbach theorem is indeed historically the first theorem which L. Bieberbach proved. After this theorem L. Bieberbach could show that there where only finitely many crystallographic groups of a given dimension, up to i s o m o r p h i s m .
Section 2.2 Almost-crystallographic groups
15
It was G. Frhbenius who reacted that one should not consider crystallographic groups up to isomorphism, but up to aft-me equivalence. Fr6benius then also proved theorem 2.1.3 as stated above. Inspired by the work of FrSbenius, L. Bieberbach then showed that an isomorphism between two crystallographic groups is always induced by an affme conjugation, i.e. he proved, as last, the Second Bieberbach Theorem. So please be aware t h a t the ordering of the Bieberbach theorems is not a historical ordering. The algebraic structure of a crystallographic group is well know, one part due to the first Bieberbach theorem and the converse part due to H. Zassen_haus ([62]). We summarize this in the following theorem: T h e o r e m 2.1.4 A l g e b r a i c c h a r a c t e r i z a t i o n Let II be a crystallographic group of dimension n, then F = H ~ R n is the unique normal, maximal abelian subgroup of H. Conversely, let Q be an abstract group such that Q contains a normal, maximal abelian group F of finite index, with r ~- ~n, then there exists a monomorphism ~ : Q --+ Aff(R n) such that ~(Q) is a crystallographic group. We will refer to such a Q as an abstract crystallographic (Bieberbach) group and even as a crystallographic (Bieberbach) group.
2.2
Almost-crystallographic groups
As an immediate generalization of the crystallographic groups, we look what happens in case G is a connected, simply connected nilpotent Lie group. Again we consider a maximal compact subgroup C of Aut (G). A uniform discrete subgroup E of G :~C is called an almost-crystallographic group (abbreviated in the sequel as AC-group). A torsion free AC-group is called an almost-Bieberbach group (abbreviated as AB-group) and the quotient-space E \ G is now called an infra-nilmanifold. (In case E C G, E \ G is said to be a nilmanifold). So the nilmanifolds take over the role of the tori and the infra-nilmanifolds the role of the flat Riemannian manifolds. Of course, the dimension of an AC-group is the same as the dimension of the corresponding Lie group. The adjective "almost" is inpired by the geometrical fact that the infra-nilmanifolds are exactly the almost flat manifolds. We are not going into detail about this geometrical point of view, but refer to [35], [31] and [57] for more information. All three Bieberbach theorems have been generalized to the nilpotent case.
16
Chapter 2: Infra-nilmanifolds and AB-groups
The first t h e o r e m was generalized by L. Auslander and published in '1960 ([1]). The formulation is as follows: T h e o r e m 2.2.1 G e n e r a l i z i n g t h e first B i e b e r b a c h t h e o r e m Let E C G ~ A u t (G) be an n-dimensional almost-crystallographic group. Then E • G is a uniform lattice of G and E / ( E n G) is finite. Although the proof of this theorem is not so difficult (but it is rather long), the techniques used do not appear elsewhere in this book and therefore we present this t h e o r e m here without proof. The interested reader should consult [1]. In his paper, Auslander also proves that N = E 5 G is the (unique) normal maximal nilpotent subgroup of E. (In fact, Auslander only shows that N is maximal amongst all n o r m a l nilpotent groups, but his arguments can be used to prove that N is maximal nilpotent amongst all groups. This was remarked by K.B. Lee and F. R a y m o n d in [48]). An alternative proof will be presented later on. We still call N the translational part of E. The finite quotient F = E / N is often referred to as the holonomy group of E. The theorem above shows that the Lie group G is precisely the Mal'cev completion of the translational group E A G. L. Auslander also formulated a generalization for the second Bieberbach theorem. However, this formulation was incorrect (even for Bieberbach groups) and a counter example was presented in ([48]). A correct generalization of this second theorem is also given in [48] by K.B. Lee and F. Raymond: T h e o r e m 2.2.2 G e n e r a l i z i n g t h e s e c o n d B i e b e r b a c h t h e o r e m Let r : E ~ E ' be an isomorphism between two almost-crystallographic groups (of a fixed Lie group G), then ~ can be realized as a conjugation by an element of Aft(G).
This t h e o r e m implies that two infra-nllmanifolds with isomorphic fund a m e n t a l groups are affinely diffeomorphic. This theorem is of significant importance since it states that a classification of infra-nilmanifolds can be done on the group level. We wrote "of a fixed Lie group G" between parentheses, since an isomorphism between two AC-groups is only possible if the corresponding simply connected, connected nilpotent Lie groups are isomorphic. Indeed, as the translational subgroups of E and E r are the unique normal and m a x i m a l nilpotent subgroups N and N r of E and E ~respectively, it follows that any isomorphism r : E -~ E ' restricts to an isomorphism of N
Section 2.2 Almost-crystallographic groups
17
onto N I. By the unique isomorphism extension property mentioned in the previous chapter, we know that r induces indeed an isomorphism b e t w e e n the two MM'cev completions. To prove theorem 2.2.2, we first define a notion of 1-cohomology with n o n - a b e l i a n coefficients. Let A and B be any groups. Suppose that a m o r p h i s m 9~ : A ~ A u t ( B ) is given. A map f : A ~ B is called a 1-cocycle (from A to B, with respect to p) if and only if w , y 9 A : f(
y) --
Two 1-cocycles f and g are equivalent if f ( x ) = (~(x)(b-1))g(z)b for all x ~ A. It is readily seen that this relation is an equivalence relation. The set of equivalence classes is denoted by H~(A, B) and is non empty since it contains at least one class~ namely the class of the map f : A -~ B : x ~ 1. If B is an abelian group, this notion of 1-cohomology coincides with the usual notion. The following lemma generalizes the fact that HI(F, IRn) = {1}, for all finite groups F . 2.2.3 Let F be any finite group and suppose that G is a connected, simply connected niIpotent Lie group. Then, for any morphism : F ~ Aut (G), we have that
Lemma
1 H~(F, G) = {1}.
Proof: We proceed by induction on the nilpotency class c of G. If c = 1, G = R n and the l e m m a is well known in this case. Any a u t o m o r p h i s m ~(x) (x E F ) restricts to an automorphism of Z(G) (which we denote by the same symbol) and induces an automorphism @ of G/Z(G). We denote the canonical projection of g 9 G in G/Z(G) by g. Suppose a 1-cocycle f : F ~ G is given. Then there is an induced 1-cocycle ] : F --* G / Z ( G ) (with respect to ~) given by f ( z ) = f(x). By the induction hypothesis, we know that f is cohomologuous to 1 and so there exists an element a 9 G such that =
=
9
F.
Therefore, we can define a map
g: F ~ Z(G) : x ~ ~(x)(a-1)a(f(z)) -1.
Chapter 2: Infra-nilmanifolds and AB-groups
18
A trivial computation shows that this map g is a 1-cocycle with respect to ~o (restricted to Z(G)). However, as Z(G) ~= IRk for some k, this 1-cocycle is cohomologuous to 1. So, there exists a z 9 Z(G) such that g(x) = ~o(m)(z-1)z, Vm 9 F. This implies that T(m)(z-1)z = ~(ae)(a-1)a(f(m)) -1, from which it follows that
Vx < F: f(x) = ~(m)((az-1)-l)az -1. In other words, f is equivalent to 1.
P r o o f o f t h e o r e m 2.2.2: As explained above, the isomorphism r : E ~ E ' induces an isomorphism : N ~ N ' between the translation subgroups, which extends (uniquely) to an a u t o m o r p h i s m of G, which we also denote by ~,. Remark that Vn 9 _N: r
1) = (p(n), 1) = (1, ~)(n, 1)(1, u) -1.
This implies that, after conjugating with (1, z~), we m a y (and do) suppose that ~b induces the identity map on the translational subgroups. Let (g,a) 9 E and denote r a) = (h,/3). For all (n, 1) 9 N
(g,
1)(g,
_-
1).
After applying ~b to both sides of the above equality, we find that
(h/3(n)h -1, 1) = (ga(n)g -], 1) ~ / 3 ( n ) = #(h-lg)a(n) from which it follows t h a t / 5 is completely determined by c~, g and h (on N and thus on G by the unique automorphism extension property). As h is determined (via ~b) by g and a, we can define a map
/:
G: (g,
H/(g,
= g-lb.
Moreover, by comparing f(g, a) with f(g', a), we see that ent of g and so, f is in fact a map of the finite group F into G. This f is a 1-cocycle with respect to ~o : F -+ Aut mapping the coset of (g, ct) to ct. The previous lemma implies that f is cohomologuous an element of G, for w h i c h / ( g , a) = c~(a-1)a, then
f is independ= E/(F__,N G) (G) defined by to 1. Let a be
~b(g, a) = (gf(g, a), #((f(g, a))-m)a) = (a, p(a)-l)(g, a)(a, p(a)-l) -1. This finishes the proof of theorem 2.2.2.
Section 2.3 How to generalize the third Bieberbach theorem
2.3
19
How to generalize the third Bieberbach theorem
The third theorem is not as straightforward to generalize as the other two. This is caused by the fact t h a t in the abelian case all n-dimensional tori are diffeomorphic, while in the nilpotent case there are infinitely many, n o n - h o m e o m o r p h i c nilmanifolds in dimensions _> 3. In fact, there can be infinitely m a n y simply connected, connected nilpotent Lie groups in a given dimension. The following considerations are due to K.B. Lee ([45]). For example, one can consider the three dimensional Heisenberg group H: H-
0 0
1 0
z 1
I]x,Y, Z E t R
.
(2.1)
For any integer k > 0, we obtain a uniform lattice rk of H , which is the subgroup generated by:
a=
( 00) ( 10) 0 0
1 0
1 1
b=
0 0
1 0
0 1
c=
0 0
1 0
.
It is easy to see t h a t I~k : < a, b, c II [b, a] = ck, [c, a] = [c, b] = 1 > . Since H l ( r k , Z) = Fk/[rk, rk] = z2| we may conclude t h a t the manifolds Fk \ H are pairwaise non-homeomorphic. So, it is certainly not exact to claim t h a t there are only firdtely m a n y infra-nilmanifolds in each dimension. Another possible way to look at the third Bieberbach theorem is the following: each torus covers only finitely m a n y fiat Riemannian manifolds. So we t r y to generalize the third theorem by fixing a nilmanifold and by looking at the infra-nilmanifolds covered by it. But r l \ H is a k-fold covering of p k \ H , and so, one nilmanifold covers infinitely m a n y other nilmanifolds. Therefore, K.B. Lee introduces the notion of an essential covering. Let M = N \ G be a nilmanifold and M t = E \ G an infra-nilmanifold. A covering p : M -* M ~ is said to be essential iff the induced map on the level of the fundamental groups is so t h a t p . ( N ) = E A G. This implies t h a t E and N have "the same" translational part, namely N. A correct formulation, due to K.B. Lee, for the generalization of the third theorem is:
20
Chapter 2: Infra-nilmanifolds and AB-groups
T h e o r e m 2.3.1 There are, up to an ajfine diffeomorphism, only finitely many infra-nilmanifolds which are essentially covered by a fixed nilman-
ifold. We postpone the proof of this theorem for a while. In fact we are going to translate the above formulation into a group theoretic language. The fact that tiffs algebraic formulation is really equivalent to the topological one will become clear in the sequel of this book. D e f i n i t i o n 2.3.2 Let N be a finitely generated, torsion free nilpotent group. A group extension 1 ~ N ~ E ~ F ~ 1, with F finite and in which N is a maximal nilpotent subgroup of E will be called "essential". At this point we whish to recall the notion of the Fitting subgroup of a polycyclic-by:finite group (e.g. see [59]). D e f i n i t i o n 2.3.3 Let F be a polycyclic-by-finite group. Then the Fitting subgroup o f F , denoted by Fitt (F) is the unique maximal normal nilpotent subgroup of F. Using this terminology, we see that the extension 1 ~ N ~ E ~ F ~ 1, with N torsion free finitely generated nilpotent and F finite, is essential if and only if N = Fitt (E) and there is no nilpotent subgroup of E, strictly containing N. It is easy to see that a covering of an infra-nilmanifold by a nihnanifold is essential if and only if the induced extension of the fundamental groups is essential. Therefore it will be enough to prove the generalization of the third theorem on the group level, which can be stated as follows: T h e o r e m 2.3.4 Let N be a finitely generated, torsion free nilpotent group. Then there are, up to isomorphism, only finitely many (essential) extensions of the form 1 ~ N ~ E ~ F ~ 1 in which N is the maximal nilpotent subgroup and F is finite. In his proof, K.B. Lee used a counting principle based on a group cohomology argument. As the example in the next section will show, this principle is unsuitable for this context and hence should be replaced. In the section thereafter we develop a different proof of the theorem, now based on the finiteness of the number of conjugacy classes of subgroups in an arithmetic group, a result which goes back to the work of A. Borel and Harish-Chandra ([8]; see also [59]).
Section 2.4 The first proof revisisted
2.4
21
The first p r o o f of the generalized third Bieberbach t h e o r e m revisited
From a conceptual point of view the idea behind the proof in [45] is that of reducing the classification of AC-groups having a fixed N as maximal nilpotent subgroup, to the known classification of crystallographic groups. It is this approach which opens the way we will follow to classify explicitly the AC-(and AB-) groups in dimensions 3 and 4. However, as we will point out in an example, it is unsuitable for counting the isomorphism types of the AC-groups obtained and consequently, it should be revised as an argument to the proof of theorem 2.3.4. Let us again focus on the situation of a group E containing a finitely generated, torsion free, c-step nilpotent group N of finite index; i.e. we consider an extension (not necessarily essential) 1 --+ N --+ E Z F --+ 1 with j: E + F: ~ H j(~)
(2.2)
_- ~.
Since the isolator of 72(N) in N is normal in E , we obtain a short exact sequence
1-+ ~/Nv/w(~N / -+ _ _ E N / ~
(N)
-+ F-+ i.
(2.3)
is a finitely generated free abelian group, and hence it becomes
an F - m o d u l e via a morphism ~N,E: F -+ a u t ( N / ~ ) . Let us write F for the isolator of 7c(N) in N. Given the canonical projection E -* ElF , we will write z r for the image of an element z E E. R e m a r k that F is contained in Z(N). In the extension 1
--+ N/F -+ E/F -+ F --+ 1.
(2.4)
N I p is torsion free, nilpotent of class c - 1 and of finite index in E/F. Similarly to (2.3), we obtain
1--~
~r/rV/.y2(N/r) + ~Nr/rV/~2(N/r)
+ F
By lemma 1.1.4 there is a canonical isomorphism
"r: ~ 2 ( N
)
--+ % 7 2 ( N / F )
-+
1.
(2.5)
Chapter 2: Infra-nilmanifolds and AB-groups
22
such t h a t the following diagram commutes : 1---+
N)
_+
z
---+ F
/~N)
--+ 1
Now it is clear t h a t , with respect to the induced action WNIr,E/r of F on ~'~
, "r/ is an isomorphism of F-inodules.
N/r~'t2(N/F) Keeping all this in mind, we establish a relation between the action ~N,E and the m a x i m a l nilpotency of N in E: P r o p o s i t i o n 2.4.1 grp(N U {x}) is nilpotent r 2. 9 ker(~N,E). Proof: Let z 9 E such that ~ N , E ( ~ ) ---- 1. Then Vn 9 N, z - l n z n -1 = r,~ 9 ~ . We use induction (on the nilpotency class of N ) to prove t h a t g r p ( g u {z}) is nilpotent. The abelian case is immediate. Assume N is of class c > 2. Since T~ is an isomorphism of F-modules, TN/r.E/r(~) = 1. Hence, by induction, g r p ( N / F u {zr}) is nilpotent and of class c - 1 (see lelmna 1.1.7). We are going to show t h a t 1 --~ Y --+ g r p ( N U {x}) --+ grp(N/r u
{xr}) -~
1
is a central extension. Remember t h a t F C Z(N) and thus, the only thing left to check is whether [x, F] = 1. By lemma 1.1.9 it is sufficient to prove t h a t x e CE(Tc(N)). So, assume n 9 N and take m 9 %_1(N). Since g r p ( N / r u {xr}) is nilpotent and m r 9 Z(N/r), it follows by lemma 1.1.5 t h a t [xr, mr] = 1, or equivalently, z - l m z = mz with z e F. Then z -1 [n, m] z = [r~n, mz] (r~ 9 ~/72(N)). An easy computation now shows t h a t
-1 In, m] ~ = [ ~ , z] n In, z] [~n, ~n] ~z In, m] z ---- In, m] implying t h a t x E CE(%(N)). Tiffs shows t h a t grp(N U {x}) is nilpotent. Conversely, suppose ~ E E such t h a t grp(N U {x}) is nilpotent. Again we use induction on the class of N. The abelian case is trivial. Assume N is of class c (c _> 2). So, grp({xr} u N/F ) is nilpotent. By induction,
Section 2.4 The/irst proof revisisted
23
we deduce that ~N/r,E/r (~) is the identity. The desired result follows at once by the isomorphism v' of F-modules above. From the above proposition, we deduce a key observation, first made by K.B. Lee in [45] allowing to start the reduction process mentioned above. For another proof (using Lie group theory) we refer the reader to the original paper of K.B. Lee. L e m m a 2.4.2 Let N be a finitely generated torsion free c - s t e p nilpotent group. Define Z = ~ c ( N ) . Then, for any finite group F, we have that
1 --+ N ~ E ~ F -~ 1 is essential
1 ~ N / Z ~ E / Z ~ F ~ 1 is essential. Proof: First of all note that N / Z is a torsion free nilpotent group. It follows directly from proposition 2.4.1, that N is maximal nilpotent in E if and only if N / Z is maximal nilpotent in E / Z . Related to this lem_ma, notice the following commutative diagram: 1
1
Z
Z
1
~
N
~
E
1
,
N/Z
,
E/Z
1
1
1
1
> F ,
F
~ 1
(2.6)
1
W i t h this observation in mind, it is natural that induction is involved to reach the crystallographic case. Indeed, assume N is of nilpotency class c > 1. Then a group E having the normal subgroup N as Fitting subgroup of finite index, induces a group E / Z in which N / Z is maximal nilpotent and of finite index. Assume there are only finitely many isom o r p h i s m types for E / Z and fix one of these, say E / Z . We look for all extensions of E / Z by Z which give rise to a group containing N as the maximal nilpotent normal subgroup. Since Z is central in N , the action of E / Z on Z factors through F. Consequently there are only finitely m a n y E / z - m o d u l e structures ~ : E / Z ~ Ant Z to consider. For each of
24
Chapter 2: Infra-nilmanifolds and A B - g r o u p s
them, there is a restriction morphism res: H 2 ( E / z , Z ) ~ H 2 ( N / z , Z). An extension < E > in H ~2( E / z , Z) will contain N as maximal nilpotent normal subgroup if and only if its restriction res (< E >) determines a group which is isomorphic to N. So far, there is no problem. However, to decide on isomorphism types, group cohomology is not the best instrument. Consequently, the argument that res only maps a finite number of elements onto the class of N , although correct, is unsecure with respect to an isomorphism type classification. Indeed, there might be more t h a n one class in H 2 ( N / z , Z ) representing a group isomorphic to N and it is not clear why a group < E > in the inverse image under res of one class should be isomorphic to some group in the inverse image of an other class. In fact, as our example will show, there will often be infmitely m a n y different cohomology classes, all representing N , and infinitely m a n y of t h e m will lie in the image of res. Among the examples we know of to illustrate this problem, the following one is of minimal dimension (in fact in dimension 3 this problem does not occur). It also shows that the final counting argument in the proof of t h e o r e m 2.3.4 as given in [45] does not count all essential extensions. E x a m p l e . Take N : < a, b, c, d I[b, a] = d, all other commutators trivial > . One verifies easily that in this case Z = grp{d}. Consequently, N / Z Z 3. Now, consider the 3-dimensional crystallographic group E / Z 23 >4Z2, where the action of Z2 on 23 is given by -1 0 1 For E / Z E/Z
~=
0 0 ) 1 0 . 0 1
we obtain a presentation : < a,b,c, al
[b,a]= [c,a]= [c,b]= 1 ct2 = 1, eta = a-lcot, orb = ba, c~c = ca > .
Take ~ the non-trivial action of E / Z on Z = grp{d}, which factors through ~2. Any extension E of E / Z by Z can be presented as E : < a , b , c , d , ct I [b, a] = d kl [c, a] = d k2 [c, b] = d k3 vta =- a - l c v t d k4 o~b = bc~dks tic = cad ks
[d, a] = 1 > [d, b] = 1 [d, c] = 1 v~d = d-lv~ ot2 = d k~
S e c t i o n 2.4 T h e first p r o o f r e v i s i s t e d
25
for some integers kl, k2,. 9 k7. H o w e v e r , we can n o t h o p e to choose the k~'s i n d e p e n d e n t l y f r o m each o t h e r (we will refer to such a s i t u a t i o n b y saying t h a t t h e r e should be " c o m p u t a t i o n a l c o n s i s t e n c y " , see c h a p t e r 5). It is not t o o h a r d to verify t h e following c o n s i s t e n c y conditions: 1. a a 2 = a2ol ~
a d k' = dkTa ~
d - k T a = dkTa :~ k7 = O.
2. a a = a - l c a d k* :~ b a a = b a - l c a d k4.
T h e left h a n d side equals a b a d -kS = a a b d -kS+k1, while t h e right h a n d side b e c o m e s a - l c b d - k l - k 3 adk4 = a - l cbadk4dkl +k3 = a - l cotdk4bdkl-ks dk~.
C o m p a r i n g b o t h sides allows us to conclude t h a t k3 = O. 3. a a = a - l c d - k * a ~ a a -1 = a c - l d k 4 - k 2 a . We use this in: ot2a = o t a - l c a d k4 ~ a = a c - l d k * - k 2 a c a d k* = a d 2k*-k2-k6
i m p l y i n g k6 = 2k4 - k2. As a c o n s e q u e n c e e v e r y e x t e n s i o n E can be p r e s e n t e d as E
[ b , a ] = d ll [c,a] = d 12
: 4 a,b,c,d,a]
[d,a]= 1 [d,b] = 1
It, hi =
[d,c] =
aa = a-lcad 5
ad = d-la
a b = b a d z*
a2 = 1
>.
(2.7)
aC = cord 213-12
T h e c o n d i t i o n s o n the ki's as given above are seen to b e sufficient b y realizing t h a t the e x t e n s i o n s (2.7) have a faithful r e p r e s e n t a t i o n in G1 (5, l~) as follows:
1 - ~ 2 -11 -12 0 ]
a
1 0 0 0 0
I /10000/ 0 0 0 0
H
1 0 0 0
0 1 0 0
0 0 1 0
1 0 0 1
bF--~
0 1 0 0 0
0 0 0 0 1 0 0 1 0 0
Oj 0 1 0 1
1 0 0 0 1
c~--~
0 0 0 0
1 0 0 0
0 1 0 0
0 0 1 0
-1 and a H
0 0 0 0
0 0 1 1
d~---~
0 0 0 0
1 0 0 0
0 1 0 0
0 0 0 0 1 0 0 1
l~-21~ -14 12 - 213 0 2 -1 0 1 0
0 1 0 0
0 0 1 0
0 0 0 1
'~ .
26
Chapter 2: lnfra-nilmanifolds and A B - g r o u p s
The restriction m o r p h i s m res : H 2 ~ ( E / z , Z ) ~ H2(Z3, Z) maps a class corresponding to the group E (2.7) to a group presented as < a,b,c,d]
[ b , a ] = d I1 [d,a]= l > . [ c , a ] = d 12 [ d , b ] = l [c,b]= 1 [d,c]= 1
(2.8)
It is well known that every couple (ll, 12) determines exactly one elem e n t in H 2 ( Z 3, Z). Indeed, recall that H 2 ( ~ 3, Z) -~ ~3, with (k, l, m ) E H 2 ( Z 3, Z) representing the group < a,b,c,d]
[b,a]=d k [ c , a ] = dI
[d,a]= I >. [d,b]= 1
[~, b]
[d, ~] = 1
= d TM
Let us now make the following observations: Observation
1: N lies in the image of res ( N ~ (1, 0, 0) E H 2 ( Z 3, •)).
O b s e r v a t i o n 2: Any couple (/1,/2) = (1,/) (e Im(res)) determines a group isomorphic to N. Indeed, taking A = a, B = b, C = blc, D = d as a new set of generators for N , we obtain as presentation N :
. [C,A]=D l [D,B]= 1
[C,B]=I
[D,C]=I
In fact, remark t h a t one can verify t h a t every 3-tuple (k, l, m) E H 2 ( Z 3, Z) for which gcd(k, l, m) = 1 determines a group isomorphic to N (we will explain this in detail in section 6.2). Anyhow, we conclude that there are infinitely m a n y cohomologically different groups, all isomorphic to N , in the image of res. Observation
3- Now take the group E' : < a,b,c,d, al
[b,a]=d
[~, a]
= d [~, b] = 1
aa -~ a - l cot ab = ba ac = cad -1
[d, a] = 1 [d, b] = 1
> .
[d, c] = 1 ad = d-la a2=l
E ' determines an extension of E / Z by Z, axtd its restricted extension is isomorphic to N.
Section 2.4 T h e first p r o o f revisisted
27
C l a i m 2.4.3 E t cannot be isomorphic to an element in res -1{(1, 0, 0)}. P r o o f : Consider a n e w set of generators for E~: A = a, B = b, C = b - l c , D = d and/3 = a. The presentation of E ~ becomes: E t : < A, B, C, D, /31
[B,A]=D
[D,A]=I
[C,A]= 1 [c, B] = 1 3A = A-1BC/3 fiB = B fl
[D,B]=I [D, C] = /3D = D - 1 3 f12 = 1
3 c = C 3 D -~
We claim that this group is not isomorphic to any group with a presentation (2.7) with ll = 1 and 12 = 0. This means that E ~ cannot be isomorphic to a group in r e s - 1 { ( 1 , 0, 0)}. In fact, assume there exists an isomorphism r : E ' ~ E E r e s - l { ( 1 , 0 , 0 ) } . Taking into account the characteristic subgroups of E and E ~ we may conclude that r has to satisfy the following conditions: r = aXlbX:'c~3d~4a for :~1,~2,23,~4 E r +1 r g?(A) = a m ' b m 2 w ( c , d ) r = ahbl2w(c,d)
where(ml m~) 11
12
E Aut :•2 and w(c, d) means a (not further specified)
word in c and d. Since r is a morphism, we have r162
(2.9)
= r162162162
In E we calculate a a ml = a - m l w ( c , d ) o ~ and orbm2 = b m 2 w ( c , d ) a . Using this result and comparing the powers of a and b in b o t h sides of (2.9) allows us to conclude that i!1 = 0 and 12 = 2m2. This implies that
(mlm2) (ml 11
I2
0
2m2
~ Aut
And so r does not exist. This example shows that not all possible AC-groups were counted in the p r o o f of the theorem in [45].
Chapter 2: Infra-nilmanifolds and AB-groups
28
2.5
A n e w p r o o f for the generalized third Bieberbach theorem
The proof we present here for theorem 2.3.4 is completely different. In fact, it is not clear to us, whether this point of view allows the same "iterative" classification approach to concrete situations, as is achieved on the basis of lemma 2.4.2. Let us quickly recall some fundamentals of the theory of group extensions with non-abel]an kernel (see [50], [38]). Let N be a group. An extension with kernel N is a short exact sequence 1 ~ N ~ E ~ F ~ 1. This exact sequence induces a hom o m o r p h i s m r 9 F ~ Out N , which is called an abstract kernel. The extension usually is called compatible with r Given an abstract kernel r : F ~ Out N , the problem of studying all extensions compatible with r is well known in literature, and gave rise to the concept of non-abelian 2-cohomology sets . Let us write E x t r N ) for the set of equivalence classes of extensions compatible with r If this set is empty, one says that there is an obstruction to the algebraic realization of the abstract kernel. Let us denote by r a fixed lifting (most likely not a morphism!) r : F ~ Aut N for r which is taken to be normalised, i.e. r = 1. Now, an extension compatible with r is determined by a map f : F • F ~ N (normal]seal, i.e. f(g, 1) = f(1, g) = 1) satisfying a cocycletype condition 1. Vg, h E F r162
= #(f(g, h))r
2. Vg, h, k E F f(g, h)f(gh, k) = r The extension E compatible with r N • F with multiplication:
( f ( h , k)) f(g, hk).
determined by f is then obtained as
Vn, m e N, Vg, h ~ F (n, g)"(~Z) ('~, h) = ( n r
h), gh).
Let us denote this particular extension by E(r If the set E x t r N) is not e m p t y it is in bijective correspondence to H2(F, Z(N)), where the F - m o d u l e structure of Z ( N ) is induced by r Let us now notice the following property. 2.5.1 Let r r : F -~ Out N be two abstract kernels which are conjugated by an element r E Out N (i.e. Vg E F, r = r162162 Then there is a bijective correspondence v between the sets Extol (F, N) and Extc2(F,N ) in such a way that corresponding extensions are isomorphic.
Lemma
29
Section 2.5 A n e w p r o o f
P r o o f : Choose a lift r : F ---+ A u t N for r (with r = 1) and a lift r E A u t N for r Take r = r 1 6 2 1 6 2 as a lift for r Consider any extension E ( r ) of Extr (F, N ) . Take ff = Cf. One checks that f ' satisfies the cocycle conditions mentioned above, with respect to r This means that we have an extension E(r Let 1} be the m a p sending E(r to E(r ). It is an elementary computation to verify that the m a p i : E(r ---+ E(r ) : (n, g) ---+ (r g) is an isomorphism of groups. Finally, remark that 1}maps equivalent extensions onto equivalent extensions and so induces the desired map v.
Let us now return to the nilpotent case. Assume again that N is torsion free, finitely generated and nilpotent, and that F is a finite group. L e m m a 2.5.2 If 1 -+ N ---+ E ---+ F ---+ 1 is an extension in which N is m a x i m a l nilpotent and F is finite, then the induced abstract kernel r : F --+ Out N is injective. P r o o f : Assume that the induced abstract kernel is not injective. This means that there exists an element in E \ N and an element no E N such that Vn E N : x n x - i = nonno 1. This implies that the morphism ~N,E, introduced in section 2.4, is not injective. Hence, by proposition 2.4.1, N is not maximal nilpotent inE.
R e m a r k 2.5.3 If N is abelian, then it is well known that the converse of the s t a t e m e n t is also true. However, in general, as the following example shows, this is not the case. I.e. r : F -+ Out N might be injective, without N being maximal nilpotent. Example
2.5.4
Consider the following torsion free virtually 2-step nilpotent group. E : < a,b,c,(xll
[b,a]= c 2 [c,a]--- 1 (~a = a-l(~
(xc= ca [c,b]-- 1 c~b = b-lot
>.
(22= c
For N = grp{a,b 2,c}, we see that E / N = Z2 x Z2 = grp{b,~}. Since conjugation with b, a and ba in E induces automorphisms of N which do
30
Chapter 2: Infra-nilmanifolds and AB-groups
not belong to Inn N , the abstract kernel of the extension 1 ~ N ~ E •2 • Z2 ~ 1 is injective. However, as N is contained in grp{a, b, c}, it is clear that it is not maximal nilpotent. A correct generalization of remark 2.5.3 towards nilpotent groups, will be given in the following chapter. A N e w p r o o f o f t h e o r e m 2.3.4: For a fixed abstract kernel r : F ~ Out N, we know that E x t r N), if non-empty, is in o n e - t o - o n e correspondence with H2(F, Z ( N ) ) which is finite. Since we only have to deal with injective abstract kernels (2.5.2) up to conjugation (2.5.1), we should show that there are only finitely m a n y conjugacy classes of finite subgroups in Out N. But Aut N is an arithmetic group ([59]) and I n n N ~ N / Z ( N ) is a torsion free, finitely generated nilpotent group. This implies that the subgroups K of Aut N such that [K : Inn N] < oo, lie in finitely m a n y conjugacy classes in Aut N ([59]). By dividing out Inn N, we obtain our result.
R e m a r k 2.5.5 The use of lemma 2.5.2 in this proof, can be replaced by information obtained in the reduction lemma 2.4.2 showing that we only have to deal with a finite number of finite groups F i.e. those occurring as holonomy group for crystallographic groups. R e m a r k 2.5.6 At the end of this chapter, I would also like to mention the work of F. Grunewald and D. Segal ([37]), who treated generalizations of Bieberbach teorems for a]:fine crystallographic groups. A group E C Aff(R '~) is an affine crystallographic group i r e acts properly discontinuously on •n, with compact quotient. It would however lead us much to far to go into any details here.
Chapter 3
Algebraic characterizations of almost-crystallographic groups 3.1
Almost-crystallographic tial e x t e n s i o n s
groups and essen-
We start this section by proving a generalization of remark 2.5.3 in case of nilpotent groups. Fix a group extension 1 ~ N ~ E ~ F ~ 1 where N is torsion free, finitely generated nilpotent and F is finite as usual. If G denotes the Mal'cev completion of N , then the group extension induces a morphism (analogous and related to the abstract kernal of the extension) ~ a : F ~ Out (G) in the following way: For any ~ E F we choose an element z E E which maps onto ~. Then conjugation in E by x induces an a u t o m o r p h i s m of N . This a u t o m o r p h i s m lifts uniquely to an automorphism ~r($) of G. It is obvious that a ( 2 ) is unique up to inner automorphisms of N (and so of G). Therefore, the map qa: F ~ Out ( G ) : ~ ~ cr(~) Inn (G) is a well defined h o m o m o r p h i s m of groups. Lemma
3.1.1
Let N be a finitely generated, torsion free nilpotent group with Mal'cev
32
Chapter 3: Algebraic characterizations of AC-groups
completion G and suppose F is a finite group. "extension 1 ~ N -~ E ~ F --~ 1 is given, then
Assume that a group
N is maximal nilpotent in E
The induced morphism ~ : F ~ Out (G) is injective. Proof: First suppose t h a t ~ : F -~ Out (G) is not injective. This means t h a t there exists an element x E E and an element g E G such t h a t Vn E N
: ~n~ -1 =
gng -1,
where the left h a n d side of the above equation is computed in E , while the right h a n d side has to be evaluated in G. But now, it is easy to see t h a t the h o m o m o r p h i s m PN,E of section 2.4 is not injective, and so, by proposition 2.4.1, N is not m a x i m a l nilpotent in E. Conversely, assume that N is not maximal nilpotent in E. Let N ~ denote a nilpotent group strictly containing N. We distinguish two cases depending on the set of torsion elements 7 ( N ~) of N~: Case 1: 7(N') # 1 T ( N I) is a characteristic subgroup of N ~, and so it is normal in N ~. As N is a torsion free normal subgroup of N ~, we must have t h a t (lemma 1.1.8) t h a t ['r(N'), N] = 1. Thus there exists an element x E E \ N such t h a t Vn E N
: xnx -1 z
n.
This implies t h a t ~ is not injective. Case 2: r ( N ' ) = 1 In this situation N ~ is a torsion free nilpotent subgroup, containing N as a subgroup of finite index. Consequently, the Mal'cev completion of N ~ also equals G (i.e. N C_ N ~ C G). Therefore, conjugating N in E with an element of N ~ is exactly conjugating with an element of G, from which it follows t h a t ~ is not injective. This l e m m a offers an alternative way to see that the translational subgroup of an AC-group is m a x i m a l nilpotent. Indeed, let us investigate the map ~ : F --~ Out (G) defined above. Let ~ E E / ( E N G), then x = ( g , ~ ) , for some g E G and (~ E Aut (G). In this case xnx -1 = (g,o~)(n, 1)(g,o~) -1 z ~(g)o~(n).
Section 3.1 Almost-crystallographic groups and essential extensions
33
Therefore, a ( $ ) = #(g)a and so =
(c) =
(a).
As the m a p F ~ Aut ( G ) : x ~ a is injective and as Inn (G) ~ G / Z ( G ) is torsion free, the composite m a p F ~ Aut (G) ~ Aut (G)/Inn (G) which coincides with ~ is injective as well, implying that N = E n G is maximal nilpotent in E. The following len~na will be needed in the proof of the algebraic characterization of A C - g r o u p s below. 3.1.2 Let G be a connected and simply connected nilpotent Lie group. Let F be a finite group. Any morphism r : F ~ Out (G) lifts to a morphism ~a : F -~ Aut (G) and consequently, any extension of G by F splits. Lemma
Proof: We proceed by induction on the uilpotency class c of G. If G is abelian then we have that Out (G) = Aut (G) and so we have ~ = r Moreover, given an abstract kernel r : F ~ Out (G) = Aut (G), we know that the set of equivalence classes of extensions stands in one to one correspondence with H~(F, G ~ R '~) = 0 (since F is finite and ]R'~ is divisible). So, there is at most one class of extensions. Moreover, as : F ---* Aut (G) is a morphism, we can form the semidirect product group G > ~ F , which is a group extension inducing the abstract kernel r So a given extension is equivalent to G ~ F , which implies that such an extension splits. Suppose now that the nilpotency class of G is c > 1. Let r : F O u t ( G ) be a morphism, and denote by 9 ( F ) C_ A u t ( G ) the inverse image of r under the canonical projection p : Aut (G) ~ Out (G). There is a short exact sequence 1 -~ Inn (G) -~ ~ ( F ) -~ r
-~ 1.
As Inn (G) ~ G/Z(G) is a connected and simply connected uilpotent group of class c - 1, the induction hypothesis implies that there is a splitting ~' : r ~ q~(F). Of course, by taking ~ = ~' o r we showed the first claim of the statement. To prove that any extension of G by a finite group splits, we can use the same argumentation as for the abelian case, since H2(F, Z(G) ~ IR'~) = 0 classifies all such extensions and a semidirect product G >1~,F inducing the given abstract kernel exists. The following theorem is a combination of the results of [48] and [21].
34
Chapter 3: Algebraic characterizations of AC-groups
T h e o r e m 3.1.3 A l g e b r a i c c h a r a c t e r i z a t i o n o f A C : g r o u p s The following are equivalent for a polycyclic-by-finite group E:
1. E is an almost-crystallographic group. 2. Fitt ( E ) is torsion free, maximal nilpotent and of finite index in E.
3. E contains a torsion free nilpotent normal subgroup N , which is of finite index in E and such that CE(N) is torsion free. 4. E contains a nilpotent subgroup of finite index and E contains no non trivial finite normal subgroups. Proof: (1 ~ 2) If E is an AC-group (of a given connected and simply connected nilpotent Lie group G), then we know that E N G is a maximal nilpotent and normal subgroup of E. It follows immediately by the first generalized Bieberbach theorem, that E n G = Fitt (E) is maximal nilpotent and is of finite index in E. (2 ~ 3) We can take N = Fitt (E). For suppose there exists a torsion element x in CE(Fitt (E)). As Fitt (E) is torsion free, grp(Fitt ( E ) U {z}) is a nilpotent group, strictly containing Fitt (E) which is a contradiction. (3 ~ 4) Let N be a torsion free nilpotent group such that CE(N) contains no torsion elements. Suppose that E has a finite, non trivial, normal subgroup Ho then, by lemma 1.1.8, we know that [N,H] = 1, which contradicts the fact that CE(N) is torsion free. (4 ~ 1) Of course, this is the real hard part of the proof. We are going to show that E can be embedded in Aft(G), in such a way that the image of this embedding is a genuine almost-crystallographic group. In [48], this was done by constructing a pushout in the category of groups. Here we formulate an alternative description of the (same) construction of this embedding. By restricting to a subgroup of finite index, there is no loss in generality in assuming that N is a normal subgroup of E. Then, the group E fits in a short exact sequence
I~N~E~F~I where F is some finite group. The nilpotent group N is torsion free, for if it where not, the set of torsion elements of N would be a non trivial finite n o r m a l subgroup of E. In the previous chapter (section 2.5), we explained
Section 3.1 Almost-crystallographic groups and essential extensions
35
that E can be described via a "2-cocycle". I.e. there exist functions r : F ~ Aut ( N ) and c : F • F ~ N , where r is a normalized lift of the abstract kernel r : F ~ Out ( N ) induced by the short exact sequence above, and such that the following cocycle conditions are fulfilled: = #(c(g, h))r
1. Vg, h E F r162
2. Vg, h,k e f
c(g,h)c(gh, k) =
hk).
Moreover, the extension E can be described as the group with underlying set N • F , and where the multiplication is given by:
Yn, m e N, Vg, h e F ( n , g ) . ( r Let us denote by r
= (nC(g)(m)c(g,h),gh).
the group h o m o m o r p h i s m which is obtained as the
composition F r Out ( N ) ~ Out (G), where the last arrow is obtained by the unique a u t o m o r p h i s m extension reults of Mal'cev. Also we use r : F r Aut ( Y ) ~ Aut (G) as a normalised lift of r If, moreover, we introduce the m a p c' : F • F ~ N ~ G, we see that the pair (r c') satisfies the cocycle conditions, and so they determine a group extension G ~ of G by F in such a way that we obtain a commutative diagram 1
~
N
~
E
~
F
~
1
I
--~
G
--~
G'
--*
F
-+
I
By the previous lemma, we know that the b o t t o m sequence splits, and so, there is an action ~o : F --* Aut (G) (not necessarily effective) of F on G, such that the group G' is isomorphic with G >~~F. This induces a morphism
k" G' ~ G > ~ F --+ G>~Aut ( G ) : ( g , f ) ~
(g,~(f)).
We claim that the map k o j : E -~ G>~Aut (G) = Aft(G) is the desired embedding. As the restriction of k o j to N is the canonical embedding of N into its Mal'cev completion, the only thing left to show is that k o j is injective. But as the kernel of the map k o j and N only have the neutral element in common, this kernel has to be a finite normal subgroup of E and hence trivial. At this point we remark that it is not true that F will, in general, be isomorphic to the holonomy group of the almost-crystallographic group k o j ( E ) . This follows from the fact that it is not true that the map k itself is injective. The holonomy group of E will be a quotient group of
Chapter 3: Algebraic characterizations of AC-groups
36 F.
As is seen in this theorem, there are several nice algebraic descriptions of almost-crystallographic groups. These descriptions inspired me to investigate the possibility of generalizing the notion of an a l m o s t crystallographic group to more general groups, in particular the polycyclic-by-fmite groups. The following sections are especially devoted to this generalization. We knew already, by the results of L. Auslander, that any A C - g r o u p E induces an essential extension 1 ~ N = F i t t (E) ~ E ~ F ~ 1. Theorem 3.1.3 shows the converse, namely any essential extension determines an almost-crystallographic group. Knowing this, it is obvious t h a t our algebraic formulation of the generalized third Bieberbach theorem is equivalent to the topological one.
3.2
Torsion in the centralizer of a finite index subgroup
One of the statements of theorem 3.1.3 deals with torsion in the centralizer C E ( N ) . Here N is a normal subgroup of finite index in E. This section is devoted to the investigation of such torsion elements. The basic result is the following: Lemma
3.2.1
Let 1 --~ Z k ~ E ---+F ~ 1 be any central extension, where F is a finite group. Then r ( E ) is a characteristic subgroup of E . P r o o f : Once we know t h a t r ( E ) is a group, it follows a u t o m a t i c a l l y t h a t it is characteristic. As E is a central extension of Z k by F , we m a y view E as being the set ~k x F , where the m u l t i p l i c a t i o n , is given by
(a,a).(b,~)=(a+bTc(a,
fl),a/3), Va, b E Z , Va, f l E F ,
(3.1)
for some 2-cocycle c : F x F ~ Z k. Since ~k is a vector space, the inclusion map i : Zk ~ Rk induces a trivial map i. : H2(F, Zk) -+ H2(F, R k) = 0 on the cohomology level. This means t h a t there is a split short exact sequence 1 --+ IRk ~ E ~ ~ F ~ 1, where E ~denotes the group determined by the cocycle i(c). (I.e. E ' = R k x F and multiplication is given by (3.1), where a and b m a y now belong to Rk.) So there is a splitting morphism s : F ~ E I. But it is now easy to see
Section 3.2 Torsion in the centralizer of a finite index subgroup
37
that s is unique, since for all f E F, s ( f ) has to be a torsion element and there is only one torsion element in E ~ mapping to f. This shows that s ( F ) = r ( E ' ) and so T(E') is a group. The proof finishes, by realizing that E C E ' and so T(E) (C_ r ( E ' ) ) has to be a group also.
Remark
3.2.2
In literature, a p r o o f for this lemma is given by means of topological arguments, which can also be used to proof analogous results in more general cases. However, they fail to be useful in the most general case, which we will state below. Remark
3.2.3
1. The proof of the lemma might suggest that the group E can be decomposed into a direct sum E = 7, k @ F ~ for some finite group F/. However, this is not true: consider the group E = (Z~)7`2)>~7`2 where 7. 2 = {8, i} and the action of 7, 2 on 7` @ 7`2 is given by i(1, 0) = (1, i) and i(0, 1) = (0, i). The group A = 27` C_ E is indeed a free abelian, central subgroup of finite index, so the conditions of the lemma are satisfied, b u t E cannot be seen as the direct sum of a free abel]an group and a finite one. . If we examine the conditions of the lemma, we quickly see that the l e m m a is false if we are not looking at central extensions (E.g. k = 1, F = Z2 and E = Z ~ Z 2 , where Z2 acts non trivially on Z ) or at extensions o f i n f m i t e index (E.g. k = 0 and F = E = Z >~7`2). However, as the following lemma shows, there is no need for a finitely generated free abel]an kernel. Lemrna 3.2.4
Let A be any abelian group. If I -+ A ~ E ~ F is any central extension, where F is finite, then r ( E ) forms a characteristic subgroup of E . P r o o f : r ( A ) is a characteristic subgroup of A, and so there is an induced short exact sequence 1 ~ A/T(A)--* E/r(A)~
F ~ 1.
Chapter 3: Algebraic characterizations of A C-groups
38
From this it follows that it will suffice to prove the theorem in case A is torsion free abelian. Suppose A is torsion free and x, y E T(E). We have to show that xy E ~-(E). Let E ~ denote the group generated by x and y, and A ~ = A A E ~. There is an induced extension 1-,A I-~E I-~F I-~1 where F ~ is some finite group. Since E t is finitely generated and F ~ is finite, we m a y conclude that A' is finitely generated too ([56~ p. 117]). Therefore, we are in the situation of lemma 2.1, which implies that v ( E ~) is a group, and so xy e T(E') C T(E).
T h e o r e m 3.2.5
Let E be any subgroup of finite index in a given group E ~, then v(CE, E ) is a subgroup o r e I. Moreover, i r e is torsion free and normal then r(CE,E) is the unique maximal normal torsion subgroup of E I. P r o o f : First, let us consider the case where E is normal in E ~. There is an exact sequence of subgroups of El:
1
Z(E)
C ,E
F
1
for some finite group F . It follows immediately from lemma 3.2.4 that r ( C E , E ) is a subgroup of E. It is normal in E , since it is characteristic in another normal subgroup (CE,E). To prove the last statement it is enough to realize that, in case E is torsion free, any normal subgroup T, containing only torsion, commutes with E (see lerarna 1.1.8). If E is not normal in E', we replace E ' by the normalizer NE,E of E in E ~. Since CE, E C NE,E we m a y apply the theorem for normal E to conclude the correctness of the theorem in the general case too.
Remark
3.2.6
If T(CE,E ) is finite, it is the maximal finite normal subgroup of E ~. This is e.g. always the case when E ~ is a polycyclic-by-finite group (see the following section). This observation will be used in the following section.
Section 3.3 Towards a generalization of A C-groups
39
It is now easy to proof the following lemma due to K.B. Lee ([45]). The original p r o o f of this lemma is based on the Seifert Fiber Space construction with typical fiber a nilmanifold. As it is our intention to avoid the topological arguments as much as possible, we use the theory developed above. L e m m a 3 . 2 . 7 Let 1 --~ N ~ E --. F ~ 1 be any extension of a torsion free nilpotent group N by a finite group F . Then, the set of torsion elements of C E ( N ) is a characteristic finite subgroup H of E and it is the unique finite normal subgroup H of E such that E / H is almostcrystallographic. Proof: H is the unique, maximal finite normal subgroup of E.
3.3
T o w a r d s a g e n e r a l i z a t i o n of A C - g r o u p s
L e m m a 3.3.1 Let r be any polycyclic-by-finite group, then r has a unique maximal finite normal subgroup. Proofi Let
E1 C_ E2 C_ E3 C . . . Fin C_ E,~+I C_ . . . be any ascending chain of finite normal subgroups of F. This chain is necessarilly finite, since every ascending chain of subgroups of r is finite (see [59]). Therefore we can choose a finite normal subgroup H o f t which is maximal among all finite normal subgroups. Now it is easy to see that H is unique, for if K was another such a normal subgroup, then H . K would contradict the maximality of H .
Remark 3.3.2
For a polycyclic-by-fmite group r , we will denote its maximal normal finite subgroup by F ( r ) . Remark
3.3.3
In general, F ( r ) is not maximal among all finite subgroups of r . For example, when r = Z :~Z2, where Z2 acts non trivially on Z, one easily sees that F ( r ) = 1, while r has subgroups of order 2.
40
Chapter 3: Algebraic characterizations of AC-groups
D e f i n i t i o n 3 . 3 . 4 Let F be any group. Then F is said to be almost torsion free if and only if F has no finite normal subgroups other then the trival one.
So, a polycyclic-by-fin_ite group F is almost torsion free if and only if F ( F ) = 1. We warn the reader that some authors use the t e r m almost torsion free, when they want to indicate groups which are virtually torsion free. This is a totally different notion from ours, since every polycyclicby-finite group is virtually torsion free. We already proved most part of the following theorem: Theorem then
3.3.5 Let E be a .finitely generated virtually nilpotent group,
E is almost torsion free E is almost-crystallographic. Proof: Let E be almost torsion free. Since E is finitely generated, there exists a finitely generated torsion free nilpotent normal subgroup N of finite index in E. This can be seen as follows: There exists a torsion free normal subgroup N of finite index in E , since E is p o l y c y c l i c - b y finite, and so also (poly-Z)-by-finite. Since E is virtually nilpotent, N = Fitt ( N ) is a characteristic subgroup of N and is of finite index in E. Therefore we have a short exact sequence
1---* N - + E--* F--+ I with F fmite. Following lemma 3.2.7, we m a y find a finite normal subgroup H <1 E for which E / H is almost-crystallographic. But the fact that E is almost torsion free implies that H = 1 or that E itself is a l m o s t crystallographic. The converse of the theorem is given by the last statement in theorem 3.1.3 Of course, this t h e o r e m also implies that a virtually abelian group Q is crystallographic if and only if Q is almost torsion free. Knowing this theorem, it is natural, at least as an algebraist, to consider those groups F which are almost torsion free as a generalization of the almost-crystallographic groups. We also show how they can be seen as a generalization from the topological point of view. But let us first examine the almost torsion free groups algebraically.
Section 3.4 The closure of the Fitting subgroup
3.4
41
The closure of the Fitting subgroup
In this section we will define a normal subgroup Fitt (F) of F, which contains the Fitting subgroup of F as a normal subgroup of finite index. In fact we take the maximal one with this property. A formal definition looks like: D e f i n i t i o n 3.4.1 The closure of Fitt (F) is denoted by Fitt (F) and satisfies Fitt (F) = < GIIG <~F and [G: Fitt (G)] < oo > . We remark that the notion of closure has no topological meaning at all! It is interesting to note that for any G <~ r , G n Fitt (F) = Fitt (G). L e m m a 3.4.2 Fitt (F) is the unique maximal subgroup G of F for which 1. G<]F 2. [G: Fitt (G)] < co. Proof: Since any ascending series of subgroups of r is finite, we can choose a group G which is maximal amongst those who satisfy 1. and 2. (G contains Fitt ( r ) and so Fitt (G) = Fitt ( r ) ) . We assert that any other group Go combining both conditions stated above is contained in G. To see this, we consider the group Go.G which is certainly a normal subgroup of F. Moreover, since (Co.a)/a = ao/(ao n a ) is finite (Go n G _D Fitt (G0)), [G0.G: Fitt (G0.G)] = [G0.G: Fitt (r)] = [G0.G: G][G: Fitt (F)] = < oo. Therefore, we m a y conclude that Go.G = G or Go C_ G. This is enough to conclude t h a t Fitt ( r ) = G.
It is well known that any polycyclic-by-finite group r has the property of being ( n i l p o t e n t - b y - a b e l i a n ) - b y - f m i t e . So there is a short exact sequence 1 - ~ X---~ F ~ F ~ 1 where F is finite and X is nilpotent-by-abelian. Fitt (F), we see that
W h e n dividing out
1 ~ Z / ( X N Fitt ( r ) ) = X / F i t t ( X ) --~ r / F i t t ( r ) ~ F ' --, 1
42
Chapter 3: Algebraic characterizations of AC-groups
where F ~ is also finite. Since X is nilpotent-by-abelian, X / F i t t (X) is abelian. Therefore, we m a y conclude that F / F i t t (F) is abelian-by-fi_nite, or F is nilpotent-by-(abelian-by-finite). Now we concentrate us on the case of almost torsion free F. So, Fitt (F) is torsion free and we have a short exact sequence l~Fitt(F)
~F~
Q ~ 1
where Q is also virtually abelian and almost torsion free. For suppose Q~ is a finite normal subgroup of Q, then G = p - l ( Q , ) would be a norreal subgroup of F satisfying [G : Fitt (G)] < co which implies that G C_ Fitt (F) or Q~ = 1. So Q is a crystallographic group. Of course, Fitt (F) is also almost torsion free. Because F ( F i t t (F)) is a characteristic subgroup of Fitt (F) and so a normal subgroup of F, which implies that F ( F i t t (F)) = 1. We summarize this in the following theorem. Theorem
3.4.3 Let F be any polycyclic-by-finite group, then
F is almost torsion free
F is (almost-crystalIographic)-by-crystallographic. Moreover, in case F is almost torsion free, the intended almost-crystallographic subgroup can be taken as Fitt (F). In [59] one can find, as an exercise, the following property: If F is a solvable group, then C r ( F i t t (F)) C_ Fitt (F). We drop the restriction of solvable groups and find the following lemma. L e m m a 3.4.4 If F is any polycyclic-by-finite group, then C r ( F i t t (Y)) c Fitt (Y). Proof: Y fits in a short exact sequence 1 --~ Fitt (F) ---, Y ~ Q ~ 1 where Q is abelian-by-finite. Let A denote an abelian normal subgroup of finite index in Q. Consider the group F' = p - l ( A ) which is solvable. Then, since F i t t ( r ' ) = F i t t ( r ) , we see that C r , ( F i t t ( r ) ) C_ F i t t ( r ) (using the exercise of [59]). Therefore we see that p ( C r ( F i t t (F))) A
Section 3.4 The closure of the Fitting subgroup
43
A = {1}. So, C r ( F i t t ( F ) ) . F i t t ( r ) is a finite extension of Fitt ( r ) , and C r ( F i t t (F)).Fitt ( r ) is a normal subgroup of F, it is even characteristic. We m a y indeed conclude that C r ( F i t t (F)) C Fitt (F).
For the following theorem we need some technical facts, which we treat now. Let N be a torsion free, fmitely generated nilpotent group of class c. We defined the groups Z~(N) (1 _< i _< c) of the upper central series of N in the first chapter. For any extension 1 --* N ~ E --* Q --* 1 the abstract kernel r : Q --. Out N induces c morphisms ~i : Q ---* Ant Zi(N)/ZI_I(N) (1 < i < c). To define the automorphism ~l(q), for q E Q, we consider a lift q E E of q and take as Tl(q) the automorphism of Z ( N ) induced by conjugation by q. For ~2(q), we first consider the extension 1 -~ N/Z(N) * E/Z(N) -* Q ~ 1 and then continue as for 9~1. Analogously we build up all ~i. Now let e be an element of E such that its projection ~ E Q has finite order and such that the group N ~ generated by N and e is nilpotent. C
We want to show that p(e) E N ker(p~). We proceed by induction on i=1
the nilpotency class c of N . If N is abelian, the assertion is immediate by lemma 1.1.5. For N of larger nilpotency class, we first consider the extension 1 --* N / Z ( N ) ---* E / Z ( N ) --* Q --* i and then again, we use r
lemma M . ~ .
Conversely, if p(e) e N k e r ( ~ ) (p(e) may be of i,finite i=1
order) then the group generated by N and e is nilpotent. This is again easy to show by induction on the nilpotency class of N . We are ready now to prove the following theorem. Theorem
3.4.5 Let F be any polycyclic-by-finite group. Then Fitt (F) is torsion free and maximal niIpotent in F
F is almost torsion free. Proof." First suppose that Fitt (F) is torsion free and maximal nilpotent in F. Consider H <3 F, a finite normal subgroup. Then H <~ Fitt ( F ) . H . But Fitt (r).H is an almost-crystallographic group (since Fitt (F) is maximal nilpotent) and so H has to be trivial. Conversely, suppose F is almost torsion free. It is easy to see that Fitt (F) has to be torsion free, since the elements of finite order of Fitt (F)
44
Chapter 3: Algebraic characterizations of A C-groups
form a characteristic subgroup of Fitt (F). So we can consider the exact sequence l~Fitt(F)~P~Q---, 1 where Fitt ( r ) can play the role of N in the discussion above. First we show that no element q of infinite order in Q can be lifted to q E E in such a way that the group generated by F i t t ( P ) and q is nilpotent. Suppose there should exist such a ~. Recall that Q is abelian-by-fm_ite, and so we can find an abelian normal subgroup A of Q of finite index n in Q. Therefore, also ~ E A can be lifted to an element qn to yield a nilpotent subgroup of E strictly containing Fitt (F). But when we look at the group F / = p - l ( A ) which fits into 1
Fitt ( r ) = Fitt
A
1
we see that Fitt (F) is maximal nilpotent in p/. Otherwise, we should find a nilpotent subgroup (normal in F') strictly containing Fitt (F'). So the desired q'~ E A cannot be found. Now suppose ~ is of finite order. Then the group generated by q and Fitt (F) is nilpotent if and only if p(q) E A ker(~ai), where the ~ai are i=1
as above. This intersection is a normal subgroup of Q and is finite, this means that q E Fitt (P). But now we use the fact that if F is almost torsion free, then Fitt (F) is almost-crystallographic. And so, Fitt (F) is maximal nllpotent in Fitt (F), implying we cannot find q outside Fitt (F). We summarize the results found so far in the following theorem. Notice the analogy with theorem 3.1.3. T h e o r e m 3.4.6 Let F be a polycyclic by-finite group, then the following are equivalent: 1. F is an almost torsion free group. 2. Fitt (F) is torsion free and maximal nilpotent in F 3. F contains a torsion free normal subgroup F ~, which is of finite index in F and such that Cr(F t) is torsion free. 4. F contains no non trivial finite normal subgroups. 5. F admits an effective and properly discontinuous action on ]Rh(r), where h(F) denotes the Hirsch number of F.
Section 3.5 Topological point of view
45
For the proof of number 5. we refer to corollary 3.5.2 (next section) and to t h e o r e m 4.2.3 of the following chapter. We r e m a r k that if N is a torsion free nilpotent group of Hirsch number n, then the Mal'cev completion G of N is homeomorphic with IRn. So the action of an almost-crystallographic group E on G can be seen as a properly discontinuous action of E on IR~. Perhaps it is also worthwhile to notice the following generalization of l e m m a 3.2.7. The p r o o f o f which is easy to deduce from the theory of this chapter. L e m m a 3.4.7 Let 1 ~ F I ~ F --~ F --* 1 be any extension of a torsion free poIycyclic-by-finite group r ~ by a finite group F. Then, the set of torsion elements of C r ( r ' ) is a characteristic finite subgroup H o f f and it is the unique finite normal subgroup H of F such that F / H is almost torsion free.
3.5
A l m o s t torsion free groups from the topological point of view
This section is intended to show that, also from a topological standpoint, the almost torsion free groups are interesting groups to investigate. The reader with little topological background should at least try to u n d e r s t a n d the formulation of corollary 3.5.2. T h e o r e m 3.5.1 Let r be any group acting on a contractible space M and containing a normal subgroup F ~ of finite index, which acts freely and properly discontinuous on M , such that the quotient space M = r l \ M is a closed aspherical manifold. Then, the kernel of the induced action of F = r / I " on M is the isomorphic image of the torsion subgroup of c r ( r ' ) inside F . In fact, the set of all torsion elements of c r ( r ' ) forms a characteristic subgroup o f F , and equals the kernel of the action o f f on M. Proof: Let t be a torsion element in Cp(F'). commutative diagram 1
~
r'
--,
1 --, I m ~ ( r ' ) ~
r
--,
hut(r')
~
This means that in the
F
Out(r')~
~
1
1
the group T generated by t maps trivially into Aut(r'). (The vertical maps are induced by conjugation in P). By [46, L e m m a 1] the image of T
46
Chapter 3: Algebraic characterizations of A C-groups
in F acts trivially on M . This image is isomorphic to T, as I~ is torsion free. Conversely, if T ~ is a subgroup of F acting trivially on M , then T ~ is the isomorphic image of a subgroup T of r (this lifting of T ~ back to P can be done already if the T ~ action on M has a fixed point). Clearly, then T commutes with r ~ so t h a t T C Cr(r~). For the last statement, note t h a t a torsion element of 1~ acts effectively on M if and only if the corresponding element of F acts effectively on M. []
As an immediate consequence, we find the following reformulation of the above theorem in the situation we're interested in. C o r o l l a r y 3.5.2 Let F be a polycyclic-by-finite group with Hirsh number K . Let Pc be a torsion free normal subgroup of finite index of F, and let F denote the set of torsion elements of C r ( r c ) . For any properly discontinuous action of p : F ~ R K, F is the kernel of p. Moreover, F is the maximal finite normal subgroup of F.
Chapter 4
Canonical type representations 4.1
Introduction
In this chapter we will study special types of actions of a group E (in most cases E will be virtually nilpotent) on some space R '~. We are not only interested in these actions from the topological point of view, but also from a c o m p u t a t i o n a l point of view. In fact, in order to be able to deal with virtually nilpotent groups, we need some way of computing formally in these kind of groups. Therefore, we want to find nice representations of (virtually) nilpotent groups. Such a "nice" representation is e.g. given by an affine representation p : N ~ Aff(I~n). As is very well known, the group of affme transformations of R n can be seen as a subgroup of Gl(n + 1, R). The embedding of Aff(R ~) into Gl(n + 1, R) is given by m a p p i n g the affine transformation with linear part A 6 Gl(n, IR) and translational part a 6 R n (seen as column vector) onto the element
(A a) 0
1
6 Gl(n + 1,]R).
Moreover, i f w e i d e n t i f y a n e l e m e n t
z 6 R~withtheelement
( x1 ) 6
[~n+l we can express the image of r under the affine transformation (A, a) 6 Aff(R n) by means of a m a t r i x multiplication. I.e.
(A, a)z -
(A a)(.) 0
1
1
"
Chapter 4: Canonical type representations
48
So, an affine representation of a group E can be seen as a matrix representation and already many computer programs can deal very well with (formal) matrices. From the geometrical point of view, we will describe in this chapter some of the latest developements concerning the conjecture of John Milnor ([52]) stating that any torsion free polycyclic-by-finite group occurs as the fundamental group of a compact, complete affinely fiat manifold. We will explain (in section 4.4.5) this notion of a compact, complete affinely flat manifold and see that Milnor's conjecture is false, even in the case of nilpotent groups. However, we will also formulate a conjecture analogous to Milnor's where we replace the word affine by polynomial and prove that this new conjecture holds in case of virtually nilpotent groups. This new conjecture uses the notion of "affine defect" of a group. This is a number which measures up to what extend a given group can serve as a counter example to Milnor's question. We finally compute that the only known counter examples to the conjecture of Milnor are of the smallest possible affine defect, namely one.
4.2
D e f i n i t i o n of canonical t y p e s t r u c t u r e s
It is well known ([59, lemma 6, page 16]) that, if r is a p o l y c y c l i c - b y finite group, then there exists an ascending sequence (or filtration) F, of normal subgroups F~ (0 < i < e + 1) of F
F,:
Fo = 1 C F I C F 2
C...CF~_I C F c C Fr
=F
for which F i / F i - 1 ~ Z ki for 1 < i < c and some hi C No and
Y/Fc is finite. Moreover, if desired, each Fi can be chosen to be a characteristic subgroup of F. Let us call such a filtration (of not necessarily characteristic subgroups) of F a t o r s i o n f r e e f i l t r a t i o n . In many cases, we will have that Fc = Fc+l in which case we do not always write the last term Fr of the torsion free filtration F, of F. We use K to denote the Hirsch number (or rank) o f F . Often, we will also use Ki = h i + k i + l + - . ' + kr and Kc+l = 0. It follows that K = K1. D e f i n i t i o n 4.2.1 Assume F is polycyclic-by-finite with a torsion free filtration F , . A set of generators A = {al,1, a l , 2 , . . . , a l , k l , a 2 , 1 ~ . . . , a c , k c , a l , . . . ,
Otr}
49
Section 4.2 Definition of canonical type structures of r will be called compatible with F. iff
ViE { 1 , 2 , . . . , c } :
al,l,al,2,...,a,,k~ generates F,.
It is clear at once t h a t any torsion free filtration F. of F admits a compatible set of generators. In the sequel, we will frequently work with quotient groups of F. In order to avoid complicated notations, we will write the same symbols for elements in F, as for their coset classes in a quotient group. E.g. we say t h a t F2/F1 is the free abelian group generated by a2,1, a2,2,... , a2,k2. Now we introduce quickly the basic building blocks for the representations we are interested in (we follow [44] and [14]): 2r
K, IRk) = {continuous maps A: R K -+ IRk}
7-/(IRK) = {homeomorphisms h : R g ~ IRK}. Jgt(~ K, IRk) is an abelian group (for the addition of maps) and can be m a d e into a Gl(k, IR) • 7-/(]RK)-module, if we define (g,h)~ _~ g o )~ o h -1, V.~ E .A/I(~ K } ~ k ) , Vg E
Gl(k,I~), Vh C ~~(RK).
We remark t h a t there is an embedding
M(R~, R~)~(GI(k, R) • ~(R~)) ~ ~ ( R k+K) with
(~,g, h)(r
(g(~) + Ah(y), h(y)),
(4.1)
VX E ]~k, Vy E ]RK, 'v')~E .A~(]~K, ]~k), Vg C Gl(]g, ]~), Vh C .~(]~K). D e f i n i t i o n 4.2.2 A s s u m e Y is polycyclic-by-finite with a torsion free filtration F . . A representation p = po : F ~ H ( R K) will be called o f c a n o n i c a l t y p e with respect to F. iff it induces a sequence of representations: p,: r / r , ~ n ( R ~'+'), (1 < i < c) such that for all i the following diagram commutes:
1 --+ Z kl = ~ Fi /F i-1 -~
tJ 1 -~ ~VI(Ii~K~+, , IRk~
r/r~_l
Pi-1
-~
r/r~
-+
1
~ Bi • Pi
~ >~(a • ~) ~ Gl(k~, R) • ~ ( R K'+') ~ 1 (4.2)
"where
Chapter 4: Canonical type representations
50
9 A4>~(G • ~ ) stands for Ad(I~K'+~,IRk')>~(GI(ki, N) • ~ ( R K ' + ' ) ) , * j(z) : R Ki+' --+ Ii@i : z ~ z, Vz E ~kl and 9 Bi : F/Fi ~ GI(k/,Z) ~-~ GI(k/,IR) denotes the action of F/Fi induced on Z ki by conjugation in F / F / _ I . This means that a canonical type representation is nothing else than an iterated Seifert construction with abelian kernel. We explain this in more detail: Let Q be a group acting properly discontinuously on a space W, via p : Q ~ 7-/(W). Let Ad(W, R k) denote the set of continuous mappings of W into R k. In the same way as above we can make ~4(W, R k) into a (Gl(k, IR) • 7-/(W))-module, and there is an embedding
M(W, Rk)~(GI(k,R) • n( W ) ) ~ n ( R k • W). Now, consider a group extension 1 ~ ~k ~ E ~ Q ~ 1 which induces an a u t o m o r p h i s m ~ : Q ~ Aut (Zk) via conjugation in E. A Seifert construction for the above data is in fact a morphism r
E ~ A4(W, ~ ) >~(Gl(k, R) • 7-/(W)) .o,~,o. A,4 >~(G x 7-l)
such that the following diagram commutes 1
--~
1
~
~k
~(w,R
--~
k)
-~
E
~x(v•
~
-~
Q
r215
~
1
~
1
The resulting quotient space is also said to be a Seifert Fiber Space with typical fiber a (k-dimensional) torus. This is because the canonical projection
p:E\(Rk•
k•
,r
is "something hke" a fiber space. Indeed, in general the inverse image p - l ( @ ) of an element @ E Q \ W will be diffeomorphic to a torus, b u t there migth be elements where the "fiber" p-l(z~) is the quotient of the torus by a finite group action. These fibers are called singular, while the first ones are said to be typical. We are now ready to summarize some important results on (general) representations of canonical type. For elements x E I~g w e w i l l label the coordinates as follows:
Section 4.2 Definition of canonical type structures
51
We Mso write xi to indicate all variables (r ff p = P0 : F -~ 7-/(• K) is any representation, then for each 7 E F there exist continuous functions h~,j : ]RK -~ R such that p ( 7 ) : s K -~ R K : x ~ ( h L ( x ) , h : , ~ ( x ) , . . . , h:,k~(x)). In a way similar to the xi5 we use h~(x) to indicate (h~,l(~), h~,2(x),..., h:,~,(z)). T h e o r e m 4.2.3 Let F be a polycyclic-by-finite group and F. be a torsion free filtration of F. Then, there exists a representation p : F --~ TI(R K) which is of canonical type with respect to F . . Moreover, any two such representations p, p~ : F -~ 7-t(R K) are conjugate to each other inside 7-l(RK). Also, for each canonical type representation p : F -+ 7-l(R K) we have: 1. F acts properly discontinuously on ]~K, via p and with compact quotient. 2. her p =the maximal finite normal subgroup of F, so p is effective if] F is almost torsion free. 3. W / c F : p ( 7 ) : S K -~ R K : ~ ~ (h:(~), h ~ ( x ) , . . . , h : ( r
is such that
h~(x) = Ui('/)zi + gT(zi+l,z~+2,...,xr for some continuous ]unction g~ : R Ki+l -~ R k~. 4. Fi acts trivially on R Ki+~ and via affine transformations on R kl . Proof: The proof of the existence and uniqueness of a canonical type representation, can be found in [47]. The reader m a y also consult [47] to see t h a t the action of F defined on I~K is properly discontinuous and that the orbit space is compact. We do not go into detail here, since we will prove the analogous results in more interesting cases later on. 2) follows from corollary 3.5.2. 3) and 4) are immediate consequences of the iterative way of building up p (see also (4.1)).
Chapter 4: Canonicaltype representations
52 R e m a r k 4.2.4
Point 3. and 4. from the theorem above can be used to define the concept of a canonical type representation. Such kind of definitions were for instance used in [20]. Having this concept of a (general) canmfical type representation, it is clear now, that to get a nicer geometric structure, one needs to use smaller subgroups of 7-/(RK). In view of the iterative set up, it is however necessary that these subgroups satisfy one crucial condition: assume we restrict AA(RK, ]~k~. to a subspace S(• K, ~k) containing the space of constant mappings R ~ and assume we restrict 7-/(IRK) to a subgroup STi(RK), then one observes that it is necessary that S(I~ K, I~k) is a (Gl(k, ~) • ST-/(Rg))-submodule such that there is a monomorphism,
s(R
Rk) (Gl(k, R) • SU(R ))
Possible examples of such situations are S m o o t h r e p r e s e n t a t i o n s : Let C~176 k) denote the vector space of functions f : ~ g --~ ~k, which are infinitely many times differentiable and denote by C~(IRK) the group, under composition of maps, of smooth diffeomorphisms of IRg . Then by considering C~176K, ~k) instead of ./kz[(~K, R k) and replacing ~.~(~g) by C~176 we find the so-called canonical type smooth representations. Here, results analogous to theorem 4.2.3 can be formulated (see [14] and [47]). A f t l n e r e p r e s e n t a t i o n s : Restrict ./Vl(]RK, I~k) to Aff(RK,R k) (the vector space of affine mappings) and 7-/(R ~) to Aff(R K) (the affine group of ]~g). We will treat this kind of representations in a few moments. P o l y n o m i a l r e p r e s e n t a t i o n s : Write p(~g, Rk) to refer to the vector space of polynomial mappings p : ~ g ---+ i~k. So p is given by k polynomials in K variables. P(R K) will be used to indicate the set of all polynomial diffeomorphisms of ] ~ g with an inverse which is also a polynomial mapping. This is a group where the multiplication is given by composition. It is not hard to verify that P(I~K, R k) is an A u t ( Z k) • P ( R g ) - m o d u l e and that the resulting semidirect product group
p(RK, irk)>~(Aut Zk • p(Rg)) C_P(R g+k)
Section 4.3 Seifert Fiber Space construction
53
since by definition V ( p , g , h ) E P(IRK,]Rk) >l(Aut Z k • P(]RK)), V ( z , y ) E R k+g
:
(,,~,h)(~,y) = (g~ + p(h(y)), h(y)). Restrict AA(IRK, ]Rk) to p ( ~ K Rk) and 7-t(R K) to P(IRK), and we will speak of canonical type polynomial representations. We will also deal with these kind of representations later in this chapter. In each of these restricted situations one now faces existence and uniquehess questions. In order to be able to solve these problems, we will first give a general t r e a t m e n t of the Seifert Fiber Space construction in the following section.
4.3
An algebraic description of the Seifert Fiber Space construction
In this section we will first prove a general algebraic lemma and then apply this l e m m a to the Seifert Construction situation. This theory was developed by K.B. Lee in [44]. Let Q and Q1 be groups and suppose that there are two abelian groups Z and S such that Z is a Q-module, while S is a Ql-mOdule. Assume moreover that two morphisms i : Z ~ S (i is an embedding) and j : Q --~ Q1 (j is not necessarily injective) are given which are compatible with the module structures. I.e.
w c Q, Vz c z : i(~z) =J(~) i(z). The aim of this section is to describe all pairs (E, f ) , where E is a group extension of Q by Z and f : E ~ S>1Q1 is a morphism of groups such t h a t the following diagram is commutative: 0
--.
Z
~i
~
E
If
~
Q
~
1
lJ
Of course, before we can start the description of all such pairs, we must know when to consider two such pairs as equal or as different. D e f i n i t i o n 4.3.1 Two pairs (E, f ) and (E', f') as above are said to be equivalent if and only if E and E I are equivalent as group extensions of Z by Q via an isomorphism 0 (inducing the identity on Z and on Q) such that there exists an element s E S with fit? = # ( s ) f .
Chapter 4: Canonical type representations
54
So if (E, f ) -,~ (E', f') via an isomorphism 0, then there exists an element s E S for which the following diagram commutes: E
0)
E~
S~Q1 "-(s2 s~Ol Let us denote the set of equivalence classes of pairs (E, F ) by H(Q; Z, S). The following l e m m a and its proof are crucial to u n d e r s t a n d most of the remaining chapter. This is because we will show a one to one correspondence between H(Q; Z, S) and a 1-cohomology group Hi(Q, ~-), where the exact correspondence will be explained during the proof. L e m m a 4.3.2 There is a bijection between H ( Q; Z, S) and Hi(Q, S) Proof: Consider the short exact sequence of Q modules 9
1-+ Z ~-~ S
S
P , -----~
Z
I,
where the Q - m o d u l e structure of S is, of course, given by
Vq E Q,Vs E S : qs =J(q) s. We will define a m a p fl : Hi(Q, S) ~ H(Q; Z, S) and prove t h a t this m a p is indeed a bijection. Construction
o f f~:
Let A: Q -+ ~- be any crossed h o m o m o r p h i s m (i.e. take A E ZI(Q, ~)). Choose any lift ~ : Q ~ S of A (so p~ = A) with ),(1) = 0. This ), induces
a pair (E(J,), f~) as follows: 1. Let E(~) be the group with underlying set Z • Q and with multiplication
Vz, y ~ z, w , ~ ~ Q: (z, .)(y, 8) = (z +" y + 6~(., ~), ~ ) where 5~((~,/3) =(~ ~(fl) - ~((~f~) + ~(a). This means in fact t h a t E ( s is the extension representing the cohomology class 5()~) E H2(Q, Z), where
S -, H2(Q,Z ) 6: HI(Q, ~)
Section 4.3 Seifert Fiber Space construction
55
is the connecting h o m o m o r p h i s m in the long exact cohomology sequence
9..-+ H I ( Q , S ) Y4 H i ( Q , 3S) s H2(Q , z ) ~ H2(Q,S) - + . . . induced by the short exact sequence 0 --+ Z --+ S --+ S/Z --~ 0 of Q-modnles. 2. Define fs : E ( ~ ) - + S>4Q1: In the sequel, we will write the fact t h a t i is injective. easily verified that fh,: E ( s
(z,~)~ (i(z)+~(a),j(a)). z in stead of i(z), which is justified by So, fs = (z + ~ ( a ) , j ( a ) ) . It is -+ S >~Q1 is a m o r p h i s m of groups.
T h e first thing to show is that the choice of lift ~ is u n i m p o r t a n t .
So
consider another lift of A, say A. T h e n we define g : Q -+ Z : c ~ ~ ) , ( a ) - ~(a). T h e fact t h a t g(a) takes images in Z follows from the fact that A and are b o t h lifts of the same )~. Using this g, we introduce a m a p O: E ( i ) -~ E ( ~ ) : ( z , a ) ~
( z - g(a), a).
Some elementary computations show that 0 is an isomorphism of groups inducing the identity on b o t h Z and Q and such that
=s oe. This shows t h a t the pairs (E(~), f~) and (E(~), f~) are equivalent. This implies t h a t the choice of lift does not play an essential role. So far, we only defined ~ on ZI(Q, ~). Therefore, we investigate what h a p p e n s in case we consider two cohomologous 1-cocycles )~ and A~. There exists an element $ E ~-- such that A'((~)-A(a) = 55(c~) = ~-~$. Let s E S be a lift of $ and suppose t h a t ~ is a lift of )~, then we can take ~ = ~ + 5s as a lift for )r R e m a r k t h a t E(~) = E(~'), since 6~' = 5~ + 55s = 5~. Moreover, f~, -- # ( s ) o fi," This shows t h a t )~ and )~' determine two equivalent pairs. Conclusion: the m a p S
a : Hi(Q, -~) -~ H(Q; Z, S ) : (A) ~ (E(~), fi,)
Chapter 4: Canonical type representations
56 is well defined.
is i n j e e t i v e : Suppose that ~{A} = ~{)~'}. So there exists an equivalence 0 : E(X) E(X') of group extensions and an element s C S such that #(s) o f~ = fX' o 0. Define the m a p g : Q ~ Z by the condition
e(o,.) = ( - g ( . ) , ~). R e m a r k that #(s) o fX(0, a) = (s + X(c~) - ~ s,j(v~)) while
f~, o 0 = ( - g ( ~ ) + ~'(~1, j(~)) which implies that s + A(~) - ~ s = - g ( ~ ) + ~'(a) ~ ,V - )~ = 5~ with ~ = p(s). Therefore, seen as elements of HI(Q, ~ ) , {A} = {A'}, which was to be shown.
~2 is surjective: Suppose ( E , f ) is a pair satisfying the necessary conditions. E is an extension of Q by Z and so there exists a 2-cocyle c : Q x Q ~ Z such that E can be seen as the set Z x Q and where the multiplication is given by Vz, y E Z;Vct, fl E Q : (z,c~)(y,/3) = (z +~ y + c(a,~),a~). Determine a map q: E ~ S by f(z, a) = (z + q(z, a), j ( a ) ) . However, as (z, ct) = (z, 1)(0, ct) and f(z, 1) = (z, 1), it follows that
q(z, ct)= q(O,a) and so, from now on, we consider q as being a map from Q to Z. Evaluating f on b o t h sides of
(z, ~)(y,Z) = (z +- y + c(~,Z), ~Z) shows that
(z + q(~) +o y +0 q(~), j(~)j(~)) = (z + . y + c(~, ~) + q ( ~ ) , j ( ~ ) )
Section 4.4 Canonical type afBne representations
57
or c(~, Z)= @(.Z). This shows that (E, f) = ~((p(q))), which finishes the last part of the proof of the lemma. We will apply this lemma in the following concrete situtation. Let Q be a group equiped with two morphisms: ~v: Q ~ Gl(k, Z) and p : Q ~ 7-/(W), W is a topological space. For Q1 we choose a subgroup of Gl(k, N) • T/(W), which contains the image of j = ~ x p. For S we shall consider a Ql-submodule S ( W , R ~) of Ad(W,]Rk), containing the constant maps (the Ql-mOdule structure is decribed in the previous section). Then, the above lemma gives a lot of information on the pairs (E, f), where E is an extension of Z/r by Q, compatible with ~ and where f : E ~ S(W, R Ir >~Q1 is a morphism making the diagram 0
-~
o ~
Zk
-+
li s ( w , R k) ~
E
--*
Q
-~
Q1
if s ( w , Rk)~Q1
-+
1
-~
1
;J
commutative. Here i(z) : W ~ N k : w ~ z. The information the lemma provides is given by the connecting homomorphism 6 : H I ( Q , s ( w-,~ Rk)) -~ H2(Q, Zk). For a given extension E, there exists a morphism f : E --+ S(W, I~k) >~Q1 as above if and only if the extension E corresponds to a cohomology class in the image of 6. So, the eventual surjectivity of the connecting homomorphism ~ guarantees the existence of a morphism f for any extension. Moreover, the number of such morphisms f, up to conjugation with an element of S ( W , •k), is measured by the kernel of the 6. So an injective connecting homomorphism implies the uniqueness of the morphism f.
4.4
C a n o n i c a l t y p e affine r e p r e s e n t a t i o n s
We will first of all concentrate on the most useful type of canonical type representations namely the affine representations. These are interesting since they can be seen as faithful matrix representations, and thus they allow a formal computational approach via very simple computer algorithms.
58
Chapter 4: Canonical type representations
However, as we will see, such a representation does not always exists, but fortunately, there is no trouble in case one considers only finite extensions of groups of low nilpotency classes. We will deal with the uniqueness problem of these representation in the following section. Let N be a finitely generated, torsion free nilpotent group and consider any c e n t r a l series with torsion free factors N.:I=N1
C_ N2 C_ N3 _D . . . D Arc = N ( : Nr
We will refer to such a filtration as a t o r s i o n free c e n t r a l series. Examples of such central series are given by the upper central series and the adapted lower central series of N. As in the general case we also denote by kl (1 < i _< c) the rank of NJN~+l. So, NJN~+I ~ 2~k'. We rewrite the definition of a canonical type affine representation with respect to the torsion free central series N,. The reader should convince himself that, in this restricted setting, this new definition is really an alternative one to the general definition 4.2.2. D e f i n i t i o n 4.4.1 A faithful representation p : N ~ A~(]~ K) will be called "of canonical type" with respect to N , if and only if 1. the matrix parts of p are blocked upper-triangular with the identity matrices of size kl , k2, . . . , kc as diagonal entries, 2. the subgroup Ni of N acts on the ith block 01~k~) as translations (= Z ki) and trivially on R gi+l .
4.4.1
Iterating
canonical type affine representations
Let us describe the Seifert construction in this context: Consider a group N with a canonical type affme representation p : N Aff(R K) : n ~ p(n) = ( A ( n ) , a ( n ) ) , where A(n) denotes the linear (K • K - m a t r i x ) and a(n) E R g the translational part of p(n). Analogously, elements of the additive group Aff(RK, R k) of affme mappings from ]~g to R k are given by a couple (D, d) where D denotes the (k x K) linear part and d (E H/c) denotes the constant part. We regard Z k as a subgroup of constant mappings in Aff(R K, R k) i.e. z (E Z k) H (0, z) (E
Aff(RK, Rk)).
There is art action of Aff(•K) (and so of N) on Aff(R K, R k) as follows: if h C Aff(R K) and A C Aff(R ~ , R k) then hA = A o h -1. Clearly Z k becomes a trivial N - m o d u l e . The semidirect product Aft(R K, R k) >~Aft(R K) acts on R k • R K = R k+g if we define for (z, y) E R k • R K, (X'h)(z, y) =
Section 4.4 Canonical type affine representations
59
(r + )~(h(y)),h(y)). It follows immediately that if h = (D, d), this action is given by +
D aa§
(A,a) and A =
)
(4.3)
and so is clearly affme. T h e i t e r a t i o n p r o b l e m . Given the previous set up, it is n a t u r a l to consider the following problem: given a central extension 1 ~ Z k ~ E ~ N ~ 1, can we extend p to a representation p': E --~ Aff(IRK, ]Rk) >~Aff(IRK) such that the following diagram is commutative: 1--+
Zk 1
~
E lp t
1 -~ Af(R ~:, R k) -~ Aft(R K, R k) >~Af(R ~)
--+
N ~p
4
1
-~ Aft(R ~:) -~ 1
If yes, clearly p' will be again canonical. As this iteration problem can be seen as a Seifert Fiber Space construction, crucial information about this iteration problem is contained in the connecting h o m o m o r p h i s m 5 of the long exact cohomology sequence
--~ HI(N, Aff(RK, Rk)/Z k) s H2(N,~ k) ---+H2(N, Aff(RK, Rk))---, ... (4.4) according to the exact sequence of N-modules
0 ~ Z k --, Aff(R K, R k) -~ Aff(R K, ]Rk)/Z k -~ 0. If E can be represented by a 2-cocycle f , with (f) E H2(N, Z k) lying in the image of 5 then the existence of an extended pt is true. In [54], Nisse announced a very general proposition stating that 5 is surjective, even in the case of non central extensions and more generally for polycyclic groups N. However, as shown in [49], a (solvable, not nilpotent) counter-example in the case of non central extensions casts doubt on this formulation. In the next section (see example 4.4.13) we will show that Nisse's formulation is incorrect, even in the case of central extensions and for nilpotent groups N. Assume we are given a cohomology class (f) E H2(N, Zk), representing an extension E = Z k x N where the multiplication in E is given by
Vz, zl e Z k, Vn, nl e N : (z,n)(zl,nl) = (z + zl + f(n, nl),nnl).
Chapter 4: Canonical type representations
60
If we can compute explicitly a 1-cochain, say 7 : N ---+ Aff(IR K, R k) killing the class (f) in H2(N, Aff(• g , R k ) ) , then p ' ( z , n ) i s given by p'(z,n) = (z + 7(n),p(n)). As (4.3)shows, this extended representation will again be of canonical type. This problem, in principle, now becomes a computational one. Indeed, we should find 7 : g --+ Aff(R K, ]t{k) : z ~ 7(z) = (D(x), d(x)), such that 57(x, y) = f ( x , y) (Vz, y E N). More explicitly, this means finding a m a t r i x part D(x) and a translational part d(x), satisfying
~(D(y), d(y) ) - ( D(xy), d(xy) ) + ( D(x), d(x) ) = (0, f ( x , y) ) or equivalently D(y)A(x -1) - D ( x y ) + D(x)
=
0
(4.5)
D(y)(a(x-1)) + d ( y ) - d(xy) + d(x)
=
f(x,y).
(4.6)
Since ~k is a trivial N - m o d u l e , this problem can be treated componentwise. Now what looks like a 2 - c o n d i t i o n problem, surprisingly is a 1 - c o n dition problem, as we point out in the following proposition: 4.4.2 Assume p : N --+ Aff(R K) is a representation of canonical type, and 1 ~ Z --+ E ~ N --+ 1 is a central extension, determined by (f) E H2(N, Z). Then, (f) lies in the image of 5: H i ( N , Aff(R K, R ) / Z ) ~ H 2 ( N , Z) iff one can find ( D, d) : N --+ Aff(IR K, IR) satisfying condition (4.6).
Proposition
Proof: We wilt show that condition (4.5) is automatically satisfied, once condition (4.6) is fulfilled. So, assume (4.6) is satisfied. Since p is of canonical type, we know that the translational parts a(x) (for :c E N ) are spanning the whole vector space R K. Thus, it will be enough to show that ( D ( y ) A ( x -1) - D(xy) + D(x))a(z) = O, Vz e Y . (4.7) = p(x-1)p(z) it follows at once that a ( x - l z ) = a(x -1) + A( x-1).a( z ). This and the assumed relation (4.6) allow us to
Now, from p ( x - l z ) write (4.7) as
D ( y ) ( a ( ~ - l z ) - a ( x - 1 ) ) - D(xy)a(z) + D(x)a(z) =
f(z-lx,y)
- f ( x , y ) - f ( z - 1 , x y ) Jr f ( z -1,x)
=
-hf(z-l,x,y) O. []
Section 4.4 Canonical type a/fine representations
61
Note that we used b o t h the fact that f is a 2-cocycle and t h a t Z k is considered as a trivial N - m o d u l e . 4.4.2
Canonical type representations polynomial rings
and
matrices
over
The examples we'll study later on, inspired us to detect an interesting property for canonical type affine representations. Basically, what we saw in all examples, were upper triangular matrices with polynomial entries and degrees going up towards the right upper corner of the matrix. We now prove this is what should happen. For any c o m m u t a t i v e ring R with identity, we write UTK (R) for the (multiplicative) group of upper-triangular ( K • K)-matrices with entries in R and l's on the diagonal. A matrix A in U T K ( R ) is called blocked upper triangular of type ( k l , . . . , kc) (with ~ = 1 ki = K ) if and only if A has identity m a t r i x blocks of size kl,. 9 kc on its diagonal. The subgroup of matrices in U T K ( R ) which are of this type is denoted B U T ~ k,(R). F r o m now on, we will speak of unitriangular and blocked unitriangular matrices. The only eventual nonzero entries (rasp. blocks) in a m a t r i x A of U T K ( R ) (rasp. B U T ~ k , ( R ) ) are the entries (rasp. blocks) ai,j (rasp. Ai,j = (ki • kj)-block) with j 7_ i. For these entries ai,j (rasp. Ai,j) we call (j - i) their distance from the diagonal. F r o m now on we take R to be the ring F[X1,. 9 Xm] of polynomials in m variables over a field F. We use this in the following definition: D e f i n i t i o n 4.4.3 A matrix A in U T K ( R ) (rasp. B U T ~ k , ( R ) ) is said to have the Diagonal Distance Degree property (ODD-property) if and only if each ai,j (j > i) (rasp. each entry in Ai,j = the (kl • kj)-block in A) is a polynomial of total degree ~ (j - i). Such a matrix will be called a DDD-matrix (rasp. a blocked ODD-matrix of type ~ ki). Lemrna 4.4.4
The set of all (blocked) DDD-matrices in UT (R) (resp. BUT forms a subgroup of U T K ( R ) (resp. B U T E k,(R)). Proof: If A and B are DDD-matrices (rasp. blocked ODD-matrices of type ~ ki), t h e n K
BI ,j =
B ,j t=l
62
Chapter 4: Canonical type representations
and so, it is easily seen that the degree of (A 9B)i,j is b o u n d e d above by (j - i). To prove that A -1 has also the required DDD-property, one can proceed by induction on g (resp. c). For K = 1 (resp. c = 1) the claim is evident. Assume now that K > I (resp. e > 1). T h e n A can be viewed as a m a t r i x (A'
A=
0
a)(resp.(A' 1
0
a )) Ikc
where A' is in U T K - 1 (R) (resp. B U T k l +...+kc_l (R)) and has the DDDproperty. It is well known that A_ 1 =
(
A '-1 0
- A '-1 9 a 1
)
(resp.
(A1 Ala) 0
).
Iko
By a s s u m p t i o n A '-1 has the DDD-property. One verifies easily that the entries in A '-1 9 a have also the required property. The following l e m m a is very i m p o r t a n t from the computational point of view! It shows that it is possible to compute A l for a formal p a r a m e t e r l, for all unitriangular matrices A. L e m m a 4.4.5 Assume A is a fixed blocked unitriangular matrix of type ( k l , . . . , kc) with entries in F . If R = F[X], then there exists a DDD-matrix B ( X ) E B U T ~ k , ( R ) such that Vt E 7/, A l = B(~).
Evidently, A t will again be unitriangular. If j > i, then
which is clearly seen to be of degree at most (j - i) in L So, it is sufficient to take B ( X ) E B U T ~ k , ( R ) with
B(x) ,j = Z t=0
((A-
*
Section 4.4 Canonical type a/fine representations
63
R e m a r k 4 . 4 . 6 The upper-bound on the degree of the polynomials in B ( X ) depends also on the nilpotency degree of the matrix ( A - I ) , as is seen directly in the proof of the lamina. E.g. if A is blocked unitriangufar of type ~ ki, and ( A - I ) has m bottom rows of blocks which are zero then ( A - i ) c + l - m = 0 and so the degree of the blocks B ( X ) ~ , j in B ( X ) will be less than or equal to M i n { j - i, c - rn}. Let us r e t u r n to nilpotent groups. Consider a finitely generated torsion free nilpotent group N of rank K = ~ = 1 k~ and a torsion free central series N . . Moreover, we fix a set of generators {al,l~ al,2~ 9 9 9~al,kl ~a2,1~ 9 9 9~ac,kc } of N , which is c o m p a t i b l e with N . . These generators are labeled w i t h two indices a n d we p r o p o s e the following ( s o m e w h a t bizarre) way of ordering these labels: label ( i , j ) is said to be less t h a n or equal to label (re, n) if a n d only i f m < i or ( ( m = i) and (j _< n)). T h e n the c o m m u t a t o r p r e s e n t a t i o n can be w r i t t e n as < ac,l~ ac,2~ 9 9 9~ac,kr ac-l,l~ 9 9 9 ac-l,kr
~9 9 9~al,l~ 9 9 9 al,kl II
[ai,tj,am,t,,] = word in at,tp's (l, tp) > (i, tj) > (m,t,~)
>.
(4.8)
R e g a r d i n g A f f ( R K) as e m b e d d e d in G I ( K + 1, I~) as usual, we are r e a d y for the following theorem: T h e o r e m 4 . 4 . 7 A s s u m e N is a group of type ( k l , . . . , k c ) with a comm u t a t o r presentation as in (4.8). Denote R for ] ~ [ ~ c , l , . . . , x l , k l ] A representation p : N --~ A f f ( R K) ~ G l ( K + 1, R ) of N is of canonical type if and only if for n = ac, 1 . . . al,kl E N , p ( n ) is a D D D - m a t r i x in B U T ( ~ k~)+l (R) combining the following properties: 1. the total degree in the variables ( x ~ , l , . . . , xi,k,) of the entries of p ( n ) is less than or equal to i, more precisely: polynomial entries containing the variables (~i,1, . . . , xi,kl) occur 9 in the linearpart only in the blocks of the r - t h row, for r <_ i - l , in t e r m s of total degree at m o s t i - r, 9 or, in the translational part in the blocks of the r - t h row, for r ~ i in terms of total degree at m o s t i + 1 - r. Moreover, the i-th block of the images of the generators ai,t (1 < t < ki) spans ]~kl ;
64
C h a p t e r 4: Canonical type representations 2. if an entry a t j (j > i) in p ( n ) is not zero, then it is a polynomial without constant term.
Proof: The basic fact in the proof is given by the definition itself of a representation of canonical type. It follows immediately, that, for each i (1 < i < c) the generators a i j (1 _< j _~ k~) are m a p p e d by p to a m a t r i x of the type
0
Ik2
*
*
:
:
:
:
0
0
Ik~
0
0
0
0
:
:
:
:
0
0
0
0
0
0
0
0
Ik~+l
"'"
*
*
:
:
999
0
Bi,5
999
0
0
:
:
...
Ik~
0
...
0
1
Here B i j E ]Rkl and •kl is spanned by {BL1 , Bi,2,...Bi,k~}. To finish the proof it is sufficient to realize that p is a homomorphism and to use the lemmas given above together with remark (4.4.6). Then one proves successively that the following matrices satisfy the conditions (1) and (2) listed in the theorem:
1 p(a, y) ;~i~2
Xl,k i \
2. p(ai~'ai,2 ...ai,k, ) 3. p ( a L ' ? .. 9a m j . . . al,k~ ) (by induction on m).
The sufficiency of the conditions listed is also easily verified. []
Let G be the Mal'cev completion of N. It is clear that the representation obtained in the theorem above, is also a Lie group representation for G. Indeed, for p : N ---* Aff(R K) as before, we get a representation of G in AiT(IRK) by allowing also reals to be substituted for the variables Via exp and log this Lie group G is in one-to-one correspondence with its Lie algebra g. It becomes natural to ask for the meaning of canonical type on the Lie algebra level. Therefore let us define the concept of a canonical representation of a nilpotent Lie algebra into a ~ (•L), the semidirect product I ~ L X ~ [ ( ] R L ) . We first introduce the concept of a central series of a Lie algebra, which is analogous to a torsion free central series of a nilpotent group.
Section 4.4 Canonical type atone representations D e f i n i t i o n 4 . 4 . 8 Let g be a Lie algebra. m e a n a series
65
B y a central series of 9, we
g, : O = go C_ gl C g2 C_ . . . of subalgebras of g which satisfies [g, gi+l] C-C-gi. The central series fl, is said to be of length c iff gc_ 1 ~ g and gc = gc+l =
~176176 For a given n i l p o t e n t Lie algebra 9, there are two well k n o w n central series of finite length, n a m e l y the lower central series a n d the upper central series. Given a central series 9, of length c we denote the dimension of 9 i / 9 i - 1 by li (for 1 < i < c). We use L = li + 12 + . ' ' + Ic to refer to the d i m e n s i o n of 9. So, we can choose a basis
Al,1, A1,2, . . ., Al,ll, A2,1,. . ., Ac&
(4.9)
of g in such a way t h a t fl~: is spanned by all vectors AI,I~
A1,2~
9 9
Al,ll
, A2,1~
9 9 .~ A i , l l
We will refer to such a basis as a basis which is c o m p a t i b l e given central series fl,.
w i t h the
D e f i n i t i o n 4 . 4 . 9 A n embedding p : fl ~ a ~ ( N L ) is of canonical type with respect to a central series g,, if and only if for an appropriate choice of a compatible basis (4.9) Oli
*
*
*
9
,
0
0 G
*
*
999
*
*
:
:
:
:
:
:
0
0
Oli
0
99
0
Ei,j
0
0
0
Oli+l
...
0
0
:
:
:
:
:
:
0
0
0
0
...
Oz~
0
0
0
0
0
...
0
0
P(Atj) =
where Ei,j = ( 0 , 0 , . . . , 0 , spot).
*
1 , 0 , . . . , 0 ) t" ((li • 1 ) - m a t r i x with 1 on the j - t h
Using a g a i n al,1, al,2, 9 9 9 ac,kc to refer to the generators of a torsion free n i l p o t e n t group N (4.8) (compatible w i t h a given torsion free central
Chapter 4: Canonical type representations
66
series), we take Ai,j = log(aid). As was explained in the first chapter, we see that the sequences of subspaces go C_ fll _C ... C_ 9~, where 9i is the subspace spanned by A1,1 ... Ai&, forms a central series fl. of 9. In particular 9 is spanned by all Ai,j's. Remark that in this situation li = kl (l
T h e o r e m 4.4.10 The map j5 = l o g p e x p is a linear representation of 9 into a~ (RK), which is of canonical type and fi(xl,lAl,l+.. "+xc,k~ A~,kr =
B(m). Here, B(x) is obtained from A(x) by replacing all the diagonal identity-blocks by zero-blocks as well as by replacing all degree > 2 parts of A(x) by zero. Proofi ~5 ( = dp, the differential of p) is a Lie algebra morphism, and so p(zi,jAi,j) = xl,j~(Ai,j). This implies that the entries of this matrix, will be of degree 1 in the variable xi,j. On the other hand we see that for all xi,j C Z, ~(xi,jAi,j) = log(p(a~,~'d)). This means that log(p(a~,~)), as matrix, has degree-1 entries in the variable xi,j. We determine these entries by observing that log(p(a~,~J)) = (p(a~,~j) - I ) +
~
(-1)k+1
"
k=2 Containing only terms of degree_> 2 So the degree-1 terms come from p(ai~) d) = A(x)ix:(0 .....0,x,,~,0.....0). In turn, this implies that p(ml,lAl,1 + ' - " +
mc,kcAc,kc) = xl,a l o g ( p ( a l , 1 ) ) + . . . + ze,kc log(p(ac,kc)) =
Now, by looking at the matrices of p(Ai,j), one concludes that t~ is of canonical type. Conversely, if one considers a canonical embedding t~ of fl into a[~ (I~K), then one can easily see that p = exptSlog : G ~ Aff(• K) induces a representation of N , which is of canonical type.
Section 4.4 Canonical type attlne representations
67
Now that we have a better picture of canonical type affme representations, we want to come back to the iteration problem as stated in the previous section. Also we can point out here that there has already been some interest in the literature for similar-looking iterative work concerning complete normal Koszul-Vinberg (KV) structures on nilpotent Lie algebras ([9]). Remark that although a canonical Lie algebra representation determines a complete KV-structure, this K V - s t r u c t u r e will not necessarily be normal. The following example, communicated to us on the Lie algebra level by Dan Segal and Fritz Grunewald (to w h o m we express our gratitude), shows however that a great amount of care is necessary with respect to the "universal" nature of both iterative approaches. For clarity, we prefer to present the example twice: once on the Lie algebra level and once on the group level. As a consequence it will follow that 1. the final theorem, called the "Lifting theorem", in [9], is incorrect as stated there. 2. the iteration problem as stated previously does generally not have a positive answer; a f o r t i o r i the announcement of a positive answer to a much more general version in [54] is incorrect. 3. one should pay attention to have a well understanding of the 3-step nilpotent case in [44].
E x a m p l e 4.4.11 (Lie a l g e b r a l e v e l ) To permit the reader an easier comparison with the situation in [9], we use the notations and terminology adopted there. Consider the 4-dimensional 3-step nilpotent Lie algebra g = < A1, A2, A3, A4 > where the brackets are defined by [A1,A2] = A3, [A1,A3] = A4, [A2, A3] = 0 = [g, A4]. It is easily seen that ~ -- g / < A4 > - - < A1, A2, A3 > = is the Heisenberg algebra. In ~, let us consider the flag of ideals F ( ~ ) , given by F(~) : ~ = g3 D ~2 =< AI,A3 >D gl =< -43 >D 0.
Chapter 4: Canonical type representations
68
Remark that this flag is finer then the lower central series of ~. It is not hard to verify that the following linear representation ~ of 9 is a complete~ normal Koszul-Vinberg structure (KV-structure);/5 is defined by
fi(A2)A1 = 43, ~(A1)A2 = 2fi~3, fS(z[i)Aj = 0 in all other cases. We now proceed to show that this KV-structure does not lift to a normal KV-structure p on 9. Taking into account the lower central series of 9, one verifies that a normal KV-structure p on 9 must satisfy
p(Ai)A4 = O, p(A3)A3 = 0, p(g)g = < A3, A4 > 9 Furthermore, a lifting of f5 must have at least the following properties:
p(A2)A1 = A3 + aA4, p(A1)A1 = 7A4, p(A1)A3 =/3A4, p(A3)A1 = (/3 - 1)A4. However, from the definition of KV-structure in [9] it follows that we should also have
p(A:)p(A1)A1 - p(A1)p(A2)A1 = p([A2, A1])A1 =~ - ~ A 4 = (1 - f~)A4 and this is clearly a contradiction. Remark 4.4.12 This situation is "exceptional", because every complete, normal KV-structure
(k # O)
~(A2)A1 = kfi.3, fi(-41)Zi2 = (k + 1).43, ~(Ai)Aj = 0 in all other cases, on 9, with k ~ 1 extends to a normal KV-structure on 9. Example
4.4.13 (group level)
We now reconsider this example on the group level. nilpotent group:
:< al, a2, a3, a4 II [a2, eli = a r
Take the 3-step
[a3, eli = a4
>.
[a3, a2] : [a4, al] : [a4, a2] = [a4, a3]---- 1
(4.10) Although this group can be given a canonical type representation, as will be described in a following section, we show that it can serve as a critical example with respect to the iteration problem.
Section 4.4 Canonical type aff/ne representations
69
N3 can be seen as a central extension of N2 :<
al,a2, a3 H [a2, at] = a 3-1 ' [a3,az] = [a3, a2] = 1 > .
Consider the following canonical type affme representation (~) of N2:
1 z2 2zl
/
_
~
~
~
P(al a2 a3 ) =
[ A(~) ~ 0
a(x)l
0
)
0 0
i
2ZlZ2 + Z3
01
0 0
zl r 1
0
I "
We show that/5 can not be lifted to a canonical type affme representation p : N3 -~ Aff(R n) (remark that N3 admits oIfly one torsion free central series of length 3). For suppose that this p exists, then it must be of the form
/IA1A2A3A4/0 102 0 p(al) =
0 0 0
0 0 0
1 0 0
0 1 0
(1 11 213 04/o0
1 0 1
p(a2) =
/1 Cl c c3 c4/
p(a3)=
0 0 0 0
1 0 0 0
0 1 0 0
0 0 1 0
1 0 0 1
0 0 0
0 0 0
1 0 0
0 1 0
(10001) 0 0 0 0
p(a4)=
1 0 0 0
0 1 0 0
0 0 1 0
0 1 1
0 0 0 1
for some real numbers A1, A2,. 9 C4. If p is a homomorphism of groups, the relations appearing in (4.10) must be satisfied if we replace al by p(al), a2 by p(a2), a3 by p(a3) and a4 by p(a4). This leads to a system of linear equations in the parameters A1, A 2 , . . . , C4: C 1 =0
- A i + C2 = 0 2Bi + C3 = 0 Ai - A3 A- B1 q- B2 - C1 + -1-AI+C2--0 --B1 + C3 = 0,
C4
=
0
which is easily seen to be inconsistent. Therefore, p cannot exist. These examples of course, do not contradict the conjecture of Milnor. But, recently Y. Benoist ([4]) and D. Burde &: F. Grunewald ([12]) proved that there exist nilmanifolds (of dimension 11) which does not admit a complete affmely fiat structure. We will come back to these examples later on.
70 4.4.3
Chapter 4: Canonical type representations Virtually
2-step nilpotent
groups
In this section we will prove that for any connected and simply connected 2-step nilpotent Lie group G, there exists a faithful aft-me representation of G;~Aut (G) letting G act simply transitively on some space R n. It will follow that any AC-group, with a 2-step nilpotent Fitting subgroup allows a canonical type affine representation. We use 9 to denote the Lie algebra of G. In the 2-step nilpotent case, the group commutators and Lie brackets are very nicely related. L e m m a 4.4.14 Va, b E G :
log[a,b] = [loga, logb].
Proof: Let us denote A = log a and B = log b. Using the CampbellBaker-Hausdorff formula we find that: [a, b] = a - l b - l a b
=
exp(-A)exp(-B)exp(A)exp(B)
=
exp(-A-
=
exp([A,B] +
=
E-A-
=
exp([A, B]).
B + ~l [ - A , - B ] ) exp( A + B + ~[A, B])
+ EA, Bt, + B + -IA, Bll)
The 2-step nilpotent group G fits in a short exact sequence 1 --* [G, G]--* G ~ G/[G, G]-* 1. Both [G, G] and G/[G, G] are real, finite dimensional vectorspaces (group operation : addition of vectors). We choose a basis {bl, b 2 , . . . , bin} of [G,G] and a basis {~l, a2,...,a,~} of G/[G,G]. We also fix a lift a l e G for ai (Vi, 1 _< i _< n). Now, any element g of G can be written, i n a unique way, in the form g = al as . . . . n "1
...bin,
xI E R , Yj E R .
(4.11)
We define an alternating, bilinear map L 9 ~'~ • ]~n ~ R,~ as follows. Let u, v E R '~ = G/[G, G] and consider any lifts ~i, ~ of u and v. Let L(u, v) = [fi,~] E [G, G] = R TM. We remark that this definition does not depend upon the chosen lifts. Suppose that L is determined by the parameters li,pd, 1 <_ i , j <_ n, 1 < p <_ m,
Section 4.4 Canonical type afline representations
71
where [ai, aj] = hli'l'Jhli'2'J .limj -1 -2 ...O,g ' . So, ll,p,j = -lj,p,i and li,p,i = O. L e m m a 4.4.15 form
With the notations above, g has a presentation of the m
9:< A1,A2,...,A,~,B1,...,BmlI[Ai,
Aj] = E l i , p , j B v
(1 _< j < i <_ n) >
p=l
[Bi, A j ] = 0 ( l < i < m , l_<j_
2. For all i E {1, 2 , . . . , n } , u~ denotes the n - c o l u m n with a 1 on the i - t h place and a 0 elsewhere. 3. For all i E {1, 2 , . . . , m}, vl denotes the m - c o l u m n with a 1 on the i - t h place and a 0 elsewhere. W i t h these symbols in mind, it is well known (and easy to check) that 9 can be faithfully represented as a Lie algebra of (m + n + 1)-matrices. This is obtained as follows: Ai ~
(0m 0) (00v) 0 0
0,~ 0
ui 01
B~ ~
0 0
0,~ 0 0 01
.
Here, 0j stands for a j x j zero matrix. From this representation, one may obtain a faithful representation of G into A f f ( ~ "~+n) C Gl(m + n + 1, R), just by exponentiating the matrices above. So, this representation is determined completely by:
bi = exp Bi ~-+ exp
I
Ota 0 0
0 0,~ 0
vi I 0 = 01
72
Chapter 4: Canonical type representations
Ira+n+1+
(0 0v) ( 0v) 0 0
On 0 0 01
=
0 0
In 0
0 1-1
(4.12)
,
o
ai = exp Ai ~-+ exp
Im+n+l +
On 0
(Oo o)(o 0
ui 01
o 0
+
gLi In 0
0 0 0 ~ I1
ui | = O1 ]
o
On 0
0 01
.
=
(4.13)
Here, Ij denotes the j • j identity matrix. This affme representation for G can be interpreted in a very nice way. We introduce the following coordinate system on G: D e f i n i t i o n 4 . 4 . 1 6 The coordinate system of G (associated to the basis B I , . . . , B m , A 1 , . . . , A n ) is the map m : G ~ R m+n which maps each element g E G to the coordinate of log(g) (with respect to the basis BI , . . . , Bin, A1, 9 9 An). It follows that the aft-me representation ~ is exactly the coordinate expression for the multiplication in G. I.e. L e m m a 4 . 4 . 1 7 Let G be a 2-step nilpotent Lie group, with a coordinate system m as above, then we have that
Vg, h e G : m(gh) = ~(g)m(h). Proof." By the C a m p b e l l - B a k e r - H a u s d o r f f formula we find that log(gh) -- logg + logh + l [ l o g g , l o g h ] . This implies that for a fixed g, the coordinate of gh is an affine function of re(h). In other words, there exists a map r
G -~ Aff(]Rm+n) : g ~ r
such that Vg, h E G : m ( g h ) = r
It is easy to see that r is a morphism of groups for which ~(ai) = r (1 < i < n) and ~(bi) = r (1 < i < m). Therefore, ~ and r coincide
Section 4.4 Canonical type atone representations
73
on the whole of G. Let us return to the discrete case now and assume that N is any torsion free finitely generated 2-step nilpotent group. N fits in a short exact sequence
1 -+ N[V/~,N ] : Z TM -+ N --+ N~ ~/[N,N] for some m, n E N. Choose a set of generators bl,. 9 brn, a l , . 9 9an of N in such a way that b l , . . . , bm generate ~/[N, N]. This means that the set b l , . . . , bin, a l , . . . a n is a compatible set of generators for the torsion free central series N,:
I=NoC_NI=
~,N]CN2=N(=Na).
As explained in the first chapter, the elements B1 = log(b1),. 9 Bm = log(bin), A1 = log(a1 ), 9 9 A,~ = log(an) form a basis for the Lie algebra fl of the Mal'cev completion G of N . It follows that the al and bi play the same role as in the continuous case. In particular, it follows that the restriction of the affme representation : G -+ Aff(R re+n) to N induces a faithful affme representation of N. Moreover, by looking at the matrices corresponding to the generators of N , we see that this representation is of canonical type with respect to the series N . . We summarize these observations in a theorem. Theorem
4 . 4 . 1 8 Let N be a 2-step nilpotent group with a presentation m
N:<
al,a2,...,an, bl,...,bmll
[ai, aj] = 1-[b~'P'J ( 1 _ j < i ~ n)
>.
p=l
Ibm,aj] = 0 (1 < i < m, 1 < j < n) [bi, b j ] = 0 ( l < _ j
Then there exists a canonical type affine representation ~ : N -+ Aff(R m+'~) with respect to the torsion free central series g,:
1 = No C_ N~ = grp { b l , b 2 , . . . , b m } C_ N2 = N ( = N3)
which is completely determined by the matrices given in (4.12) and (4.13). Proof: Everything is already shown above except for one thing: From the presentation of N above, we are not allowed to conclude that the group
74
Chapter 4: Canonical type representations
generated by bl,b2,...,b,~ is really the subgroup N [ x / ~ N ] of N. Indeed, we can only assume that ~ C_ grp{bl, b2,..., bin}. However, this does not play an essential role in the theory developped above and everything works well in this more general case too. (One just has to use G ' = the Mal'cev completion of grp { b l , b 2 , . . . , b m } in stead of [G, G]).
D e f i n i t i o n 4.4.19 We call a canonical type affine representation of a 2 step nilpotent group N as obtained in the theorem above a s t a b l e a ffine representation. A 2-step nilpotent group N can have more than one stable affme representation, since such a representation is determined by a particular choice of generators of N. Nevertheless, all these stable representations have one thing in common: they extend to a canonical type affme representation of any AC-group having this nilpotent group as Fitting subgroup. To prove this, we look at the continuous case again. P r o p o s i t i o n 4.4.20 Let G be a connected and simply connected m + n dimensional 2-step nilpotent Lie group. Then there exists a faithful affine representation ~5 : G>~Aut (G) -~ Aff(I~m+'~) which restricts to !a (given
by (4.12) and (4.1S)) on C. Proof: By expressing the natural action of G ~ A u t (G) on G in terms of the coordinate system m introduced before, we can define a homomorphism r : G ~ i u t (G) --~ ~(Rm+n), where 7-/(JRm+'~) denotes the group of homeomorphisms of ]Rm+n. To be precise, this homomorphism is given by Vg, h E G, Va E Aut (G): r
a ) m ( h ) = m((g'~)h) = m(ga(h)).
We already know that r 1) = ~(g) (see lemma 4.4.17). This implies that we will finish the proof, provided we can show that r a) is an aft-me map. But r a ) m ( h ) = m ( a ( h ) ) = the coordinate of log(a(h)) with respect to the Lie algebra basis introduced above. However, by the commutative diagram 1.2, we know that log(a(h)) = da(log h), where da, the differential of a is a linear map. It follows that the coordinate expression for da(log h) is linear in re(h). This implies that the homeomorphism r a) of ]~m+,~ is linear and so a fortiori aiTme. We return to the discrete case again and reformulate the previous result for AC-groups.
Section 4.4 Canonical type affine representations
75
T h e o r e m 4.4.21 Let E be any AC-group with a 2-step nilpotent Fitting subgroup N with a presentation m
N :< a l , a 2 , . . . , a n , b l , . . . , b m II [ai,aj] = 1-I b~'p'j (1 < j < i < n)
> .
p=l
[bi, aj] = 0 ( 1 < i <
m, 1 < j < n)
lb. bj] = o (1 < j < i < m) If grp {bl, b 2 , . . . , bin} is a normal subgroup of E, then there exists a canonical type affine representation ~ : E ~ Aff(R re+n) with respect to the torsion free filtration E , : 1 = Eo C_ E1 = g r p { b l , b 2 , . . . , b m } C_ E2 = N C_ E3 = E
and where the induced representation of the Fitting subgroup is completely determined by the matrices given in (4.12) and (4.13). Proof." The existence of a faithful affine representation is immediate now, since we know that any AC-group E embeds into G>~Aut (G) in such a way that N is m a p p e d identically onto itself. The reader in invited to see that this representation is of canonical type. (One can also use lemma 4.5.10)
Remark
4.4.22
For any AC-group E, we can always find a subgroup grp {bl, b2,..., bin} which is normal in E. It suffices to choose for it a characteristic subgroup of N e.g. Z ( N ) or N [ X / ~ , N ]. So, any AC-group with a 2-step nilpotent Fitting subgroup admits a canonical type affme representation which restricts to a stable representation of the Fitting subgroup. D e f i n i t i o n 4.4.23 Let E be an AC-group with a 2-step nilpotent Fitting subgroup N . An a]:fine representation of E is said to be a stable representation iff the restricted representation of N is stable. 4.4.4
Virtually
3-step nilpotent
groups
The reason why we are so interested in stable affine representations is that they can be used to build up affme representations for AC-groups with a 3-step nilpotent Fitting subgroup. We make this clear in the following theorem, which is a more concrete version of theorem 2.5 of [44]:
Chapter 4: Canonical type representations
76
4.4.24 Let Q be any AC-group of a 2-step nilpotent Lie group G, and let A : Q --+ Aff(R '~) be a stable representation of Q. Then for any Theorem
extension 1--+Z'~__~ELQ__+I inducing a morphism ~ : Q -~ h u t ~m with ~(Fitt(Q)) = 1, there exists an embedding ~ : E --+ h f f ( R re+n) with Ve E E : 7~(e) = ( ~(p(e)) 0
m o r e o v e r Vz
gin:
s
=
* ) )~(p(e)) ' Ik
0
z)
0
I~
0
0
0
11
.
Proof: Following sections 4.3 and 4.4.1, the statement is equivalent to proving that the connecting homomorphism
5: Hi(Q, Aff(R '~, Item)/2~TM) ~ H2(Q, 7,TM) with respect to the short exact sequence of Q-modules + zm-
A r(R , R TM)-+ Afr(Rn, R m ) / Z m + 1
is surjective. We recall that the Q-module structure on the abelian group of affme mappings of n-space into m-space is given by Vq E Q, Vh E Aft(JRn, R ' ~ ) : qh = ~(q)hA(q) -1. The group Z TM is to be considered as a submodule of Aff(IR'~, II~TM) by identifying z E Z TM with the constant map R '~ ~ R TM : x ~ z. We prove the theorem in two steps. First we show it for torsion free nilpotent groups of class 2, thereafter we look at the general case. Case 1: Q = N is 2-step nilpotent So, let N be a uniform lattice of a 2-step uilpotent connected and simply connected Lie group G. We have to show that any central extension 1 ---+ Z m ---+E ~ N ---+ 1 can be given an affme representation into Aff(R m+'~) build up from the stable representation of N. Such an extension E can be seen as the uniform lattice of a connected, simply connected nilpotent Lie group G', which itself is a central extension of G by ]RTM. The stable embedding of N , is the restriction of an embedding of G. By the work of Scheuneman [58] the embedding of G (to be exact of its Lie algebra fl) is of such a kind that it can be lifted to an embedding of G' into Aff(IRm+'~).
Section 4.4 Canonical type af/ine representations
77
The restriction of this embedding to E gives the desired result in the nilpotent case. Case 2: General Q To see what happens in the virtual nilpotent case, we look at this problem from the cohomological point of view (in fact, this is the first place where we really need the cohomological approach). Consider an AC-group Q with a Fitting subgroup F i t t ( Q ) -- N of class 2. Note the following commutative diagram with exact columns:
HI(Q,Aff(~',Rm)/z H2(Q, Urn)
TM)
Z~
TM)
res
H2(N, Um)
res
H2(N, Aff(Rn, Rm))
il ~
H2(Q ' Aff(Rn, Rm))
HI(N, A f f ( R ' , R m ) / Z
i2
We have to show that ~I is surjective or that il = 0. By the first case we know that 62 is surjective or that i2 = 0. By the fact that A f t ( R ' , R m) is a vectorspace, we know that the restriction map at the b o t t o m of the diagram is injective. Now, it follows easily that also il = 0, which was to be shown. []
R e m a r k 4.4.25 The previous t h e o r e m is afortiori also valid in case Q is crystallographic, i.e. Q has an abelian Fitting subgroup Z n. In this case stable means that any element z 6 Z " of the Fitting subgroup acts on R" exactly as translation by z C o r o l l a r y 4.4.26 Let E be an A C-group, with a 3-step nilpotent Fitting
subgroup, then E admits a canonical type a]fine representation. Proof: Let N denote the Fitting subgroup of E. By, lemma 2.4.2, there is a short exact sequence
where Z = ~/[N, [N, N]] and Q = E / Z is an AC-group with a 2-step nilpotent Fitting subgroup. By the previous section we know that E / Z
Chapter 4: Canonical type representations
78
admits a stable affine representation, which can be lifted, due to the theorem above, to an aft-me representation of E. Moreover, a close look to the matrix form of this representation shows that we can assume that this representation is of canonical type with respect to the following torsion free filtration E.:
E~ Eo = 1 C_ E1 = Z C_ E2 = ~/[N,N] C_ E3 = N C E4 = E.
4.4.5
What
about
the general
case?
Our study of canonical type affine representations is related to the so called affinely flat manifolds. An affmely flat manifold of dimension n is a (smooth) manifold M provided with an atlas .4 = {#~ : Us --* ]Rn [[ a E I} of coordinate homeomorphisms, such that the transition functions
t~,~ : #~
o#~ -1 : #~(U~ n U ~ ) ~
#z(U~, n Uz), for a,/J EI, U~nU~ r r
extend (uniquely) to an affme transformation: ta,Z : IR'~ ---+ R '~ : x ~-+ A~,~ 9 + a~,~, where A~,Z is an invertible (n • n ) - m a t r i x and a~,~ E ]R'~. If one imposes even more requirements on the affme transformations s one finds special classes of affinely flat manifolds. For example, if all A~,~ are orthogonal matrices, so if all t~,~ are rigid motions of Euclidian n-space, one obtains the flat Riemannian manifolds M . Another interesting subclass of these aff-mely fiat manifolds are the Lorentz-fiat manifolds, i.e. those affmely flat manifolds for which each A~,~ belongs to the Lorentz group O(n - 1, 1). A geodesic in an affmely flat manifold M is a curve ~ : [a, b] -+ M which is locally a straight line, traversed at constant speed. The manifold M is said to be complete if any geodesic can be defined on R. This means that every "partial" geodesic ~ : [a, b] ---+ M can be extended to a "full" geodesic ~ : IR -+ M. It is known that any n-dimensional connected complete affinely fiat manifold can be constructed as a quotient space M -- E \ R n where E acts freely and properly discontinuous, via affine transibrmations on IR'~ ([3]). We are especially interested in the case where the manifold M is compact. Here, we should mention the famous Auslander conjecture stating that if a complete affmely flat manifold M -- E \ R '~ is compact, then E necessarily has to be a p o l y c y c l i c - b y finite group. In fact, Auslander formulated this as a theorem ([2]), but unfortunately his proof contained a gap. So, the problem is still open and
Section 4.4 Canonical type af//ne representations
79
is therefore nowadays referred to as the Auslander conjecture. Nevertheless, it follows that the class of polycyclic-by-finite groups is an interesting class of groups to study. At this point we also want to mention the paper of F. Grunewald and G. Margulis ([36]) in which a general classification scheme for fundamental groups of compact, complete Lorentz-flat manifolds is given. In 1977 ([52]), John Mi]_nor showed that every torsion free polycyclicby-finite group F can be realized as the fundamental group of a (not necessarily compact) complete affinely flat manifold M. Moreover, it is known that any such F appears as the fundamental group of a compact manifold (e.g. consider the quotient space p ( F ) \ R h(r), where h(r) denotes the Hirsch number of F and p : F --+ 7-/(R h(r)) is any canonical type representation, see theorem 4.2.3). Therefore, J. Milnor wondered whether these two results could be joined and formulated the following question (often referred to as Milnor's conjecture): Given a torsion free polycyclic-by-finite group F, is it possible to construct a manifold M which is both compact and complete affmely flat with fundamental group equal to F? For a long time only a few special cases (e.g. r is a virtually 3-step nilpotent group), giving positive answer to this question in these special cases, were known, but nevertheless m a n y people started to believe that the answer to Milnor's question had to be yes. However, in 1992 V. Benoist ([4]) constructed an example of a 10-step nilpotent group of rank 11, which contradicted the question of Milnor. This example was generalized to a family of examples by D. Burde &: F. Grunewald ([12]). In fact Benoist and Burde &: Grunewald do not work with groups but r a t h e r on the Lie algebra level. It is possible to understand this change of language by the work of D. Fried, W.M. Goldman and M. Hirsch ([31]) and [30]. Indeed, in case we restrict ourselves to nilpotent groups we have the following results, where we use N to denote a finitely generated, torsion free nilpotent group of rank K, G is meant to be the Mal'cev completion of N and g is the Lie algebra of G. Suppose N acts properly discontinuously on ~ g via affine motions. Refer to this action by ~ : N - + Aff(Rg). It follows from the proof of t h e o r e m 7.1 of [30], that 9~ extends uniquely to a simply transitive and affine action 95 : G ---+ Aff(R K) of G on ]Rg . Conversely, given a simply transitive action 95 of G on R K, one obtains a properly discontinuous action of N on R g by considering the restriction ~ of 95 to N. Denote d95 : 9 ~ a~[ (R K) is the differential of 95. As mentioned before, any
Chapter 4: Canonical type representations
80
element of a~(R K) consists of a linear part and a translational part, "determined by the two maps:
lin:
ff(R n)
A0 a)~_+ A 0
(K,R):
and
tr:aff(R K)
R K:
(A
0
a) 0
~--+ a.
It follows from [31], that 1. the linear parts of dq5 consist of nilpotent matrices and 2. the translational parts of dqb(g) form all of n-space. More precise, t r o d~3 : g ---+ R n is an isomorphism of vector spaces. Conversely, given a Lie algebra homomorphism p : g ---+ a ~ (I~K), such that tr o p : g ~ R K is an isomorphism of vectorspaces and such that the linear parts of p consist of nilpotent matrices, there exists a unique simply transitive action 9~ : G ~ AiT(RK), with d~ = p. We call such a p a c o m p l e t e afflne s t r u c t u r e on g. We remark that one also speaks of an affine structure of a polycyclic-by-finite group F, by which one means a representation p : F ---+ Aff(R g ) , which lets p act properly discontinuously on R K and with compact quotient. We conclude that there are the following one to one correspondences (See also [43]) Affine structures of N . Simply transitive affine actions of G. I-I Complete afline structures on g. This shows that it is indeed possible that, at least in case of nilpotent groups, one approaches Milnor's problem at the Lie algebra level as was done by Y. Benoist (and later by D. Burde and F. Grunewald). The Lie algebra of Y. Benoist which does not admit a complete affme structure is given in the last section of this chapter. Finally, we want to mention that, we were able to construct a complete afl~ne structure on a broad class of 4-step nilpotent Lie algebras ([19]).
Section 4.5 Canonical type polynomial representations
4.5
81
Canonical type polynomial representations
Now, that we know that not all finitely generated nilpotent (and so a fortiori not all polycyclic-by-finite) groups admit an affine structure the following notions make sense: Let us call a quotient manifold E \ R K, where E acts freely, properly discontinuously, via polynomial diffeomorphisms on •K a polynomial manifold. Moreover, if all diffeomorphisms involved are of degree ~ s, we say that the (polynomial) manifold is of degree ~ s.
D e f i n i t i o n 4.5.1 Let E be a poIycyclic-by-finite group with Hirsch rank h(E). The affine d e f e c t o f E, denoted by d(E) is defined as
Min { s e Nll E acts properly discontinuous and via poly nomial } diffeomorphisms of degree ~ s + 1 on ~h(E) if there exists such an action for E; d(E) = c~ if E does not allow any properly discontinuous action via polynomial diffeomorphisms of bounded degree on ~h(E) As is easily seen, a torsion free, polycyclic-by-fmite group E has affine defect zero iff E occurs as the fundamental group of a compact, complete affinely flat manifold. Therefore, this affine defect number somehow measures the obstruction for E to be realized as the fundamental group of a compact, complete affmely fiat manifold. Analogously to the notion of an affine structure, we introduce the concept of a polynomial structure of a polycyclic-by-finite group F, which is nothing else than a representation p : r ~ P(RK), letting F act properly discontinuous and with compact quotient. It is our intention to show in this book that the affine defect of a virtually e-step nilpotent group is always finite. In fact there exists an u p p e r b o u n d depending (in a linear way) only on the nilpotency class c. More generally and very recently P. Igodt and myself obtained some nice existence results of polynomial structures for polycyclic-by-finite groups. It would lead us to far away from our subject to discuss these results here in detail but let me at least state the two main facts: 1. As a first result ([22]) we obtained that any polycyclic group F, for which F / F i t t (F) is free abelian, has an affine defect satisfying: d(r) < h(r). This is an importatnt result as we know that any polycyclic-by-finite group F ~ has a subgroup F of finite index satisfying the condition above. (The way to prove this first result was analogous to "the first approach" we follow below.)
82
Chapter 4: Canonical type representations . As a second result ([23]), we very recently obtained t h a t any polycyclic-by-finite group F admits a polynomial structure of finite degree. However the way of proving this was not of that kind t h a t we could deduce any information on how small/large this finite degree really is. Remark t h a t this result implies that there is a sentence to much now in de definition of affine defect. Indeed, it cannot happen t h a t a polycyclic-by-fmite group has an infinite affme defect. (The way to prove this second result was analogous to "the second approach" we follow below.)
These two results together lead to the following conjecture: C o n j e c t u r e There exists a linear function ~ : N --* N, such t h a t for any polycyclic-by-finite group r we have t h a t the affine defect d ( r ) satisfies d ( r ) < , ( h ( r ) ) , where h ( r ) as usual denotes the Hirsch length of F. Stated otherwise, it is reasonable to believe t h a t any polycyclic-by-finite group F has a polynomial structure of degree _< . ( h ( r ) ) , for some linear function ~,. It even seems to be possible to assume t h a t we can take y(n) = n - 1 (at least if we ignore the case n = 1). 4.5.1
The first approach
As always, N denotes a torsion free, finitely generated nilpotent group of rank K and class c. Although, everything we will say in the sequel works for general torsion free central series of a group N, we will restrict ourselves to central series which are as short as possible. So, the torsion free fi]trations we are looking at are central series of length c: N,:
No = I C N 1 C N2 c . . . c
Nc_I c N c = N.
From now on, if we speak of a polynomial representation p : N --~ P ( R K) which is of canonical type, we mean of canonical type with respect to some torsion free central series of length c. Referring to theorem 4.2.3 and to the fact that we are dealing with a central series, we can state the following corollary, which provides us an alternative definition of a canonical type polynomial representation.
Corollary 4.5.2 A s s u m e that N is as above and that N , is some torsion .free central series. A representation p : N --~ p ( • g ) is of canonical type with respect to IV,
Section 4.5 Canonical type polynomial representations
Vn ~ N :
p(n):
I~ K --+ ] ~ K : x
~
83
( P ")' l ( )~E , P 2 ( ~' ~E)," . . , p ~ ( x ) ) is such that
pT(~) = ~ + Q ~ ( ~ + I , ~ + : , . . . , ~ o ) , for some polynomial mapping Q'~ : ~Ki+I ~ ~kl. Moreover, Ni acts trivially on R g~+~ and as translations on the i-th block R k~, in such a way that these translations form the whole subgroup Zkl. E x a m p l e 4.5.3 Let N :< a1,1, al,2, a2,1, a2,2, a3,1, a3,2 II
a3,2, a3,1 ] = al,la2,1a2, -1 23
>
[a2,1, a3,1] = al, 2
[a~,~, a3,~] ---- a1,1 [a2,~, a~,2] = a1,1 [a2,~, a2,1] = a 21,1 all other commutators trivial. One can check that there is a canonical type representation p : N p(]~6) of this N given in terms of the images of the generators (the central series used, is suggested by the labeling of the generators):
p(al,1)(~) = (1 + x1,1, z2,1, z2,2, z2,3, z3,1, z3,2) p(a2,1)(x) = (~rl,1 - 253,1, 1 + x2,1, x2,2, x2,3, x3,1, x3,2)
p(a2,2)(x)
= (~1,1 + 2x2,1 § x3,2, x2,1, 1 § x2,2, x2,3, z3,1, x3,2)
p(a2,3)(x) = (z1,1 + z~,2, Z~,l, z m , 1 + z2,3, z3,1, z3,2) p(a3,1)(x) = (Xl,1, x2,1, x2,2, x2,3, 1 § x3,1, x3,2) 2 X2,1 ~- ~3,1, ~2,2, ~2,3 p(a3,2)(x) ---- (Xl,1 ~- 2~2,2Z3,1 -- X3,1'
~- 2~r3,1,
z3,1, 1 + z3,2). The following theorem, intrinsically due to Mal'cev ([51]), will imply the existence of canonical type polynomial representations for all A C groups. T h e o r e m 4.5.4 Let G be any connected, simply connected c-step nilpotent Lie group G of dimension K . Then G>~Aut (G) embeds into P ( R K) in such a way that the image of G:~Aut (G) consists of polynomials of degree <_ s for s = m a x ( l , c - 1) and so that the action of G on ~ g is simply transitive. Proof: Write 9 for the Lie algebra of G. We know that there is a one to one
Chapter 4: Canonical type representations
84
correspondence between 9 and G via the exponential mapping. We follow Mal'cev's ideas to describe the group structure of G. Consider any central series of length c of 9: 9. : 0 -- 90 c_ 9~ c_ 92 c_ . . . _c 9~ and choose a basis AI,1, Ai,2, 9 9 Al,ki, A2,1,. -., A2,k2, A3,1 99 9 Ac,kr of 9, which is compatible with 9.. Now, introduce a system of coordinates on G by defining for all g of G its coordinate re(g) to be the coordinate of log(g) with respect to the basis A1,1,..., A~,k~. Consider 9 (resp. y) E G, with logx = ~ x i , j A i , j (resp. l o g y = yisAij). The Campbell-Baker-Hausdorff formula implies that
xy
=
e x P ( E x i , j A i , j ) e x p ( E yi,jAi,j)
=
exp
(z
+ y
,j)A,,j +
'
+ ..-
)
.
A close examination of the previous expression now shows that the (i, j ) th coordinate of the product zy satisfies: rn(xy)i5 = xi,j + yi,j + Pi,j(xi+l,1,..., xc,k~, Yi+1,1,. 9 Yc,k~) where Pi5 is a polynomial of total degree _< c - i + 1 in all the variables Xp,q and Y,,s (use the nilpotency of 9). Moreover, when regarded as a polynomial in the Xp,q-variables (resp. y~,s-variables) alone, Pi,j has degree < c - i. So, in terms of the coordinates introduced for G, the product is expressed by polynomial functions. We use these functions to define the embedding of G in P ( R K) as follows:
p : G ~ P(]R K) : g ~ p(g) with Vy E ~ g : (P(g)(Y) )i,j : m(g)i,j § Yi,j + Pi,j(m(g)i+l,1, ... , m(g)c,kr Yi+1,1,..., Yc,kr It is now easy to see that this p indeed defines an action of G on R K (in fact, it is just left translation in G!), which is simply transitive. We can extend this embedding of G to an embedding of G >~Aut (G) --+ P ( R K) by defining: p : G>~Aut (G) ~ P(IRK): ( g , a ) ~
p(g,a) with
Section 4.5 Canonical type polynomial representations
85
Vy E RK : p(g, oi)(y)= m(goi(m-l(y))). (This is just the coordinate expression for the usual action of G :~Aut (G) on itself.) In order to show that the image of p consists indeed of polynomials of degree _~ s it is enough to show that p(1, a) is polynomial of degree _~ 1 forall a E Ant (G). So, consider a E Ant (G) and assume that Y = (Yl,1,...,Y~,kr Let h E G be determined by h = m - l ( y ) , then there is a commutative diagram h T exp Yl,IAI,1 + ' " +
y~,k~A~,k~
C ")C T exp T exp
9 de)
9
1"exp
da(Yl,lAl,1 + ' " + y~,k~Ac,k~)
where da denotes the differential of a. As da is a linear map, the coordinate expression for a ( h ) -- a ( m - l ( y ) ) is linear in the components of y as well.
Corollary 4.5.5 If G admits a lattice N, then we consider any torsion free central series N , of characteristic subgroups of N. Ira1,1, a 1 2 , . . . , ar denotes a set of generators of N which is compatible with N , , we may choose the basis used in the theorem above to be determined by A1,1 = log a1,1, A1,2 = log al,2, 9 9 Ac,kr = log ac,kc.
Restricting the representation p : G -~ P ( R K) (based on the chosen basis of 9) to N , we see that we obtain a canonical type representation of N into p ( • K ). Moreover, i r E is any A C-group containing N as its Fitting subgroup, we see that the restriction of p to E is a canonical type polynomial representation with respect to the torsion Fee .filtration E . (see also lemma 4.5.10):
E. :Eo = N o = l cEI =NI C...cEr
=N~-I cE~=N~CE~+I =E.
C o r o l l a r y 4.5.6 Let E be any A C-group with a c-step nilpotent Fitting subgroup, then the affine defect of E is ~_ Max(0, c - 2).
4.5.2
The second approach
For technical reasons we need the following (rather well known) lemma.
Chapter 4: Canonical type representations
86
L e m m a 4 . 5 . 7 Let F C R k be any set of elements which span R k as a vector space, for some k E No. Let p ( x l , x 2 , . . . , x k , x k + l , . . . , x ~ ) be a polynomial in n variables for n ~ k, with real coefficients. Suppose
P(~I"~Z1, T2~-Z2,..., ~k"~Zk, ~ k + l , ' ' ' , Xn) =P(;~I, ;~2,''', ;~k,~k-bl,''', Xn) for all (Zl, z 2 , . . . , Zk) E r and all (~el, x 2 , . . . , xn) E R '~. Then, the polynomial P(zl, x 2 , . . . , xk, X k + l , . . . , xn) does not depend on the variables z l , x 2 , . . . , xk. Proof: First of all we remark that we m a y suppose that {(1,0,...,0),(0,1,...,0),...,(0,0,...,1)}
C_ F.
We prove this lemma by induction on the number k. * Suppose k = 1. - W e start with the investigation of the case n = 1. So p(x) is a polynomial in one variable for which p(z) = p(z + 1). We proceed by induction on the degree d of the polynomial p(r For d = 0 there is nothing to show. So suppose d > 0 and the
dp
l e m m a holds for lower degrees. Since ~xx is a polynomial of degree d -
1 for which ~-~P(z) = d~(X + 1),
dp
we conclude that ~
= r for some r E R. Therefore p(x) =
f rda = ra + s for some r,s E I~. The relation p(x) = p(r + 1) now easily implies that r = 0 or that p(a) is independent of
- F o r n > 1, we have to consider polynomials of the form p(r z 2 , . . . , z , ) f o r which
P(~l,x2,...,xn) =P(r
1, x 2 , . . . , x n )
V(Xl,r162
We arrange the polynomial p ( x l , . . . , ~,~) in the following form: P(Zl,-..,z~)
--
q0(x2,...,z~)+zlql(z2,...,z~)+
9"" + x~ql(z2,...,x,~)
Section 4.5 Canonical type polynomial representations
87
where the q i ( x 2 , . . . , x n ) denote some polynomials in n - 1 variables. W h e n we fix real numbers X2,X3~...,$n~ 0 o 0 we see t h a t p(xl, x ~ ~ is a polynomial in one variable xl satisfying the conditions of the lemma with k = 1. Therefore we m a y conclude that
x0
0
= 0, v i > 0.
Since this happens for all choices of x2,~x3,0 . . . , xn,0 it follows that qi(x2, x 3 , . . . , x n ) = 0, Vi > 07 implying that p(xl, x 2 , . . . , x,~) is independent of xl. 9 Next, we suppose that k > 1 and the l e m m a holds for smaller values of k. By first considering the fact that = p(
l + 1, x : , . . ., , n )
we m a y conclude that p ( x l , . . . , xn) is independent of xl. Now, we use the induction hypothesis to say that p(xl, x 2 , . . . , x,~) is independent of X1,~2~...~X k .
L e m m a 4.5.8 Suppose N and N. are as before. For any canonical type polynomial representation of N with respect to N . we have
p(~Ki,
]i~k)Ni/Ni_, ~
p(~Ki+l, Rk)
for all i, 1 <_ i < c, as N/Ni-module. Proof: Since p ( • K , , R k ) is an N/Ni_l-module via hp = p o pi_l(h) -1, where 9 pi-1 : g/Ni_~ -+ P ( N Ki ), it makes sense to speak of P ( N Ki , R k ) Ni/Ni - .1 Let p(x) E P ( R g', Rk). Then it is easy to see that Zp(x) = p(x) for all z e N~/NI-1 and x C R if and only i f p ( x ) i s independent of the variables xi.1, xl,2,.., zi,k, (see l e m m a 4.5.7). Therefore, we m a y identify p(x) with an element of p(]~gi+l, Rk). This identification is of course compatible with the N/Ni-action.
T h e o r e m 4.5.9 Let E be a group containing a finitely generated torsion free nilpotent group N of rank K as a subgroup of finite index. Let ~ :
Chapter 4: Canonical type representations
88
be any morphism with ~(N) = I and let p : E P(I~ K) be any representation restricting to a canonical type polynomial representation of N, with respect to some torsion free central series. With a E-module structure on P(IR K, ]Rk) via E
, A u t ( Z k)
hp= w(h) opop(h) -1, pE P(R K,]Rk), hE E, we have that Hi(E, P(IRK, Rk)) = 0 for all i >_1. Proof: First we consider any torsion free abelian group Zl with a canonical type polynonual representation into P ( R ) and we assume l < K. We consider P(I~ z) _C P ( R K ) , by lettting P(]R l) act on the l first components of I~g and by leaving the other K - t components fixed. If l = 1 then HI(•,P(RK,Rk)) = 0 by theorem 2.5 of [39] (notice that for an abelian group Z , the concept of a canomcal type polynomial representation is the same as a "special affine" representation in [39].) For i > 1 H i ( Z , P(R K, R k)) = 0 because Z has cohomological dimension one. Suppose Hi(Z z-l, P(R K, Rk)) = 0 for all i > 1. Without loss of generality t w e assume that ~ has a normal subgroup Z , with torsion free quotient ~ I / Z and acting only on the first component o f R g . As in lelimla 4.5.8 we find t h a t P(R K, Rk) Z ~ P(R K-l, R k) as Z z / Z = Zz-1 modules. Since Hi(Z, P(]RK, Rk)) = 0 for all i, the restriction-inflation exact sequence yields isomorphisms: .
9
9
.
/
l
0 = H i ( Z z-l,
P(R K-l, Rk)) ~ Hi(Z z, P(R K, ]~k)).
This proves (something more than) the theorem for torsion free abelian groups. Now we consider any finitely generated nilpotent group N of nilpotency class c > 1 and we assume the theorem holds for groups of lower nilpotency class. By the above we know that Hi(N1, p(•g, Rk)) = 0 for i _> 1. Therefore, the restriction-inflation exact sequence now provides us isomorphisms:
0 = Hi(N/N1, P(]R g2, Rk)) ~ Hi(N, P(]RK, Rk)). The generalization for virtually nilpotent groups can be obtained as follows: Because P(R K, R k) is a vector space and N is of finite index in E , we know that the restriction map
Hi(E, p(RK, Rk)) ~ Hi(N, p(]~K, ]~k)) = 0
Section 4.5 Canonical type polynomial representations
89
is injective, which ends the proof.
4 . 5 . 1 0 Let E be a group containing a torsion free, finitely generated nilpotent group N as a normal subgroup of finite index. If p : E --~ P(IR K) is a representation of E, restricting to a canonical type polynomial representation of N , with respect to a torsion free central series N . consisting of normal subgroups of E, then p itself is canonical, with respect to E., where E . is given by: Lemma
E. : Eo=No=ICEI=N1c...cEc-1
=Nr162162162
=E.
Proof: It suffices to show that for e E E, p(e) is such that p(e) : R K
, I~ K
: ~ = (~1,1, ~ , ~ , . . . ,
x~,~,, ~ , ~ , . . . ,
~,j,...,
~o,k~) ~ p ( e ) ( x ) ,
with p(e)(x.) = (P~,l($),...,P~d(x),...,P~:r , and P~j(x) is a polynomial depending only on the variables Xp,q with p > i. First we show that P~,j(z) does not depend on Xl,q, if i > 1. Let z be any element of N1, then there exists a z' 6 N1 such that
ez = z'e.
(4.14)
We will look at the effect of b o t h sides of (4.14) on a general element with coordinates (Zl,1, Zl,2,. 9 ~e,k~). R e m e m b e r that z (resp. z') acts o n R g by some translation on the first kl components, say by (z1,1,..., zl,k~) (resp. ( z ~ , l , . . . , z~,k,)). On the one hand we have that
=
e(zl,1 + ~1,1,...,zl,k~
=
(P;,l(Zl,1 +
+ xl,k~,x2,1,~2,2,...,~c,k~)
~l,1,...,~o,~o),...,P:,~o(zl,1 + ~1,~,...,~o,~,)).
While on the other hand, we see that ~zx =~'~ x
=
~'(~x)
=
(z~,l,...,
z'~,~, 0. ... ,0) + (ff,~(~),..., P;:,o(x)) I
---- (P;,I(;~) + Z l , 1 , . . . ,
~Oe
/
1,kl + Zl,kl,P~,l(~),
e
"'',Pc,kr
These two computations result in Vi > 1, Vz E N1 :
P~,j(z,,, + ~ 1 , 1 , . . . , zl,kl + x,,k,, x 2 , 1 , . . . , ~c,kc) =
9
Chapter 4: Canonical type representations
90
P~,j(
Xl,l,
. . . , Xl,kl
, X2,1,
. . . , Xc,kr
Since the set of all (Zl,l,..., gl,kl ) forms a lattice of R kt, we conclude that the polynomial P [ 5 ( x l , 1 , . . . , x~,kr has to be independent of the coo~nates ~1,, (1 < q < kl) ( . s e le,uma 4.a.7). The same technique can now be used to proceed this proof. One shows by induction on p that P~j is independent of Xp.q if i > p.
R e m a r k 4.5.11 We will call a filtration E , as in the lemma above, a torsion free pseudo-central series of E. If E is virtually nilpotent as above, then, if we consider a canonical type representation we will always mean "canonical with respect to a torsion free pseudo-central series of E", We finish this section with the following lemma: L e m m a 4 . 5 . 1 2 Let E be a virtually abelian group containing a group Z k as a normal subgroup of finite index. Then a canonical type polynomial representation is an affine representation. Proof: Let p : E ~ P(IR k) be the canonical type polynomial representation. We know that the abelian normal subgroup acts on IRk via translations: Zx=z+x VzE~kandVxEl~ k. There is a h o m o m o r p h i s m ~o : E --+ h u t (Z k) such that eze -~ = ~o(e)z. Fix an element e and assume ~(e) is given by a matrix (aid)l<_iS<_k. We denote p(e) = ( P l ( x ) , P 2 ( x ) , . . . , P k ( x ) ) , where the Pi(x) are some polynomials in k variables. For any element x = (xl, ~ 2 , . . . , xk) and any z = (Zl, z 2 , . . . , z k ) E Zk we have that eza
=
(~,(e)z)~ x
~(~ + z l , . . . , ~ k
+ zk)
=
~(e)z+
Pi(ml + z l , . . . , z k
+ z~)
=
(~i,lzl + " " + a i , k z k + P / ( x l , . . - , ~ k ) .
(P~(~),...,Ph(~))
(1 < i < k ) Now, consider the polynomial
Q ( ~ ) = P~(~) - ~,1~1 . . . . .
~,k~k -- P(O).
Section 4.5 Canonical type polynomial representations
91
The previous computation shows that Q(z) = o Vz c Z k. This implies that Q(ix) -- 0 or that Pi(ix) is a polynomial of degree 1.
4.5.3
Existence and uniqueness of Polynomial Manifolds
In this section we make intensive use of the basic facts of the theory of Seifert Fiber Space constructions. T h e o r e m 4.5.13 Let E be a group containing a normal subgroup N , which is torsion free, finitely generated of rank K , nilpotent of class c and of finite index in E and fix a pseudo-central series E , of E . Then there exists a canonical type polynomial representation of E with respect to E , . Moreover, if Pl~ P2 are two canonical type polynomial representations of E into P ( R K ) , with respect to E , , then there exists a polynomial map p E P ( R K) such that P2 = p-1 o Pl o p. In case E is torsion free this means that the manifolds p l ( E )_~ and
P2(
E ) ~ K are "polynomially diffeomorphic".
Proof: Existence: We will proceed by induction on the nilpotency class c of N. If N is abelian (or 2- or 3-step nilpotent), then the existence is known~ since we already obtained even a canonical type affme representation for these groups. Now, suppose that N is of class c > I and the existence is guaranteed for lower nilpotency classes. Using the induction hypothesis, the group E / N 1 can be furnished with a canonical type polynomial representation ~ : E / N 1 ~ P(RK2). We obtain an embedding i : N1 ~- Z kl --~ P ( R K2 , R kl ) if we define i(z) : ~ g 2 ~ Rkl : ix ~ Z. We are looking for a map p making the following diagram commutative: (P :~ (A • P) ~ t p ( R K 2 , Rk,) ~ ( h u t ~k, • p ( R g 2 )))
1 ~
N1
1 ~ P(I~ K 2 , R k~)
--+
E
--~
-*
P ~ ( A • P) N P ( R K)
-*
E/N1
A u t Z k' •
--* 1
K2) -* 1
Chapter 4: Canonical type representations
92
where ~ : E/N1 ~ Aut Z k~ = Aut N1 denotes the morphism induced by the extension 1 ~ N1 ---* E ~ E/N1 ~ 1. The existence of such a map p is now guaranteed by the surjectiveness of 5 in the long exact cohomology sequence
9..---* 0 --ttl(E/N1, P(E g~, ]Ek')/Z kl) L H2(E/N1, Zkl}-, 0 --~...
II
II
HI(E/N1, p(]Eg~, ]~k,))
(theorem 4.5.9)
H2(E/N1, P(R g2' Rk'))
(4.15) Uniqueness: Again we proceed by induction on the nilpotency class c of N. For c = 1 the result is well known. Indeed, two canonical type representations of a virtually abelian group are even known to be affmely conjugated. (To see this, we can use the generalized second Bieberbach theorem). So we suppose that N is of class c > 1 and that the theorem holds for smaller nilpotency classes. The representations Pl, P2 induce two canonical type polynomial representations
ill, fi2 : E/N1
~
R K2 9
By the induction hypothesis~ there exists a polynomial diffeomorphism 9 : ~K2 ----+~K2 such that P2 --~ 9 -1 0 Pl 0 q"
Lift this q to a polynomial diffeomorphism q of ]~K1 aS follows:
Let us denote r = q-1 o p l o q . Then we see that r and p2 are two canonical type polynomial representations of E~ which induce the same representation of E/N1. This means that r and P2 can be seen as the result of a Seifert construction with respect to the same data (cfr. the c o m m u t a t i v e diagram above). Now we use the injectiveness of 5 in (4.15), which implies that r and P2 are conjugated to each other by an element r E P ( R g2, ]Rkl) (seen as an element of P(I~K)!). So we may conclude that P2 = r - 1
o~)or
if we take p = q o r.
= r -1 oq -1 opl
oqor
=p-1
op 1 op
Section 4.5 Canonical type polynomial representations
93
C o r o l l a r y 4.5.14 Since any canonical type polynomial representation p of E is polynomially conjugated to a polynomial representation of finite degree by the previous section, p itself has to be of finite degree. R e m a r k 4.5.15 Theorem 4.5.13 is a generalization of theorem 1.20 in [31] and of theorem 4.1 in [39]. In this second paper, the authors are concerned with canonical type a]fine representations (called "special" there), with respect to the upper central series. They claim that we can take the polynomial to conjugate with, of degree _< c!. Certainly there is a polynomial conjugation, but the degree does not seem to be right. We indicate a gap in the proof of theorem 4.1 of [39]. This gap appears in the induction argument in [39]. Indeed, an induction argument assumes that, after conjugation by a polynomial on the N/N1 = N/Z(N)-level, the result on the N-level is again a canonical type a]fine representation. Of course, the conjugation does not touch the canonical type character of the representation. However, as the subsequent example shows, one might obtain a polynomial representation which stays outside the affme group. E x a m p l e 4.5.16 Let N be the following finitely generated torsion free 3-step nilpotent group: N : < al,a2, b, cll [a2, a l ] = b, [b, al] = c, >. [b, a2] = [c, al] = [c, a2] = [c,b] = 1 We propose the following two canonical type aft-me representations of N (upper central series), where we indicate only the images of the generators: l/I pl(al):
-1 1 0
1
0
0
I
'
C1oo) 0
ol(b)=
0 1 / 0 -1
o
1
o
0 0
0 0
1 0
-1'a2'
pl(c)=
0 1 0 0 0 0
o 0
and P2(al) :
/i ~176176 001 u1/i)) 2a2 ((100 00 0 0~ 1 10i)(i/
Chapter 4: Canonical type representations
94 1
P2(b)=
0 1
~ 0
0 0
1 0
~
1 0
0 1
0 0
0 0
0 0
1 0
i)(1)) ({ '
p,.(c)=
0
1
'
"
The presentation for N/Z(N) can be read from the presentation of N , by simply replacing all c's by 1. The induced canonical type aft-me representations Pl, P2 : N/Z(N) ---+Aff(]~3) are also easy to write down: 9 In the linear part of pi (i = 1, 2), we discard the first column and the first row. 9 In the translational part, we omit the first term. Following the general theory of Seifert Fiber Spaces and the p r o o f of theorem 4.1 in [39] or of theorem 4.5.13 we find that there must exist a polynomial mapping p(x) in P(]R 2, R) such that
P2 _ p - 1 o~ 1 o p when p ( x ) i s seen as an element of P(IR2, R):~P(R 2) or as an element of p(•3). Some elementary computations show that the only possible p's are of the form: p : IR3 _+ IR3 :
where r can be any real number. If we now lift this p to a polynomial mapping of ]R4, say
I t q : R 4 _, R 4 :
Z F--a,
Y
z-~-+r Y
we find that
t+z-z+~--r q-lpl(al)q : R 4 _~ ]R4 :
Z
Z
Y
y+l
2
2g
which is clearly not affme. Moreover, the same can be said about any lift of p of the form
q : ]R 4 _~ ]R 4 :
z Y g~
z-~-+r Y T,
Section 4.5 Canonical type polynomial representations
95
where m ( z , y , z,t) denotes some polynomial of degree _< 2 in the four variables. Finally, we indicate how pl and p2 can be seen to be conjugated by a polynomial. Consider
/ t)
P : ~ 4 --+ I~4 :
z
t+~+ ~-~Z - 2~ - ~3 ) 6 xy 3 z -- --~-
H
Y
Y
T h e n one can see t h a t
z+~-
2:
Y
Y
x
x
from which some easy calculations result in p~(n) = p -1 o p l ( , , ) o p
VnEN.
We r e m a r k that, in spite of this example, we were able, using totally different techniques, to determine an upper bound for the degree of p, again in terms of the nilpotency class c ([16]). 4.5.4
Groups
of affine defect one
The only known connected, simply connected nilpotent Lie groups not acting simply transitively and affmely on Euclidean n-space, are those constructed by Senoist ([4]) and Burde and Grunewald ([12]). As we told already, it is the situation on the Lie algebra level which is investigated in b o t h papers. The Lie algebras in question are ll-dimensional; an interesting family of t h e m is denoted by a ( - 2 , s, t) and has a basis el, e 2 , . . . , e n . Moreover, the brackets of these Lie algebras are completely determined by [el,el]=e~+l 2
[e2, e3] = es and [e2, es] = - 2 e 7
+ se8 + te9.
The results of Benoist, Burde and Grunewald imply that, in case s ~ 0, ct(-2, s, t) has no faithful 12- dimensional linear representation. Therefore,
Chapter 4: Canonical type representations
96
no lattice of the corresponding Lie group , 4 ( - 2 , s,t), can act properly "discontinuously on IRn , via affi_ne motions and with compact quotient. In this section we indicate, how one can construct a uniform lattice F of the Lie group .2[(-2, s, t), acting properly discontinuous on R n via polynomial diffeomorphisms, in such a way that the polynomial mappings used are of degree _~ 2. It will follow that these lattices F have affine defect 1. To do so, we will embed this F into a connected, simply connected Lie group P (of finite dimension) of polynomial diffeomorphisms of IRu . In fact, let P be the group consisting of all mappings Yl: Yl0
Yll +P11(Y:, Y2,..., Y:0) Yl0 +Pl0(Y:, Y2,..-, Y9)
Y2 Y:
Y2 -]-P2(Yl) Yl + P:
f : ~11 ~ ~11 :
where pi(yl, Y2,..., yi-1) is a polynomial of degree _~ 1 in the variables Y:,...yi-1, except P n which is of degree _~ 2. In view of the results mentioned above, an embedding F ~ P is really the best (non affme) thing we could expect. The first step to clear this job, was to construct a unitriangular matrix representation of , 4 ( - 2 , s, t). This was not difficult since Burde and Grunewald ([12]) give an explicit way to construct a 34-dimensional (and with some more trouble, a 22-dimensional) matrix representation for the Lie algebra a ( - 2 , s,t), using its universal enveloping algebra. These matrices are upper triangular, with 0's on the diagonal, and so by exponentiating we find the desired representation on the Lie group level. For the sake of simplicity we will restrict ourselves to the cases where s, t E 7~, however, we remark here that things work out for rational s and t too. Let F denote the subgroup o f . A ( - 2 , s, t) generated by A:, A2,. 9 A n , where A1 - exp(el) A4 = exp(l:e4) A7 = exp( i ~1 e T ) A10 = e x p ( ~ e l10 )
A2 = exp(e2) As = e x p ( l e s ) As = exp(72~6es) A n = exp( ~ e: l : )
A3 : exp(e3) A6 : e x p ( l e 6 )
= exp(
- ooe )
Making use of the m a t r i x representation of A ( - 2 , s, t), and so of F, mentioned above, one checks that F is a uniform lattice of , 4 ( - 2 , s, t) with
Section 4.5 Canonical type poIynomial representations
1~
-----
(AI,...,
97
A~lll
[A2, A1] = . - 1 . . 2
a-5.109.4(fg-60,).6(-llO8-525a-1400t)
.t-x3 .t~4~5.t-~ 6 Alg267555§
2-x7
2~18 126000$
219
A2(203230225-12930435J § 11
~+4372200t)
[Aa, All = a-2a~A-4alS~A6(-91-60')A35(S20+lS0'-3600 A80(45515-3066, +31 sot) 10
ALSO(-5202055+135870m +188166~- 190050t)
11 [A3, A21 = a-6~-120A360(3+,)~s40(-26-15~+15t)
~ 5 x=~7 ~8 ~=~9 A 6 7 2 0 ( - 5 8 5 - 28~ - 7 5 t ) A 3 7 8 0 ( - 4 1 9 7 5 - 1 3 0 3 5 J ~ 2688~ ~+ 5150t)
10 ~Ii d--3 A6A--10A15AI6905AI40(--2425--S64J) A OO(836225+6o48o, + 28224~2- 222o75t ) 11 [A4, A2] --A-12A-900A12600''~8400(341+60t)A13494600* --~6 "~8 "~9 ~10 "~11
[A4, A31 --a-ls~176176176176176176176 --'~7 "18 "=9 ~'I0 "Xll [As, A1] --~J--4AlOA--20ZI175A--1400A21708750
A 40 A--120t a 840(13- St ) a 94080, a1260(--12350+4032* ~-- 7725t ) [A5, A3] --A-360~4200'A168000ta91003500 - - "'18 "x9 "~I0 ":t 11 [As, Aa] A - - 113't0 A 8 0 6 4 0 # A 3 7 8 0 ( - 4 4 8 ' zq"3525t) --"~-7
""-8 ""~'9
"'~I0
;'~11
[A~, A~] _~o~-~oo,~-ooo(-~a-~t)~-~o~5~o, - - -"~8 -'~'9
"r~lO
"-'-11
A2520A1680~ A1890(448~-1525t)
[A~, A,~] _ a - ~ l v ~ o o ---~Xl0
,, lS:~,~4oo,
"-~ 1 1
[A6, As] ----All~064~875 [AT, A1] -- - nA-~ Al~ '~-~4~176176176 8 "~9 " ~ I 0 ~ll JAr, A 2 ] : n
[Az, A3]
A 546 A -- 17136~, A 189( - 6175-- 896~ $-- 4950t ) 9 ~tO "1-11
a655~oa-~9s,~o,
[A;, A4] -~'1t-~4651~5 [As, A~] ~-~s~7oo~-~ooo ~---"~9 "~ 10 "~iI [As, A2] ----A[o79S~ 411~~
[As,A3] --:At?05575
[A~, A~] ---~I ~-4oa~8oo 0 ~11 [Ao, A2] = A [ # 275 [Alo, A l l = ~ [A{,Aj] = O for l < j < i <_11and i § j > 1 1
In fact, after having taken A1 = exp(et) and A2 = exp(ez), the others have been chosen carefully so that they generate a lattice. It is obvious that this choice is not unique. Now that we have enough information about the group structure, we are going to build up an embedding F --* P. In order to do so we will
Chapter 4: Canonical type representations
98
first realize the rank 10 group r/z(r) = r / ( A n ) as a properly discontinuous group of affine motions of ]R1~ The idea behind this is the fact t h a t the adjoint representation ad : a ( - 2 , s, t) --+ 9 [ ( a ( - 2 , s, t)) is an 11dimensional representation factoring through a ( - 2 , s, t)/Z(a(-2, s, t)) = a ( - 2 , s, t)/(en). By exponentiating this representation, we m a y realize A ( - 2 , s, t)/Z(.A(-2, s, t)) as a group of affme motions of R 1~ However, in this way the group A ( - 2 , s, t)/Z(A(-2, s, t)) does not act simply transitive on R l~ as we would like. Therefore we slightly modify the adjoint representation in order to obtain, on the Lie group level, a simply transitive action. This altered adjoint representation, denoted by r is determined completely by the images of el and e2:
r
0 0 0 0 0 0 0 0 0 0 0
--
1 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 I 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 -1 0
and 0
r
=
0
19 16
28 s 25
448 s 2 -[-2475 t 2000
0
0
0
0
{}
0
0
0
0
2~ 8
~ls 25
0 0 0
0 0 0
0 0 0
0 0 0
13 0 0
2t 2s -5 0
0
0
0
0
0
t s -2
0 0 0
0 0 0
0 0 0
0 0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0 0
0 0
0 0
0 0
0 0
0
0
0
0
0
0
0 0
0 0
0 0
0 0
0 0
-1 0
0
0
0
0
0
0
0
0
0
0
0
Let us define the m o r p h i s m ~ as the composite map: .A(--2, s, t) ~-~ a ( - 2 , 8, t ) - +
Ct(-2, s,t)/<en) r ~I[(11, IR) ~ G I ( l l , •)
Section 4.5 Canonical type polynomial representations
99
As the images of T consist of unitriangular matrices, we can look at the Lie group .2,(-2, s, t ) / Z ( A ( - 2 , s, t)) as being a simply transitive group of affme motions of R l~ In the next step we lift the map ~ : F ~ G I ( l l , IR) to a faithful morphism r : F ~ P. We present r by giving the images of the generators:
Yl0
r
: R 11 ~
[r
:
Y2 Yl Here, we see T(Ai) of degree
/
as an affme mapping
o f ]~10 a n d t h e
fl a r e p o l y n o m i a l s
_( 2 g i v e n b y :
f l ( Y l , . . . , Ylo)
=
1
I
9 9 5 6 9 9 2 5 t -~- 1 7 0 1 3 0 2 4 s 2 - 250 90720000 Y2 1
1
1
50~Y4 - 7-~y~ - 1-~y6 - ~y7 -5458475 -142755-
Y2Y4
84000 1617-320s-960t
1920 -195 + 46s -~ 50 Y:Y7
800+40s+448s Y2 Y5 -
189
Y2 Y6
321
3 8 0 2 5 -- 2 5 6 0 s + 1792 s 2 - 1 4 1 0 0 t
4000 1377
2-1525t
2000
+ --~Y~.Ys
16000 1 5 0 7 5 - - 1920 s + 896 s 2 -- 7 0 5 0 t
-
Y2Y8
1680000 6860 s d- 3136 s 2 - 3 8 6 7 5 t
-~
-
1
- ~ y 8 - ~ y 9 - ylo
+ 7392820 s + 1937376 s 2 + 10623200t
-~
-[-
1 40320Y8
YaY4 ~ 3 (45-- 16s)
Y3Ys -t-
50
Ya Y6
1377
I--~YaY7 + -i-~-Y4Ys -t- ' - ~ Y 4 Y 6 --
6 4 8 6 1 5 0 -- 2 2 2 8 7 1 0 0 s -- 5 7 2 7 4 5 6 s : -- 1 1 2 8 9 6 0 s a -- 3 2 6 6 1 8 2 5 t 11881800st
+ ~ y ~ + 459
2
1-~Y4
2
10080000 -62761513440s + 12544s 2 - 98700t 336000
y~
yl +
Chapter 4: Canonical type representations
100
=
- 6 9 5 0 0 - 3 2 6 3 9 0 4 s ~ - 376320 s 3 § 6 0 4 4 3 2 5 t -
3960600 st Ya +
3360000 21465 s - 6272 s 8 - 66010st Y4+
28000
-8843500 - 2685984s 2 § 7797825t
1680000 15775-
198 s
Y5 § 8 - - ~ Y 8 §
6272s 2 - 34650t -28s 28000 Y~ + - - ~ y s
-19 + -~Y9
+
- 7 6 7 8 0 0 + 210464 s 2 - 120825 t 160000 - - 4 3 3 6 4 9 s + 6 2 7 2 s 3 + 66010 st
Y2Y3 + Y2Y4 +
28000 519000 § 8 3 1 0 4 s 2 -- 2 1 8 8 2 5 t 517s 4089 40000 Y2Ys § 1-~~Y~Y6 - ~ - Y 2 Y 7 § - 1 1 3 9 1 3 0 0 § 309792 s 2 + 2217775 t 1671 s 560000 Y3Y4 + ~ - Y 3 Y 5
+
21087 5913 140~--o-Y3Y6 -- - ~ Y 4 Y s + - - 1 6 5 5 4 3 7 5 - - 339872 s + 150528 s 3 § 1584240 Sty~ 1671s 2
~-~ y3+ f3(y,,...,y,o)
2-~
y4
=
3 (2187- 3584s) 2800 Y4 321 -
+
1344000 --33039s 2
-
-
80
-41175Y8
+
62901 448s 2 - 1525t -23s 2 8 0 ~ Y5 + 2000 Y6 + --~--Y7 6272s 3 - 66010st
Y2Y3 +
28000
11 ( - 9 1 8 4 s : § 2 9 0 7 5 t ) -19407s 549 80000 Y2Y4 + 14000 Y:Y5 - -~Y2Y~ + 3573 s 76383 - 2 6 3 3 6 s : - 457825tya2 _ 2673 2 4 0 - - - ~ y3y4 + 1---4~ yay5 § 80000 '~~Y4
f4(yl,
. . . , yio)
=
--448 s 2 + 3525 t 12 s 189 26336 s 2 + 457825 t 4000 Y~ § -~-Y6 + -~-Y7 + 160000 Y2Y3 123183s
2 8 0 0 ~ Y2Y4
35397
6561
1120 Y2Y5 + l-~Y3Y4 +
448 s 3 + 4715 s t 2
4000
Y2
Section 4.5 Canonical type polynomial representations fs(Yl,...,Ylo)
=
3157s 15000 2 Y2: -
-
12793t
2
19200 Y2 -
39s
1--~Y2Y3 -
6469 s 2
8073
-
ll-Kb-6-dy:
+ 2~Y2Ya
fT(Yl, . . . , Ylo)
=
183 2 22400 y2
Myl,...,ylo)
=
0
=
0
=
0
f6(yl,...,
101
yl0)
2187 2
~Y3
891
+ ~Y2Y4
--
459 ~Y6
1 y
o)
-
90720000
T h e o r e m 4.5.17 Let A ( - 2 , s, t) be the Lie group which stands in oneto-one correspondence with the Lie algebra a ( - 2 , s,t). If s, z C Z , then A ( - 2 , s, t) acts simply transitively and via polynomial diffeomorphisms of degree <_ 2 on R u . So, if s ~ O, then any lattice of A ( - 2 , s,t) is a group of affine defect one. Proof: By the computations above we know that A ( - 2 , s, t) contains a lattice F which embeds nicely into a Lie group P of polynomial diffeomorphisms of degree _< 2. Observe that this group P , introduced above, is nilpotent. This can be seen as follows. Let P1 be the normal subgroup of P consisting of those polynomial diffeomorphisms for which p~ -= 0 (1 < i _< 10). So P1 consists of those polynomials of I~ 11 which only affect 1 coordinate. Denote by P2 the subgroup of P determined by those polynomials for which P n = 0. So P2 is isomorphic to the group of unitriangular 11 • l l - m a t r i c e s . There is a split short exact sequence 1--* P1--* P--* P2--+ I. P2 is a nilpotent group, and so 7n(P) C P1 if n is big enough. It is now an exercise to see that further c o m m u t a t o r subgroups will gradually get smaller and smaller until, at the end, they vanish. Using [33, Proposition 2.5], the embedding F ~ P extends uniquely to an embedding of the Lie group , 4 ( - 2 , s,t) ~ P, making it acting properly discontinuously on I~11, which was to be shown.
102
Chapter 4: Canonical type representations
Let us finally point out that the use of MATHEMATICA | [61] was quite crucial in setting up and performing the computations above. A set of procedures enabling the reader to verify these results is available on request from the author.
Chapter 5
The Cohomology of virtually nilpotent groups 5.1
The need of cohomology computations
For the classification of the infra-nilmanifolds (or AB-groups) we will make intensive use of the reduction lemma 2.4.2 (see section 2.4 "The first proof revisited"). This lemma tells us that an AC-group E having a Fitting subgroup N of nilpotency class c can be obtained as an extension
1 ~ Z~ E ~ E/Z~
1.
Z is the free abelian group of finite rank defined as Z = ~/7r = % ( G ) N N , where G denotes the Mal'cev completion of N. E / Z is also an AC-group, with Fitting subgroup N/Z of nilpotency class c - 1. The action of E / Z on Z, induced by the extension, is trivial when restricted to N/Z. This motivates the importance of having a way to compute H2(E/Z, Z) for actions of E / Z on Z which factor through E/N.
5.2
The c o h o m o l o g y for some virtually nilpotent groups
First, we develop the theory for crystallographic groups (i.e. the A C groups with abelian Fitting subgroup) and at the end of the section, we make some considerations about how to generalize this theory to a l m o s t crystallographic groups. We already explained that an abstract crystallographic group Q is characterized algebraically by the fact that it contains a free abelian nor-
Chapter 5: The Cohomology of virtually nilpotent groups
104
mal subgroup of fmite rank, say n, which is maximal abelian and of finite index in Q, where n is the dimension of Q. So Q fits in a short exact sequence 0 ) Z '~ >Q ) F ~ 11 where F is finite. Also the F - m o d u l e structure of Z n is faithful (i.e. F embeds in Gl(n, 7~)). Every group Q has a faithful affme representation p : Q -, Aff(R '~) mapping Z n into the group of pure translations. From now on, we fix such a representation p. F is generated by a finite number of elements, which we denote by a l , a 2 , . . . , ak. Elements of F will be written as a word in these generators; for each element o~ o f F we fix a unique word u ( a ) = a ~ la+li2 " " " O~-I-li~ which represents it (take u(1) = 1). So, F can be looked at as
F:
< al,a2,...,ak
II w i ( a l , ( 1 2 , . . . , O~k) = 1 (1 < i < l) > .
2~n, the translation subgroup of Q, will be considered as determined by Zn:
< al,a2,...,an
[[ [ai, aj]---- 1 (1 < j
< i < n) > .
As a consequence, Q can be presented as
Q : < a l , . . . , an, a l , . . . , ak [I [ai, aj] = 1 (1 < j < i < n) w i ( a l , a 2 , . . . , o ~ k ) = a Iii l ' a 12~ 2' . . . a nl '~ ( l < i < l ) aiajot7~ 1 = w i s ( a l , a 2 , . . . , a n )
(1
>
k, 1 < j < n)
and elements q of Q can be written uniquely as words
_ql_q2 q = ~1 ~2 9..aq~u(a) (qj E Z, 1 < - j < - n,
a E F).
We are interested in computing H v2( Q , z m ) , for an action ~ : Q A u t ( Z TM) which factors through the finite group F . From a conceptual point of view, let us observe the following result: P r o p o s i t i o n 5.2.1 Assume Q is a crystallographic group acting on Z TM via 9~ : Q --~ A u t ( Z TM) such that ~lZ,~ = 1. Write res : H2~(Q,Z TM) -~
H~r(Z" , Z TM) for the restriction morphism. part of H ~2( Q , Z m ) and we can write
Then ker(res) is the torsion
H v2 ( Q , Z m ) ~- I m ( r e s ) ~ ker(res), where the torsion free part has rank <_ m n ( n - 1)/2.
105
Section 5.2 T h e c o h o m o l o g y for s o m e virtually nilpotent groups
Proofi The proof is ilmnediate once we realize that H ~ ( Z r~, 7/,,TM) Z m'~(n-1)/2 is torsion free and that Z " is of finite index in Q. In view of computing H2v(Q, Z m) in practice, let us consider an extension 0 )Z ~E ~Q )1 TM
compatible with ~. E has a presentation E:
< al...,ak, al,...,a,,bl,...,bm [bi, bj]-- 1 ( l _ < j < i _ < m )
l[
>
.b~'''''j (1 -< j < i < ~) [ai, aj] = ~k,,,,~,,,~ "1 "2 "" [b~, a / = 1 (1 < i < m, 1 < i -< ~) ~ b j ~ -1 -- v ( ~ ) ( b j ) (1 < i < k, 1 < j ___m) -
wi(al,Ot2,...,ak) aiajoti
I
=
lii
w i j ( al , a 2 ,
(l
lr, i t k i i
.km /
= a i ' . . . a n ' v 1 ' ...Ore' k' . . . ,
9 9
(I
k'
a n ) b l l ' ' ~ . . . b m~ ' ' ' J
<_j <_n)
(5.1) which is completely determined by the integers k,.,ij, k,.,i and k~,.,tj. As we already explained in the example of section 2.4 these integers cannot be chosen completely free, since there should be c o m p u t a t i o n a l cons i s t e n c y . Let us look at another example to make this more clear. E x a m p l e 5.2.2 Take Q ~ Z 2 ~ Z2 given as Q : < al,a2, ot U [a2, al] = 1, ctalct -1 = al, cta2a -1 = a21, ot2 = 1 > Here F = Z~. Take ~ : Q ~ A u t Z such that it induces the non-trivial action of F on Z. Then, an extension E has a presentation: E : < el, a2, b, a I[ [a2, al] ---- bk~, [b, eli = [b, a2] = 1, aalot -1 cta2ct -1 _- a21bka, a b a -1 = b-1 ct 2 - bk4
----
alb k2 >
Now, note (in E ) that a ( a 2) = (a2)a ~ ab k' = b k ' a ~ b - k ' a = bk4ct from which it follows that k4 = 0 showing that not all choices of ( k l , k 2 , ka, k4) can be accepted. A 4-tuple ( k l , k 2 , k3, k4), which really determines an extension of Q by Z is said to be a computational consistent 4-tuple. We r e t u r n to the general set up. Fix an ordering for the parameters kr,~,j, kr,~ and k~,,~,j in (5.1). Assume there are p parameters. This means
Chapter 5: The Cohomology of virtually nilpotent groups
106
that E is determined completely by a p-tuple of integers which we denote from now on by K. We refer to E ( K ) as the group E determined by K. As in the example, we refer to a p-tuple K for which there exists a group E ( K ) (as an extension of Z TM by Q) as a c o m p u t a t i o n a l c o n s i s t e n t p-tuple. The elements of E ( K ) can be written uniquely as words b~l 9 . vhprn m ,,ql ~1 " " a ~ u ( ~ ) , .
(p~,qj e
Z).
Take a section s : Q ~ E ( K ) : q = ~_ql_q~ ~2 9. . a y u ( a ) which we will refer to as t h e s t a n d a r d s e c t i o n .
~
b~ " . b ~ q ,
The cocycle f g : Q • Q ~ Z TM, determined by the standard section is called a s t a n d a r d c o c y c l e . The set of standard cocycles {fgll K computational consistent } will be denoted by SZ~(Q, Zm). Since every extension of ~'~ by Q is equivalent to some E ( K ) , every cocycle in Z~(Q, ~m) is cohomologous to a standard cocycle. So it will be sufficient to work with SZ~(Q, Z TM) to obtain the cohomology group. The 7,TM) based on these following proposition shows that computing H~(Q, 2 standard cocycles is possible without too much difficulties and, in fact, reduces to a linear problem. Proposition
5 . 2 . 3 H~( 2 Q, Z TM) can be computed algorithmically, since
2 1. SZ~(Q, Z m ) is a subgroup of Z2(Q, Zm). Moreover, if K1 and K2 are computational consistent, then K1 + K2 is computational consistent and fKl+g2 = fgl + fg2.
2. A standard cocycle fK is a coboundary if and only if (a) the parameters k,,i,i (in 5.1) (1 <_ j < i <_ n, 1 < r <_ rn) are zero (b) K is an integral solution of a well determined finite set of linear equations. P r o o f of 1: By r e m a r k 4.4.25 we know that every E ( K ) has a faithful representation into Aff(]~m+n). Although not unique, this representation is of canonical type. We can now exploit this information to u n d e r s t a n d the proposition. Elements of Aff(R m+'~) will as usual be represented by a ( m + n + 1) • ( m + n + 1 ) m a t r i x with b o t t o m row ( 0 , 0 , . . . , 0 , 1 ) .
Section 5.2 The cohomology for some virtually nilpotent groups
107
Now, assume A: E(K) ---+Aff(e re+n) is a canonical type representation for E(K). Then, 0
I,~+1
0 0
I,~ 0
0 1
~OtBj
where uj = (0, 0 , . . . , 0, 1, 0,..., 0) tr having 1 on the j - t h spot. Furthermore, Aj ~ n_ot Aj and
A(aj) = ( ~(aJ)o: Im
pia~) ]
dj ~
~("J) = ( ~(~J)O p(~j) ]
not
cj.
In order to know A, we compute the entries of Aj and Cj by requiring that the matrices Aj, B 5 and Cj satisfy all the relations of (5.1) when substituted for as, b5 and aj respectively. Remark that the relations in (5.1), not involving components of K, are satisfied automatically (because of the type of A). The other relations can be written in a form
w(al,a2,...,an, al,...,ot k) = b~11...b ink., where k l , . . . , km are some of the components of K. Since A j l = ( /m -~{~P(a~l) )p(aj ) and
o
0(~21)
'
we see that
w(AI'A2"" I 'Ck)= r a "(
P(.Al,fl2,...,Ck)
0
where P is a matrix whose entries are linear combinations of the entries of -4i (1 < i _< n) and Cj (1 _< j _< k) with coefficients determined by ~(aj), p(aj) and p(ai). We have to claim that
P(A~I, A~2,9 9 Ok) --
0 0 ... 0 0 ...
0 0
:
:
"_
0 0 ...
kl / k2 :
0 k~
108
Chapter 5: The Cohomology of virtually nilpotent groups
Hence every relation of (5.1) gives rise to a set of linear equations, the variables being the entries of .4j and Cj. Each of these equations is of the form
+E j,l,m
= h(K)
(5.2)
j,l,m
with h(K) = 0 or h(K) = ki (1 < i < m). (The coefficients fj,z,m and gj,z,m, independent of Aj and Cj, are well determined by ~ and p.) It is not hard to see that this system of equations has a solution for a particular choice of K if and only if this K is computational consistent. In fact, this is a practical way to determine all computational consistent g's,
Assume that -4j,1 (resp. -4j,2) and Cj,1 (resp. Cj,2) are solutions to (5.2) with g = K~ (resp. K = K2). Then it is clear that Aj,1 + Ai,2 and Ci,1 + Cj,2 form a solution to (5.2) for g = K1 + K2. As a consequence the set of computational consistent K's forms a subgroup of ~P and hence is a free abelian group of rank s _< p. Choose s generators K1, K2,. 99 K~ for this group and assume Aj(Ki) and Cj(Ki) are solutions to (5.2) for K = K~ (1 < i < s). One can now write any computational consistent p-tuple K as
K = llK~ + 12K2 + "" + l,K,
for some (unique) 11,12," ",l, E Z
$
S
according to which Aj(K) = ~ li.4j(Ki) and Cj(K) = ~ i=1
liCj(Ki) are
i=1
solutions to (5.2). Having obtained a faithful representation A for each E ( K ) , we use A to compute the corresponding special 2-cocycle f g ( x , y ) , defined by s(z).s(y).s(xy) -1 = fg(r Vx, y e Q. For x , y E Q, the left side of this equality is a word in al, a2,. 9 an, a l , . 9 9 ak and so
=(
)•
where R(A1, A 2 , . . . , Ck) is a matrix having entries which are linear in 11,12,..., l,. However since R(.41, A 2 , . - . , Ck) -- (0, 0 , . . . , 0, fg(z, y)) this implies that f g ( z , y) is linear in 11,..., 1,. And so we m a y conclude that SZ~(Q,~ TM) is a subgroup of Z~(Q,Z TM) generated by s elements. 2 P r o o f of 2: To understand part a), note that if < fK > : 0 6 H~(Q, Zm), 2 n then also res(< fK >) = 0 E Ht~(Z , Z'~). It is well known that the m -
109
S e c t i o n 5.2 T h e c o h o m o l o g y for s o m e v i r t u a l l y n i l p o t e n t g r o u p s
i)
parameters k , , i j uniquely identify the elements of H~ (Z n, Z " ) . 2 This proves the claim. Let us now try to understand b). Assume f K ( m , y ) 67(z,y) =~ "~(y) - "f(my) + ~,(m) for some 1-cochain 7 : Q ~ Zm. It is clear that, for each a E Q, =
0 = fK(aqlaq2...aq",u(oO)=
7(a~' . . . a q " ) + ~ / ( u ( a ) ) - - 7 ( a q l . . . a q ~ ' u ( o L ) ) ,
which implies that 7(a~' . . .a~u(a)
) = 7(a~ 1 . . .ay ) +
"l(u(a)).
In a similar way, one verifies easily that 7 ( a ~ ' . . . a q') = qlT(al) + " " + qnT(an). So, we have "y(a~' . . . a y u ( a ) )
=
ql"I(al) + " " +
q,n(a,,) +
-~(u(a))
(5.3)
showing that 7 is completely determined by a finite subset of Z m i.e. and
7(ai),7(a2),...,7(an)
7(u(a)), V a E F.
Moreover, the affme representation A for E ( K ) , allows us to compute the expression f g ( a ~ . . , a ~ " u ( a ) , aYll.., a~"u(/~)) for fixed a and/3 and for arbitrary Z l , . . . , z n , y l , . . . y n . Here it will be necessary to use the identity
/=0
l
(Im+,~+m is the (m + n + 1)-identity-matrix). Now, the problem of finding all K such that f K ~ 0 is transformed to finding all K for which the finite set of linear equations (in the variables 7 ( a l ) , . . . , 7 ( a n ) and 7 ( u ( a ) ) Y a E F) fK(a~
. . . a~"u(o~), a~l . . . aYn"u(fl) ) = 67(a~1. . . a ~ " u ( a ) , a~' . . . aYn"U(fl) ).
(5.4)
has an integral solution. We have one equation for each pair (u(a), u(/3)). Notice also that both sides of (5.4) are polynomials in the variables Xl,
9 9 9
Yn.
Although this problem seems quite difficult to solve in general, it is often possible to simplify a lot. Also the key information of the above observations is that the problem has been linearized. Let us continue this section with two examples:
110
C h a p t e r 5: T h e C o h o m o l o g y
of virtually nilpotent groups
E x a m p l e 5.2.4 Take Q ~ Z 2 x Z 2 as Q:
< a,b,a
II [b,a] = 1, a a = a - l a ,
ab = b - l a ,
a 2 = 1 >.
F = Z2 and generated by a. Take u ( a ) = a. Consider the trivial action of Q on Z. T h e n an extension E has a presentation of the form E:
< a, b, c, a II [b, a] = c a`
ac = ca
=
>.
b] = 1
a a = a-lo~c k2
ab = b - l o l c ka
0~2 = Ck4
Every group E is determined by four parameters (kl, k2, k3, k4) -- K. In this case there are no consistency conditions on K since we know a faithful representation A: E ( K ) --+Aft(JR3):
A(a) =
)~(c) =
1 0 0 1 0 0 0 0 1 0 0 0
o)
-kl 2 0 1 0
1 0 1
0 0 1 1 0 0 0 1 0 0 0 1
.,r,=
1 0
~2 1
0 0
O) 0
0 0
0 0
1 0
1 1
o-1
o
o o
0
0
-1
0
0
0
0
1
We could proceed by computing special cocycles via this representation, but in fact we can as well calculate f g ( r Y) in a direct way. Let z = aZlbZ2ct z3 (resp. y = aU~bY~a y~), where r (resp. Y3)is calculated modulo 2. Verifying
aZ~b=2 ot=3 a m by2 oly3
=
c(_1)~3 =2y~kl +z~ul k2 a =, +(-1) ~'~v~ b=2 az~ bU2 au3
=
C(-1)=3a~zy'kl+X3Ylk2+:r'3Y2 k3
aZ, +(-1)~3u, bZ2+(-1)X3W aZ3+y3 =
C(-- 1) =3 =2Yl kl -}-a~3Yl k2 -[-=3 Y2 k3 -'}-x3 Y3 k4
aZl + ( - 1 )=3u, b=2 + ( - 1) =3u2 a(=3 +u3 )rood2,
we find f ( r y) = (-1)=3r + zzylk2 + r + r Now we investigate when f ( z , y ) = ~7(r y), for 7(aZlb=2a =s) = x l T ( a ) + r + r (see (5.3), r is either 0 or 1). First we calculate ~7(r
---- ( r
a)+(r
b)+(r
S e c t i o n 5.2 T h e c o h o m o l o g y for s o m e v i r t u a l l y n i l p o t e n t g r o u p s
111
- ( x l + (-1)~3y1)7(a)- (x2 + ( - 1 ) Z 3 y 2 ) 7 ( b ) -((x3 + Y3) rood 2)7(a) =
(Yl - ( - 1 ) Z 3 Y l ) 7 ( a ) +
(Y2 - ( - 1 ) Z 3 Y 2 ) 7 ( b )
+(~3 + Y3 - (x3 + Y3) rood 2)7(a). In this case we get four equations of the type described in (5.4), namely 1. X 3 - - = O , y3 = 0 : : ~ k l = O.
2. z a = l ,
y3=O~7(a)=
3. x3 = 1, Y3 = 1 ~
7(a)
~2 a n d T ( b ) =
~2 ~ k 2 , k 3 E 2 Z .
= k~t 2 =~ k 4 E 2 ~ .
4. z3 = 0, Y3 = 1. This equation does not imply new conditions. This allows us to conclude with: H2(Q,Z)
= Z @ (Z2) 3.
E x a m p l e 5.2.5 Take Q -~ Z 2 :~Z~ as in the previous example. C~
( 01 01)" E x t e n s i ~
are of the form E(K): < a,b,c,d,a
II [d,a]= 1 [d, c] = 1 [c,b] = i
[d,b]= 1 [c, a] = 1
> .
a a = a - l a c k 2 d ~2
[b,a] = c k ' d h ab = b - l a c k 3 d z3
a 2 = ck4d 14
OtC = d a
ad = ca
However, not every K = (kl, k2, k3, k4, 11, 12,13,14) is computational consistent. It is not difficult to get the following consistency conditions: 1. From the above presentation we deduce that ba = a b c k l d 11
~
a b a a -1 = a a b c k l d l l a -1
::::k b - l a - l c t 2 + t ~ d
k~§
= a-lb-lcll+12+13d kl§247
and so it follows that kl = 11 since b - l a -1 = a - l b - l c k l d h . 2. Notice that a a a -1 = a - l c ~ 2 d k2 and thus a a - l a -1 = ac-12d -k2. Use this in a a = a - l o t c k 2 d l~ :=:k otaaoL - 1 = a a - l a - l ( : t 2 c k 2 d l 2 a :==ka c k 4 d 14 = a c - l ~ + k 4 + k 2 d -k2+14+12 w h i c h
implies
-1 k2 = 12.
Chapter 5: The Cohomology of virtually nilpotent groups
112
3. S i m i l a r l y one finds t h a t k3 = 13. 4. Also a . a 2 --
a2.a ~ ack~d14 = ck4dZ*a ~
k 4 ~-
14.
T h e s e f o u r c o n d i t i o n s are sufficient for K to b e c o m p u t a t i o n a l consistent, since for e a c h K we h a v e t h e following f a i t h f u l r e p r e s e n t a t i o n A : E ---+Aft(R4):
l oo o) /lo oo/
a
0
1
0
0 0 0
0 0 0
1 0 0
0
0 1 0
1 0 1
b~--~
0
1
~2
0
0
0 0 0
0 0 0
1 0 0
0 1 0
0 1 1
0 0 0 0
1 0 0 0
(10000//10001) c~-*
0 0 0 0
1 0 0 0
0 1 0 0
0 0 1 0
1 0 0 1
/o1 1 0 0 0
a~---,
0 0 0 0
d~-+
k2 -1 0 0
ks 0 -1 0
0 1 0 0
0 0 1 0
0 0 0 1
2
0 0 1
9
L e t x = a~lbXUot x3 (resp. y = ambY2aV3), w h e r e ~3 (resp. Y3) is equal t o 0 or 1. fK(X, y) c a n b e c a l c u l a t e d f r o m t h e m a t r i x r e p r e s e n t a t i o n a b o v e . T h i s r e s u l t s in
f : QxQ___~Z2 : (~,y) ~.__~( ( - l f f 3x2ylk~ § x3ylk2 -4:-z3y2k3 § ~3y3k4 ) ( - l f f 3 z 2 Y l k l + x3Ylk2 + x3yzk3 + z3y3k4 " S e a r c h i n g w h i c h f = 67 for 7 : Q ~ ~2 easily implies kl = 0 (see also p a r t 2.(a) in p r o p o s i t i o n 5.2.3). M o r e o v e r , if kl = 0 t h e n fK = ~7 for
7 : Q --* 7"2 : a~lbX2a~3 ~
~1
+ x2
0
+ x3
0
"
W e conclude: 2 Hv(Q,
Z 2) = Z .
N o w , we do n o t longer r e s t r i c t ourselves to the v i r t u a l l y a b e l i a n case, b u t we s u p p o s e t h a t Q is a n a l m o s t - c r y s t a l l o g r a p h i c g r o u p , w i t h a g r o u p N o f r a n k n as its n o r m a l , m a x i m a l n i l p o t e n t s u b g r o u p . T h e w a y t o
Section 5.2 The cohomology for some virtually nilpotent groups
113
compute H2(Q, •) will be completely the same as the way we followed in the proof of proposition 5.2.3. We suppose that a canonical type representation p : Q ~ Aff(R '~) is provided for Q. (This is certainly possible if N is of class <_ 3.) Again, we assume t h a t an action ~ : Q ~ Aut (~m) is given, with ~ ( N ) = 1. Moreover, we suppose that the following, rather restrictive, property is fulfilled, namely we assume that the homomorphism
H2(Q, Z TM)
~ H2(Q, Afr(R ~, R'~))
is trivial. This assumption is needed, in order to be able to lift the canonical representation p of Q to any extension of Q by Z TM, compatible with the given action. This condition is very complicated to investigate and involves the computational consistency conditions. We r e m a r k here however, that there is one fact which simplifies things quite a bit. We write this down in the following lamina. L a m i n a 5.2.6 Let G be a group, containing a normal subgroup H of
finite index. Suppose there is an exact sequence of G-modules O---* Z ~ A---~ A / Z---~ O where A is also a vector space (over a field of characteristic 0). Then, i , : H2(H,Z) ~ H2(H,A) is trivial
i . : H~(G, Z) -~ H~(G,A) is trivial. Proof: The proof is based on the following commutative diagram:
H2(G, Z) res H2(H,Z) cor H2(G,Z)
'" J~ -~
H2(G, A) I res H2(H,A) ; cor H2(G,A)
Since cor o res=multiplication by the index [G : HI, we see that [G : H] i, = 0. Now we use the fact that A is a vector space to conclude that i, itself is 0.
So in order to check the condition needed, we m a y restrict ourselves to the nilpotent group N. Once we have a group Q, satisfying the conditions
114
Chapter 5: The Cohomology of virtually nilpotent groups
above we can now repeat the procedure of proposition 5.2.3 to compute the group H2(Q, ~m). As we know that any virtually nilpotent group admits a stable affme representation and by t h e o r e m 4.4.24, we m a y conclude t h a t T h e o r e m 5.2.7 Let Q be an almost-crystallographic group with a 2-step nilpotent Fitting subgroup. I f !a : Q ~ Aut ( z m ) is any representation factoring through the holonomy group of Q, then there exists an algorithm to compute H v2( Q , Zm).
In stead of writing out very carefully the proof of this theorem, it is more helpful to the reader to present an instructive example. Example
5.2.8
Consider Q : < a, b, c, a II [b, a] = c 2 aa = a - l a Ct 2 ~
C
[c, a] = [c, b] = 1 > . ab = b - i a OLC ---- COt
One can check t h a t Q is an almost-crystallographic group and we propose the following stable representation p for Q:
p(a) =
p(c)=
1 0 0 1 0 0 0 0
-1 0 1 0
1 0 0 1 / 0 1 0 0 0 0 i 0 0 0 0 1
0 1 0 1
J
p(b)
1 1 0 O~ 0 1 0 0 0 0 1 1
=
0
1 o 0 -1 P(a)=
0
0
o 0
1
1/2 0
0
0
-1
0
0
0
0
1
J
"
We choose an action of Q on ~ by letting a, b, c act trivially on Z and a non trivially. As p is a stable affme representation, we know (theorem 4.4.24) t h a t the m a p H 2 ( Q , Z) ~ H2(Q, Aff(]~ 3, R1)) is trivial.
S e c t i o n 5.2 T h e c o h o m o l o g y for s o m e v i r t u a l l y n i l p o t e n t g r o u p s
115
An extension of Q by Z inducing the action proposed above, has a presentation of the form: E:
< a,b,c,d, all
[ b , a ] = c 2 d k~ [c, a] = d k2 [c,b] = d k3 a a = a - l o r d k4 ab = b - l a d k5
[d,a]= 1 [d, b] = 1 [d,c] = 1 oL2 = cd k~ ad = d-la
>.
c~c = c a d k~
The first thing to do is to search the consistency conditions. This is done by checking for which ki's we can construct a canonical type aft-me representation for E , based on p. The fact t h a t we know t h a t the m o r p h i s m H2(Q, •) ~ H2(Q, Aff(]~3, R1)) is zero, implies that we are able to construct a canonical type affine representation for a group E if and only if the parameters (kl, k 2 , . . . , kT) are computational consistent. Using this procedure, we fred the following equations, which should be satisfied: k6 = - 2 k 7 k2 2k4 k3 2k5 kl 4(k4 + k5 + k7). We introduce the new symbols 11 = k4, 12 = ks and 13 = kT. Now, an extension is determined by the three parameters ll, 12, 13 and has a presentation E : < a, b, c, d, a [] [b, a] = c2d 2(11+/2+13) [c, b] = d 212 [d,a] = 1 [d, b] = 1 a a = a - l o l d 11 a 2 = cdl3
[c, a] = d 2ll
> .
[d,c] = 1
ab = b - l a d h otc _w_ cord -213
ad = d-la
If-412
A representation for such a group E looks like 1 0
A(a)=
0 0 0
-4ll 3
1 0 0 0
0
0
1 0 0
1 0 0 1 0 0 0 0 0 0
2/1 3
-13 -1 0 1 0
~r 3 0 1 0 0
0/ 0 1 0 1 _~ 3 0 0 1 0
A(b)=
0 1
0 0 1
A(d)=
-212 3 + 13 0 1 0 1 0 0 1 0 0
3 1
0 0 0 0
0 0 0 1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
1\ 0 0 0 1
J
0 0
0 1 1
116
C h a p t e r 5: The Cohomology of virtually nilpotent groups
O
~(~) :
1
0
o
0 0
0 0
3
3
0
1
0
0 0
0
0 1
Now we are ready to compute the cohomology group H ~(Q, Z). Of course, we work with the section s : Q ~ E : a~bUcZa ~ ~-* a~bUc~d~ ~, x , y , z ~ ~ , ~ ~ {0, 1}.
We compute the cocycles f ( m , n), m, n ~ Q with respect to this section and compare t h e m with 57(m, n), where 7 : Q ~ Z : aZbYcZa ~ ~ xT(a ) -k yT(b) § zT(c) q- eT(a). This computation is done with the m a t r i x representation and the result is summarized in the table below: f ( m , n) m
it
f ( m , n) - 57(m, n) 2laxzyl + 2l:x~yl + 21~xzy~ + 2llxzy~ + 21:xzy: + 412x ~yl y~. aY~bY~cya
2.~(c)~yl + 2 t ~ : y l + 21:~y~ + 2~ ~ + ~ ~ + 2l:xsy~ + 4l:x:y~y: 2lzx2y~ + 2l~x~yl + 2l~xay~ + 2l~x~y~ + 21~xay~ + 4l~ x ~y~ y:
aZ~b~cZ~
a y~ b y~ c ya a
27(c)x2yl + 213x2yl + 212x2yl + 2/1$3Yl + 21lx2y~ + 212x3y2 + 412x2yly2 --(Z~ Yl ) -- 213x:yl -- 21~ xiY~ -- 21~ x3y~ + 2Zl x : y ~ --
12y2 -- 212x3y2 + 412x:yly2 + 2l~ya aZabZ~cZaa
aY~bYacya
- ( t l y~ ) - 2 z ( c ) x : y ~
- 213 x~y~ - 2t: ~ i y ~ - 2z~ ~ 3 y l +
2ll x2y~-12y2-212 xay2 +4/2 x2yl Y2+27(c)y3 +21aya 13 - It Yl - 213x2yl - 212x~yl - 211~3Yl + 211x2y~ 12y: - 212xay2 + 41:x2yly2 + 21ay3 aZ,bZ~cZa~
aY~bY~cya ~
~ ( c ) + t~ - l ~ y l
- 2z(c)x~yl
- 2l~x~y~ - 2z~gy~
-
2llzayl + 21lx2y 2 - 12y2 - 212z3y~ + 41~x2yxy2 + 2"y(c)y3 + 2lay3
The group of standard cocycles SZ2(Q, Z ) is the free abelian group on three generators 11,12 and 13. The above table shows that a standard
Section 5.3 The c o h o m o l o g y o f virtually abelian groups
117
cocycle f of an extension determined by (ll, 12, 13) is cohomologous to zero iff l,~ = 0 and 12 = 0. (We have to take 7(c) = -13 to find a coborder 57 which equals f ) . Therefore, we conclude that H : ( Q , Z ) = Z a / g r p {l~} ~ Z ~
5.3
M o r e a b o u t the c o h o m o l o g y of virtually ab e l i a n groups
Again, let Q be any virtually finitely generated torsion free abelian group. (e.g. Q is a crystallographic group). Q fits in a short exact sequence 0..
~Z n
~Q--.F
~I
such that F is a finite group. We suppose that F is generated by the elements a~, a2,. 9 9 ak. We denote the action of F on Z ~ via conjugation in Q, by p : F ~ A u t Z ~. T h e o r e m 5.3.1 For any action ~ : Q --~ Aut Z T M which factors through F (i.e. ~ ( Z n) = 1), the rank of H ~ ( Q , Z m ) is completely determined by p and ~. In other words, for any group < Q1 >E H2(F, ~n) we have that rank(H~(Q, Zm)) = r a n k ( H v2( Q ~ , Z m )). Proof: Whenever we use " a ' , we will m e a n one of the al's of F. Any extension E of Q by Z'~, compatible with ~ has a presentation of the form E :<
a l , a 2 , . . . , a n , bl, b 2 , . . . , b , ~ , a l , a 2 , . . . , a k I] [al, aj] = vhkl,l,jhk2.i,j 1 v2 . .b~m . "~''J. (1 . < j < i < n) [bl, b j ] = l ( l _ < j < i _ < m ) [bi:aj] = i (1 < i < m, 1 < j < n) a a i a -1 = a 1 a 2 . . . a n ' b 1' . . . b i n ' a b i a -1 h~x'~ h~'~'~ (1 < i < m)
>
(1 < i < n)
o . ,
where p ( a ) = (/~i,j)l
=
ab1-k,.i,j . .. bkm.~,i.ja _ l m
:
bF
m
m
...
Chapter 5: The Cohomology of virtually nilpotent groups
118 and
a[al, aj]a -1
=
[aaia - 1 , a a j a -1]
~--
[a I
=
9
9
H
n
, ~1
j
9 9 .an
[ap,aq](flv,'~qO-/3q,d3v,j)
n>p>q>l
(/3q,j /3q,~) ~p,j
det
n
= H In ~]p > q ~ l
~p,i
kl,p,q
bt
/=1
By comparison we find that the kl,i,j must satisfy a system of linear equations, completely determined by p and ~, namely Vi, j (1 < j < i _~ n), Yl = 1 , . . . , m ,
~
det
3q,j 3~,~ kz~,~= ~ 7~,tk~,~,S /3v,J
n~_p>q~_l
Va:
[3v,~
'
(5.5)
t=l
This s y s t e m contains krnn(n2 - 1) equations in m n ( n ~ - 1) unknowns kl,l,j.
Although we did not describe all consistency conditions involved, we n ( n - 1) will be able to show that the rank of the subgroup of Z T M ~ which forms the set of solutions of (5.5) is equal to the rank of H~(Q, zm). Therefore it is enough to show that for every solution (kt,~,j) of (5.5), there exists an extension E determined by a cocycle g, < g >E H~(Q, Zm), such that res < g >E H 2 ( Z n, Z TM) determines the group <
al, a 2 , . . . , a n , bl, b 2 , . . . , b m 1]
~ ' ' ~" [ai, aj] = b "~kl"'~b~"'" 1 9. 9bm
[b.bs]=l (l<j
>.
(5.6)
(1 _< j < i _< n)
m, 1 < j < n)
for some l E N0. We call N the group with a presentation (5.6) in which l = 1. If we choose the section s to be defined as
s : Z "~ --. N : a~ 1 . .a~" . . .~ . a~ 1
a n~,.,o Ol 9
o bm
then the corresponding 2-cocycle f satisfies
f(a~' ... anz.. , a y, .
a~~) . = (. ~ n~i>j>l
. kl.l,jxiyj,.. ,
E n>i>j>l
Section 5.4 Application to the construction of AC-groups
119
W i t h means of the group N we define the following function g: g:Z"•
(x,y)~[s(x),s(y)]eZmCN.
This functions g has the following properties: 1. g is a normalized 2-cocycle from TJ~ to ~'~. 2. g is cohomologous to 2 f , because g = 2 f + 57 for 7 : 2~n ---~ z m : a z1 ' ' " a ~
"~'-+ ( Z
kl,i,jxixj,. 99 ~
n>_i>j>l
km,l,jxixj).
n>i>j>l
3. Vf~ E F : g(~z, 13y) =~3 g(x, y). This is seen because the exactness of (5.5) guarantees this property for/3 = c~. 4. res o corg ~ ]Fig ~ 21Fir We m a y explain this as follows. By [11, Prop. 9.5, p. 83] we know that r e s o c o r g ~ ~ 1 3 g . Since for x , y E Z m we have that
13EF
(f~g)(x,y) =f~ g(f~-lx,/3-1 y) = g(x,y), we m a y indeed conclude that IFIg.
res o corg ~
Conclusion: The cocycle cor g is the cocycle we were looking for.
5.4
A p p l i c a t i o n to the construction of a l m o s t crystallographic groups
9 Suppose Q is a 2-dimensional crystallographic group and we try to compute Hv(Q, 2 ~) for art action which factors through the holonomy group of Q. We are interested in those cases where rank H2(Q, ~) > O. For if the rank H2(Q, Z) = 0, then all extensions will be virtually abelian and we really whish to construct virtually 2-step nilpotent groups (see section 5.1). So the only possibility is a rank = 1. Let c~ be any element of the holonomy group F of Q. The system of equations (5.5) now consists of only one equation. We write
ll 12 ) 13 /4
and ~ ( a ) = e = •
W i t h these notations (5.5) becomes dee ( l l
/3
12) 14 k2,1 -- 6k2,1.
120
Chapter 5: The Cohomology of virtually nilpotent groups
Or we may conclude that rank(H2(Q, 2)) = 1 r
= det
11 12
* Now suppose that Q is a three-dimensional crystallographic group with holonomy group F and write
=
kl 11 ml ) k2 I2 m2 and ~(ct) = e. k3 I3 m3
Now the system of equations (5.5) reads as det det
k2 12
k2,1 §
ll) 31 dot( 2 z2)
k3 13
k3 13
k3,2 : sk2,1
(kl ml) (kl ml)k3,1+det(k2 m2)k32=ek3,1 k2 m2 k2,1 + det k3 m3 k3 m3 '
And so the dimension of the vectorspace of solutions of this system of equations equals the rank of H2(Q, 7.). Moreover, one can check that for crystallographic groups with holonomy F = Z4 or F = Z6 one has to take e = 1 in order to have a non zero rank. Therefore, we may conclude this section with the following theorem: T h e o r e m 5.4.1 Let Q be a 3-dimensional crystallographic group. Suppose Q acts on Z via a morphism ~ which factors through the holonomy group F of Q. If there exists an element (x E F of order ~ 2, acting non trivially on Z, then rank(H~(Q, ~)) = O.
Chapter 6
Infra-nilmanifolds and their topological invariants 6.1
3-dimensional Almost-Bieberbach
groups
As indicated already, a 3-dimensional AC-group E (which is not a crystallographic group) fits into a short exact sequence 1 ~ N ~ E ~ F --* 1, where N is the unique m a x i m a l nilpotent normal subgroup of E , and N is a lattice of the Heisenberg group H (see (2.1)). It is known t h a t such N is isomorphic to one of the following groups, for some k:
Yk: < a,b, cll [b,a] = c k, [c,a] = It, b] = 1 >, k r 0.
(100/ (110)(101jk)
This group is realized as a uniform lattice of H if one takes
a
--
0 0
1 0
1 1
, b=
0 0
1 0
0 1
, c=
0 0
1 0
0 1
.
Remark t h a t Nk is isomorphic to N - k and is nilpotent of class 2. It is easy to see t h a t for each k # 0, Z = ~ / [ N , N ] = grp{c} '~ Z . This implies t h a t N k / Z ~- ~2. Suppose we have a 3-dimensional AC-group E containing Ark as its F i t t i n g subgroup. Applying proposition 2.4.2, we know t h a t E / Z is a 2-dimensional crystallographic group (i.e. wallpaper group) containing N k / Z ~ Z 2 as a m a x i m a l abelian normal subgroup (i.e. translation subgroup). Conversely, fix Nk and assume we are given any 2-dimensional crystallographic group E / Z not. Q, i.e. 1 ~ N k / Z ~- Z 2 ~ Q ---+ F ~ 1.
122
Chapter 6: Infra-nilmanifolds and their topological invariants
Assume the F - m o d u l e structure of Z 2 is given by r : F ~ Gl(2, Z). We can try to build up all possible AC-groups E , giving rise to a commutative diagram (2.6), with N = Nk. Here we make the following observations which will be of great practical impact. O b s e r v a t i o n 1. Since Z lies in the center of Nk, the action of Q = E / Z on Z (induced by the sequence 1 -* Z -+ E ~ Q = E / Z ~ 1) must factor through the finite group F and so is completely determined by an F - a c t i o n on Z. O b s e r v a t i o n 2. An extension E of Q = E / Z by Z, which is compatible with such an action, will be an AC-group containing Nk as maximal normal nilpotent group if and only if the restricted extension of N k / Z ~ Z 2 by Z is a group isomorphic to N~. O b s e r v a t i o n 3. On principle, the F-action on Z can be chosen. Since F is finite and Ant Z ~ Z2, the choice is (very) limited. However, by (5.7) there is no choice at all. In our search for the torsion free AC-groups (= the AB-groups), we need a criterion to detect torsion. Here we use the following well known lemma: L e m m a 6.1.1 Let Q be any group. Assume that a Q-module structure on z m is given by ~ : Q --. Aut(~,m). Write n(q) for the order of a torsion element q in Q. Then, an element < f >E H~(Q, Z TM) determines a group with torsion if and only if there exists a torsion element q E Q and an element z E Zm, such that (1 + ~(q) + . . . + ~(q)n(q)-l)z = f ( q , q ) + f(q, q2) + . . . + f(q,q,~(q)-l).
Moreover, the order of a torsion element in the extension determined by < f > equals the order of its image in Q.
As a consequence, we notice that the order of an element in a crystallographic group (and in an AC-group) is always a divisor of the order of its holonomy group F . In our context of AC-groups, we apply the lemma with m = 1. C o r o l l a r y 6.1.2 Consider an extension 1 ---* ~ ~ E J-~ Q -+ 1 where Q is a wallpaper group or an A C-group of dimension 3.
Section 6.2 Classitlcation of rank 4 nilpotent groups
123
1. If there exists a torsion element in Q acting non-trivially on Z then E has torsion. 2. If every torsion element in Q acts trivially on Z, then checking the presence of torsion in E can be done in a finite number of steps. Proofi Take ~ E E such that q = j(~) E Q is of order n(q) and acts non-trivially on 2 . Let ~ be generated by z. Then, ~z = z - l ~ . Since ~n(q) = z z E Z, we can verify that qq~(q) = ~ ( q ) q ~ qz z = zZq ~ z - l O = z~(t ~ l = o. This implies that ~ has order equal to n(q). Now, assume that every torsion element in Q acts trivially on Z and that the extension E is determined by a cocycle f : Q x Q --+ 2 . Checking for torsion in E is equivalent to looking for an element q of finite order in Q such that
f(q, q) § f(q, q2) + . . . + f(q, q(n(q)-~)) = 0 rood n(q). As noticed before, in some cases (e.g. when Q is crystallographic or AC of dimension 3) f can be chosen to be a standard cocycle and consequently it is a linear expression in a finite number of parameters K = (k~, k 2 , . . . , ks). Now, the condition above shows that E will be torsion free or not, depending only on the values of the ki m o d n (1 _< i < s), where n is the order of the holonomy group F associated with Q. []
There will be more information on how to detect torsion in section 6.6 and in the appendix. At this point, we are ready to start the computations to obtain all AC-groups of dimension 3. We refer to section 6.3 for an outline of the steps to follow in such a computation. For each of the 17 wallpaper groups Q, and each of the Nk, we determined the corresponding AC-groups. The results are summarized in chapter 7.
6.2
Classification of rank 4 nilpotent groups
Let N be a finitely generated, rank 4, torsion free uilpotent group. Then N is of class 1, 2 or 3. We will show that this nilpotency class completely determines the Mal'cev completion of N . In fact we give a standard comm u t a t o r presentation for each isomorphism type. Also we show for more
124
Chapter 6: Infra-nilmanifolds and their topological invariants
general c o m m u t a t o r presentations how to obtain this standard commutator presentation. During this section we will use the convention that in a c o m m u t a t o r presentation all commutators which are not explicitely written down are trivial. E.g. the presentation for the group Nk of the previous section is given by Nk : < a, b, c ]l [b, a] = c k > .
6.2.1
N is a b e l i a n
( c l a s s 1)
N ~ - Z 4,
N :< a , b , c , dII
>.
N is a uniform lattice of ]R4. 6.2.2
N is o f c l a s s 2
In order to show that any group N of rank 4 and of nilpotency class 2 has a center of rank 2, we prove the following proposition: P r o p o s i t i o n 6.2.1 Suppose N is a nilpotent group with a presentation g:<
a , b , c , d [ ] [ b , a ] = d ~, [c,a]=d/~, [ c , b ] = d e >
(6.1)
then N can be given a presentation
g :< a,b,c,d[[ [b,a] = d (~'~'~) >, where (a,/3, 7) denotes the (positive) greatest common divisor of a, /3 and 7.
Proof: Suppose N is given by a presentation (6.1). By eventually renaming the generators we m a y assume that a > 0. (a) Suppose/3 ~ 0 (If/3 = 0 go to part ( b ) ) . ( a l ) We choose a new set of generators for N, namely a' = a, b~ = b, c ~ = bxc, d ~ = d for some ~ C Z .
We determine the new presentation of N by computing all the commutators of the generators (use [56, lemma 4.1 p.93]): [d', a'] = [d', b'] = [d', c'] = 1 [b',a'] -- [b,a] = d ~ = d '~
Section 6.2 Classification o f rank 4 nilpotent groups
125
[c',a'] = [b~c,a] = [b,a]~[c,a] = d '/3+~x [c', b'] = [bZc, b] = [b, b]~[c, b] = d "~. By choosing an a p p r o p r i a t e z we can o b t a i n / 3 + zc~ = (/3 m o d a). Therefore N can also be presented as: N :< a, b, c, d [l [b, a] = d ~ [ c , a ] = d ~m~
[ c , b ] = d 7 >,
where we deleted the accents. (a2) If/3 rood a r 0 we again choose a new set of generators:
at=a,
b~= cZb, c ' = c, d ' = d f o r
s o m e z C Z.
C o m p u t i n g the c o m m u t a t o r s gives: [d', a'] = [d', b'] = [d', c'] = 1
[b', a'] = [cXb, a] = [c, a]X[b, a] = d a+x(~m~ [c', a'] = [c, a] = d '~r"~ [c', b'] = [c, c~b] = [c, c]~[c, b] = d '7. This shows t h a t we can reduce a to a m o d (/3 rood a). B y r e p e a t i n g steps ( a l ) and (a2) we finally get a p r e s e n t a t i o n for N of the form:
Situation 1: N :< a,b,c, dll [b,a] = d (~'z), [c,b] = d v > or
Situation 2: g :< a,b,c,d[I [c,a] = d (~'~), [c,b] = d r > .
In Situation 2, now consider the new set of generators: a'=
a, b ' = c ,
c ' - - b -1, d I = d ,
a n d a new p r e s e n t a t i o n of N is obtained: g :< a,b,c, dll [b,a] = d (a'z), [c,b] = d ~ > which is Situation 1.
126
Chapter 6: Infra-nilmanifolds and their topological invariants
(b) The starting point for step (b) is a presentation for N as in Situation 1 of step (a). By repeating simular steps as (al) and (a2) (now adjusting generators a and c in stead of b and c) one finally finds: N :< a,b,c,d]] [b,a] = d (~'z'7) > .
C o r o l l a r y 6.2.2 Every 2-step nilpotent group of rank 4, has a center of rank 2. P r o p o s i t i o n 6.2.3 Let N be a group with a presentation N :< a,b,c,d]] [b,a] = c~d ~ >
then N can also be presented as N :< a,b,e,d]] [b,a] = d (a'~) > . Proof: Let (a, fl) = ka -4- lfl for k, l E Z. We consider the following set of new generators for N:
a t = a, b' = b, c' = cl d -k, d' = c ~ - Y d C ~ . It is now easy to see that N :< a',b',c',d' II [b',a'] = d '<~'z> > .
C o r o l l a r y 6.2.4 Every finitely generated 2-step nilpotent group of rank 4 has a presentation of the form N :< a,b,c,d]] [b,a] = d k >,
for some k E ~ , k > O. Moreover this presentation is uniquely determined by N and therefore we call it the s t a n d a r d p r e s e n t a t i o n of N . Proof: The existence of such a presentation is guaranteed by the preceeding propositions. The uniqueness is also clear since such a presentation shows that N ~ Ark x Z, where Nk : < a,b, d ]] [b, a] = d k >. By e.g. looking at the quotient Z ( N ) / [ N , N] of characteristic subgroups of N , we see that Nk x Z -~ N1 x Z iff k = • C o r o l l a r y 6.2.5 Every 2-step nilpotent group of rank 4 can be seen as a uniform lattice of H • •, where H denotes the 3-dimensional Heisenberg group.
Section 6.2 Classitication o f rank 4 nilpotent groups 6.2.3
127
N is o f c l a s s 3
Given a group N of n i l p o t e n c y class 3 and rank 4~ we m a y a s s u m e t h a t N has a p r e s e n t a t i o n of the form: N:
dl] [ b , a ] = c ~ d ~ > [c, a] = a" [c, b] = d ~
(6.2)
w i t h a > 0 and 7 # 0 or 5 # 0. P r o p o s i t i o n 6 . 2 . 6 A group N with a presentation given a n y of the two presentations: N :< a, b, c, d I] [b, a] = c~d +f3m~ [c, a] = d("~'~) b] = 1 Proof:
(6.2) can also be
> .
Suppose N is given by a p r e s e n t a t i o n (6.2).
(a) Suppose 7 7~ 0 a n d 5 # 0. ( a l ) We choose a new set of generators:
a' = bX a, b' = b, c I = c, d' = d, for which [d', a'] = [d', b'] = [d', c'] = 1
[c', a'] = [c, b~a] = [c, b]Z[c, a] = d ' ~ + ~ It', b'] = It, b] = [b', a'] = [b, b~a] = b - l a -1 b-~bb~a = [b, a] -- c'ad t~. B y an a p p r o p r i a t e choice of x we m a y reduce 7 to 7 rood 5.
(a2)
Suppose "y m o d 5 ~ 0. Take as new generators:
at=a,
b~ = a ~ b , e ' = c ,
d'=d
so t h a t [d', a'] -- [d', b'] -- [d', c'] = 1 [c', a'] -- d "rm~ [c', b'] = [c, a]~[c, b] = d '~+~(~m~ [b',a'] = [aXb, a] = b - l a - x a - l a X b a
= [b,a]-= c'ad '~,
showing we can reduce 5 m o d u l o 7 m o d 5.
128
C h a p t e r 6: I n f r a - n i l m a n i f o l d s a n d their t o p o l o g i c a l i n v a r i a n t s
B y r e p e a t i n g steps ( a l ) a n d (a2) a finite n u m b e r of times we find S i t u a t i o n 1:
N :<
a,b,c, dll
>
[b,a] = c~d ~ [c, a] = d (~'~)
[~,b] = 1 or
S i t u a t i o n 2:
N :< a , b , c , dll [b,a] = c~'d z > . [c, a] = 1 [c, b] = d(~'~) In case of S i t u a t i o n 2, consider t h e g e n e r a t o r s aI
~-
b, b ~ ~ a, c I
z
c -1,
d !
= d-l~
w h i c h t r a n s f o r m S i t u a t i o n 2 into S i t u a t i o n 1. (b) In S i t u a t i o n 1 consider t h e g e n e r a t o r s a I
=
a -1
'
b'
=
b, c'
=
c -1
,
d'
--
d
[b', a'] = [b, a-l]. Since ba = abc~d f~, a - l b = c~d~ba -1, a n d so ba -1 = c - a a - l b d - ~ , implying: [b I, a'] = c'~d I-~+~('n6), while [c', a'] = d r
[c', b'] = [d', a'] = [d', b'] = [d', c'] = 1.
This leads to S i t u a t i o n 1 ': N :< a, b, c, d II
[b, a3 =
c~d-~+~< ~,~) > . [c, a] = d(~,~)
[~, b] = 1 (c) ( c l ) T h e s t a r t i n g point for this step is either S i t u a t i o n 1 (call/3' = fl) or S i t u a t i o n / ' ( c a l l fl' = -t3 + a ( 7 , 5)). The new generators: a' = a, b' = c~b, c' = c, d' = d
give:
Section 6.2 Classification o f r a n k 4 nilpotent groups
129
[d',aq = [d',b'] = [d',c'] = [c',b'] = 1, [c',a'] = d !('~'6) [b', a !] = [c%, a] -- b - l c - Z a - l c z a a - l b a = [e, ~]~[b, a] = c"~d!~'+~('Y,~)
,
s h o w i n g t h a t we can r e d u c e / 3 ' m o d u l o (7, 5). (c2) T h e s t a r t i n g point is t h e result of (cl). We consider
a ! = a , b! = b , c ' = cd x, d ! = d to o b t a i n [d', a'] = [d', b'] = [d', c'] = [c', b'] = 1 [c', a'] = [c, a] = d '(7'6) [b', a'] = [b, a] = c'"d '~'-z". So it is possible to r e d u c e / 3 ! m o d u l o a. B y a g o o d r e d u c t i o n in ( c l ) a n d (c2) one m a y r e d u c e /3' m o d u l o ( a , 7, 5). This shows N :< a, b, c, d [] [b, a] = c~d +~m~
> .
[c,a] = d ( 7 , 5 ) [~, b] -- 1 Conversely, Proposition N:<
6 . 2 . 7 Suppose
a,b,c, dl] [b, a] = c~d ~3 > ~ M :< a,b,c,d[I [c, a] = d r [c, b] = 1
with a, 7, a',
[b, a] = c~'d ~' > [e, a] -- de [c, b] = 1 (6.3)
> O, then a'=a,
7'=7,
Proofi S u p p o s e T : N --+ M is by: ~(d) ~(c) ~(a)
and/3'= • a n i s o m o r p h i s m of groups, t h e n W is given = d• = c+ld ~ = am'bm2w(c,d)
~(b) = a~lb~w(c, d)
130
Chapter 6: Infra-nilmanifolds and their topological invariants
where w(c, d) denotes a not further specified word in c and d and
11
[~(c),~(b)]
12
E Aut (E2).
=
1,
=
[c~=ldk, allbl~w(c,d)]
_-
It, a] ll
=
d-t-7'll.
Therefore 11 = 0 ~ m l = =kl, 12 -- :J=l. [c, a]
~(a rr) = d •
=
[~(c), ~(a)]
[c~ l , a m l b m2] d• ' = d• '.
=
= ::~ 7 = 71 . ~[b,a] [bl2w(c, d), amlbm2w(c, d)]
~
= =
~v(c~d ~) = c+'~dk~'+~, [b12cv, a T M bin2 c q]
=
c-Pb Tlc-qb-m2 a ~1 b=t:lcPa+l bin2 c q
= =
c-%-m2[b ~=l,a• c•177
O~I ---- 0~.
=kfl = fl' + s(7, a) ~ fl = =kfl' rood (% a).
We call a presentation for a rank 4, 3-step nilpotent group N of the form (6.3), with t3 < ( a , 7 ) / 2 the S t a n d a r d p r e s e n t a t i o n for N . By playing the same game as in proposition 6.2.6 on the continuous level, one can see t h a t every torsion free 3-step nilpotent group of rank 4 is a uniform lattice of •3 >4]~ where the action of 1 is given by: 1
1 0 /
0
1
1
0
0
1
Section 6.3 4-dimensional Almost-Bieberbach groups
6.3
4-dimensional
Almost-Bieberbach
131
groups
9First we look at the 4-dimensional A l m o s t - B i e b e r b a c h groups E whose Fitting subgroup N is a group of nilpotency class 2. In view of corollary 6.2.4, N has a presentation of the form N :< a, b, c, d I] [b, a] = d k, c and d central > for some integer k r 0. Since ~ / [ N , N ] = grp{d}, such a group E fits in a short exact sequence 1 --* 2~ -~ E ~ Q --+ 1, with Q 3-dimensional crystallographic. The possible actions of Q on Z are described in section 5.4. The construction of all possible E ' s is analogous to the 3-dimensional case. For more specific information, we refer to the appendix, where all computations involved in dimension 4 are carried out for a concrete case. At this point we limit ourselves to an indication of the steps involved in the construction of AB-groups: 1. C h o o s e a n A C - g r o u p Q. The basic idea behind our theory was the reduction lemma 2.4.2. This shows that A C - g r o u p s E having a normal maximal uilpotent subgroup of nilpotency class c > 1, are built up from an A C - g r o u p Q with a normal maximal nilpotent subgroup N of class c - 1. Fix such a Q. We have to look at all extensions of Q by some torsion free abelian group Z compatible with an action of Q / N = F on Z. The choice of the action can be limited by the results of section 5.4, corollary 6.1.2 and by the theory developed in the rest of this section. 2. D e t e r m i n a t i o n o f t h e c o m p u t a t i o n a l c o n s i s t e n t p r e s e n t a tions. Once the building blocks for the application of the reduction lemma are provided we have to determine the computational consistent presentations of the extensions of Q by Z, compatible with the chosen action. The way to do this is explained and illustrated in section 5.2. These computations also realize these extensions as matrix groups. 3. C o m p u t a t i o n o f H2(Q, Z). Still following section 5.2, we now compute H2(Q, Z), which is interesting as being a first (rough) indication of the isomorphism types of the A C - g r o u p s obtained. 4. I n v e s t i g a t i o n o f t o r s i o n . The next step is to determine which of the extensions E are torsion
132
C h a p t e r 6: Infra-niImanifolds and their topological invariants
free. Of great help for this is corollary 6.1.2, section 6.6 and the discussion made in the appendix. . D e t e r m i n a t i o n of the i s o m o r p h i s m types. In many cases we will find several torsion free extensions E of Q by Z. In order to really classify AB-groups, we will have to search the isomorphism classes of the groups obtained. This can be done fairly easy using the matrix representations obtained for the groups E. We give some more information in chapter 7. 9 Now we look at the other possibility. The standard presentation for a 3step nilpotent group N of rank 4, shows us that v/[N, [N, N]] = g r p { d } = Z. Therefore, if a group E is an AB-group of dimension 4, having such a N as maximal nilpotent subgroup, then E / Z is an AC-group of dimension 3 (and not a crystallographic group). Hence, we are looking for extensions of 3-dimensional A C - g r o u p s Q by ~ inducing a (restricted) extension 1 ~ Z ~- ~ --+ N ~ N / Z --+ 1. Again, we m a y derive some information concerning the actions we have to consider. Let Q be such an AC-group of dim. 3, then, Q = < a , b , c , ot(,fl)II
[b,a] = c k It, a] = 1 [c, b] = 1
>
(6.4)
c ~ a ~ - 1 = al~bl2cl3 ozbo~ - 1 _: a m x b m 2 c m 3 OteOL- 1
_~ c l l m 2 - 1 2 m l
with k > 0. Consider an action of Q on ~ = g r p { d } , such that a,b and c act trivially, say ~d = d e with e = +1. An extension compatible with this action can be presented as E=<
a , b , c , d , a ( , t 3 ) II [ b , a ] = c k d ~1, [d,a]= l [c, a] = d r~ [d,b] = 1 [c,b] = d ~3 [d,c] = l ~aot -1 = allbl2el3dr4
(~b~ 1
= arnl b m 2 c m 3 d r 5
otcot - 1 = c l l m 2 - 1 2 r n l d r S
a d = dea
>
Section 6.34-dimensional A1most-Bieberbachgroups
133
We are only interested in extensions for which (r2, r3) # (0, 0). Let us first of all notice that: :
[CtCO~- 1 ,
~ac~ -1]
= [Chm~--Z2ml,ahbl2cZ3] =
d(,2h +'3z~)(zl"~-t2ml)
while on the other hand
c~dr2ct-1 = der2. This leads to
(r211§
det( llml
m212) = er2.
(6.5)
From ct[c, b]~ -1 = ~d~3~ -1 we deduce
(r2ml+r3m2) det( llml m212) =er3.
(6.6)
We distinguish two cases
Casel'e=det( llml m212)" Equation (6.5) and equation (6.6) then look hke v2(/1- 1)+r312 = 0 r2ml + r3(m2 - 1) = 0
havingan~176176176
iffdet( ll-lml
m212- 1 ) = 0 .
Remark: the solution can be seen as an eigenvector with eigenvalue 1. aso
Equation (6.5) and equation (6.6) then look like
=0 r2ml + r3(rnz + 1) = r2(/1+ 1) +r312
havingan~176176176
0
iffdet( ll+lml m212+1 ) =0.
Remark: the solution can be seen as an eigenvector with eigenvalue --1.
Chapter 6: Infra-nilmanifolds and their topological invariants
134
P r o p o s i t i o n 6.3.1 An AC-group with a presentation (6.4) can be used to build up an AC-group of dimension 4, containing a 3-step niIpotent group as maximal nilpotent subgroup, only if
ml
m2
has an eigenvector with eigenvalue=• C o r o l l a r y 6.3.2 The A C - g r o w s listed in section 7.1 with number 10,11,...,17 cannot be used as building blocks for 4-dimensional A C-groups, containing a 3-step nilpotent group as maximal nilpotent subgroup.
6.4
On the B e t t i n u m b e r s of I n f r a - n i l m a n i f o l d s
In this section, we involve ourselves with the computation of all Betti numbers of an infra-nilmanifold of dimension _~ 4. Since we are dealing with aspherical manifolds~ any computation can be done on the level of the fundamental group and so we will actually determine Betti numbers of AB-groups. The Euler characteristic of an (infra-nilmanifold with fundamental group the) A B - g r o u p E m a y be computed as dim E
i=O
where Z is to be considered as a trivial E - m o d u l e . R e m a r k 6.4.1 In this section, we will frequently speak about the rank (Hirsh number) of a group. This number will always be defined since we only deal with polycyclic-by-finite groups. The following theorem seems to be well known. Theorem
6.4.2 Let E be any AB-group, then
x ( E ) = O. Proof: We first p r o o f this theorem for any finitely generated, torsion free nilpotent group N . Note that X(Z) = 0. Now let N be of rank k >_ 2, then N fits in a short exact sequence
1--~ Z---~ N - ~ N'---~ I ,
Section 6.4 On the Betti numbers of lnfra-nilmanifolds
135
with N ~ a torsion free nilpotent group of rank k - 1. It is known t h a t x ( N ) = X ( Z ) x ( N ' ) (e.g. see [11]) from which it follows that x ( N ) = O. Now, we consider a general AB-group E, with m a x i m a l normal nilpotent group N , and use the property that x ( E ) = X ( N ) / [ E : g ] to see t h a t indeed x ( E ) = O.
The i-th Betti number/3i of an AB-group E is defined as
ill(E) = rank Hi(E, Z) and so the Euler Characteristic is in fact the alternating sum of the Betti numbers. For the rest of this section we will be concerned with some of the Betti numbers of an AB-group. However, before we continue this investigation let us recall some facts needed further on. For more details, the reader is referred to [11]. Let E be any group for which there is a compact K ( E , 1) manifold Y. We call X the universal covering space of Y. X is an orientable space, and we denote by D, the orientation module of X. Thus D is an infinite cyclic group in which I and - 1 correspond to the two possible orientations of X . The action of E on X induces an action of E on D in the following way: an element e E E acts as +1 (resp. - 1 ) if the action of e on X is orientation-preserving (resp. orientation-reversing). E acts trivially on D iff Y is an orientable space. In case Y is orientable we will also say t h a t E is orientable. D is generally considered as a right E-module. T h e o r e m 6.4.3 [11, p.
220] For all integers i, all E-modules A and using the conventions introduced above we have that H i ( E , A) % H,~_i(E, D @ A).
We remark t h a t D | A is meant to be D |
A equiped with the diagonal
E - a c t i o n e(d | a) = d e-1 | Ca. We m a y as well consider D as a left E - m o d u l e , and in this case we fred the following: L e m m a 6.4.4 For all integers i and all E-modules A we have
H I ( E , D) ~ H,~_i(E,
Z)
where Z is to be considered as a trivial E-module.
136
Chapter 6: Infra-nilmanifolds and their topological invariants
Proof: We use the previous theorem with A = D considered as a left E - m o d u l e . This implies that H i ( E , D) ~- H n _ i ( E , D | D). Now we claim that as E - m o d u l e s D | D -~ 2 . Indeed, consider the map
A : D | D ~ Z : dl | d2 ~ dl d2. This m a p is an E - m o r p h i s m since, A(~(dl | d2)) = A(d~ -1 |
ed2)
=
A(dld2 | 1)
=
did2 =~ A(dl | d2)
which was to be shown.
We now concentrate on the case where X = G, a connected and simply connected nilpotent Lie group of class c. For any diffeomorphism : G ~ G we define
or (~) =
+1 -1
)~ is an orientation-preserving map, )~ is an orientation-reversing map.
We are especially interested in the action of G:~Aut G on G. Remember that an element ( g , ~ ) E G:~Aut G acts on G, in the following way: for any h e G : (a,~)h = g~(h). Since ( g , ~ ) - - (g, 1)(1,~) and or ( g , 1 ) = +1, it is seen that o r ( g , ~ ) = or(1,~o) = o r ( ~ ) ( A u t G acts in a natural way on G). Therefore we fix a ~ E Aut G and we try to provide a way to conclude whether ~ is orientation-preserving or not. Note that for each integer i (1 < i < c), ~ induces a morphism on 7i(G) and so on 71(G)/Ti+l(G) ~- R k' for some kl. So, the induced morphism can, after a choice of basis for R ki , be represented by means of a ki • k l - m a t r i x A~(~). This means that we can attach to each automorphism T of G, a series of matrices, A I ( ~ ) , A 2 ( ~ ) , . . . ] Ac(~). Each matrix Ai(~) is well determined up to conjugation in Aut R i. Now, if we define for a 6 R0, sgn(a) = • according to the sign of a, we have the following theorem:
T h e o r e m 6.4.5 Using the notations above: or(~)=sgnlI(detAi(~))
V~EAutG.
i=1
Proof: We will prove this by induction on the nilpotency class c of G. obvious that for c = 1, G ~ R kl and the theorem holds.
It is
137
Section 6.4 On the B e t t i numbers of lnfra-nilmanifolds
Now suppose c > 1, so we get a commutative diagram with induced morphisms ~r and ~: 1
---* % ( a )
~
G
--*
a/%(a)
~
1
1
~
-*
a
-~
a/%(a)
~
1
%(a)
It is obvious that or (~) = or (~o~).or(~). Together with the fact that A i ( ~ ) = A i ( ~ ) (1 < i < c - 1) this allows us to conclude that or (qo) = sgn(det Al(~o)).sgn(det A2(~)) . . . . . sgn(det Ac(~)).
Now we return to infra-nilmanifolds. Let E be any A B - g r o u p , with maximal normal nilpotent group N , s.t. E fits in a short exact sequence I~N~E--+F~I.
We define a sequence of free abe]Jan groups Zi (1 < i < c) by
Zi-
~/Ti(N)
All these abe]jan groups may be considered as F - m o d u l e s , where the action is induced by conjugation in E. Since Zi ~ Zk~, for some ki, we m a y choose a set of kl generators for Z~, and then represent the action of F on Zi via a morphism Bi : F ~ Aut Zi : ~ ~ Bi(~). Thus, according to each ~ E F there is a set of matrices B1(6), B2(~), 9 9 Bc((~) such that each matrix Bi(~) is well determined up to conjugation in Aut Z k~. T h e o r e m 6 . 4 . 6 Using the notations introduced above, E acts on the orientation module D in the following way: c
VaEE:
o r ( a ) = H det B i ( ~ ) i=1
where ~ denotes the projection of a in F .
138
Chapter 6: Infra-nilmanifolds and their topological invariants
Proof: Since E is an AB-group we can realize it as a subgroup of G>4Aut G, where G is the Mal'cev completion of N , more precisely, there exists a c o m m u t a t i v e diagram, with a monomorphism r 1
--~ N
+
E
+
F
+
1
1
~
~
G>1AutG
~
AutG
~
1
G
Recall t h a t r = N. For any a E E (projecting onto ~ E F ) , we can write r = (g~, ~ ) . As before, we m a y associate to ~ a set of matrices Ai(~a) (1 < i < c), such t h a t o r ( a ) = sgnl-I~=l d e t A i ( ~ ) . It is enough to show t h a t we can take A i ( ~ ) = Bi(~). By an induction argument it is sufficient to deal with the case i = c. Since Zc is a uniform lattice of 7r we can take as a basis for %(G) the same set of kc generators of Z~ as we used to establish the m a t r i x Bc(~). Let z E Zc be any of these generators then we have
~3(aza -1) = ~3(Bc((~)z ) =- (Bc(Ot)Z, 1)
(6.7)
and
r162
-1)
1 -1 -I ~-1 (go), ) 1,1)
= = =
i).
(6.8)
Comparing (6.7) and (6.8) for any generator of Zc allows us to conclude t h a t for this basis of %(G), A r = B~(~), which finishes the proof.
R e m a r k 6.4.7 This theorem also shows that if there exists a canonical representation )~ : E ~ A f f ( I ~ ) , then the action of a E E on D, corresponds to the determinant of the linear part of )~(a). Now, we really start computing Betti numbers for an infra-nilmanifold.
T h e o r e m 6.4.8 Let E be any AB-group, then rio(E) = 1. Proof: H o ( E , ~) = Z, for any group E and a trivial E - m o d u l e Z.
Section 6.4 On the Betti numbers of lnfra-nilmanifolds
139
T h e o r e m 6 . 4 . 9 Let E be any AB-group of dimension n, then
1 0
fin(E) =
~ E is orientable, ~ E is non-orientable.
Proof: We are searching the rank of Hn(E, Z). By lemma 6.4.4 Hn(E, Z) = HO(E, D) = D E. And so, if D is a trivial E - m o d u l e (E orientable) then /3n(E) = 1 else/3n(E) = O. []
6.4.10 The preceeding 2 theorems are true for any group E for which there exists a K ( E , 1)-manifold.
Remark
In order to find i l l ( E ) we have to calculate r a n k ( H i ( E , Z ) ) = rank(E/[E,E]). T h e o r e m 6.4.11 Let E be any AB-gvoup, with a maximal normal nilpotent subgroup N , then
rankHl(E/~,Z)
= r a n k H l ( E , Z).
Proof: Consider the commutative diagram 1
--*
N
-+
J.p
~p
1 --+ N/.ff-~2(U) If we denote E ' =
E
E~ ~ ( N ) ,
~
E/~/~2(N)
--+ F
~
1
~.P --+ F
--* 1
then p induces an epimorphism E
q : [E,E]
E'
~[E',E'--]"
Of course, torsion elements are mapped onto torsion elements, and conversely, suppose ~ e E / [ E , E ] (e E E ) s . t . q(~) is a torsion element. So there exists a k E No such that q(~)k = 1. This is equivalent to p(e) k e [E',E']. It follows t h a t e k E [E,E] ~/v2(N). We write e k = e l n l with el E [E, E] and nl E N such t h a t there exists an I E No for which t I for some n~ E [N,N]. We compute e kl = e l n l e l n l . . . e l n l = eanle2, e2 E [E, E]. Therefore we m a y conclude t h a t e m E [E, E] and so ~ is a torsion element in E / [ E , E].
Chapter 6: Infra-nilmanifolds and their topological invariants
140
Because q is an epimorphism, and q maps only torsion elements onto torsion elements, we are allowed to conclude that the rank of E / [ E , E] equals the rank of E I / [ E ', E'].
We remark here that
E
is a crystallographic group in which the
maximal abelian group is exactly ~ N.
So we reduced the problem
of finding i l l ( E ) to computing the r a n k ( H i ( Q , Z)) for a crystallographic group Q. This is strongly simplified by the following theorem: T h e o r e m 6.4.12 Let 1 ~ Z k ~ Q -+ F ---+ 1 be any extension of groups in which F is finite. Then
Zk 0 rank [-'Q~" = rank Q] [Q, Z k] Proof: The short exact sequence 1 ---+ Z k ~ Q ~ sequence Zk Q H2(F,Z)---* [Q,Zk-----~ ~ [Q,Q]
o
F ---+ 1 leads to the exact F ~ [F,F-----] ~ 1.
Since H2(F, Z) and F/[F, F] are finite, we must have the same ranks for Zk/[Q, Z k] and Q/[Q, Q].
T h e o r e m 6 . 4 . 1 3 Let 1 ~ Z k -~ Q -~ F ~ 1 be any extension of groups in which F is finite. Suppose this extension induces an action : F --* A u t Z k. Let c q , ( ~ 2 , . . . , a s be a set of generators for F. Then
~k rank
-[ Q ,-
zk]
:
k - rank
- I,
- L...,
- I)
in which I is the k • k-identity matrix. We remark here that we used the term "rank" to indicate two different concepts: at the left-hand side of the equality we mean the rank of a group, while at the right-hand side we want to indicate the rank of a matrix, obtained by the juxtaposition of s square matrices. Proof:
141
Section 6.4 On the B e t t i numbers of lnfra-nilmanifolds ~k
rank[Q,Zk] - k - r a n k ( [ Q , Z k ] ) . We denote by al, a 2 , . . . , a k the k generators for Z k and we choose s elements a l , a 2 , . . . , a~ E Q which project respectively to 5~1,...,(~ E F. [Q, Z k] is the subgroup of Zk generated by all elements [q, z] (q E Q, z E Zk). We remark first of all that (6.9)
Vq e Q, Vzl, z2 E Z/~: [q, ZlZ2] -= [q, Zl][q, z2].
This means that [Q, Z k] is generated by all elements of the form [q, a~-1] (q E Q, 1 < i < k). We also see that
w , 21 ~ s ~, Vq c Q: [zq, Zl] --
q-lz~z;~zqzl
= [q, Zl]
(6.10)
and
v ~ , z e Q, Vz ~ zk: [,Z,z]
=
Z-I[,,z]Z[Z,z]
=
[~,z][Z,[a,z]]-l[Z,z]. (6.11)
(6.9),(6.10) and (6.11) show that [Q,Z k] is generated by all elements [Oti- 1 ,aj -1] (1 < i _< s, 1 _< j _< k). When we use the notation aj = ( 0 , . . . , 0 , 1 , 0 , . . . , 0 ) w~ (1 on the j - t h spot) then
I~ ~ 9
[Oti- 1 ,aj -1] = a i a j a ~ l a j 1 = ~(ai)
1
0
~
-
1
0
So for a fixed i, the rank of the subgroup generated by all [Oti- 1 , a~-1] is equal to the maximal number of linear independent columns of ~((~i) - I. Now letting i vary from 1 to s, leads to the desired result.
Remark
6.4.14 We notice that the rank of H i ( Q , Z) depends only on the action ~ : F ~ Aut Z k and not on the group Q itself.
T h e o r e m 6.4.15 Let E be an n-dimensional AB-group with Fitting subgroup N and let F = E / N be generated by (~l,...(~s. by ~ the action induced by the extension 1 ~ N ~ E ~ N~ ~ ( N ) ~ Z k (for some k). Then ~n-l(E)
= k - rank (or (al)~(C~l)
- I,...,
or (~ts)~(~s)
We denote F ~ 1 on
- I)
where I is the k • k-identity matrix and ai is a lift of (~i to E.
142
Chapter 6: Infra-nilmanifolds and their topological invariants
Proof:
fln-l(E) = rankHn_l(E, •) = r a n k H l ( E , D) (lemma 6.4.4). Now we use the restriction-inflation 5-term exact sequence
0 ---+HI(F,D N) ~ HI(E,D) --+ HI(N,D) F --+ H2(F,D N) --+ H2(E,D). Notice that HI(F,D N) and H2(F,D N) are finite, and so the rank of H~(E, D) equals the rank of H~(N, D) f. We remark also that D ~ is a trivial N - m o d u l e , and so Hi(N, D) is the set of morphisms of N to Z. We take a closer look at this set. Let A : N ~ ~ be a morphism of groups. Then, for nl, n2 C N we have that A([nl, n2]) = 0. Therefore we see that each morphism A leads to a unique morphism N
Conversely, each A has a lift A : N -~ Z.
Therefore we m a y identify to 7/,. Such a m o r p h i s m A can be seen as a k-tuple of integers (A1, A 2 , . . . , A/~), in a way that for the natural set of generators al, a 2 , . . . , ak of N~ ~/-~(-N) ~- Z k
HI(N,D) with the set of morphisms from N / ~
A(ai) = (A1,A2,...,An)
= Ai.
Since for a 1-cocycle A : N --* • the action of ~-1 on iX is defined as (a71A)(n) = defined as
or(a~l)A(aina~l), the corresponding action of F on A is =
or
=
or
Therefore the rank of the elements which are fixed under this action equals the dimension of the space of solutions ( A 1 , . . . , Ak) such that for every i =
or
Therefore the rank of this group is k - rank(or ( a l ) ~ o ( a l ) - I , . . . , or ( % ) ~ ( a , ) -
I).
143
Section 6.4 On the B e t t i numbers of lnfra-nilmanifolds
R e m a r k 6 . 4 . 1 6 The morphism ~ : F -+ Aut Z k used in the theorem 6.4.13 and theorem 6.4.15 is in fact determined by the matrices B I ( ~ ) introduced above. Therefore, we can summarize the results obtained in this section as follows: Let E be any A B - g r o u p of dimension n. Suppose t h a t the h o l o n o m y group F of E is generated by a l , . 9 as , then:
fio(E)
=
1.
fin(E)
:
ill(E)
--
r a n k Z1 - r a n k ( B l ( ~ l )
iln-l(E)
=
rankZ1 -rank(or(al)Bl(~l)-I,...,or(ots)Bl(~,)-[).
1 r 0 r
E E: 3ct @ E :
or(or) = 1, or(or) = - 1 . - I, B l ( a 2 ) - I , . . . , B l ( a s )
- I).
R e m a r k 6 . 4 . 1 7 Note that indeed, for an orientable AB-group E , we do have ilo(E) = fin(E) and i l l ( E ) = i l n - l ( E ) . Note also that we now have enough information to compute all Betti numbers of a 4-dimensional AB-group E , because il2(E) can be found from the identity x ( E ) = 0 = ilo(E) -- i l l ( E ) + il2(E) -- il3(E) + il4(E).
E x a m p l e 6.4.18 Consider the following A B - g r o u p E: E : < a , b , c , d , ot II [b,a] = d 2 [c,a] = 1 [c,b] = 1 ota = a-lord ab = bot
[d,a] = 1
> .
[d,b] = 1
[d,c] = 1 ot2 = b otd = d-lot
ot C -~- C - I (~
N = g r p { a , b, c, d} is nilpotent of class 2 and F = {1, ~} -~ Z2. Z2 =
~/[N, N] ~/[N,[N,N]]
grp{d} ~ ~ . 1
The relation a d : d - l o t implies t h a t B2(~) = ( - 1 ) .
Z1 -
~ ~/[N,N]
__N ~= Z3. grp{d}
144
Chapter 6: Infra-nilmanifolds and their topological invariants
From aa = a - l a d , ab = ba and ac
BI((~) =
=
c-lo~ we deduce t h a t
-1 0 0
0 1 0
0 ) 0 . -1
We compute or(a) = det B l ( a ) . d e t B 2 ( a ) = - 1 . non-orientable. So ri0(E) = 1 and ri4(E) = 0.
ril(E)= 3-rank
while
ri3(E)=3-rank
This shows t h a t E is
((_10 0)/100)) 0 0
1 0
0 -1
-
0 0
1 0
0 1
((loo) (1 o o)) 0 0
-1 0
0 1
-
0 0
1 0
0 1
= 1
=2.
So, ri2(E) has to be equal to 2 in order to have x ( E ) = 1 - 1 § ri2(E) 2+0=0.
6.5
Seifert invariants of 3 - d i m e n s i o n a l infra-nilmanifolds
In this section we will compute the Seifert invariants for all 3-dimensional infra-nilmanifolds. This result is not new, since L. Moser [53] obtained these invariants already by a topological construction. Nevertheless, we wanted to include tiffs section, because we are able to recover the Seifert invariants in an algebraic way by manipulating the fundamental group. In the following sections, we are then able to extend this algebraic approach to more general classes of groups. Let M be a 3-dimensional infra-nilmanifold, with I I I ( M ) = E. We denote by N the m a x i m a l nilpotent subgroup of E and Z = Z ( N ) = ~/[N, N]. We know t h a t Q = E / Z is a 2-dimensional crystallographic group. Following Conner and R a y m o n d [14], we may look at M as being
M = R 3 / E = (T 1 • R 2 ) / Q after first deviding out the Z - p a r t of the action. As a result of this we find an (injective) Seifert Fibering M = (T 1 • R 2 ) / Q ) IR2/Q with base orbifold IR2/Q and typical fiber the circle T 1. It is known t h a t the
145
Section 6.5 Seifert invariants of 3-dimensional infra-nilmanifolds
fiber above the orbit of w E ]R2 ( = a point of the orbit space R 2 / Q ) is homeomorphic to (T 1 • { w } ) / Q ~ , where Q~, the isotropy group of w, acts freely on T 1. The following lemma is interesting to note. L e m m a 6.5.1 Let F be a finite group acting freely on a k - d i m e n s i o n a l torus T k, then Tk / F is a manifold with fundamental group isomorphic to a k-dimensional Bieberbach group. Proof: It is obvious that the orbit space is a manifold M. The group of covering transformations A ( T k , p ) corresponding to the covering p : T k ~ T k / F is isomorphic to F and moreover, it is known that A ( T k , p ) ~I I l ( M ) / p , ( I I l ( T k ) ) where p, is a monomorphism. Since ]Rk is the universal covering space of M , M is a K ( H I ( M ) , 1) manifold, we may conclude that I I I ( M ) is torsion free. As a conclusion, we see that I I I ( M ) is torsion free, finitely generated virtually free abelian, and therefore H i ( M ) is a Bieberbach group. []
C o r o l l a r y 6.5.2 Let F be a finite group acting freely on T 1, then F is cyclic. Proofi F ~ II1 ( M ) / Z where II 1(M) is a 1-dimensional Bieberbach group. So I I I ( M ) ~ Z which implies F being cyclic. []
If we think of a 2-dimensional crystallographic group Q as being a subgroup of the rigid motions of R 2, then we have the following possibilities for the finite cyclic subgroups F of Q: - F ~ 7~m and acts on ]~2 as rotations over an angle of 2 I I / m . - F ~ Z2 and acts on R 2 as reflexion through a line. L e m m a 6.5.3 Suppose Q is a 2-dimensional crystallographic group, and suppose there exists a point x E IR2 such that Q~ ~ Z2 acts on IR2 as reflexion through an axis. There is no extension of the f o r m i --+ Z --~ E --+ Q --* 1 where E is an Almost-Bieberbach group (and no Bieberbach group). P r o o f Since Q~ ~ Z2 acts orientation reversing on ]R2, the only action of Q on 7/, to be considered is one for which Q~ acts non-trivially on Z. But then there exists a torsion element of Q acting non-trivially, and so any E will have torsion. []
Chapter 6: Infra-nilmanifolds and their topological invariants
146
Using the t e r m i n o l o g y of [55] we m a y conclude t h a t we are in the s i t u a t i o n in which there are no fixed points (this should m e a n t h a t some fiber reduces to a point) a n d there are no special exceptional orbits (because all singular points are isolated). In particular, this m e a n s t h a t we deal w i t h Seifert Fiber Spaces as in Seifert's original definition. This is a 3 - m a n i f o l d which satisfies: 1. T h e m a n i f o l d decomposes into a collection of simple closed curves called fibers so t h a t each point lies on a unique fiber. 2. E a c h fiber has a t u b u l a r n e i g h b o u r h o 0 d V consisting entirely of fibers so t h a t V is h o m e o m o r p h i c to a "fibered solid t o r u s " . A trivial fibered solid torus is defined as being S 1 • D 2, where the fibers are the circles S 1 • {y} for a n y y E D 2. A general fibered solid torus is o b t a i n e d f r o m a trivial one as follows: choose a n u m b e r p E No a n d a n u m b e r 0 < q < p relative prime to p. Cut a trivial fibered solid torus open along {x} • D 2 for some ~ E S 1, r o t a t e one of the discs so o b t a i n e d t r o u g h q/p of a full twist a n d glue b o t h ends together. This fibered solid torus is said to have local Seifert invariants ( a , f l ) , where a = p a n d 0 < / 3 < a, flq = l m o d a . We also know t h a t all 3-dimensional i n f r a - n i l m a n i f o l d s are orientable, therefore we m a y a t t a c h to each one a series of Seifert invariants
{b; (E, g); (o~1,/31),... , (O~n,/~n) }
(6.12)
where 9 c = ol if IR2/Q is orientable a n d c = n2 otherwise. 9 g denotes the genus of the surface I~2/Q. 9 n is the n u m b e r of singular orbits. 9 (ai,/3i) are pairs of relatively prime integers, w i t h 0 < fli < ai, called the local Seifert invariants a n d i n d i c a t i n g to which fibered solid torus the e n v i r o n m e n t of a singular fiber is h o m e o m o r p h i c to. 9 bEZ. These invariant s d e p e n d on the chosen o r i e n t a t i o n of the i n f r a - n i l m a n i f o l d a n d a change of o r i e n t a t i o n t r a n s f o r m s the above set of invariants (6.12) into { - b - n; (~, g); ( a l , a l - i l l ) , . . . , (an, a~ - ~n)}.
Section 6.5 Seifert invariants os 3-dimensional infra-nilmanifolds
147
The structure of the fundamental group of a Seifert Fiber Space with invariants (6.12) is well known and a standard presentation for this fundamental group is given by: 1. Case e
=
o1
II :< a l , b l , . . . , a a ,
> . bg, q l , . . . , q n , h II [ai, h] = [bi, h] = 1 (1 < i < g) [qi, h ] = 1 ( 1 < i < n ) qi h = 1 qlq2 . . . q n [ a l , b l ] . . . [ a a , bg] = h8
2. Case e = n2 II :< v l , . . . , v g , q l , . . . , q , ~ , h II vihv~ 1 = h -1 ( l < i < g ) [qi, h] = 1 ( 1 < i < n) ~ (1 < i < n ) qi h = 1 2 = h b qlq2 9 9 9qnv~.. 9vg
> .
A Seifert Fiber Space with "enough" singular fibers is called a large manifold and is completely determined by its invariants (up to a change of orientation). More precisely, an orientable Seifert Fiber Space is said to be small if it satisfies one of the conditions below: 1. ol, g = O,n < 2. 2. o l , g = O , n = 3 , -
1T+ 1 ~ + - - > 11 . ~ 3
3. The set of invariants equals { - 2 ; ( O l , 0 ) ; ( 2 , 1 ) , ( 2 , 1 ) , ( 2 , 1 ) , ( 2 , 1 ) } . 4. o l ~ g = l , n = O . 5. n2, g = l~ n <_ 1 If an orientable Seifert Fiber Space is not small, it is called large. T h e o r e m 6 . 5 . 4 Let M and M ' be large Seifert manifolds. The following s t a t e m e n t s are equivalent: 1. M and M ~ have the s a m e set of invariants (possibly after a change of orientation). 2. M and M ~ are h o m e o m o r p h i c
148
C h a p t e r 6: I n f r a - n i l m a n i f o l d s and their topological invariants
3. M and M ~ have isomorphic f u n d a m e n t a l groups.
This t h e o r e m provides a n o t h e r way to classify (non equivalent) A l m o s t B i e b e r b a c h groups in H 2 ( Q , Z ) up to isomorphism. We now s t a r t comp u t i n g the Seifert invariants of the 3-dimensional i n f r a - n i l m a n i f o l d s by t r a n s f o r m i n g the f u n d a m e n t a l groups o b t a i n e d i_n the table of section 7.1, into s t a n d a r d ones. We use the n u m b e r s a n d n o t a t i o n s of our table. 1. • 2 / Q = T 2 w i t h no singular points. So 9 = ol, g = l, n = 0. For k > 0 we t a k e the following new set of < (k) >:
al 7_ a~ bI = b~ h 7_ c. It is obvious t h a t < (k) > has the p r e s e n t a t i o n < (k) > : < a l , a 2 , h I] [al,bl] = h -k > . Conclusion: the invariants of < (k) > are
[ 2. ~ 2 / Q = S 2 w i t h 4 singular points. (We leave the verification of this to the r e a d e r or refer to [63]). So e = ol, g = 1, n = 0. For k > 0 we t a k e the following new set of generators for < (2k, 0, 0, 1 ) > : ql = olc-l~ q2 -= aotc-l~ q3 = aback-l~ q4 : bcec-l~ h = c.
It is obvious t h a t this set generates < (2k, 0, 0, 1) > since a = q2qzh, b = q4qlh, c = h, a = qxh.
T h e following relations are fulfilled for the new generators: [ql, h] = [q2, hi = [q3, h] = [q4, hi = 1, =
=
=
=
1,
qlq2q3q4
=
h
(6.13)
This can be seen by calculating these by h a n d or by using the m a t r i x r e p r e s e n t a t i o n of < (2k, 0, 0, 1) >. These relations d e t e r m i n e the group
Section 6.5 Seifert inv--;arr
149
o f 3-dimensional i n f r a - n i l m a n i f o l d s
structure of < (2k, 0, 0, 1) > completely since all other relations can be computed for the relations (6.13), e.g.: [b,a]
=
h-1q{lq41h-lq{lq~lq4qlhq2qlh
=
h-lqlhq4hh-lqlhq2hq4qlhq2qlh
=
qlq4qlq2q4qlq2qlh 4
=
qlq4 ql q2q4qlq2 q3q4q41 q31 ql h4
=
qlq4qlq2q3qlh k+3
=
qlq4q41qlh 2k+1 h 2k
(q2h = 1 ~ q~l = qih)
([q~,h] = 1) (qlq2q3q4 = h k-2)
r
The other relations are proved in the same way. Therefore we may conclude that the invariants of < (2k, 0, 0, 1) > equal [ { k - 2;(Ol,0);(2,1),(2,1),(2,1),(2,1)}] For all other cases, we restrict ourselves to the results and omit all (almost trivial) calculations.
4. R 2 / Q = K (the Klein Bottle) without singular points.
As a new set of generators for < (k, 0) > (k > 0), we take Vl = a, v2 = a - l bc~, h = c
and find that 2 2=h-k < (k, 0) > : < v l , v 2 , h II v l h V l 1 = h - l , v2hv~ 1 = h - l , vlv2
Conclusion: The set of Sefeirt Invariants of < (k, 0) > equals { - k ; (n2, 2)} ]
8. I~2/Q = p2 (the projective plane) with two singular points. As a new set of generators for < (2k, 0, 0, 1) ) (k > 0), we take ql = ~ - 1 ,
q2 : a c ~ c - l - k , Vl = ~, h --- c
>"
150
C h a p t e r 6: I n f r a - n i l m a n i f o I d s
and their topological invariants
and find that < (2k,0,0, 1 ) > : < v l , q l , q 2 , h II v l h v ~ 1 = h - 1
> .
[ql, hi = [q~, hi = 1 ql~h = q~h = 1 ql q2 v2 = hk- 1 Conclusion: The invariants for < (2k, 0, 0, 1) > are {k-
1;(n2,1);(2,1ii(2,1)}
10. I ~ 2 / Q = S 2 with 3 singular points. 9 As a new set of generators for < (4k, 0, 0, 1) > (k > 0), we take ql = a c - ~ , q2 = a a 2c-1, q3 = ac~ck-1, h = c
and find that < (4k, 0 , 0 , 1 ) > : < q l , q 2 ,
q3, h[[ [ql,h]=[q2, h ] = [ q 3 , h ] = 1
>. q4h3 = q~h = q4h3 = 1, q, q2q3 = h k - 2
Conclusion: The invariants for < (4k, 0, 0, 1) > are [ { k - 2; (o1, 0); (4, 3), (4, 3), (2, 1)} 9 For < (4k,0,0,3) > (k > 0), we take ql = C~C-1, q2 = act2c-2, q3 -= a a c k - 1 , h = c and find that < (4k,0,0,3)>:<
>. ql,q2, q3, h H [q~, h] = [q2, h] = [q3, h] = 1 q4h = q~h = q~h = 1, qlq2q3 = h k-1
Conclusion: The invariants for < (4k, 0, 0, 3) > are
{k- 1; (ol, 01; (4,1), (4,1), (2,1)}] 9 For < (4k-t- 2 , 0 , 0 , 1 ) > (k > 0), we take ql = otc-1, q2 = aot2c-1, q3 -= ao~ck, h-= c
S e c t i o n 6.5 Seifert i n v a r i a n t s o f 3 - d i m e n s i o n a l i n f r a - n i l m a n i f o l d s
151
and find that < ( 4 k + 2 , 0 , 0 , 1 ) > : < ql,q2, q3, h II [ql, h] = [q2, h] = [q3, h] = 1 > q~h 3 = q2h : q'~h = 1, qlq2q3 : h k - 1 and for < (4k + 2, 0, 0, 3) > (k > 0), we take ql = a c - 1 , q2 = a a X c - 2 , q3 = aolck-1, h = c
and find that < (4k+2, 0, 0, 3) > = < ql, q2, q3, h ]] [ql, h] : [q2, hi = [q3, h] = I >. q'~h : q22h : q~h 3 = 1, qlq2q3 = h k - 1 Conclusion: The invariants for < (4k + 2, 0, 0, 1) > and for < (4k + 2, 0, 0, 3) > are [ { k - 1; (ol, 0); (4, 3), (4, 1), (2, 1)} Remark that this implies indeed that < (4k + 2, 0, 0, 1) > ~ < 2, 0, 0, 3) > as indicated in the table of section 7.1.
(4k +
13. R 2 / Q = S 2 with 3 singular points. * As a new set of generators for < (3k, 0, 0, 1) > (k > 0), we take ql = a c - 1 , q2 : a a c k - l ~
<(3k, 0,0,1)>=<
q3 : a b a c 3 k - 1 , h = c
ql,q2, q3, h H [ql,h] = [q2, h] = [q3, h] = l >. q3h2 = q3h2 = q3h2 = 1, qlq2q3 = h k - 2
Conclusion: The invariants for < (3k, 0, 0, 1) > are ] { k - 2; (ol, 0); (3, 2), (3, 2), (3, 2)} ] 9 For < (3k, 0, 0, 2) > (k > 0), we take ql = ctc-1, q2 : a a c k - 1 ,
q3 : a b a c 3 k - 1 ,
h = c
< (3k, 0, 0, 2) > = < ql, q2, q3, hll [ q l , h ] = [ q 2 , h ] = [ q 3 , h ] = l >. q3h = q3h = q3h = 1, qlq2q3 = h k-1 Conclusion: The invariants for < (3k, 0, 0, 2) > are
1; (ol, 0); (z, 1), (z, 1), (3,1)} 1
152
C h a p t e r 6: Infra-nilmanifolds and their topological invariants
9 For < (3k + 1, 1,0, 1) > (k > 0), we take ql = ac -1, q2 = aa2c k-~, q3 = abac 3~, h = c <(3k+l,l,O,1)>=
q3, h[]
[ql,h]=[q2,h]=[q3, h]=l >. q3he = q3h = q3ah = 1, qlq2q3 = h k-1
Conclusion: The invariants for < (3k + 1, 1, 0, 1) > are
[{k- 1; (o~, 0); (3, :),ia, 1), (3, i)} Remark: the same set of invariants is found for < (3k+1, 0, 0, 2) > and for < (3k § 2, 0, 2) >. 9 < (3k+2,1,0,1)> (k> 0),wetake ql :
ac-1,
q2 ---=a o t 2 c k - 1 ,
q3 = abo~c3k+l, h = c
q~,q~,q3,h II [ql, hi = [q~,h] = [q~,h]-- 1
< (3k+2,1,0,1)>:<
>.
qalh2 = q3h2 = q]h = 1, qlq2q3 = h k-1
Conclusiom The invariants for < (3k + 2, 1, 0, 1) > are
[{k-
1; (o~, 0); (z, 21, (3, 2),(3, t)}]
Remark: the same set of invariants is found for < (3k+2, 0, 0, l) > and for <(3k+2,2,0,2)>.
16. R 2 / Q = S 2 with 3 singular points. 9 As a new set of generators for < (6k, 0, 0, 1) > (k > 0), we take ql ~- OLC--1, ([2 -: ao~3r
([3 ~- aol2c4k-l~
h ~. c
< (~k, 0, 0,1) > : < ql, q~,q3, h I} [ql, hi = [q~,h] = [q3, hi = 1
>.
q~h 5 = q~h = q~h ~ = 1, qlq2q3 = h k-2
Conclusion: The invariants for < (6k, 0, 0,1) > are
* For < ( 6 k , 0 , 0 , 5 ) > (k > 0), we take ql = a c - 1 ,
q2 -~ aoL3 c 3 k - 3 , q3 = a~2 c 4 k - 2 , h -~. c
< (6k,0,0,5) > : < ql,q~,q3,h LI [q~, hi : [q~, hi : [q~, hi : 1 q~h = q~h = q~h = 1, qlq2q3 = h k-1
>.
Section 6.5 Seifert invariants of 3-dimensional infra-nilmanifolds
153
Conclusion: The invariants for < (6k, 0, 0, 5) > are
I{k- 1; (ol, 0); (6,1), (3,1), (2,1)} 9 As a new set of generators for < (6k + 2,0,0, 5) > (k > 0), we take ql = ac -1, < (6k+2,0,0,5)>
q2 = a o ~ 3 c 3 k - 2 ,
: < ql,q2, q3, h[l
q3 = ao~2c4k-1, h : c
[ql,h]=[q2, h ] = [q3, h ] = l >. q~h = q2h = q~h 2 = 1, qlq2q3 = h k-1
Conclusion: The invariants for < (6k + 2, 0, 0, 5) > are {k- 1;(Ol,0);(6,1),(3,2),(2,1)}] 9 For < (6k + 4 , 0 , 0 , 1 ) > (k _> 0), we take ql ---- ~ c - 1 , q2 -= a~3c3k-f-l~ q3 -= aoL2c4k+2, h = c
< (6k+4,0,0,1)>:
< ql,q2, q3, hll
[ql,h] = [q2, h ] = [qs, h]= 1 >. q6h5 = q2h = q3h = 1, qlq2q3 = h k-1
Conclusion: The invariants for < (6k + 4, 0, 0, 1) > are
I{k- 1; (ol, 0); (6, 5), (3,11, (2,1)}]
Chapter 6: Infra-nilmanifolds and their topological invariants
154
We summarize the computations of this section in the following theorem: Theorem
6.5.5
Let M be any 3-dimensional infra-nilmanifold with fundamental group E, holonomy group F and with underlying crystallographic group Q, then M has a set of Seifert invariants according to the table below (k > 0): I~2/Q Fitt(E)
Set of Seifert Invariants
F
I~
{-k; (ol, 1)}
{1}
T2
Nk
II.
(k - 2; (ol, 0); (2,1), (2,1), (2, I), (2,1)}
Z2
S2
N2k
III.
{-k; (~2, 2)}
Z:
K
N2k
IV.
{k - 1; (n2, 1); (2, 1), (2, 1)}
Z2 • Z~
I?2
N4k
{k - 2; (ol, 0); (4, 3), (4, 3), (2, 1)} V.
VI.
VII.
{k - 1; (Ol, 0); (4, 1), (4, 1), (2, 1)}
g4k
Z4
S2
N4~
{k - 1; (Ol, 0); (4, 3), (4, 1), (2, 1)}
N4k+2
(k - 2; (Ol, 0); (3, 2), (3, 2), (3, 2)}
N3k
(k - 1; (ol, 0); (3, 1), (3, 1), (3, 1)}
Z3
S2
Nsk
(k -- 1; (Ol, 0); (3, 2), (3, 1), (3, 1)}
N3h+l
(k - 1; (Ol, 0); (3, 2), (z, 2), (3,1)}
g3k+2
(k - 2; (Ol, 0); (6, ~), (3, 2), (2,1)}
N6h
(k - 1; (ol, 0); (6,1), (3,1), (2,1)}
Z6
S~
N6k
(k - 1; (ol, 0); (6, 1), (3, 2), (2, 1)}
g6h+2
{k - 1; (Ol, 0); (6, 5), (3, 1), (2, 1)}
N6k+4
where Nk : < a,b, cl[ [b,a] = ck, [c,a] - - I t , b] = 1 > .
6.6
Investigation
of torsion
In this section we will define a set of invariants for a large class of Almost-crystallographic (and other) groups, which can be seen as a kind of generalization of the local Seifert invariants computed in the previous section. These invariants no longer suffice to determine the isomorphism
Section 6.6 Investigation of torsion
155
t y p e of an almost-crystallographic group (even not in dimension 3) but are a very nice tool to detect whether there is torsion or not. To compute the local Seifert invariants of a 3-dimensinonal A B - g r o u p E , we needed for each singular point of the quotient space Q\IR 2 (Q = E / Z ) a generator q~ of the group E. The projection of qi in Q is a generator of the (cyclic) isotropy group Z,~ at the singular point. We were then especially interested in the result of q.~i which was of the form h #, where h was the generator of a (unique) normal, infinite cyclic subgroup of the fundamental group. We now try to generalize this situation as much as possible. Let Q be any group acting properly discontinuously (on the left) on a topological space X . For x E X , we will write Q~ for the (finite) isotropy group at x. The orbit type of Qx is the set of all isotropy groups which can be found along the orbit and is denoted by (Q~). Of course = {qq II q e q } = {qO q -1 II q e q } , the r class of Q~ in Q. We will assume that there are only finitely many orbit types. This is e.g. always the case when Q is a polycyclic-by-fmite group, since those groups only have a finite number of conjugacy classes of finite subgroups [59]. By a set of representatives for the orbit types, we will m e a n a collection of finite subgroups Q~I, Q x 2 , . . . , Q ~ such that any orbit type can uniquely be written as a (Q~,) for a (unique) zi. We want to investigate extensions E of the form 1
)Z
)E
)Q
~1
compatible with a chosen action of Q on Z = grp{h}. Fix such axL extension E and let q be an element of E which projects onto q E Q~i for some xi E X . By c~4 we will denote the order of ~. So q ~ = h ~ for some integer ft. We use the notation f14 = fl m o d a~, so 0 < flq < aq. L e m m a 6.6.1 The ordered pair ( aq, flq) is independent of the chosen lift q E E , used in computing it. Proof: Let q' = qh k be another lift of q. We distinguish two cases. First let us suppose that the action of q on h is trivial, then q ' ~ = (qhk) ~ = h~h k~q, showing that flq is independent of the choice of lift of q. If the action of q on Z is nontrivial, then aq is even and so q ~ (q2
= (, q ,n k,22x ) 2 =
= q~r = h ~, which proves the lemma. []
R e m a r k 6.6.2 In section 6.1, we even showed that in case the action of q on Z is nontrivial, then ~ =/34 = O.
156
Chapter 6: Infra-nilmanifolds and their topological invariants
For each Q ~ : {1, ql, q 2 , . . . , qk} we now define the local invariants of this orbit type to be
SE(Q~, ) = < (c~4~,/341), (a~2,/342 ) , ' " , (~4k,/34~) > 9 This SE(Q~,) is to be considered as a set of ordered 2-tuples, the order in which they appear not being important, but identical 2-tuples are listed more then ones. The following lemma shows that the local invariants of an orbit type do not depend nor on the choice of the point xi of the orbit, nor on the generator chosen for Z. L e m m a 6.6.3 If (Q~,) = (Qy,) for some yi E x , then SE(Q~,) = SE(Qu~). Moreover, Q ~ does not depend upon the generator chosen for
Z. Proof: Suppose (Q~,) = (Qy,) (This does not mean that xi and yi belong to the same orbit), then Qy. = g Q ~ g - 1 for some g E Q. So a 4 = ag4g-1 for all q E Q ~ . Take any lift ~ of g in E, then ( ~ q 9 - 1 ) ~ = 9 q ~ - 1 = ~qZ~-l. So, ifg acts trivial on Z we instantly see that we have a one to one correspondence between SE(Q~:,) and SE(Qy,). But if g acts non trivial, then there is also a one to one correspondence, since in this case one sees that (a4-~,/34-~) = (a94g-1,/39~g-~). Moreover, a change of generator (h ) h -1) for Z only changes (a4, f14) into (a4, a 4 - /34) for each ~. But this does not effect the whole set SE(Q~,) since (c~4,(x4 - / 3 4 ) -
D e f i n i t i o n 6 . 6 . 4 Let Q be any group acting properly discontinuously on a topological space X with finitely many orbit types (Q~I), (Q~2),'' ", (Q:~)" Let E be any extension of Q by Z , then we define a set of invariants S ( E ) as
S(E) :
{SE(Q
I),
Remark that this definition is independent of the chosen representants of the orbit types and of the chosen generator for ~. The following theorem shows the importance of these invariants. T h e o r e m 6.6.5 Let E be an extension of a group Q as described above where the action of Q is on a space X .-~ Y('~ for some n, then E is torsion free f i e n d only if]or every 2-tuple (c~,~3) o r S ( E ) , (x and~3 are mutually prime.
Section 6.6 Investigation of torsion
157
Proof: Suppose that there exists a pair (a,/3) in S ( E ) , such that a and /3 are not mutually prime. We suppose that they came from q E Q. So there is a lift q E E o f q for which q~ = h ~, where h denotes the generator of Z as before. We remark that if q acts non trivially on •, fl = 0 and so there is torsion. Therefore we may assume that the action of q on is trivial. Because of the fact that a a n d / 3 are not mutually prime there exists a 7 , 5 E Z, with 0 < 7 < a such that ~/3 + 5a = 0 and so (q'~h~) ~ = h z~+6~ = 1. This shows that E has torsion. Conversely, suppose E has torsion. Then there is an element q E E of order c~, with a a prime number. The projection ~ of q in Q is also of order a and therefore we m a y conclude that the action of q on ~'~ has some fixed point x Thus q C Q~ and (a, 0) belongs to S ( E ) .
Remark
6.6.6
In stead of using representatives of orbit types, one can also avoid the topological aspect by considering all (or all maximal) finite subgroups of Q. Analogous results can be formulated.
Chapter 7
Classification survey 7.1
3-dimensional A C - g r o u p s
The table we present here describes all possible 3-dimensional AC-groups E. One table entry contains several items, which we explain now. First of all, the entries are ordened according to the ordening of the wallpaper groups (Q) as found in ([10]) (and so as found in the International Tables for Crystallography). Each table-entry contains a presentation for the family of 3-dimensional AC-groups corresponding to the indicated group Q. This presentation depends of at most 4 parameters kl, k2, k3 and k4. Always, the subgroup generated by the symbols a, b and c is the maximal nilpotent subgroup Nk contained in E. In each entry we present a faithful affme representation /k : E Aff(]~3). /k is given by its images of the generators. However, as we use stable affme representation we have for every AC-group E in the table that /k(a)=
/10o0) /100/ /1001) 0
1
0 0
0 0
1
1 0
0 1
A(b)=
0 0 0
0 0
0 1 0
0 1 1
)~(c)=
0 0 0
1 0 0
0 1 0
0 0 1
where the values for k is determined by [b, a] = ck. Each table-entry also contains H~(Q, Z) (~, as indicated in observation 3 in section 6.1). This group has been computed using the m e t h o d indicated in chapter 5 and depends on the number of parameters k~ in the presentation of E. An element (cohomology class) in this group is written as < (kl, k2, k3, k4) >. Furthermore, in each entry we show which
160
Chapter
7: C l a s s i t ~ c a t i o n s u r v e y
groups are AB-groups and we indicate the isomorphism classes of these AB-groups. The general set up of one table entry is as follows: Number of Q Symbol of Q Presentation for E depending on kl, k2, k3, k4 The images under A for the generators other then a, b, c H 2 ( Q , 7,) in terms of kl, k2, k3, k4
Eventually: AB-groups: The cohomology classes corresponding to AB-groups and isomorphism type information for these. 1. Q = p l E : < a, b, c [l [b, a] = c k' [c, a] = [c, b] = 1 >
H2(Q, z) = z
AB-groups: k >0, E = < (k) > (~Yk) 2. Q = p 2 E:
< a, b, c, o~ ll
[b, a] = c kl
o~c = c a
[c, a] = 1
A(~)=
H2(Q,Z)=TZ~(Z2)
tea ~
a-l~c
OL2 ~
C k4
k~
[c, b] -= 1 o~b = b-lo~c k~
1 0
k2 -1
k3 0
0
o
o
0
0
-1 0
o 1
2
3 = z 4 / A , A={(kl,k2, k3, k4) l]kl--0, k2, k3, k4E2Z}
AB-groups: k>0, k z0mod2, E-<
(k,0,0,1) >
3. Q : p m E:
< a,b,c,~]]
[b,a] = c kl [c, a] =- 1 c~a = ao~c k~
O~2:1
o~c = c - l e x [c, b] = 1 ab = b-lo~
>
161
S e c t i o n 7.1 3 - d i m e n s i o n a l A C - g r o u p s
--1 0 0 0
At}''a' =
-k2 1 0 0
0 0 -1 0
0 ) 0 0 1
H ~ ( Q , Z ) = Z G Z2 = Z 2 / A , A = {(kl,k2) II kl = 0, k2 E 2Z}
4. Q = p g E:
[b, a] = c 2kl
ac = c-la
[~, ~]
[~, b] = 1
= 1
o:a = ao~c - 2 k ~
>
o~b = b - l o ~ c - k l
0:2 • ac k2
A(a) =
-1 0 0 0
2k~ 1 0 0
~ 2 0 -1 0
0 i
0 1
H 2 ( Q , Z ) = Z = Z 2 / A , A = {(kl, k2)II k~ = 0, k~ c z }
AB-groups: k >0, k_~0mod2,
E=<
(k/2,0) >
5. Q = c m
E:
[ b , a ] = c k~ [c, a] = 1 a a = bac k~
o~c=c-lo~ > [c, b] = 1 ab = ao~c k~
a2=l
A(a) =
-1 0 0 0
k2 0 1 0
-k2 1 0 0
0 0 0 1
H 2 ( Q , Z) = Z = Z 2 / A , A = {(kl, k2) II k~ = o, k~ c z }
6. Q = p 2 m m E:
< a,b,c,a,~
>
[b, a] = c kl
[c, a] = 1
[c, b] = 1
OLC ~ CO:
a a = a-lo~c k2
ab = b-lo~c k3
OL2 = Ck~
j32=1
Za = a~c ~ ~ = ~ c -k,
C h a p t e r 7: Classification s u r v e y
162
1 0 0 0
A(a) =
H:(Q, ~)
= E ~
k2 -1 0 0
k3 0 -1 0
~2 '~ 0 0 1
-1 0 0 0
%[f2~
^~']=
-k: 1 0 0
0 0 -1 0
0 0 0 1
(E~) 3 -- E4/A, A = {(kl, k2, k3, k4) II kl : 0, k2, k3, k 4 ~ 2~}
7. Q = p 2 m g E:
< a,b,c, oqj3[[
>
[b, a] : c TM [c,a]: 1 ~c : c,~ o~a = a-lo~c k2
[c,b]: 1 ~'c : c - 1 ~ oLb : b-lc~c -2(hs+k~)
c~~ : c k~
fl~ : 1
Za =
aZc k~-k'
Zb : b-1Z
o~j3 = b - l i l a c k"
l k2 23 k
0 0 0
A(.) :
-1 0 0
0 -1 0
(1 o o / 2
0 0 1
0 0 0
A._.:
1 0 0
0 -1 0
0
1
1
H 2 ( Q , Z ) : Z ~ (Z2) 2 = Z 4 / A ,
8. Q -= p2gg E:
[b, a] = c TM [c, a] = 1
< a,b,c,a,fl[[
=
o~a -: a - l a c h~+2ha
(~b = b-lo~c -hl+2k~-2ha
o~2 : ck4 j3a = aflc :ka
~2 = ac-ka Zb = b - l flc - k ' a Z = a - l b - l j 3 a c -(k'+h~+k~)
1
k l + 2k3
2k2 - kl - 2k3 0
k-i 2 0
0 0
0 0
-1
0
0
1
I ~,(~)
> [c, b] = 1
-1
I
- ~ -22 k 3
h_r 2
o
1
0
0 0
0 0
--1 0
H 2 ( Q , Z ) = Z ~ Z2 = Z4 / A,
0 "~ i 1
163
S e c t i o n 7.1 3 - d i m e n s i o n a l A C - g r o u p s
A = {(kl,kz, kz, k4)II kl = 0, k4 E 2Z, k2,ka @Z} AB-groups: k>0, k-0mod4, .
E - - < ( k / 2 , 0 , O, 1 ) > .
Q, = c 2 m m
E:
< a,b,c,a,/3[[
>
[b, a] = c t'l [c, a] --- 1
[c, b] = 1
~
Z~
= c-~
ab
.~ b - l o t c 2 k ~ - k z
= ~
ota :
a - l o t c k2
a 2 = c k4 fla = b/3ck" ~ Z = Zo, c - ~"
~] ' t a ~ =
1 k2 0 --1 0 0 0 0
-k2+2k3 0 --1 0
~2 ) 0 0 1
132 = 1 ~3b = aflc ~*
a(~) =
-1 0 0
-ka 0 1
-ka 1 0
0
0
0
0 "~ 0 0
1
)
H 2 ( Q , Z) = Z @(Z2)2 = Z 4 / A , A = {(kl, k2, k3, k4) II kl = 0, k4, k2 ~ 2Z, k3 ~ Z}
10. Q = p 4 E:
[ b , a ] = c k~ [~, a] = 1 a a = bac ~ OL4 :
A(a) =
ac=co~ > [~, b] = o~b : a - l o t c k~
Ck4
1 k2
k3
~4 '~
0 0 0
-I 0 0
0 0 I
0 1 0
)
HZ(Q, Z) -- Z @Z2 ~ Z4 -- Z4/A,
A = { ( k l , . . . , k 4 ) [I kl = 0, (k2 + k3) E 2Z, k4 ~ 4Z} AB-groups: k>0, k-0mod2, E = < (k, 0, 0,1) >. k > 0 , k _ - - 0 m o d 4 , E = < (k, 0, 0, 3) >. Remark: if k - - 4 1 + 2 , f o r s o m e l E Z t h e n < (k, 0, 0,1) > ~ < (k, 0, 0, 3) >. k -- 4l, for some 1 E Z then < (k,0,0, 1) > ~ < (k,0,0,3) >.
164
C h a p t e r 7: C l a s s i f i c a t i o n s u r v e y
11. Q = p 4 m m E:
< a,b,c,c~,~[[
1 0 0 0
A(a) =
k2 0 1 0
k3 -1 0 0
[ ba] •:
>
Ck l
[c,a]= 1
[c,b]= 1
~c = r
~c = c - 1 /
o:a = bo~c k2
o:b = a - l oLc k3
a4 = c k.
/2 = 1
fla = a / c ~+k~
fib = b - l ~
-1 0 o 0
4
0 0 1
A(/) =
oo)
-(ks+k3) 1 o 0
0 -1 0
0 0 1
H 2 ( Q , Z ) = Z G Z2 @ Z4 = Z 4 / A ,
A.= {(k~,..., 12. Q =
k~) II k~ = O, (k~ + k~) ~ 2Z, k 4 ~ 4G}
p4gm E:
>
[b, a] = c TM
It, a] = 1
[c, b] = 1
~C =
/c = c-1/
CO:
ab = a - l a c kl-k~-2ka / 2 = acka
c~a = bceck2 ~4
=
Ck4
/ a : a / c -2ka
~(~) =
1
ks
0 0 0
0 1 0
4
kl-k2-2ka -1 0 0
/ b = b - 1 / c -k~ o~/ = a - 1 / a a c k 3 - ~
~ "~ 0 0 A(/) = 1
-1 0 o 0
)
H 2 ( Q , Z) = Z G ~4
----
_ 2 2 + 2ka 1
o 0
~4/A,
A = { ( k l , k 2 , ka, k4)11 kl = O, k2, ka E Z, k4 E 4Z}
13. Q = p 3 E:
< a,b,c,~[]
~(~) =
>
[b, a] = c k~
o~c = ca
[c,a]= 1
[c,b]= 1
(xa = bac k2 Ol3 = Ck4
~b = a - l b - l a
I1o k2o 0 1 0
0
4-k3 -~ -1
-1 0
~3o 1 0 1
c ka
2
0 -1 0
o) 1
Section 7.1 3-dimensional A C-groups
165
H 2 ( Q , Z) = Z @ (Za) 2 : Z 4 / A ,
A = {(kl, k2, ka, k4)
II k~ = 0, (k~ - k~) ~
AB-groups: k > 0 , k _ = 0 m o d 3 , E = < (k,O,O, 1 ) > k >0, k-0mod3, E=< (k,0,0,2)> k > 0 , k ~ 0 m o d 3 , E = < (k, 1 , 0 , 1 ) > Remark: If k - 1 mod 3, E = < (k,0,0,2) >-~< (k, Ifk-2mod3, E = < (k, 0, 0,1) > ~ < (k,
3Z
]g4
~ 3Z}
1, o, 1) >~< (k, 2, o, 2) >. 1, o, 1) >~< (k, 2, o, 2) >.
14. Q = p 3 m l
E:
a(~) = (
1 k2 0 0 0 1 0 0
[b,~]=c ~
>
[c,a]= 1
[c,b]= 1
a a = bc~c~2 a 3 = c k4
~b = a - l b - l o L c ks r2= 1
r a = b - l r ca=
r b = a - l f l c -a=
_a__Z+ka 2 -1 -1 0
•3 ~ 0 ) 0 1
[ -1 / 0 A(~) = 0 0
-k2 0 -1 0
k2 -1 0 0
0 0 0 1
H2(Q,Z) = Z@ (Za) 2 = G 4 / A , A = { ( k l , k 2 , ka, k4)II kl = 0,(k2 - ka) E 3Z, k4 E 3Z}
15. Q = p 3 1 m E:
[ b , a ] = c kl
>
[c, a] = 1 ac = ca
[c, b] = 1 rc = c- l r
a a = bac kl-2k~+ak~
ab = a - l b - l a c ks
a a = c k~ r a = bric k=
r2= 1 fib = a r c k=
~ f l = rc'= c -~"
~(~)=(
1 kl -- 2k3 +3~2 0 0 0 1 0 0
--k-12~ k a -1 -1 0
k43 ~ / --1 0 J / 0 0 A(fl) = 0 1 0
--k2 0 1 0
H 2 ( Q , Z ) = Z G Za = Z 4 / A ,
A = {(kl, k2, ka, k4) I] kl = 0, k2, ka G Z, k4 E 3Z}
--k2 1 0 0
0 0 0 1
C h a p t e r 7: C l a s s i f i c a t i o n s u r v e y
166 16. Q = p6 E:
< ~,b,c, all
[ ba]• =
Ckl
[c, a] = 1 o~a = abac kz
>
~ C -~ COL
[c, b] -= 1 o~b - ~ a-lo~c ks
Ot 6 = C k~
1 - ~2 + k2 k3 ~6) 0 0 0
A(a) =
H2(Q,Z)
1 1 0
-1 0 0
0 0 1
= Z @ Z~ = Z 4 / A ,
A = {(kl, k=, ks, k4) II kl = 0, k~, ks ~ Z, k, c 6Z]. AB-groups: k > 0, k = 0 k > 0, k _= 4 k > 0, k _= 0 k > 0, k - 2
mod mod rood mod
6, 6, 6, 6,
E E E E
=< =< =< =<
(k,0,0, (k,0,0, (k, 0, 0, (k,O,O,
1) 1) 5) 5)
>E >E >E >E
H~(Q,Z). H2(Q,Z). H 2 ( Q , Z). H2(Q,Z).
17. Q = p 6 m m
E:
1 12
0 0 0
1 1 0
-1 0 0
>
[b, a] = c ~ [~,~J-- 1
I~,bl = 1
aa = abac k~ ar = ck, 13a = bflc*"
o~b = a-lo~c k~ f12 = 1 /3b = a/3c I*~
0 0 1
A(fl)
(_1._,o / 0 0 0
0 1 0
H2(Q, Z) = Z ~ Z6 = Z4/A, A = {(k~, k2, k3, k4) ]] k~ = o, k~, k~ ~ Z, k4 E 6Z}
1 0 0
0 0 1
Section 7.1 3-dimensional A C-groups
167
I n t h e n e x t t a b l e we s u m m a r i z e the a b o v e r e s u l t s b y i n d i c a t i n g h o w m a n y i n f r a - n i l m a n i f o l d s are essentially covered b y t h e n i l m a n i f o l d w i t h fundamental group
Nk
: < a , b , cll
[b,a]---
ck > .
T h i s n u m b e r d e p e n d s o f t h e value of k m o d 12.
A l m o s t - B i e b e r b a c h g r o u p s of d i m = 3
~
1
1
1
1
1
1
1
1
1
1
1
1
2. 4.
1 1
0 0
1 1
0 0
1 1
0 0
1 1
0 0
1 1
0 0
1 1
0 0
8. 10.
1
0
0
0
1
0
0
0
1
0
0
0
2
0
1
0
2
0
1 0
1 1
2 0
1 I
1 0
0 1
2 1
0 2
1 1
0 1
16.
2 2
1 2 2
0
1
0
1
0
Total
10
2
6
3
8
2
8
2
8
3
6
2
13.
Chapter
168
7.2
7:
Classification survey
4 - d i m e n s i o n a l A B - g r o u p s , w i t h 2 - s t e p nilp o t e n t F i t t i n g subgroup
The table we present here describes all possible 4-dimensional AB-groups E with a maximal normal nilpotent group of class 2. For many 3dimensional crystallographic groups, we know in advance, e.g. by application of section 5.4 and of corollary 6.1.2, that they cannot give rise to an A B - g r o u p . So we did only include one entry for each "interesting" group Q. There are only few Q which cannot be excluded in advance and which are not the underlying crystallographic group of a 4-dimensional AB-group. One table entry contains several items, which we explain now. First of all, the entries are listed according to the ordering of the crystallographic groups (Q) as found in ([10]). We also indicate the number of Q as in the International Tables for Crystallography (I.T.). Each table-entry contains a presentation for the family of 4-dimensional AC-groups corresponding to the indicated group Q. This presentation depends of at most 7 parameters kl,k2,...kT. Always, the subgroup generated by the symbols a, b, c and d is the maximal nilpotent subgroup N contained in E. In each entry we present a faithful affme representation A : E Aft(R4). A is given by its images of the generators. However, as we use stable representations we have for every AC-group E in the table that
=
1
o
0
1
0
0
0/ (1 02 /
o o
1
0
0
0 0
o o
0 0
1 0
0 1
1 0 0 0 0
h2 1 0 0 0
h2 0 1 0 0
0 0 0 1 0
1
0 0 0 1 1
A(b)--
and A(d) --
0
1
0
0
0
0
1
0
1
0 0
0 0
0 0
1 0
0 1
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
0
1 0 0 0 1
,
where the values for I1, 12 and 13 are determined by
[b,a]=d l',
[ c , a ] = d 12, [ c , b ] = d 13.
Each table-entry also contains H~(Q, 2 ~) (~, can be read of the presentation of the extensions E ) . This group depends on the parameters kl in the presentation of E. An element (cohomology class) in this group
Section
7.2 4 - d i m .
w i t h 2-step F i t t i n g
AB-groups,
subgroup
169
is written as < (kl, k 2 , . . . , kT) >. Furthermore, in each entry we show which groups are AB-groups and we indicate the isomorphism classes of these AB-groups. The general set up of one table entry is as follows: N u m b e r of Q as found in I.T. Symbol of Q Presentation for E depending on kl, k 2 , . . . , k7 The images under )~ for the generators other t h e n a, b, c, d H2(Q, Z) in terms of kl, k 2 , . . . , k7 AB-groups: The cohomology classes corresponding to AB-groups and isomorphism type information for these. 1. Q = P 1 E:
dll
[ b , a ] = d k' [c, a] = d k~ [c, b] = d k~
[d,a]=l
>
[d, b] = 1 [d, c] = 1
H 2 ( Q , Z) = Z a
AB-groups: w > o, < ( k , o , o ) >
Remark: Vkl, k2, k3 E Z : < (kl, k2, ks) > ~ < ((kl, k2, ka), 0, 0) >. 2. Q = P i E:
A(c~) =
[ b , a ] = d k' [c, a] = d ~2 [c, b] = d k~ ola = a-lo~d k* o~b = b - l a d k~ otc = c - l o~dke
l10 0 0 0
k4 ks -1 0 k8 0 0 -1 0 0 0 -1 0 0 0
[d,a]=l [d, b] = 1 [d, c] = 1 o~2 -- d k~" otd = do~
>
k,02 1 0 0 1
H 2 ( Q , Z ) = Z 3 @ (Z2) 4 = Z T / A , A = { ( k l , , , , , kT)llkl =ks = k3 = O, k4, ks, k6, k7 E 2Z}
AB-groups: Vk > 0 , k - 0 m o d 2 , < (k, 0, 0, 0, 0, 0,1) > Remark: < (2k, 21, 2m, O, O, O, 1) > ~ < (2(k, 1, m), O, O, O, O, O, 1) > Vk, 1, m E Z
170
Chapter
7: C l a s s i f i c a t i o n s u r v e y
3. Q = P 2 E: < a,b,c,d,a
II
[b,a]: 1 [c, a] : d ~
[d,a]= 1 [d,b] : 1
It, b] = 1
[d, c] = t
o~a = a-lo~d k= c~b = bo~ o~c : c-lo~d k3
o~2 = d k" old = d a
1 ks
0
k3
k--x 2
0 -i 0 0 0
~(~)=
o
o
1
o
o
0 0
0 0
0 -1 0 0
0 1
>
]
H2(Q,Z) = 7~q~ (Z2) 3 = Z 4 / A , A = { ( k l , . . . , k,)ltkl = 0, ks, k3, k, ~ 2 z } AB-groups:
Vk>0, k = 0 r o o d 2 , < (k,0,0,1) > 4.
Q =
P21
Trivial Action: E : < a , b , c , d , o ~ II [ b , a ] : 1 [c, a] = d kl [c, b] = 1 o~a = a-lo~d k= ab = bo~
[d,a]= 1 [d, b] = 1 [d, c] = 1 o~2 = bd k" o~d = d a
>
otc ~ c - l o r d k~
l1 A(a) =
H2(Q,
A
=
k2 0 -1
0 0
kz 0
0 0 0
1 0 0
0 -1 0
0 0 0
~2 ) 0 -~ 0 1
Z) = Z ~ (Z2) 2 = Z 4 / A ,
{ ( k l , . . . , k,)llk~ = 0, k~, k3 ~ 2~, k4 ~ Z}
AB-groups:
w > o , < (k,0,0, o) > V k > 0 , k - 0 m o d 2 , < (k, l, 0, 0) > Remark: Ifk ~_ 0 mod 2, < (k, O, O, O) > ~ < (k, 1, 0, 0) >-~< (k, 0, 1, 0) >-~< (k, 1, 1, 0) > Ifk ~ 0 moo 2, < (k, 0, 0, 0) > ~ < (k, 1, 0, 0) > ~ < (k, 0, 1, 0) > ~ < (k, 1, 1, 0) >
S e c t i o n 7.2 4 - d i m . A B - g r o u p s ,
with 2-step Fitting subgroup
Action: ~d = d -1
II [b, a] = d TM
E : < a,b,c,d,a
[c, b] = d sk2
[d, a] = 1 [d,b]= 1 [d, c] = 1
aa = a-tad ~
a s = bd k~
ab = bad -sk~ ac = c - t a d -k~
ad = d-la
[c,a]= 1
-1 0
~(~) =
o
0 0
-
~2 -1 0 0 0
2k3 0 1 0 0
h2 0 0 -1 0
>
0 '~ 0 1
0 1
H 2 ( Q , Z ) = Z 2 = Z 3 / A , A = {(k~, ks, k~)llk~ = k= = O, k3 ~ Z}
AB-groups: vk >
o,
(~,o,o)
<
Remark:
>
< (k, l, 0) > ~ <
((k, l), 0, 0) >
5. Q = C 2 E: < a,b,c,d,a
II [b, a] = 1 [c, a] = d k~ [c, b] = d kl a a = b - l a d It: ab = a - l a d ~2 ac = c - l a d k3
~(a) =
0 0 0 0
ks 0 -1 0 0
ks -1 0 0 0
k3 0 0 -1 0
[d, a] [d, b] [d, 4 as =
= 1 = 1 -- 1 d ~,
>
a d = da
2
0 0 0 1
H2(Q, Z) = Z @ (Z2) 2 = E 4 / A ,
A = {(kl,...,k4)llkl
= O, ks, k4 C 2Z, ks E Z }
AB-groups:
v k > o , < (k,o,o, 1) > 6. Q - - P m E:
all
[b,a]=l [c, a] : d kl [c, b] = 1 a a = act
[d,a]= 1 [d, b] : 1 [d, c] = 1 a 2 = d k~
ab = b - l a d k~
a d = da
OLC ~
COt
>
171
Chapter 7: Classification survey
172
A(a) =
1
0
k2
0
~2 '~
0 0
1 0
0 -1
0 0
0 0
0 0
0 0
0 0
1 0
0 1
J
HS(Q, Z) = Z ~ (Zs) s = Za/A, A = {(kl, ks, k3)llkl = 0, ks, k3 9 2 z } AB-groups: V k > O , < (k,O, 1) >
7. Q = P c Trivial action:
E:
[b,a] = 1 [c, a] = d TM [c,b]= 1
[d,a] = 1 [d, b] = 1 [d,c]= 1
>
o~a = aad h' a s = cd hs ab = b- l ad k~ ad = da O[.C ~
1
~(~)
=
2
ca[.
2
0
1
0
0
0
o
0 0 0
-1 0 0
0 1 0
0
0 0
1
1
J
HS(Q, Z) = Z @ Zs = Za/A, A = {(kx, k=, ka)llk~ = O, ks ~ 2Z, ka ~ Z} AB-groups:
Vk>O, < ( k , O , O ) > v k > 0 , < (k, 1,0) > Action ~d = d - l :
[d, a] = 1 E : < a , b , c , d , a ]1 [b, a] = d kl [c,a]= 1 [d,b]= l [c, b] = d sk~ [d, c] = 1 aa = aad k~ a 2 = cd h" ab = b - t a d k~ ad = d - t a OtC ~
I -1 0 A(a) = 0 0 0
co'd -sk4
-k3 1 0
-~ 2 0 -1
2k4 0 0
0 0
0 0
1 0
H 2 ( Q , Z ) = Z 2 ~ Z2 = Z4/A,
0 I 0 0 1
1
>
Section
7.2 4 - d i m .
AB-groups,
with 2-step
Fitting
subgroup
A = { ( k l , . . . , k 4 ) H k l = k2 = O, k3 E 2Z, k4 E Z} AB-groups: Vk > 0, < ( k , 0 , 0 , 0 ) > Vk > 0 ,
k = O m o d 2, < ( k , 0 , 1 , 0 ) >
vk>o, < ( o , k , o , o ) > Vk>O, < (0, k, 1,0) > Remark: If k ~ 0 m o d 2 < (k, 0, 0, 0) >-~< (k, 0, 1, 0) >. 8. Q = C m E : < a,b,c,d,a
II [ b , a ] = 1
[d,a]= 1 [d, b] = 1
[c, a] = d ~1
[c, ~]
=
d ~,
[d, c]
~ a = bc~d k~ ab = a a d -k~
=
a 2 = d k~ o~d = do~
OtC ~ - COL
A(a) =
H2(Q,Z)
=
Z G
Z2 =
1 k2 -k2 0 0 1 0 1 0 0 0 0
0 ~2 / 0 0 0 0 1 0
0
0
0
0
1
Z a / A , A = {(kl,k2, ka)[[kl = 0, k2 C Z, ka E 2Z}
AB-groups: Vk>0, < (k,0,1) >
9. Q = C c Trivial action: E:
al[
[b,a]= 1 [c, a] = d kl [c, b] = d ~I a a = b a d k~ ab = a a d al-k~ OLr z
H2(Q,Z)
>
COL
4
A(a) =
[d,a]=l [d, b] = 1 [d, c] = 1 a 2 = cd k~ ad = da
4
2
0 0
0 1
1 0
0 0
0 0
0
0
0
1
1
0
0
0
0
1
= Z = Z 3 / A , A = {(kl, k2, k3)Hkl = 0, k~,k~ E Z}
173
174
Chapter
7: C l a s s i f i c a t i o n s u r v e y
AB-groups: vk > o, < ( k , o , o ) > Action ~d = d-Z: E:
H [ b , a ] = d kl [c, a] = d k~ [c, b] = d-k~ a a = bad k3 a b = a a d ~:~-k2
[d,a] = 1
>
[d, b] = 1 [d, c] = 1 a 2 = cd k~ ad = d-la
a c = c a d -2k"
1 o o 0 0
A(~) =
~-k3 ~ 0 ~ ; 0
~-k3 1 o 0 0
2k4 0 o 1 0
0 '~ 0 o 1
)
H 2 ( Q , Z) = Z 2 = Z 4 / A , A = {(kl, k2, k3, k4)llk~ = k= = 0, k3, k4 ~ ~}
AB-groups: Vk>0, < (k,0,0,0) > Vk > O, < (O,k,O,O) > 10. Q = P 2 / m E : < a,b,c,d,a,/311
[b, a] = 1 [c, a] = d kl [c,b]= !
[d, a] = 1 [d, b] = 1 [d,c]= 1
a a = a - l a d k~
a 2 = d k"
a b = ba
ad = da
>
O~C ---- c - l o ~ d k~
fla = a - 1 ~ d k~ ~b = b - l ~ d k~ fie = c - l f l d ks
A(a) =
1 0 0 0 0
k2 -1 0 0 0
0 0 1 0 0
k3 0 0 -1 0
k-i 2 0 0 0 1
a(Z) =
f12 = dh~ ~d = d~ o~/3 = ~ a
1 0 0 0 0
k2 -1 0 0 0
H 2 ( Q , Z) = Z q~ (Z2) s = Z 6 / A , A = { ( k l , . . . , k ~ ) l l k l =
AB-groups: None
ks 0 -1 0 0
k3 0 0 -1 0
k_~/ 2
0 0 0 1
0, k~,..., k6 ~ 2 z }
Section
7.2 4 - d i m .
AB-groups,
with 2-step
Fitting
175
subgroup
13. Q = P 2 / c
II
E : < a,b,c,d,a,fl
[b,a]= 1 [c, a] = d TM [c,b]= 1
>
[d,a]= 1 [d, b] = 1 [d,c]: 1
a a = a - l a d k2
a 2 = d k~
a b = ba
ad = da
O~c = c - l o ~ d - 2 k r
fla = a - l f l d l'~+k=
~2 = dk~
fib = b - l fld k* /3c = c - l f l d -2k~
fld = dfl a f t = c f l a d k~
--2kB A(a)=
0
-1
0
0
0
0
-1
0
0
0
1
0
0
0
0
-1
0 0
0 0
0 0
-i 0
1
0
0
0
1
~0
0
0
A(e)=
0 0
-1 0
H 2 ( Q , Z) = Z G (Z2) 4 = Z 6 / A ,
A = { ( k ~ , . . . , k~)ll~ = o, k : , . . . , k~ c 2 z , k, ~ AB-groups: Vk > 0 , k = 0 m o d 2 , 11.
Q
i)
z}
< (k, 0,1, 0,1, 0) >
= P21/m
E: < a,b,c,d,a,fl
.,,(o,) :
II
>
[b,a]= 1 [c, a] = d k~ [c, b] = 1
[d,a]= 1 [d, b] = 1 [d, c] = 1
tea = a - l a d k~
ce2 = bd ~
a b = ba r = c-lo~d k3
ad = da
fla = a - t l3d k~
t32 = d ~
fib = b - l fld - ~k~ tic = c - l fld k~
fld = dfl a13 = bfloed k~
1 0
k2 -1
0 0
k8 0
2 0
0
0
1
0
1
0 0
0 0
0 0
-1 0
0 1
)
A(e)=
(1t 122k6 o o o3 o2 o 0
o 0
-1 0
o -1
o 0
0
0
0
0
1
H~(O, Z) = ~ 9 ( z ~ ) ~ = Z ~ / A , A = { ( k ~ , . . . , k~)ttk~ = O, k2, ka, ks, (k4 - k6) ~ 2Z} AB-groups: Vk > 0 , k = 0 m o d 2 ,
<(k,0,0,0,
1,0) >
176
Chapter
7: C l a s s i f i c a t i o n s u r v e y
14. Q = P 2 1 / c Trivial Action: E : < a , b , c , d , o ~ , J II [ b , a ] = 1 [c, a] = d TM [c,b]= 1 ota = a-lo~d ~2
>
[d,a]= 1 [d, b] = 1 [d,c]= l o~2 = bd k4
a b = ba
~d = da
O~C = c - l o r d k3
1 ~r+k2 0
A(a) =
k3
0
-1
0
0
0 0 1 0
o
o
o-1
0
0
0 0
~ 4+
/32 = d k5
fib = b-1/3d - k s - 2 ~ ~ c = c - l ~ d k"
~d = d~ a/3 = bc/~ad k~
2 [11
kl +k2 - k z - 2 k s
0
A(Z)=
I_
1
H2(Q, A =
/3a = a - l f l d k ' + ~
0 0 0
-i 0 0 0
~0
Z) = Z @ (Z2) 2 q~ ~ 4
=
0 -I 0 0
k3 k~_ 2 0 0 -I 0
0 0 0 1
Z6/A,
{ ( k l , . . . , k6)Nk~ = 0, k=, k~ e 2Z, (-k~ + 2k, - 2k6) Z 4Z}
AB-groups: Vk>0, k - 0 m o d 2 , < (k, 0, 0, 0,1, 0) > Vk >0, k - 0 m o d 2 , < (k, 0, 0,1,1, 0) > Yk>0, k ~ 0 m o d 2 , < (k, l, 0, 0,1, 0) > Remark: Vk ~ 0 mod 2, < (k, 1,0, 0, 1,0) > ~ < (k, 1, 0, 1, 1,0) > Action ~d = d -1, ~d = d: E: < a,b,c,d,a,fl
II
[b, a] = d TM [c, a] = 1 [c, b] = d 2k~ a a = a - l a d k'
[d, a] : 1 [d, b] = 1 [d, c] = 1 a 2 = bd k~
a b = b a d -2k~ a c = c - i c e d -k~
ad = d-la
j3a = a - l fld ~4 [32 = d k~ fib = b - l ~ d k~- 2k~ ~d = d~ tic = c-lfld k~+2k~-2kS+2k~ aft = bc/3ad k~
I -0I A(a) =
2
k_.t -I
2
- ~ + 20 k 3
2
/r0
0
0
i
0
0 0
0 0
0 0
-1 0
0i 1
1
>
Section
7.2 4 - d i m .
AB-groups, 1 0
A(~) =
k4
with 2-step
k2
-1 0 0 0
0 0 0
H2(Q,
2k3
-
Fitting
k2 + 2k3
2k5 + 2k6
-
~2
0
0
0
-1 0 0
0 -1 0
0 0 1
Z)=
Z2 G
177
subgroup
J
(Z2) 2= Z 6 / A ,
k2 : O, k4, ks E 2Z, k3, k6 E Z}
A : {(kl,...,k6)llkl=
AB-groups:
vk>o, < (k,o,o,o, Lo) > Yk > 0 , 12.
q
k-Omod2,
<(0, k,0,0,1,0)>
= c2/m
E : < a , b , c , d , oq/311
l
1
k2 0 -i 0 0
0
;~(~)
=
o 0 0
k2 -I 0 0 0
k3 0 0 -I 0
[b,a]=l [c, a] = d ~ [c, b] = d k~
[d,a]=l [d, b] = 1 [d, c] -- 1
o~a = b - l ~ d k2
oL2 = d k"
~b = a - l ~ d k~ ~ c = c - l a d k~
~ d =- d a
fla = a-1]3d k~ fib = b-Xfld 2k~-k~ tic = c - l f l d ~
f12 _= dk~ fld = dfl aft =/3a
2
0 0 0 I
>
[ 1 0 A(fl) = 0 0
ks -1 0 0
2k2-k5 0 -1 0
ka 0 0 -1
0
0
0
0
H ~ ( 0 , Z ) = Z 9 (Z2) 4 = Z e / A , d = { ( k l , . . . , k 6 ) l ] k l = 0, ka, k4, k~,ks 6 2Z, k2 6 Z}
AB-groups:
None 15. Q = C 2 / c E : < a,b,c,d,c~,/3][
[b, a] :- 1 [c, a] = d k~
[d, a] : 1 [d, b] = 1
[~, b] = d ~
[d, c] = 1
&a = b-lo~d k~ ab = a - l ~ d k~
o~2 : d k~ ~d : d~
O~C =
c - l o~d - 2k 6
~ a = a - l fld ~ fib = b - t f l d k'+2k~-k" tic = c - Xfld - 2k~
f12 = dk~ fld = dfl exl3 :- c/~ad k~
>
2
0 0 0 1
178
Chapter
A(a) =
A(Z) :
7: C l a s s i f i c a t i o n s u r v e y
1
~4 + k 2
~4 + k 2
-2k6
0 0
0 -1
-1 0
0 0
o o
0 0
0 0
0 0
-1 0
1
k1+2k2-k4
~2+
~2 )
1
1
k4
-2k6
~2
0 0 0
-1 0 0
0 -I 0
0 0 -l
0 0 0
0
0
0
0
1
H : ( Q , Z ) = ~ @ (~2) 3 = Z 6 / A ,
A
=
J
{ ( k l , . . . , ko)llk~ = 0, k~, k~, k~ ~ 2Z, k=, ko ~ ~ }
AB-groups: Vk > 0 k = mod2, < (k, 0,1, 0,1, 0) > 18. Q = P21212 E : < a,b,c,d,a,flll
[b, a] = d TM [c,a]= 1 [c,b]= 1
[d, a] = 1 [d,b]= 1 [d,c]= 1
oLa = a - l o ~ d - k ~ - 2 ~ s + 2 k ~ + 2 k ~
o~2 = d k~ ad = da
ab = b - l a d k l - 2 ~ ~C
1 0 A(a) =
=
COt
fla = a - l fld k~
;2: = bd ks
fib = bfld -2k~ tic = c - I fl
fld = d-1/3 a f t = a b - l f l a d k4
-kl+2(k2-k3+k4) -1
2
0 0
2ks
2+
_k_i
0
0
-1
0
0
0 0
0 0
0 0
1 0
0 1
A(fl) =
-I
-~ 2
2k3
0
0
0
-1
0
0
0
0 0 0
0 0 0
1 0 0
0 -I 0
0 1
I
H2(Q, Z) -- Z (~ Z2 = Z 4 / A , A = { ( k l , . . . , AB-groups: gk > 0 k - r o o d 2 ,
0
-
< (k,l,0, O)>
k4)llk~
1
= O, k2 E 2~, k3, k4 E E}
Section 7.2 4-dim. AB-groups, with 2-step Fitting subgroup
179
19. Q = P212121 Action ad : d -1, ~d = d: ]] [b, a] = 1
E: < a,b,c,d,o~,~
[d, a] [d, b] [d, c] o~2 = ad :
[c, a] = d TM
[~, b]
= 1
o~a = a - l o r d k~ ab = b - l a oLc = co~d - 2k~
f12 = bd k, fld -- d/3 a/3 = ab-lc/3o~d k"
fla = a - l fld 3k'+ 2(k~-~+k~) fib = bfl
Zc = c-lZd -k~-~k~ l
-1
-~
0 A(a) =
l1 0 0 0 0
A(Z) =
2
-I 0 0 0
0 0 0
0 0
~ + 2k~.
-I 0 0
0
0
1
1 2
0
1
3kl + 2(k2 - k3 + k4) -1 0 0 0
H2(Q, Z) -- Z = •4/A,
2
0 "~ 1
0
0 0 1 0 0
>
= 1 = 1 = 1 cd k~ d-la
J
- k l - 2k2 0 0 -1 0
A =- { ( k l , . . . , k4)[lkl = 0,
~2 ] 0 !2 0 1
k2, k3, k4 E 25}
AB-groups: vk>o < (k,o,o,o) >
Action ~d = d, ~d = d - l : E : < a , b , c , d , ol,fl [[ [b, a] = d TM [c, a] = 1 It, b] = 1
O~Ct: a-lo~d -kl-2k3+2k~+2k~ ab : b - l a d k~-2k~ OLC ~
C~
~2 = bdk~ fld = d - l ~ aft = a b - l c13ad k"
/3a = a - l ~ d ~ ~b =
l
1
0
A(a) =
0 0 0
[d, a] = 1 [d, b] = 1 [d, c] = 1 o~2 : cd k~ ad = da
b~d - 2k~
~176OooO1
- k l + 2 ( k ~ - k3 +k4) -1 0 0 0
>
180
Chapter
A(fl) =
HS(Q,Z)
7: C l a s s i f i c a t i o n s u r v e y
-1
- Ms
2kz
0
0 )
0
-I
0
0
0
0 0 0
0 0 0
1 0 0
0 -1 0
89 0 1
= Z = Z 4 / A , A = { ( k l , . . . , k 4 ) l l k l = 0, ks, ka, ka E Z}
AB-groups: All groups are isomorphic to one of the previous ease. Action ~d = d -1, ~d = d - I : E: < a,b,c,d,a,fl
N [b,a]= 1
-1 0 A(oo) =
[d,a]= 1
It, a] = 1
o o 0
~(~) = /
>
[d, b] = 1
[c, b] = d TM
[d, c] = 1
Ooa .= a - l o o
Oo2 = c d k 2
Oob -= b - l Ood k l Ooc = cOod-2k2 Za = a- l ~
Ood = d - l Oo /3 2 = bd k~
fib = b~3d- 2 ~ t3c = c - l fld - ~ '
fld = d - l f l veil = a b - l cflood k"
o
_as
2ks
-I 0
0 -i
0 0
0 0
0 0
1 0
- 1 0 0 0 0
0 -1 0 0 0
2ka 0 1 0 0
H 2 ( Q , Z) = Z = Z 4 / A , A = { ( k l , . . . ,
~+~,
s 0 I 5 1
~s 0 0 -1 0
k4)llkl
)
0 "~ 0 1
0 1 =
J
0, ks, ka, k4 E Z}
AB-groups: All groups are isomorphic to one of the previous case. 27. Q = P c c 2 E : < a , b , c , d , o ~ , f l II [b, a] : d k'
[c, a] -- 1 [c, b] -- 1
[d, a] -- 1 [d, b] --=- 1 [d, c] -- 1
Ooa = a - l o o d ka
Oo2 = d ~,
a b = b-lood k~
ad=
dol
OOC = COo
fla = a/3d k~
j3 2 = cd k"
fib = b - l fl tic = c/3d -2k~
l~d = d - lj3 Oo13 = ~Ood -k"
>
Section
7.2 4 - d i m .
AB-groups,
with 2-step
o1 o -1OOo / o ~
,x(o~)= o 0 0
0 0
0 0
A(~)=
1 0
Fitting
oo o1 0 0
0 0
181
subgroup
0 0 -1
2k5 0 0
0 '~ 0 0
o
1
89
0
0
1
H 2 ( Q , Z) = Z 9 (Z2) a = Z S / A ,
J
A = { ( k l , . . . , ks)Ilk1 = 0, ks, k3, k4 E 2Z, ks E Z} AB-groups: Vk>0k=mod2,
< (k, 0, 0,1, 0) >
30. Q = P n c 2 E : < a,b,c,d,a,fl
ll [b, a] = d TM
[c,a]= 1 [c, b] = 1 aa
= a-lad
o~2 = d k" ad = da
k~
ab = b - l a d ~k~+2k~ ~C ~
COt
~ a = afld ~'~+k2 fib = b - l ~ /3c = c/3d-2k~
-1
0
o
-1
0 -~ 2 0 0 1 0
0 0
0 1
10 ~2+ k 2
~(~)=
0 ~00
2(kz+ks)
0 0
1 0
H2(Q,Z)
A
=
>
[d, a] = 1 [d,b]= 1 [d, c] = 1
f12 = cd k. ~d = d-lfl a f t = bj3ad k~
-1 0
A(Z)=
-kl-k2 1
0 0
2k4 0'~ 0 0
0
0
-1
0
0 0
0 0
0 0
1 0
o
Z G (Z2) 2 = Z S / A ,
{ ( k l , . . . , ks)llk~ = 0, k2, k~ ~ 2Z, k,, k~ ~ Z}
AB-groups: V k > 0 < (k,0, 1,0,0) > 32. Q = P b a 2 E: < a,b,c,d,a,
flll
[b, a] = d TM [c,a]= 1 [c, b] = 1
[d, a] = 1 [d,b]= 1 [d, c] = 1
aa = a-tad -~-2k4
ot 2 = d It2
a b = b - l a d -3k~-2k'+2k~+2k~
a d = do~
OLC ~
COL
jOa = arid - : k " /3b = b - l f l d -k~ /3c = c~d k ~
f12 = ad k, fld = d - l f l a f t = a - l bflo~d k~
>
J
182
C h a p t e r 7: C l a s s i f i c a t i o n s u r v e y
1 _ ~_r 2 _ 2k4 - 3 k l + 2 ( k 2 - k 4 + k 5 ) 0 -1 0 0 0 -1 0 0 0 0 0 0
~(~) =
2
( -1 ) 02k4 1000 0 -10002 ~0 -k3 0 0001 00~11
~(Z) =
= Z @ (Z2) 2 =
H2(Q,Z) A --- { ( k ~ . . . .
0 ~4+ ~ - ~2 0 0 0 !2 1 0 0 1
,
z~)llk~
ZS/A,
~}
= 0, k2, k~ c 2~, k4, k~ c
AB-groups: Yk > 0 k _ = 0 m o d 2 , < (k, l, 0, 0, 0) > Yk > O k = _ O m o d 2 , <(k,l,l,0,0)> 34. Q =
Pnn2
E : < a,b,c,d,o~,flI[
[b, a] = d TM [c, a] = 1 [c,b]= 1
[d, a] = 1 [d, b] = 1 [d,c]= 1
a a = a-lc~d ~
c~2 = d k~
>
ab = b-tad -2k~+2k~+2k~+~; a d = da ~ C ~ CO~
fla = aj3d k~+k~ ~b = b - l f l d -k~ ~c = c/3d -k~-k~-2k"
~(~) =
1 ~t+k2 0 -1 0 0 0 0 0 0
~(Z)= (
j3~ = acd k~ fld = d - t ~ o ~ = a - l b t 3 a d 1'~
k2+2(k3-kl+ks) o -1 0 0
0 ~2 - ~ 0 0 1 0
-1 0 0
- - k l -- k2
k~t 2
k t + k2 + 2k4
1 0
0 -1
0 0
0 0
0 0
0 0
1 0
H2(Q,
Z) : Z q~ (Z2) 2 :
4
0 "~
1 2
1
ZS/A,
A : { ( k ~ , . . . , k~)llk, : 0, kz, k~ c 2Z, k,, k~ ~ z }
AB-groups: V k > O < (k,O,l,0, O) >
2
0 12 0 1
J
/
S e c t i o n 7.2 4 - d i m . A B - g r o u p s ,
183
with 2-step Fitting subgroup
26. Q = P m c 2 1 E: < a,b,c,d,a,fl
II
[b, a] = 1 [c, a] = d TM [c, b] = 1
[d, a] = 1 [d, b] = 1 [d, c] = 1
aa ab ac /3a
a 2 = cd I'2 ad = d-la
= = = =
a - l a d ka b-la cad -2h2 a/3
fl: = d k"
fib = b- l fld ks Zc = cZ
-1 0 0 0 0
A(a) =
_2 2 -1 0 0 0
0 0 -1 0 0
2k2 0 0 1 0
fld = dt3 , ~ = Z,~d k"
l1
0 0 0
>
0 0 0 0
a(Z) =
1
1
0 1 0 0 0
ka 0 -1 0 0
0 0 0 1 0
h2) 0 0 0 1
H ~ ( Q , Z) = Z @ (Z2) 2 = Z 4 / A
A = {(kx,...,k4)llkl = O, k3, k4 E 2Z, k2 E Z} AB-groups:
Vk>0
< (k,0,0, 1) >
31. Q = P m n 2 1 E : < a,b,c,d,a,1311
-1 - ~2 0 -1
~(~)-=
o 0 0
o 0 0
[b, a] = 1 [c, a] = d TM [c, b] = 1
[d, a] = 1 [d, b] = 1 [d, c] = 1
o~a = a - l a d k'
a 2 = cd I'~
ab = b - l a otc = cord -2k~
ad = d-la
/3a = aft ~b = b-lj3d -2k~+2k" tic = c[3
f12 = dks ~ d = dfl at3 = blSad k*
0 2k2 0 ) 0 0 0 -1 0 0
o 1 0
1
1
~(~) =
1
0
0 0 0 0
1 0 0 0
H 2 ( Q , 25) = Z ~ Z2 = Z 4 / A , A = ( ( k ~ , . . . , k , ) l l k ~
AB-groups:
Vk>O
< (k,O, 1,0) >
-2kz+2k4 0 -1 0 0
>
0 0 0 1 0
~2 '~ 0 0 0 1
= o, k~ E 2 z , k~., k, ~ z].
184
Chapter
7: C l a s s i f i c a t i o n s u r v e y
29. Q = Pca21 Action a d = d - t , ~d = d: E: < a,b,c,d,a,flll
i ~(Ot)=
[b, a] = 1 [c, a] = d TM [c, b] = 1 ota = a-lo~d I~ o~b = b - t a otc = co~d -2k~ ~ a = a ~ d k~ j3b = b - l ~ d 2k~-2k~+2k" tic = c~3
[d, a] = 1 [d, b] = 1 [d, c] = 1 a s = cd h~ c~d = d-tOt
>
~2 = cd k, fld = d~ c ~ = bj3ad ~
o o) o) iil oo2-2k3 o 2o o
o
o
-1
o
'
A(Z)= o
o
-1
o
0 0
0 0
0 0
1 0
~ 1
0 ~0
0 0
0 0
1 0
H S ( Q , Z) --- Z r Z : -- Z 4 / A , A = { ( k l , . . . ,
k,)llkl -- 0, (k3 - ks) e 2z, ka e z }
AB-groups:
Vk>0
< (k,0,0, o) >
Vk>0k=0mod2 < (k, l, 0, 0) > R e m a r k : If k ~ 0 m o d 2 < (k, 0, 0, 0) > ~ < Action ~d = d, ~d = d - i : E : < a,b,c,d,o~,13
[I
(k, 1, 0, 0) >
[b, a] = d TM [c, a] = 1 [c, b] -- I om =- a - f o r d k~ ab = b - l ~ d sk~-2~'+2h~ OtC ~
1 ~(~) =
0 0 0
S
M+ks -I 0 0 0 -I
~(~) =
I
0 0 0 0
>
COt
~ a = a ~ d k~+~2 f~b = b - l ~ j3c = cj3d -sk"
0
[d, a] = 1 [d, b] = 1 [d, c] = 1 o~s = cd k~ ad = da 13s = cd k4 fld = d - l ~ a/3 = b ~ a d k~
2(kz-ka+ks)
-kt-ks 1 0 0 0
0
0
0
-I 0 0
0 1 0 0
0 -1 0 0
2k4 0 0 1 0
k~-kaO_~2121 I
S e c t i o n 7.2 4 - d i m . A B - g r o u p s ,
H s (Q,
Z) =Z @ (Zs)2=ZS/A,
185
with 2-step Fitting subgroup
A = {(kl, 9 9
= o, (k~ - k~), ks e 2z, k~ c z}
ks)llk~
AB-groups: V k > 0 < (k, 1 , 0 , 0 , 0 ) >
V k > O < (k,0,0,0, o) > V k > 0 < (k,o, Lo, o) > R e m a r k : < (k, 1, 0, 0, 0) > ~ < Action ~d = d -1, ~d = d - l :
(k, 1, 1, 0, 0) >.
[b, a] = 1
E:
[c, a]
-1 0
~(~)
~(f) =
=
-1 0 0 0 0
>
[d, a] = 1 [d, b] = 1
= 1
[c, b] = d 4k~
[d, c] = I
a a = a-lo~ ab = b - l a d TM ore = cad -sl~2
a s = cdh2 ad = d-lot
f a = a f d ~3 fib = b- l f d TM f c = c f d -sk~-2k2
f s ~_ cdk~+k~
0 -1
o
o
0 0
0 0
-kl 0 -1 0 0
-k3 1 0 0 0
-kl 0 -1 0 0
H 2 ( Q , Z) = Z (~ Z2 = Z 4 / A , A = { ( k l , . . . ,
fd = d-~f a f = b f a d k4
k~ + 2k2 - k l 0 0 0 1 1
!2
0
1
0 0 1 0
0 0 1
1
k4)llkx =
J
o, k~ ~
2Z, ks, k4
~ z}
AB-groups:
vk>o Vk>0
< (k,0,0,0) > < (k,0,1,0) >
33. Q = Pna21 A c t i o n ~ d = d -1 ~ d = d : E: < a,b,c,d,a,f]]
[b, a] = 1 [c, a] = d TM [c, b] : 1
[d, a] = 1 [d, b] -- 1 [d, c] -- 1
a a = a - l a d k~ ab = b - t a a c = cord -2k~
a 2 = cd k~ ad = d-la
fa = af fib = b - l f d -kl-2k~+2k" fc = cfd -~
f12 = adks fd = df a f = a - l b f a d k"
>
186
C h a p t e r 7: C l a s s i f i c a t i o n s u r v e y
A(.)=(
H 2 ( Q , Z ) -- Z -- Z 4 / A , A : {(kl,...,k4)llkl : O, k2, ks, k4 E
Z}
AB-groups: vk>o
< (k,o,o,o) >
Action ~d --- d, Zd = d-l: >
[d, a] = 1 [d,b] = 1
II [b, a] = d TM [c, a] = 1 [~, b] = 1
E: < a,b,c,d,a,fl
[ 4 c] = 1
o~a = a-lo~d -kl-2k~ ab = b-lo~d -3kl+2k~+ka-2k4+2k~
c~ 2 = c d k~
a d = da
O~C = CO~
A(.) =
~ a = a Z d - ~k"
~2 = ad k,
fib = b- l ~ d -k~ t3c = cfld k"
fl d = d - l fl aj3 = a - l bflad k~
1
k _ _2 l _ 2 k 4
0 0 0 0
-1 0 0 0
0001 3hl-'k~1+2ka
-3kl +2(kz-k4+ks)+k3 0 -1 0 0
a(~) =
--1
2k4
0 0 0 0
1 0 0 0
n:(Q, z) = z,
0
~2
0 -1 0 0
-k3 0 0 1 0
01 ) 0 0 1
z , = ZS/A,
A = ( ( k l , . . . , ks)Ilk1 = 0, (ks + 2k2) ~ 4~, k4, ks C Z)
AB-groups: vk > o < ( k , o , o , o , o ) >
Vk>O, k_=Omod2, < ( k , l , 0 , O,O)> V k > 0 < (k,0, 1,0,0) > Remark: I f k ~ 0 m o d 2 < ( k , 0 , 0 , 0 , 0 ) > ~ < < (k,o, 1,o,o) >z-< (k, 1,1,o,o) >
(k, l, 0, 0, 0) >.
1
Section
7.2 4 - d i m .
AB-groups,
with 2-step
Fitting
187
subgroup
Action ~d : d -1, ~d = d - l : E : < a,b,c,d,o~,13 [[ [b, a] = 1 [c, a] = 1 [c, b] = d TM o~a : a - l o ~ vtb = b-lo~d k~ ~ c = c ~ d -2k2 fla = at3d -2k~
[d, a] [d, b] [d, c] O~2 = ~d =
132 = ad ks
13b = b-l~3 13c : c13d -k'-2~t2
13d = d-113 c~13= a-lb13o,.d k`
-1 0
A(~):
0 -1
0
0 0
1 0
0 0
A(Z):
-
>
= 1 = 1 = 1 cd ~ d-lc~
0
2k3 0 1 0 0 -1
0 0
0 0
kl+2k2 0 0
0 0
- ~ -2~ 1
0
1 0
0 1
H 2 ( Q , Z) = Z = Z 4 / A , A = { ( k l , . . . , k~)llk~ = o, k~, k~, k~ c ~} AB-groups: Vk>o < (k,0,o,0) > 37. Q = C c c 2 E: < a,b,c,d,~,~l
I [b, a] = d k~
[c, a] = 1 [c, b] : 1
[d, a] : 1 [d, b] : 1 [d, c] = 1
o~a = a - l a d h~ c~b = b - i c e d -k~+2k"
o~2 = d k" otd = do~
OLC :
COt
~ a : b13dk" fib = aj3d k" ~ c = cfld -2k5
A(a)=
>
1 0
k~ -1
-k2+2k4 0
0 0
~2 0
0 0 0
0 0 0
-1 0 0
0 1 0
0 0 1
J
A(~):
fl~ : cd k~ fld = d-t13 a f t = 13ad -k" -1
-k4
-k4
2k5
0
0
0
1
0
0
0
1
0
0
0
0 0
0 0
0 0
1 0
89 1
H~(Q, z ) = ~ 9 (~:): : Z~/A, A : { ( k ~ , . . . , k~)llk~ : o, k:, k~ ~ 27~, k,, k~ ~ ~:} AB-groups: Vk>0, k~0mod2
< (k, 0,1, 0, 0) >
J
188
Chapter
7: C l a s s i f i c a t i o n s u r v e y
36. Q = C m c 2 1 E : < a , b , c , d , a , fl II [ b , a ] = 1 [c, a] = 4 TM It, b] = d TM a a = a - l a d kl o~b = b - l a d k~
>
[d,a]= 1 [d, b] = 1
[4 ~] = 1 ct 2 = cd k2 ad = d-Xa
otc = co~d - 2 k 2
fa=bfld ~ fib = a f d -k~ tic = cfl
-1 0 0 0 0
A(a) =
-~ 2 -1 0 0 0
-~ 2 0 -1 0 0
2k2 0 0 1 0
0 0 0
f 2 = d k. fd = df ~fl = f a d k"
1 0 0
a(f) =
k3 0 1
-k3 1 0
0 0 0
h2 ) 0 0
1
0
0
0
1
0
1
0
0
0
0
1
H 2 ( Q , Z) = Z $ Z : = Z4/A, A = { ( k l , . . . , k 4 ) [ l k l : 0, k4 E 2Z, k2, k3 E El.
AB-groups:
vk>o
< (k,o,o, 1) >
41. Q = A b a 2 E: < a,b,c,d,~,f
1[ [b, a] = 1 [c, a] = d TM [c, b] = d TM a a = b - l a d k~ v~b = a - l a d k~
>
[d, a] = l [d, b] = 1 [d, c] = 1 a 2 = d k~ otd -= d~
OtC = c - l o r d 2 k ~ - 2 k 5
f a = b-tfldk" f b = a - l f d 2kl-k" f c = c f d -2k~
-
~(~)=
0 -1 0 0 0 0
0 0 0
0 -1 0
~ 0 1
~(~)=
f2 = cdk~ fd = d-if c~f = a b c - t f ~ d k~+k2-k,+k~
0 0
0 -1
-1 0
0 0
0 01
0 ~0
0 0
0 0
1 0
1
H 2 ( Q , Z) = Z (3 (Z2) 2 = Z S / A ,
A -- ( ( ~ , . . . , AB-groups:
V k > O < (k,o, 1,0, o) > V k > O < (k, 1, 1,0,0) >
k~)llk~ -- 0, k~, (k~ + k,) Z 2Z, k~ C Z}
Section
7.2 4 - d i m .
AB-groups,
with 2-step
Fitting
189
subgroup
43. Q = F d d 2 E : < a , b , c , d , o ~ , f l N [b, a] = d k'
1 A(a) =
[c, a] = d kl [c, b] = d - k l
[d, a] = 1 [d, b] = 1 [d, c] = 1
a a = b c - t a d k~+2ks+2k~
a 2 = d ks
ab = ac-l ad ~ aC = c - l a d 2(k~+hs+kS)
ad = da
13a = b c - t / 3 d -k2-~k"
/3 9" = bd it"
fib = b/3d - 2k"
fld = d - l fl
tic = a - l b ~ d ~''+~'2
a f t = a b - l f l o t d I`"
_ 2 +4 k 2 + 2 ( k a + k s )
~4+ k 2
0 0
0 1
1 0
0 0
-I
-I 0
A(~)
0
=
H ~ ( Q , Z) = Z @ Z 2
--1 0 0 0 0
>
_ 2 4 -4- k2 -4- 2k4 0 1 -1 0
= ZS/A,A
=
2(k2 + k~ + k~)
_k2+k~ 2
0 0 -1 0
0 0 89 I
2k 4 0
- - ~4 -- k 2 --1
1
1
1_ 2
0 0
0 0
0 1
0
0
{(kx,...,ks)llkx = O, k3 9 2Z, k2,ka, k5 ~ Z}
AB-groups: Vk>0 < (k,0, 1 , 0 , 0 ) > 45. Q = I b a 2 E: < a,b,c,d,a,
flll
[b, a] = 1 [c, a] = d k' [c, b] = d - ~'
[d, a] = 1 [d, b] = 1 [d, c] = 1
a a = b a d k2
a 2 = d~
c~b = a a d -k~ ~C = a - l b - l c - l o t d
)~(a)=
1 k2 - k 2 0 0 1 0 1 0 0 0 0 0 0 0
k2-
ad = da k2-2;t4-2k~
fla = c - l ~ d k"
/3 2 = abd ~
fib = abcfld - k ' - 2 ~ tic = a-1/3d - k"
~ d = d-1/3 a f t = j3ad - k s
2k251o (ilk4o -1
o
-1 0
0 1
H~(Q, Z) =
>
~(Z)=
Z @ (Z2) 2 =
o
-1 0
ZS/A,
k4 + 2k5 --1 1 1 0 1 0 0
1
o!
0
1
190
Chapter
7: C l a s s i f i c a t i o n s u r v e y
= O, k 3 , ( k : + k s ) ~ 2Z, k4 ~ E}
A = {(k~,...,ks)llk~
AB-groups: Vk > 0 , k - 0 m o d 2 , < (k, 0,1, 0, 0) > Vk > 0, k -- 0 mod 2, < (k, 1, 1, 0, 0) > 60. Q =
p~:~ b ~,~
E : < a , b , c , d , ot,f~,7 I[ [b, a] = d 4k~
[c,a]= 1 [c, b] = 1 ota = a-totd -2(k~-k2+k3-k')
[d, a] = 1 [d,b]: 1 [d, c] = 1 ot2 = dk~
otb : ot7 : t3a = fib = ~c =
otd otc /32 ~d ot~
dot cot bd k" d-1/3 a b - l f l o t d k"
7 a = a - 1 7 d -2(k~-k~+k,-k4)
72 = d k~ 7 d = d7 t37 = bcTfld k~
kt-2k3 0 -1
0 0 0
k~+k~-k, 0
0
0
0
1
0
0
0
0
0
1
=
1 0 0 0
: = : = :
"/b = b - 1 7 d -2hs 7c = c - 1 7 d 2(k~+k~-k~)
1 2(k2-kl+k4-ks) 0 -1 0 0
A(~) =
=
b - l o t d 2(k~-k') a~ otdk~-k:+ka-~4 a-1/3d 2kl b~d - 2k" c-lfl
-1 0 0 0
-kt -1 0 0
2k3 0 1 0
0 0 0 -1
0
0
0
0
-2(kl-k2+ka-k4) -1 0 0
0
0
-2k3 0 -1 0 0
H 2 ( Q , Z) = E ~ (Z2) 2 =
AB-groups: Vk > o < (k, 1, o, O, O, 1) >
>
J
0 ) 0 1 1
2ka+2k~-2k6 0 0 -1 0
Z /A,
h2 "~ 0 0 0 1
J
Section
7.2 4 - d i m .
AB-groups,
with 2-step Fitting subgroup
191
56. Q = p ~ ! E:<
a , b , c , d , a , / 3 , Tll
1
[d, a] = 1 [d, b] = 1 [d, c] = 1 o~2 = d ~ ad = da a 7 = abTo~d k" f12 = bd k, ~ d = d - t fl tic = c - t ~ ,.[2 = dk,
7b = b-17d-2k~ 7c = c - 1 7 d 2('~+/t~-k~)
7d = d 7 ,27 = bcT~d h~
~+2(k3-k4)
0 0 0 0
~(~) =
[b, a] = d TM [c, a] = 1 [c, b] = 1 ota = a - l a d k~+2k~-2k~ ab = b-tc~d k ~ - 2 ~ o~c = co~ fla = a - l fld k~ fib = b13d- 2k~ a f t = a b - l f l a d hl-k~+2ks-k~ 7a = a - 1 7 d 2(kl+ha-h')
2
~(~) =
=
1 0 0 0 0
~-2k3
0
k~+k,-k~
0 -1 0 0
0 0 1 0
-2 I2 0 1
--1 0 0 0
0 0 0 0
-1 0 0 0
2(kl + k3 - k4) -1 0 0 0
2k3 0 1 0 0 -2k3 o -1 0 o
0 0 0 -1 0
0 '~ 0 i 1
2(k3 + k5 - k~) 0 0 -1 0
H 2 ( Q , Z ) = Z G (Z2) 2 = Z 6 / A , A - - { ( k l , . . . , k 6 ) l l k l = O, k : , k e E 2Z, k3, k4, k5 e Z } AB-groups: Yk>O,k-Omod2
< (k, l, 0, 0, 0,1) >
2
0 0 0 1
>
192
C h a p t e r 7: C l a s s i f i c a t i o n s u r v e y 2
55. Q = P ~ E:<
[b, a] = d =kl [c, a] = 1 [c, b] = 1
a,b,c,d,(~,~,Tll
ota, ~ a - l o ~ d - k x - 2 k 2 + 2 k 3 - 2 k 4
ab = b - l a d - k ' - 2 k ~
1
A(a) =
o~c
=
co~
oc'{
~a ~b tic 7a 7b 7c
= = = = = =
a-1~d k~ bfld -2k, c-1/~ a - 1 7 d -k~-2(ka-k~+k') b-13'd - ~ ' - 2 k ~ c - I 7 d k~
~2 = bdk~ /3d = d - l f l a/3 = a - l b - 1 / 3 a d ~'~ 7 2 ---- d 2k~+k2+k4+k6 7d = d7 ~ 7 = ab'yfld k~
-k 1 --
0 0 0 0
2(k2-ka+k4) -1 0 0 0
(_1
_ k_~ 2
A(fl) =
1
a(u)=
0 0 0 0
>
[d, a] = 1 [d, b] = 1 [d, c] = 1 Ot2 = d kz a d = da
0 0 0 0
-1 0 0 0
- k l - 2(k2 - ka + k4) -1 0 0 0
-kl-2ka 0 -1 0 0
~2 + 2k3 0 1 0 0
0 0 0 -1 0
- k l -- 2k3 0 -1 0 0
=
"fa
2
0 0
0 0
1
0
0
1
0 I 0 t ks 0 0 -1 0
2k~+k2+k4+h6 2
0 0 0 1
H 2 ( Q , Z ) - - Z @ (Z2)3= Z6/A,
A = { ( k l , . . . , k e ) I l k l = 0, k2, ks, (k4 + k6) C 2Z, k3 C Z} AB-groups: None
S e c t i o n 7.2 4 - d i m . A B - g r o u p s , w i t h 2 - s t e p F i t t i n g s u b g r o u p
58. Q = E:<
193
p ,~~ , ~ 2 [b, a] = d TM
a,b,c,d,~,~,7]]
>
[d, a] = 1 [d,b] =
[~, ~] -- 1 [~, b] = 1
[ 4 ~] = 1 ot 2 = dk=
eta = a - t a d -k~-2(~=-k~+~') ab = b - l a d -k~-~k,
ad = da a7 = 7a
/32 = bd k, ~d = d - l f l o~fl = a - l b - l t3ad k~ 7 2 = dk6
13a = a - l ~d t'~ #b = b/3d- 2k~ ~c = c - 1 # 7 a = a - t 7 d-k~-2(/r
7b = b - t T d - k ~ - 2 ~ 7C =
1 0
~(.) =
o 0 0
- k l - 2(k2 - ka + k4) -1 0 0 0
~(~) =
A(-~) =
1 0 0 0 0
C-17d2(2k1+ka+k4+l~--h~)
o 0 0
-kt-2(k2-k3+k4) -1 0 0
0
o 0 0
1 0 0
-kt-2ka 0 -1 0
-kl
-
Nolle
2kz
2
0 0 1 0
0 -1 0 0
o -I 0
0 0 0 1
1
1
2(2kt+k2+k4+ks-ke)
0
H2(Q, Z) = Z @ (Z2) 2 = ZS/A,
AB-groups:
7d = d7
~7 = abcTfl d ~
0 0
-1 0
~2 0 0 0
1
194
Chapter
62. Q = P ~ Mm
7: C l a s s i f i c a t i o n
CZ
E: < a,b,c,d,a,~,Tll
[b, a] = 1
Ot2 = cdk~
ab = b - l a
ad = d-la a')" = a c T a d h~ f12 = bd k,
OtC~ cord-2k~ fla = a - l l3d 3k'+ 2(~2-1'~+k*)
~b = b~ ~c = c - l ~ d -~'-2k~ 7 a = a - 1 7 d 3k~+2(k~-k~+~')
fld = dZ a13 = a b - l c f l a d k"
,T2 --_dh~-h4+k~
7b = b - 1 7 d -2k~
7d = d 7
"[c = c-17d -1~1-2k2
A(a) =
1 a(~) =
A(7 ) =
Z'r = b~Zd k~
-1
-ka 2
0
0
-1
0
0
!2
0
0
-1
0
0
0
0
0
1
1
0
0
0
0
1
~2 + 2 k 2
3kl+2(k~-k3+k4)
0 0
-1 0
0 0
>
[d, a] = 1 [d, b] = 1 [ 4 ~] = 1
[c, a] = d TM [c, b] = 1 ota = a - l a d k~
I
survey
0
0
-kl-2k2
0
~ 0 1_ 2
0
1 0
0 0
-1
0
0
0
0
1
1 0 0
3kl + 2(k2 - k3 + k4) -1 0
-2k6 0 -1
- k t - 2k2 0 0
0 0
0 0
0 0
-1 0
J
2
0 ka-k4+k5 I 0
0 1
H 2 ( Q , Z ) = Z ~ (Z2): = Z 6 / A , A = { ( k l , . . . , k e ) H k l = 0, (k3 - ke), (kz - k4 + k~) E 2Z, k2 E Z} AB-groups: Vk>0, k_=0mod2,
< (k, 0, 0, 0,1, 0) >
Section 7.2 4-dim. A B - g r o u p s , w i t h 2 - s t e p F i t t i n g subgroup
195
61. Q = p ~ 2__re Action ad = d - l a , f d = d~: E : < a , b , c , d , ~ , f , 3"]] [b, a] = 1 [c, a] = d 4~ [c, b] -- 1 aa = a - l o l d TM ab = b - l a ac --- cad -2k~ fa = a-lfd r fib = bf tic = c- l f d - 2(k~+k2) 3'a = a-13'd 8k~+2(k~-ks+k~) 3'b = b-13'd 2(k~+k2-k~ 7c = c-13'd -2(k'+k~) -1 0 0 0 0
A(a) =
1 0
a(f) =
-41 -1 0 0 0
0 0 -1 0 0
7kl+k2-248+2k4 --1
0 0 0
I 1 2(441+k2-k3+k4) 0 --1 a(3")= 0 0 0 0 0 0
kl+2k2 0 0 1 0 0 0
0 0 0
[d, a] -- 1 [d, b] = 1 [d, c] = 1 a 2 = cd k~ ad = d - l a a3" = ac3"ad k~ f12 = bdk~ fld = d f a n = ab- l cflctd k* 3'2 = d -~+k~-Jt'+k5 3"d -- d3' f3" = bc3"fldk~
1 0 0
2(kl+k2-k6) 0 --1 0 0
0 / 1 !2 1
-2(kl+k2) 0 0 -1 0
-2(kl+k2) 0 0 --1 0
ka+~+k,
1
-k1+ks-k,+k5 2 0 0 0 1
H2(Q, Z) = Z ~ (Z2) 2 = Z~/A, A = {(kl,...,
k6)llkl = 0, (42 + 48 - 46), (48 - 4, + 45) e 2 Z )
AB-groups:
Vk > 0 , Vk>0, Vk > 0 , Vk > 0 ,
k=0mod2, k=0mod2, k~0mod2, k~0mod2,
< < < <
(k, (k, (k, (k,
0, 0, 0,1, 0) 0, 0, 0,1,1) 0, 0, 0, 0, 0) 0, 0, 0, 0,1)
> > > >
>
C h a p t e r 7: Classification s u r v e y
196
A c t i o n a d = da, fld = d-~fl:
OLC ~ CO:
[d, a] = 1 [d, b] = [d, c] = 1 o~2 = cdk~ a d = dc~ a 7 = acTad k5
Za = a - 1 ~ d 2k~ fib = bfld -2k3
f12 = bdk~ fl d = d - l fl
E : < a , b , c , d , a , f l , 7 II [b, a] = d 4k'
[c, a] = 1 [c, b] = 1 aa = a-tad ~(-~+~-~+~') ab = b - l a d 2(k~-k~)
7 2 =
& = ~-1~
d-k~+k2+k4+k~+k6
7a = a-17d2(-k~+k~-k~+k4) 7b = b - 1 7 d -2k~ 7c = c - 1 7 d 2(kx-k2+k~-k~-k~)
~(~) =
1 0 o 0
I 0
A(~) =
2(-k~ + k2-k3+k4) -1 0 0 0 -1 0 0 0 0
-kt -1 0 0 0
i 2(-kl + k2 - k3 + k,) -2k, -1 0 0 0
0 -1 0 0
2k3 0 1 0 0
kl-2kz 0 -1 0 0
0 0 0 -1 0
aft = ab- l c/3ad k4 7d = d7 /37 = bcTfld k~ 0 0 0 1 0
kl+as-k, 5 0 !2 1
J
i 0 a n d ,~(7) =
1
2(kl - k2 + k3 - k, - k~) -kl+k~+~,+~o+~o 2 0 0 -1 0
0 0 0 1
H ' ( Q , Z ) = Z ~ (Z2) ~ : Z~/A,
A:{(kl,
,k6)llkl=0,
>
(k2 + k4 + k5 + ks), (k3 - k4 - ks) ~ 2Z}
AB-groups: A l l g r o u p s a r e i s o m o r p h i c to one of the p r e v i o u s case.
J
Section
7.2 4 - d i m .
AB-groups,
with 2-step Fitting subgroup
197
Action o~d = d-lc~, fld = d-l,~: E : < a,b,c,d,~,~,7
~(~)=
[b,a]= 1 [c, a] = 1 [c, b] = d 4kl oea = a - l o t o& = b - l o~d TM o~c = co~d - 2 ~ 13a = a - l fl fib = bl3d - ~ tic = c-1t3d - 2~' 7 a = a - 1 7 d 2(-k=-ka+kS-k~) 7b = b-13'd 2 ( k ' - k ' ) 7 c = c - I T d -2ka
o o0
o
1
~i
0
1
o
o
-1
0 0
0 0
0 0
1
II
2(-kl-ka+ks-k6)
0
A(7 ) =
~(~)=
/io
01
0 0
0 0
o
o
-I
0
-2k2
k1+k2+k'+~ 2
0
0 0 -1
0 0 0
0
0
1
-1
0
0
0
0
o
2(kl-k3)
-I 0 0
0
H 2 ( Q , Z) --- Z @ (Z2) 2 = :Be~A,
A:
{(kl,...,k6)llkl
: 0, (ks - k4 - k~ + k ~ ) , ( k = + k 3 + k 6 )
AB-groups: All groups are isomorphic to one of the previous case. 75. Q = P 4 E : < a , b , c , d , oell
[ b , a ] = d kl [c, a] = 1 [c, b] = 1 o~a = botd ~ o~b = a-lo~d k~ OLC ~
A(~) =
>
[d,a]= 1 [d, b] = 1 [d, c] = 1 ot 2 = cd k~ e~d = d - l v~ a 7 = a c T a d k~ f12 = bd~,, l~d = d - l fl a]3 = ab- t cl3otd It" 7 2 = d k~+k'+ka+k" 7d = d7 f17 = bcT~ dl'~
[d,a]=l [d, b] = 1 [d, c] = 1 o~4 = d k~ o~d = do~
CO:
1
k=
ka
0
M4 )
0 0 0 0
0 1 0 0
-1 0 0 0
0 0 1 0
0 0 0 1
>
e 2Z}
1
J
198
Chapter
7: C l a s s i f i c a t i o n s u r v e y
H2(Q, Z) -- Z G Z2 @Z4 -- Z4/A,
A = {(k~,..., k~)llkl = 0, (k~ + k~) e 2z, k~ ~ 4z} AB-groups:
Vk > 0 , k = 0 m o d 2 , < (k, 0, 0,1) > Vk > 0 , k = - 0 m o d 4 , < ( k , 0 , 0 , 3 ) > Remark: < (k, 0, 0,1) > = < (k, 0, 0, 3) > V k = 4 1 + 2 ,
IEZ.
76. Q = P4t E:
[ b , a ] = d k~ [c, a] = 1 [c, b] = 1 a a = b a d k~ ab=a-lad ks OtC ~
a(~) =
[d,a] = 1
>
[d, b] = 1 [d, c] = 1 a 4 = cd k~ ad=da
COL
1
k2
ka
0
~4
0 0
0 1
-1 0
0 0
0 0
0 0
0 0
0 0
1 0
88 1
H2(Q, Z) = 25 @ Z2 = Z 4 / A ,
J
A = {(k~,..., k,)llk~ = o, (k2 + k3) E 2z, k4 ~ ~ } AB-groups: V k > 0 , < (k,0,0,0) > Vk>0, k-0mod2, < (k, l, 0, 0) > Remark: < (k, 0, 0, 0) > ~ < ( k , l , 0 , 0 ) >
Vk, k ~ 0 m o d 2
77. Q = P42 E:
[ b , a ] = d k~ [c,a]= 1 [c, b] = 1 a a = b a d k~ ab = a - l o t d ks OtC ~
A(~)=
[d,a] = 1
>
[d,b]= 1 [d, c] = 1 a 4 = c2d It~ ad = da
COt
h4 '~
1
k2
k3
0
0
0
-1
0
o
1
o
o
o
0
0
0
I
1
0
0
0
0
1
0
)
H~(Q,Z) = Z ~ (Z~) ~ = Z 4 / A , a - - {(kt,...,k4)llkl = 0, (k~ + k~),k4 e 2Z}
S e c t i o n 7.2 4 - d h n .
AB-groups,
with 2-step Fitting subgroup
199
AB-groups: Vk > O, k =_ O m o d 2, < ( k , 0 , 0 , 1 ) >
79. Q = I 4 E:
all
[b,a]=l [c, a] = d k~ [c, b] = d -k~
[d,a]=l [d, b] = 1 [d, c] = 1
a a = c - l a d k~
a 4 = d k~
ab = abcad -k~ a c = b - l a d ks
ad = da
>
1 k2 -k2 k3 ~4 / 0 1 0 0
0 0 0 0
~(~) =
0 -1 0
1 1 0
-1 0 0
0 0 1
H2(Q, E) = Z @ Z4 = Z4/A, A = { ( k l , . . . , k4)llk~ = o, k=, k3 ~ Z, k4 c 4 z } AB-groups: Vk > O, k =_ 0 m o d 2, < (k, 0, 0, 1) > Vk > 0 , k = O m o d 2 , <(k,0,0,3)>
80. Q = I41 H [b,a] = 1
E:
A(a) =
[c, a] = d k' [c, b] = d -k~
[d,a] = 1 [d, b] = 1 [d, c] = 1
o~a = c - l o ~ d k2
o~4 = a b d k4
o~b = abcc~d -k~ o~c = b - l a d k~
o~d = da
>
1 0
-k~ 4 -4- k2 0
~4 - k2 1
k3 0
~d6 + k*+k4~-k:
0
0
1
-I
0
0 0
-1 0
1 0
0 0
0 1
2
H2(Q, Z) = Z ~ Z2 = Z4/A, A = { ( k t , . . . , k~)llk~ = o, (k~ + k~ + k~) c 2Z} AB-groups:
V k > 0 , < (k,0, o,D >
200
Chapter
7: C l a s s i f i c a t i o n s u r v e y
81. Q = pTt [d, a] = 1 [d, b] :
E : < a , b , c , d , a H [b, a] = d ~'
[~, a] = 1 [~,~] : 1
[d, c] :-
c~a = b-Zc~d k~ o~b = ao~d ~ ~ c = c - l c ~ d k4
A(~) :
1 0
k2 0
0 0 0
-1 0 0
k3 1 0 0 0
>
~4 = dk~
~d : da
k4 0 0 -1 0
4
0 0 0 1
H 2 ( Q , ~ ) = Z | (Z2) 2 ~ Z4 = Z S / A , A = {(kl,...,
k~)llk~ -- 0, (k2 -4- k3), k4 E 2Z, ks E 4Z}
AB-groups: Vk > 0 , k = 0 m o d 2 < (k, 0, 0, 0,1) > Vk > 0 , k ~ 0 m o d 2 < (k, 0, 0,1,1) > Vk>0, k=0mod4<(k,0,0,0,3)> Remark: V k > 0 , k = 0 m o d 2 , < (k, 0, 0,1,1) > ~ < (k, 0, 0,1, 3) > Vk = 4 / + 2 , l E E , < (k, 0, 0, 0,1) > ~ < (k, 0, 0, 0, 3) > 82. Q = ITt E:
[b,a]=l
[d,a]:l [d, b] : 1
[c, a] : d k'
It, b] = d -k~
[4 cl =
a a = c ~ d k~
c~4 : d k~
ab = a - l b - l c - l a d
~
ad :dct
c~c = b a d ~~
l1 A(~)=
0 0 0
k2 0 0 1
ka -1 -1 -1
k4 0 1 0
4 0 0 0
0
0
0
0
1
H2(Q, 25) = Z G (Z4) 2 = ZS/A, A = {(kl, . . . , ks)Ilk1 = 0, (2k4 - k2 + k~), k5 e 4 Z } AB-groups:
Vk>0, Vk > 0 , Vk>0, Vk > 0 ,
k_=0mod2< k_--0mod4< k-0mod2< k=4/+2, lE
(k, O, O, O, 1) > (k,o,o,o,z)
>
(k, 0,0, 1, 1) > Z, < (k,0,0, 1,3) >
>
Section
7.2 4 - d i m .
AB-groups,
with 2-step
Fitting
201
subgroup
Vk > 0 , k ~ 0 m o d 2 < (k, l, 0, 0,1) > Remark: Vk = 4 l + 2 , / E Z , < (k, 0, 0, 0,1) > ~ < (k, 0, 0, 0, 3) > Vk=41, I E Z , < (k, 0, 0,1,1) > = < (k, 0, 0,1, 3) > Vk ~ modO, < (k, 1,0,0, 1) > ~ < (k, 1, 0,0, 3) > ~ < (k, 1, 0, 1, 1) > ~ < (k, 1, 0, 1,3) > 83. Q = P 4 / m E : < a , b , c , d , cq13 II [b, a] = d ~' [c,a]= 1
[d, a] = 1 [d,b]= 1
[c,b]= 1
[d,c]= 1
a a = bo~d k~
a 4 = d k4
ab = a - I a d k~
a d = do~
OtC ~
CO:
fla = a-113d k~+k~ 13b = b - l fld k3-k~ ~ c = c-113d ~
1 k2 0 0 0 1 0 0 0 0
=
ka -1 0 0 0
0
M "~ 4
0
0 J
1 0
0 1
0
0
>
~2 = dk8 13d = &3 a ~ = 13o~
1 ks + ka -1
0 0 0 0
A@)=
k3 -- k2
0 -1 0 0
0 0 0
0 0 -1 0
0 0 0 1
H 2 ( Q , Z) = 25 9 (252)3 9 Z4 = Z 6 / A ,
k )llk
A = {(kl,...,
= o,
(ks -k ka), ks, k6 6 2Z, k4 E 4Z}
AB-groups: None 84. Q = P 4 2 / m H [b, a] : d k l [c,a]= 1 [c,b] = 1 oea =- hoed k2
E: < a,b,c,d,~,~
[d, a] = 1 [d,b]= 1 [d,c]= 1 o~4 = c2d k4
ab = a-lo~d k~ O:C z
A(a) =
ted = dot
COt
13a = a-t13d ~+k~
132 = d k5
13b = b-113d k:-k~ 13c = c-113d - 2k~
/3d = d13 ~ = c13~d k~
1
k2
kz
0
~4 "~
1
0 0
0 1
-1 0
0 0
0 0
0 0
0 0
0 0
1 0
1
0 o 0 0
1
J
>
k2+kz -1 0 0 0
ka-k2 0 -1 0 0
-2k~ 0 0 -1
~2 0 0 0
0
1
J
202
C h a p t e r 7: Classification s u r v e y H~(Q, Z) = Z @ (E2) 2 ~ 7/.4 = Z 6 / A , A = {(k~,...,
None
k6)llkl = 0, (k= + k~), k~ ~ 2~, (k4 - 2ko) ~ 4Z}
AB-groups:
85. Q = P 4 / n
[c, b] = 1 a a = bad k~ ab = a-lo~d -k~+h2-2k* OLC ~
1 0
~(a)
=
o
0 0
~(Z)
=
>
[d, a] = 1 [d, b] = 1 [d, ~] = 1 a 4 = dk~
E : < a , b , c , d , a , ~ ] ] [b, a] = d TM [c, a] = 1
ad = da
COt
~ a = a - l ~ d 2(k~-ko) ~b = b - l ~ d - 2k~
f12 = d ~
tic = c - l ~d k"
a ~ = b~ad k8
k2
_k_x 2 + k2 - 2ko
0 1 0 0
-I 0 0 0
1 2 ( k 2 - k6) 0 -1 0 0 0 0 0 0
-2k6 0 -1 0 0
fld = d/3
kl+2ka+4k601_0812
0 0 0 1 0
]
k4 0 0 -1 0
~2 ] 0 0 0 1
H 2 ( Q , Z) = Z $ (g2) 2 @ Z4 = Z 6 / A ,
= O, k4, k5 C 2Z, k3 C4Z, k2, k6 ~ Z}
A = {(kl,...,k6)llkl
AB-groups: Vk >O, k = - O m o d 2 , < ( k , 0 , 1 , 0 , 1 , 0 ) >
Vk>0, k~0mod2, 86.
< (k, 0, 3, 0,1, 0) >
Q = P42/n
E : < a , b , c , d , a , flll
[b, a] -- d TM [c, a] = 1 Iv, b] = 1
[d, a] = 1 [d, b] = 1 [d, c] = 1
o~a = bad k~ ab = a - l a d ks
o~4 = c2d k* o~d = da
f~a
f12 = dk~
= a - t f ~ d kl+k2+ks
fib = b - l fld ~l-k~+ks /3c = c - t ~ d - k ' + k ~ - k ~ - 2 ~
fld = dfl a ~ = bc~ad k~
>
S e c t i o n 7.2 4 - d i m . A B - g r o u p s , 1 0 A(a) =
0
0 0
~(z)
1 0 0
=
k~ 0 1 0 0
kl+k2+k3 -1 0
0 0
with 2-step Fitting subgroup ~S + k3 -1
0
0
0
0 0
1 0
-k,+s(,'~2-h,+k,) 8 0 1
0
~.
~ i 1
kl-ks+k3 0 -1
0 0
203
-kl+ks-k3-2k6 0 0
0 0
2
0 0
-I 0
0 1
H2(Q, Z) : Z @ (Z2) 2 @ Z4 : Z 6 / A ,
A = { ( k ~ , . . . , ko)llk~ = 0, (ks + k~), k~ ~ 2~, (ks - k~ + k4 - 2ko) e 4 ~ } AB-groups: Vk > 0 , k - 0 m o d 2 , Vk>0, k ~0mod2,
< (k, 0, 0,1,1, 0) > < (k, 0, 0,1,1, l) >
sT. Q = I4/,,~ E : < a,b,c,d,o~,l~lJ
[b, a] -- 1 [c, a] = d k~
[d, a] = 1 [d, b] -- 1
[~, b] = d - ~
[d, ~] --- 1
o~a = o~b = ac = 13a = Zb =
a 4 = d ~,
c-lo~d k2 abcold -~2 b - l a d kz a - l fld k~ b-lfld-2k~+sk,+k~
>
a d = da fls =
dk~
fld = dfl
tic = c- l fld s~-k~
~(.)=
0 0
1 1
0 -1
-1
1
0
0
0
0
0
0
A(f0 ---
z-/~(Q, z ) = z A = ((k~,...,
AB-groups: Nolle
-1 0
-2k2-+2k3+k5 0
2k~-ks 0
0
-I 0 0
0 -i 0
0 0 I
( z : ) ~ 9 ~ , = Z~/A,
ko)llkx = O, ks, ko ~ 2Z, k4 C 4Z, k2,
k3 E ~]'
2
204
Chapter
7: C l a s s i f i c a t i o n s u r v e y
88. Q = I 4 1 / a
E:<
a,b,c,d,a,
[b, a] = 1
flt
[c, b] = d -kl
l l 0 ~(~) = 0 0 0
A(f) =
c~a = c - l crd k~ o~b = a b c a d - ~ O:C -~ fa = a-lfd:k~+:h~
f 2 = dk~
fib= b - l f d -k'+2k~+~k'
fd = df
fc = c- l fd - :~
a f = c l a d k"
k2 0 0 -1 0
- k l + k3
-k2 1 1 1 0
1
2(k2 + k~)
0
-1
0 0 0
0 0 0 H2(Q,Z)
>
[d, a] = 1 [d, b] = 1 [d, c] = 1 c~4 = a - l b - l d k 4 czd = d a
[c, a] ---- d ~1
kl-4(J~a+h,-lt.) "~
4
16
0 -1 0 0
0 0 $1 1
-kl+2ka+2k6 0
-2k6
J
M= '~
0
0
-1
0
0
0 0
-1 0
0 1
= Z @ Za @ Z4 = Z S / A ,
A = { ( k l , . . . , k~)llk, = 0, k~ ~ 2Z, ( - k = - ka + k~ - 2k~) 9 4Z}
AB-groups: Vk > 0 , k ~ 0 m o d 2 , Vk > 0 , k ~ 0 m o d 2 ,
< (k,0,0,1,1,0)> < (k,0,0,1,1,1)>
103. Q = P 4 c c E: < a,b,c,d,a,fll
>
[d, a] = 1
[b, a] = d k' [c, a] = 1 It, b] = 1 ~ a = bad k~ ab -=- a - t a d k~
[d, b] = [< ~] = 1 o~ 4 =
dk4
ad = da
C~C ~- CO~
~2 ~_ cdh5
f a = arid ~+k~
~ d ~ d 1 1~
fib = b-~p & = cfd-2ko
A(~) =
1 0 0 0 0
k2
k3
0 1 0 0
-1 0 0 0
0 0 0 1 0
~4 '~ 0 0 0 1
J
A(f) =
~ 0 = O~ad -k' -1 0 0
0 0
- k 2 - ka 1 0 0 0
0 0 -1 0 0
2ks 0 0 1 0
0 o 0 1
1
J
Section
7.2 4 - d i m .
with 2-step Fitting subgroup
AB-groups,
205
= Z @ Z= @ Z4 = Z S / A ,
H2(Q,Z)
AB-groups: Vk > 0 , k ~ 0 m o d 2 , < (k, 0, 0,1, 0) > gk > 0 , k ~ = 0 m o d 4 , < (k, 0, 0, 3, 0) > Remark: If k = 41 + 2 then < (k, 0, 0, 1, 0) > ~ < (k,0, O, 3, 0) >. 106. Q = P42bc
E : < a,b,c,d,c~,f
I] [b, a] = d 2k'
[~, ~] = 1 [c, b] -- 1
[d, bl= 1 [d, c] -= 1
o~a = bo~dk~ oeb = a - l o ~ d -k~-~a-2k~
o~4 = c2dk~ ~ d = dc~
OLC ~
COL
f 2 _= a d ~ fd = d-if o~f
f a = arid - 2k~ fib = b - l fld -~'' fc = cfd 2~
1 ~-+k2 ~(~) = (
0 0 0 0
0 1 0 0
0
~(f) =
0 0 0
-= Z ~ E 4 = Z S / A , A
AB-groups: Vk > 0 , k - z 0 m o d 2 , Vk > 0 , k ~ 0 m o d 2 ,
= bc-l fo~ad-~-k'-ak~-k~
-kl-k~-2ks -1 0 0 0
-1
H2(Q,Z)
>
[d, a] = 1
2ks
2
1
0
0 0 0
-1 0 0
0 O
-2 0
1 0
g 1 1
- 2k 07001)t
= { ( k l , . . . , k s ) ] ] k l = 0 , (ka+2k4) E 4Z, ks, ks C Z}
< (k, 0,1, 0, 0) > <(k,0,1,1,0)>
104. Q = P 4 n c E : < a,b,c,d, oe,fll
[b, a] = d ~k~ [c, a] = 1 [c, b] : 1
[d, a] = 1 [d, b] = 1 [d, c] -- 1
a a -- botd k~ orb = a - l a d -k'+k2+2k3+2k5
a 4 : d k~ a d -- dc~
0~ r - ~ C Ct
f a = arid 2k~+ 2k~+ 2k~ fib = b - l f d - k ' f c = cfld - 2k~- 2 ~ - 2 k ' - 2k~
/3 ~ -= acd k" fd = d-lfl oe/3 = bl3aa d h~
>
206
C h a p t e r 7: C l a s s i f i c a t i o n s u r v e y
A(a) =
A(f) =
1
~ + k2
- k t + k2 + 2k3 + 2ks
0
kl-4k2-2k3-4k~
0 0 0 0
0 1 0 0
-1 0 0 0
0 0 1 0
0 0 1
/
-~
-2(~+~+~)
~
2
2(~+~+~+~)
0 o
1 o
0 -1
0 o
0 0
0 0
0 0
1 0
0 1
o 2
1
H ~ ( Q , Z ) = Z ~ Z4 = Z S / A , A = { ( k t , . . . , k s ) l l k l = 0, k3 E 4Z, k2, k4, k5 E Z} AB-groups: Vk > 0 , k = 0 m o d 2 , Vk>0, k-0mod2,
< (k, 0,1, 0, 0) > < (k, 0, 3, 0, 0) >
110. Q = I 4 t c d E: < a,b,c,d,a,f
[d, a] -- 1 [d,b] -- 1 [d, c] = 1 ot4 = a-lb-ldl~s ad = da
II [b, a] = 1 [c, a] = d TM [c, b] = d - 2 ~ a a = - c - l o~d41~x-4k2+21~s-3k4+2k~ ab = abcotd -4k~+4kz-2kz+3k4-21% o~c = fa = fib = /3c =
b-lord k~ c - l f d ~" abc~d -2k~+2k~+~4 a - l f d -k4
/32
_= abd~l-k2-k4 fld = d - t fl (~fl = a b c f a a d k5
~(.) = i
4 k l -- 4k~ + 2 k s + 2 k 5 -- 3 k 4 0 0 --1 0
~(Z) =
4k~ -- 2 k 3 -- 2 k 5 ~ 3 k 4 -- 4 k 1
1
-1 0 0 0 0
-k4 0 0 -1 0
~
+ k~
o
1
--1
0
0
0
1
2(kl-k2)-k4 1 1 1 0
k4 -1 0 0 0
0 { I
0 1
H 2 ( Q , Z ) -- Z @ Z4 = Z S / A , A = { ( k x , . . . , ks)Ilk1 = o, (k~ - k3 + k4) C 4Z, k5 c Z}
AB-groups:
vk>o,
< (k,o, 1,o,o) >
>
S e c t i o n 7.2 4 - d i m . A B - g r o u p s ,
207
with 2-step Fitting subgroup
v k > o , < (k,o,3,o,o) > 114. Q = P/t21c E : < a,b,c,d,a,fl
[I
[b, a] = d TM [c, a] = i [c, b] = 1 aa = b-lctd k2 ab = a a d k~-k~- 2k~ O~C ~-
c-lo~d 2(k'-k~+k3-k4+l%)
t3a = a - l fld k' fib = b~d -2k"
/32 = bd k, l~d = d-tt3 0~]~ -= ac~a3 d h~
ZC "~ c-- l fl
l i
~-t-k2 0 -1 0 0
>
[d, a] = 1 [d, b] = I [d, c] = 1 a 4 = d~ ad =dct
2(kl - k2 + k3 - k4 + k~) 0 0 -1 0
kl-k2-2k4 1 0 0 0 -1 0 0 0 O
=
-
~2 -1 0
2k4 0 1
0 0 0
0 0
0 0
-I 0
_ kl+2~,+4k~ ,~ 18 0 !2 1
)
0 '~ 0 1
0 1
H ~ ( Q , Z ) = Z ~ Z4 = Z S / A , A = { ( k l , . . . , k s ) [ [ k l = 0, ks E 4Z, ks, k4, ks E E}
AB-groups: Vk > 0 , k - - 0 m o d 2 , Vk > 0 , k - 0 m o d 2 ,
< (k, 0,1, 0, 0) > < (k, 0, 3, 0, 0) >
146. Q = R3 E:
all
[ b , a ] = d k~ [c, a] -- d-k1 [c, b] = d ~1 aa = bctd k~ ab = cad k~ ac = a a d -}~-}~
1 k2 0 0
A(a) =
k3 0
-k2-}3 I
[d,a]--1 [d, b] = 1 [d, c] = 1 a 3 = d k~ a d = da
k3l '~ 0
0
1
0
0
0
0
0
1
0
0
0
0
0
0
1
>
208
7: C l a s s i f i c a t i o n s u r v e y
Chapter
AB-groups: V k > 0 , < ( k , 0 , 0 , 1) > Vk>0, < (k,0,0,2) > 144. Q = P31 E : < a , b , c , d , o ~ I] [b,a] = d k~
[~, bl = 1
[d,a] = 1 [d, b] = 1 [d, ~] =
a a = bad ~
a 3 = cd k~
a b = a - l b - l a d k~
ad = da
[c, a] = 1
OtC ~
CO~
1 k2 0 1 0 0 0
=ktk2+k3 -1 --1 0 0
0 0 0
H2(Q,Z)
>
0
~3 \ 0 0
0 0 1
1
0
1
: Z G Z3 : Z 4 / A ,
A : {(kl,k2, ks, k4)[lkl = 0, (k3 - k2) ~ 3Z, k4 E Z} AB-groups:
vk>o,
< (k,o,o,o) >
Vk > 0 , k - 0 m o d 3 , < (k, l, 0, 0) > Remark: Yk < (k, l, 0, 0) > ~ < ( k , 2 , 0 , 0 ) > Vk~0mod3:<(k,0,0,0)>5< (k,l,0,0)>
148. Q = R3 E:
[d, b] -- 1
[c, b] = d ~
Id, ~] = 1
a a : b - l a d k~ a b = c - l a d k3
a 6 = d k~ ad = da
OLC ~
A(a) =
[d,a]-I
N [ b , a ] = d kl [c, a] = d - k '
a-lord
k4
1
k2
k3
k4
k~6
0 0 0 0
0 -1 0 0
0 0 -1 0
-1 0 0 0
0 0 0 1
H2(Q, Z) = Z ~ Z2 ~ 75~ = Z S / A ,
AB-groups: Vk > 0 , k _ - - 0 m o d 2
< (k, 0, 0, 0,1) >
>
Section
7.2 4 - d i m .
Vk > 0 , k ~ - 0 m o d 2
AB-groups,
209
with 2-step Fitting subgroup
< (k, 0, 0, 0, 5) >
147. Q = P3 E:
all
a a = b - l a d k~ a b = a b a d k3 a c = c-lo~d k*
0 0 0 0
A(oL) =
H2(Q,Z)
A = {(kl,...,
~ + k3 1 1 0 0
0 --1 0 0
>
[d,a] = 1 [d, b] = 1 [d, c] = 1
[ b , a ] = d k'
[c, a] = 1 [c, b] = 1
a ~ = d k~ (~d = d a
k4 0 0 -1 0
6 0 0 0 1
= Z @ Z2 $ Ze = Z S / A ,
ks)llk~ = 0, k~ C 2~, k~ e 6Z, k2, k3 ~ ~,}
AB-groups: Vk>O,k=_Omod6 Yk >0, k_--0mod6
Yk > 0 , k ~ 2 m o d 6 Vk > 0 , k = 4 m o d 6
< < < <
(k,0,0,0, 1) (k, 0, 0, 0, 5) (k, 0, 0, 0,1) (k, 0, 0, 0, 5)
> > > >
161. Q = R 3 c [c, a] = d - k ' [c, b] = d k' oLa = bad k2
c~3 = abcdk~ ad = da
ab = c a d ~" ac = aad -k~-k,
~3a =
b~d ~+=~3-2k~
fib =
a~d k'+k~+:k3-2k~
tic = cfld - 2k~
I 1 k~_ 4 k2 k3 +
X(a) =
~(~) =
/ -1 0 0 0 0
0 0 0 0
0 1 0 0
_h_
0 0 1 0
>
[d, a] = 1 [d, b] = 1 [d, ~] -- 1
E : < a , b , c , d , o ~ , ~ [I [b, a] = d k'
--
)3d = d-l,G fla = b - l a 2 f l d ~+k~+k'-2k~
4 k2 ks
k~t
--
k ~ - 2k3 + 2k5 0 1 0 0
--
24
61
1 0 0 o
3
4
0 1 _ 3 k4~ _ k ~ - 2 k 3 + 2 k s 1
2k5
0 '~
o
0
0 0
o 1
0
0
o
1
1
J
210
C h a p t e r 7: C l a s s i f i c a t i o n s u r v e y
H2(Q, ~.) = Z q~ ~3 = 7~S/A, A = { ( k ~ , . . . , ks)[Ik~ : 0, (ks + 2k3 + k4) E 32, ks E Z} AB-groups: Vk>0, k=0mod3, Yk > 0 , k - = 0 m o d 3 , Vk>0, k-lmod3, Yk > 0 , k - l m o d 3 , Vk > 0 , k : = 2 m o d 3 , Vk>0, k-=2mod3,
< < < < < <
(k, 1,0,0,0) > (k,2,o,o,o) > (k,o,o,o,o) > (k,2,o,o,o) > (k,o,o,o,o) > (k, l, O, O, O) >
158. Q = P 3 c l E : < a , b , c , d , a , ~ ][ [b, a] = d ~'
[c, a] = 1 [c, b] = 1
[d, a] = 1 [d, b] = 1 [d, c] = 1
a a = bad k~ ab = a - l b - l a d k3
a 3 = d k~ ad = da
>
O~C ---- COt.
A(a) :
1 0 0 0 0
~ a = b- l ~dk~ ~b = a - t ~ d -k~
~ : = cdk~ j2d = d - t ~
tic = cfld - sk~
~ a = a2 fld k"
ks
- ~2 + k3
0
3
0 1 0 0
-2 -1 0 0
0 0 1 0
0 0 0 1
~(~) =
-2 0 0 0 0
-k2 0 -1 0 0
k2 -1 0 0 0
2k5 0 0 1 0
H2(Q, E) = z ~ (~3) s = ZS/A,
A --- { ( k l , . . . , AB-groups: Vk > 0, k = 0 mod 3, Vk>0, k=0mod3, Vk>0, k ~0mod3, Remark: Vk = 1 mod 3, < (k, Vk = 2 rood 3, < (k,
k d l l k x = 0, (k3 - ks), k4 E 3E, ks E Z}
< (k, 0, 0, 1, 0) > < (k, 0, 0, 2, 0) > < (k, l, 0,1, 0) > 0, 0, 2, 0) > ~ < (k, 1,o, 2,o) >-~< (k, 2, o, 2, o) > 0, 0, 1, 0) > ~ < (k, 1,0, 1,0) > ~ < (k, 2, 0, 2, 0) >
o)
0 0 1
1
Section 7.2 4-dim. A B - g r o u p s , with 2-step Fitting subgroup
211
159. Q = P31c
E : < a,b,c,d,o~,fl[[
[b, a] = d k~ [c, a] = 1 [c, b] = 1 ota = botd kl-2k2+3k~
o~b = a-lb-lo~d k~ 0r
-~-
COt
~a = bfld k" I% = afld k~ ~c = c/3d - 2k~
f12 = cdk5 fld = d- l fl flot = c~2/3d ks iI 1 - k 4 - k 4 2k5 0 0 1 0 0 0
2 + k2 0 k__~ 3 ) I! k 1 - 2k2 + 3k4 -~-X 0 -1 O 0
A(,~)=
1
~00
-1
0 0
>
[d, a] = 1 [d, b] = 1 [d, c] = 1 Ot3 = d k~ o~d = do~
0
0
1
0
0
0
:~(~)=
0 1
0
1
0
0 ~0
0
0
0
0
0 1 0
0 1
1
H2(Q, Z) = Z ~ 253 = ZS/A,A = ((kl, 9 .., ks)Ilk1 = 0,k3 e 3z, k=, k4, k5 ~ z } AB-groups: Vk > 0 , k = 0 m o d 3 , Vk > 0 , k - 0 m o d 3 , Vk > 0 , k = _ l m o d 3 , Vk > 0 , k - 2 m o d 3 ,
< < < <
(k, 0,1, 0, 0) (k, 0, 2, 0, 0) (k, 0,1, 0, 0) (k, 0, 2, 0, 0)
> > > >
168. Q = P 6
E : < a,b,c,d,o~ II [b, a] = d kl [c,a]= 1 [c, b] - 1
[d, a] = 1 [d,b]= 1 [d, c] = 1
ab = abad k3
o~d = do~
>
OlC = COt
,x(o,)=
1 k2 0 0 o-1 0 0
0 0
-~+k3 0 ~ ) 2
6
1 1
0 o
0 o
0 0
1 0
0 1
H2(Q,25) = Z @ Z6 = Z4/A, A = { ( k l , . . . , k,)llkx = 0, ~, ~ 6z, k~, k3 ~ z } AB-groups: Vk > 0 , k = 0 m o d 6 Vk > 0 , k - 0 m o d 6 Vk > 0 , k - 2 m o d 6 Vk > 0 , k - 4 m o d 6
< < < <
(k,0,0,1) > (k, 0, 0, 5) > (k,0,0,1)> (k,0,0,5) >
212
Chapter
7: C l a s s i f i c a t i o n s u r v e y
172. Q = P64 E : < a,b,c,d,a[[
[d, a] = 1 [d, b] z 1
[b, a] = d k l
[~, ~]
= 1
O~a = b - l o l d h2
[d, c] = 1 oz6 = c2dk4
a b = abc~d k~
a d = do~
It, hi = 1
>
OLC -~- CO:
1
~(~)
=
o
i~ 0 ~0
k2 0 -1 0 0
_ 2 2 -4-k3 1 1 0 0
H 2 ( Q , Z) = Z @ Z2 = Z4/A, A = { ( k ~ , . . . ,
0 0 0 1 0
k4)llkl
6
0 0 1 g 1 = 0, k 4 C 2Z, k2, ka E Z}
AB-groups: Vk > O , k = _ O m o d 2
< (k,0,0,1)>
173. Q = P 6 s E: < a,b,c,d,a
]] [b, a] = d kl
[d, a] = 1 [d, b] = 1 [ 4 c] = 1
[c, a] = 1
[~, b]
A(a) =
1 0 0 0 0
= 1
a a = b - t a d k~
a 6 = c3dk4
a b = a b a d ~3
ad = da
O~C z
COL
k2
_22 +ka
0
1
0 0
-i 0 0
1 0 0
0 1 0
H 2 ( Q , Z ) = Z @ Za = Z 4 / A , A = { ( k l , . . . ,
>
6
0 0 1
1
~4)Jlkl = O, k4 E 3Z, k2, ka E Z}
AB-groups: Yk>O,k-Omod3
Vk > 0 , k _ = 2 m o d 3 Vk > 0 , k ~ 0 m o d 3 Vk > 0 , k = l m o d 3 169. Q
=
< < < <
(k,0,0,1)> (k,0,0,1)> (k,0,0,2)> (k,0,0,2)>
P61 E : < a,b,c,d,o~ll
[b, a] = d k~
[c, al = 1 It, b] = 1 o~a = b-lo~d k2 o~b : abo~d k~ OLC =
CO~
[d, a] = 1 [d, bI = 1 [d, c] = 1 a 6 = cSd h, otd = do~
>
Section
7.2 4 - d i m .
AB-groups,
with 2-step
1 k2 0 0 0 0
A(a) =
Fitting
- 2
0 -1 0 0
213
subgroup
6 1 1 0 0
0 0 1 0
0 0 5
1
H 2 ( Q , Z ) = Z = 7~4/A, A = { ( k l , . . . , k~)ll~ = 0, k~, k3, k4 9 Z}
AB-groups: v~>0, < (k,0,0,0) > 174. Q = P6 E:
all
[ b , a ] = d k' [c, a] = 1 It, b] = 1 c~a -- bc~d k~ ab = a - l b - l a d
[d,a]=l
k~
>
[d, b] = 1 [d, c] : 1 c~6 -- d k~ ad = da
~ C ~ c - l o ~ d k~
l1 A(a) =
0
k2 - ~ 2 q- k3 k4 0 -1 0
6 0
0 0 0
1 0 0
0 0 1
-1 0 0
0 -1 0
H 2 ( Q , ~ ) =- ~ G E2 @ 75a @ 7~6 --- Z ~ / A ,
A = { ( k l , . . . , k s ) ] ] k l = 0, (k3 - k2) 9 3E, k4 9 27~, ks 9 6Z} AB-groups: Vk > 0 , k - 0 m o d 3 , Vk > 0 , k - - 0 m o d 3 , Vk > 0 , k ~ 0 m o d 3 , Remark: Vk-lmod3 < (k, V k - 2 m o d 3 < (k,
< (k, 0, 0, 0,1) > < (k, 0, 0, 0, 5) > < (k, l, 0, 0, 5) > 0, 0, 0,1) > ~ < (k, l, 0, 0, 5) > ~ < (k, 2, 0, 0,1) > 0, 0, 0, 5) > ~ < (k, l, 0, 0, 5) > ~ < (k, 2, 0, 0,1) >
175. Q = P 6 / m
E : < a,b,c,d,a,~lt
[b, a] = d ~1 It, ~] = 1 [c, b] -- 1
[d, a] = 1 [d, b] = 1 [d, c] -- 1
c~a ~ b - l a d k~ (~b = a b a d k3
ct 6 = d k~ ad = da
~C
~
COL
/3a = a-1/3d k'-2k~
/32 = d ~6
fib = b - t ~ d -k~+2k:+2k~
~d = dfl
]3c = c - l j3d k6
f l a = a/3
>
Chapter 7: Classification survey
214
! k2 - ~ 2 + k a 0 1 -1
1 ~
0
0
0 ~6
['1 k l - 2 k a
0 0
I~~
0 0 1 0 0 1
A(13)=
2k2+2kz-kl
-i
0
0
-1
0
0
0
0
0 0 -1 0
0 0 0 1
H2(Q, Z) = Z @ (Z2) 2 @ Z6 = ZS/A, A = {(k~,...,ke)llkl = 0, ks, k6 E 2Z, k4 E 6Z, k2,k3 E Z} AB-groups: None 176. Q = P 6 3 / m
E : < a,b,c,d,a,13 11 [b, a] = d k' [c, a] = 1 [c, b] = I aa = b - l a d k2 ab = abad k~ OLC =
COt
fla = a-113d k~-2k" fib = b-lfld -kl+zk~+2k" /3c = c-113d 2k~ Ii
k2 - ~ + k 3 0 1 -1 1 0 0 0 0
0 0 0 1 0
s
l/
00 ~1 1
H2(Q,Z)
A = {(kl,...,k6)llkl AB-groups: Vk > 0 , k ~ 0 m o d 6 , Vk>0, k~0mod6, Vk>0, k~2mod6, Vk > 0 , k ~ 4 m o d 6 ,
<(k,0, <(k,0, <(k,0, <(k,0,
=
>
[d, a] = 1 [d, b] = 1 [d, c] = 1 a 6 ~- c3d k* ad = da
A(13)= ( i
kl-2k3
t32 = d ks 13d = d13 /3a = c- t afld k~ 2k2+2k3-kl
- 01 0 0
- 01 0 0
2ks
00 -1 0
=Z ~ Z2@Zs =Z6/A, 0, k5 E 2Z, (k4 -I- 3k6) C 6Z, k2, k3 E Z}
O, 2, O, 4, O, 4, 0, 2,
1, O) > 1, O) > 1, O) > 1, 0) >
2
0 0 0 1
/
S e c t i o n 7.2 4 - d i m . A B - g r o u p s ,
215
with 2-step Fitting subgroup
184. Q = P 6 c c E : < a,b,c,d,o:,/311
[b, a] = d kl [c, a] -- 1 [c, b] = 1
[d, a] -- 1 [d, b] -- 1 [d, c] = 1
O:a = b-lo:d k~ O:b = abo:d k"
oL6 d ~" O:d = do:
O:C =
CO:
~ a = b - l ~ d k~-k~-2k~ 1% = a - l f l d -h'+k~+~ks 13c = cfld - 2k~
1 0 0 0 0
a(o:) =
-1 0 =
HZ(Q,Z) = Z~Z6 AB-groups: Vk > 0 , k - 0 m o d 6 , Vk>0, k-0mod6, Vk>O, k=2mod6, Vk>O, k=4mod6,
>
k2 0 -1 0 0
-kl+k2+2k3 0
-~ 2 +k3 1 1 0 0
0 0 0 1 0
kl-k2-2k3 -1
~2 = c d ~ fld = d - l j 3 f l a = aS~d k"
6 0 0 0 1 2k5 0
0 "~ 0
o
-1
o
o
o
0 0
0 0
0 0
1 0
1
1
)
= Z ~ / A , A = { ( k l , . . . , k s ) l [ k l = O , k , C 6Z, k2,ka, k5 e Z}
< < < <
(k, (k, (k, (k,
0, 0,1, 0) > 0, 0, 5, 0) > O, O, l, O) > O, O, 5, 0) >
Chapter 7: Classification survey
216
Comments
and
proofs
For the first groups we give detailed information on the choices of the actions and on the search for isomorphism types. Since the methods used are the same in all cases we do not give explicit information for most of the cases. 1. See classification of rank 4 nilpotent groups. 2. Since a 2 = 1 in Q we only have to investigate the trivial action of Q On ~.
The only torsion free groups E are parametrized as (2k, 2I, 2m, 0, 0, 0, 1). One can see that, by doing the analogous changes of generators as we did during the classification of rank 4 nilpotent groups, any extension (kl, k2, k3, k4, ks, k6, kT) is isomorphic to the extension with parameters 1,/ b t h / k~). Therefore (2k,21,2m, O,O,O,1) is isomor((kl,k2, k3),O,O,,~4,,~5,..6, phic to (2(k,l,m),O,O,k~4, ks, ' k6, ' kT) ' which has to be torsion free and so is equivalent with (2(k, l, m), 0, 0, 0, 0, 0, 1). 3. a 2 = 1 in Q =* only the trivial action of Q on Z has to be investigated. 4. Here we consider b o t h the trivial and the non-trivial action. It is obvious that two extensions inducing a different action are not isomorphic. (e.g. In one group the center= Z ( N ) while in the other extension the center will be trivial.) Trivial action: Isomorphism types: Suppose k ~ 0 m o d 2, so k : 2 l + 1 for some l E Z. We have the following isomorphisms:
~1 :< (k, 1, 0, 0) > 4 < (k, 0, 0, 0) >, with ~ol(a) = ad -z, ~l(b) = b, ~1(c) = c, ~l(d) = d, ~ l ( a ) = c - i n ~2 :< (k,o, 1,o) >-~< (k,o,o,o) >, with ~2(a) --: a, ~2(b) = b, ~ 2 ( c ) = cd -z, ~2(d) = d, ~2(a) = a a ~3 :< (k, 1, 1, o) > 4 < (k,o,o,o) >, with w3(a) = ad 1+/, ~ 3 ( b ) = bd -2l-1, ~3(c) = cd -l, ~3(d) = d, ~3(a) = aca. Suppose k - 0 m o d 2, so k = 21 for some 1 E Z. We have the following isomorphisms:
~1 :< (k,o, 1,o) > - ~ < (k, 1, o, o) >, with ~ol(a) = c, ~l(b) = b, ~1(c) : ad -1, ~ l ( d ) = d -1, ~1(o~)
~a~ :< (k, 1,1,0) > 4 <
(k,l,O,O) >,
:
O~
Section 7.2 4-dim. AB-groups, with 2-step Fitting subgroup
217
with ~2(a) = a, ~2(b) = bd, ~2(c) = ac, ~ 2 ( d ) = d, ~2(a) = aa. There is no isomorphism between < (2/, 0, 0, 0) > and < (2/, 1, 0, 0) >. Suppose ~ :< (2/, 1, 0, 0) > ~ < (2/, 0, 0, 0) > is an isomorphism of groups, then ~ ( a ) = a ~ b ~ 2 c a 3 d a4 (7.1) ~(b) = a ~ bz~c z~ d ~' ~(c)
= a ~ b~ c ~ d ~
~ ( d ) = d ~, 5 = + 1 ~(a) = amlbm2cm3dm4ot now one has to see if this ~ is compatible with the given relations, e.g. the following should be satisfied: ~(aa) = p ( a - l a d ) or
am1 bin2 cm3 dm4 aa~,~ be,2c,~2d,~4 = ( a,~l b,~2ca3 dC,4)- 1a,nl bin2 cm3 d~4 ad ~ inducing 5 - 0 m o d 2 which is impossible. Non-trivial action: I s o m o r p h i s m types: Consider the group < (kl, k2, k3) >. Suppose (kl, k2) = pkl + qk2 for some p , q E Z. There is an isomorphism ~ :< ((kl,k2),O, k3) > 4 < (kl, k2, k3) > given by ~(a) =aPc - q ~(b) = b, ~p(c) = ak2/(k~'k2)c k*/(k~'k2), ~(d) = d, ~(a) = a. 7. Trivial action: The groups < (k, 0, 0) > and < (k, 1, 0) > are not isomorphic: Suppose ~ :< (k, 1, 0) >---~< (k, 0, 0) > is an isomorphism, then ~v satisfies (7.1) and one checks that the relation ~(ab) = ~ ( b - l a d ) can not be satisfied. Non-trivial action: Consider any extension < (kl, k2, k3, k4) >. By a new choice of generators we can transform the presentation for < (kl, k2, k3, k4) > to show that this group is isomorphic to another one. Some situations:
218
Chapter 7: Classification survey
1. a / = a-l~ b' = b, c ~ = c, d ~ = d, c~' = < (kl, k2, k3, k4) > ~< (-kl, k~, k'~, k'~) > . 2. a I = a, b I = b, c ~ = c, d ~ = d -1, ct / = a < (kl, k2, k3, k4) > '< (-kl,-k2,-k3,-k4)
>.
3. a' = a, b' = b, c I = a 2 m c , d ~ = d, a ~ = a m a < (kl, k2, k3, k4) > ,< (kl,k2' k4) ' >. m k l , k3, 4. a ~ = ac m, bI = b , c I = c , d I = d , c J = v ~ < (kl, k2, k3, k4) > ~< (kl - 2ink2, k2, k'~,k~) > .
This shows t h a t one can reduce ]gl to kl rood (2k2) or - kl rood (2k2). One of these two values will be < k2/2. One can also reduce k2 m o d kl. This shows t h a t after a finite n u m b e r of reductions of these two kinds one finds: S i t u a t i o n 1: (If ( k l , k 2 ) = (kl,2k2)) < (kl,/g2, k3, ]g4) > ~ . ( ((]gl, k2),0, k~, k~) >. S i t u a t i o n 2: (If 2 ( k l , k 2 ) = ( k l , 2 k 2 ) )
< (kl, k2, k3, k4) > ~ < (0,(kl, k2), k'3,k~)
>.
One also proves < (2k, O, k3, k4) > ~ < (O,k,k~3, kt4) > by presenting a general f o r m for a possible i s o m o r p h i s m (7.1) and see things do not work out. For k = 2 l + 1 , we take the following set of new generators in < (k, 0, 0, 0) > a I = a d -z, bI = b , c I = c ,
dI = d ,
a I=b-la
to see t h a t < (k, 0 , 0 , 0 ) >-~< ( k , 0 , 1 , 0 ) >. W h e n k = 2 1 + l i t i s seen t h a t this is not the case. Also < (0, k, 0, 0) > ~ < (0, k, 1, 0) > is checked by trying a general f o r m for the isomorphism. 13. a 2 = /32 = 1 in Q. Therefore, we only have to look at the trivial action. 16. T h e rank of the torsion free p a r t of H 2 ( Q , •) is 0.
S e c t i o n 7.3 4 - d i m . A B - g r o u p s ,
7.3
4-dimensional
AB-groups,
potent
subgroup
Fitting
219
with 3-step Fitting subgroup
with
3-step
nil-
For this class, it seemed adequate to present two tables: one containing the presentations of the AB-groups, the other containing the affine representations. This is done because, in the representation, we cannot longer use the same matrices for the generators a, b and c in all cases. First, we list the presentations of all AB-groups E of this class. The order in which the groups appear is determined by the order of the underlying AC-group in the table of section 7.1. Again, the group generated by a, b, c and d corresponds to the Fitting subgroup of E. Under the table, we added some comments concerning the establishment of this table. The general set up of one table entry is as follows: Number of Q as found in section 7.1 Cohomology class corresponding to Q Presentation for E depending on kl, k2, k3, k4 H 2 ( Q , Z ) in terms of kl, k2, k3, k4 AB-groups: The cohomology classes corresponding to AB-groups and isomorphism type information for these. 1. Q = < (k) > E : < a , b , c , dll [ b , a ] = c kdkl
[d,a]= 1 >
[c,a] = d k2
[d,b] = 1
[c, b] = d ~
[d, c] = 1
H 2 ( Q , Z) = Z 2 ~ Zk = Z 3 / A , A = {(kl, k2, k3) l] k2 : k3 -~ 0, kl E k~}
AB-groups:
< (kl, k2, ka) >. Remark: see classification of rank 4, 3-step nilpotent groups. 2. Q =< (k, o, o, 1) > E : < a,b,c,d,o~ [I [b,a] = c2kd 2k(k~+k~+~) [c, a] = d TM [c, b] = d 2k~ aa = a - l a d kl ab = b - l a d k~
[d,a] = 1 [d, b] = 1 [d, c] = 1 a 2 = cd ~3 ad = d-ta
>
a c -= co~d - 2 k 3
H~(Q, z) = z: = Z~/A, A = {(k~, k:, k~) ll k~ = k: = O, k~ ~ z}
220
Chapter
7: C l a s s i f i c a t i o n
survey
AB-groups: V m > 0 , < (m, 0 , 0 ) > . Remark: < (kl, ks, k3) > ~ < ( g c d ( k l , k2), O, O) > 3.
Q = < (21 + 1, 1) >: E : < a , b , c , d , c ~ II [b, a] -- c2t+ld zk~+(2t+l)k~
[c, a] -- 1 [c, b] = d k~ ~ a = a a c d k~ ab = b - l a d k~ OtC ~ c - l o r d -sk~
[d, a] -- 1 [d, b] = 1 [d, c] = 1
>
~2 = d k, ad = da
H S ( Q , Z) = Z | (Z2) s = Z 4 / A ,
A = {(kl, k2, k3, k4) I[ kl : 0, k3, k4 E 2~, k s C ~} AB-groups: Vm>0, m_=0mod2,
< (m, 0,0,1) >.
Q = < (2z, 1) >: E: < a,b,c,d,a
II [b, a] = cSld (2z-1)k~+2lk:
>
[d, a] = 1 [d,b]= 1 [d, c] = 1 a 2 = d k" ad = da
[c,a]= 1 [c, b] = d TM a a = a a c d k~ ab = b-lo~d k" otc ~ c-lo~d -2kz H 2 ( Q , Z) = Z G (Zs) 2 = Z4/A,
A -- {(kl,ks, k3, k4)II kl = 0, k3, k4 E 2Z, ks E Z} AB-groups: Vm > 0, < (m, 0, 0, 1) >.
Q = < (2/, 0) >: E : < a, b, c, d, a II [b, a] = c2Zd l(kl-k~) [c, a] -- 1 [c, b] = d ~ ~a = ao~ ab = b - l a d k~ otc = c - l o~d k~
[d, a] = 1
>
[d, b] = 1 [d, c] = 1 o~2 = d ~' ad = da
H 2 ( Q , Z) -- Z @ (Z2) 3 = Z 4 / A , A = { ( k l , k2, kz, k4) II k~ = 0, k2, k3, k4 ~ 2 z }
AB-groups: Vm>0, m-0mod2,
< (m, 0 , 0 , 1 ) >.
S e c t i o n 7.3 4 - d i m . A B - g r o u p s ,
221
with 3-step Fitting subgroup
4. Q = < (k, 0) > Trivial action: E:
all
[ b , a ] = c 2 k d h(k~-h~) [c,a]= 1 [c, b] = d k' o~a = ao~ ab = b - l a c - k d k~ otc : c-lord ks
[d,a]=l
>
[d,b]= 1 [d, e] = 1 o~s = ad ~" ad = da
If k ~ 0 mod 2, then H 2 ( Q , Z) -- Z (9 (Z2) s = Z 4 / A ,
A = {(kl, ks, ks, k4) II kx = 0, k~, k3 ~ 2~, k, ~ Z} AB-groups: V m > 0 , < (m, 0 , 0 , 0 ) > . Y m > O, m =_ O m o d 2, < ( r e , l , 0 , 0 ) > .
Vm>0, m_0mod2, <(m, 0,1,0)>. Remark: If k~ ~ 0 mod 2, < (kl, ks, ks, k4) > ~ ' ( (kl, 0, 0, 0) >, Yk2, ks, k4 E Z Ifm~0mod2,<(m, 0,1,0)>=< (m,l,l,0)> I f k ~ 0 mod 2, then H S ( Q , Z) = Z @ Z4 = Z 4 / A ,
A = {(kl,k2, ka, k4) H kl = 0, (2k: - k3) E 4Z, k4 E Z} AB-groups: V m > 0 , < (m, 0 , 0 , 0 ) > . Vm>0, m=0mod4, < (m, l, 0, 0) >. Vm>0, m=0mod2, <(m, 0,1,0)>. Remark: If kl ~ 0 mod 2, < (kt,ks, k3, k4) > ~ < (kt,0,0,0) >, Vk2, ks, k4 E Z I f m = 4 n § 2, n C Z, < (m, 0,0,0) > ~ < (m, 1,0,0) > Ifm=0mod2, < (m, 0,1, 0) > ~ < (m, l, l, 0) > Non-trivial action: E : < a,b,c,d,a
II [b,a] = c2kd - 2 ~ - k k ~
[d,a] = 1
[c, a] = d - 2 ~ [c, b] = 1
[d, b] = 1 [d, c] = 1
a a = a a d -2k~
a 2 = ad ~s
ab = b - l a c - k d k: a c = c - l a d k~
ad = d-la
3>
222
Chapter
7: C l a s s i f i c a t i o n s u r v e y
H 2 ( Q , Z) = Z @ Zk : Z3/A, A = {(kl, k2, ka) II k~ = 0, k: 9 kZ, k~ 9 ~}
AB-groups: All groups with the remark that groups, containing isomorphic maximal normal nilpotent groups, are isomorphic themselves. 5.
Q=<
(2z, o) > E : < a , b , c , d , otH
[ b , a ] = c 2 1 d -~k~
[d,a]~- 1
[c, a] = d k' [c, b] = d-k,
[d, b] = 1 [d, c] = 1
ota = botd k~ otb = aotd -k~ otc = c - l o t d ka
ot2 = d~4 otd = dot
>
H 2 ( Q , Z) = Z 9 (Z2) 2 : Z 4 / A ,
A = {(kl,ks, k3, k4) II kl : 0, k3, k4 9 2Z, k2 9 Z} AB-groups: V m > 0 , < (m, 0,0,1) > q=< (2l+1,0)> E : < a , b , c , d , ot H [b,a] = c2l+ld -(21+l)k~ [c, a] = d k~ [c, b] = d - k ' ota = botd k~ otb = aotd -k~ otc = c - l o t d 2k~
[d,a] = 1 [d, b] = 1 [d, c] = 1 ot2 = dk~ otd = dot
>
HS(Q, Z) = Z @ Zs : Z4/A,
A = {(kl, k2, k3, k4) H kl : 0, k4 e 2z, k~, k3 9 z } AB-groups: V m > 0 , < (m, 0 , 0 , 1 ) > .
q --< (21, 0,1, 0) > E : < a , b , c , d , ot,~ II [b, a] = c4Zd41(k'+k~)
[c, a] = 1
[d, a] = 1 [d, b] -- 1
It, b] = d TM
[d, r = 1
aa otb otc fla /3b /3c
ot2 = cdk~ otd = d - l o t
= = = = : =
a-lot b - t o t d k' cotd - 2 ~ a/3c-2Zd l(kl-2k=) b - l f l d ~'l-2k~+2k~-2k" c - l fld k l - 2k~
~2 = d ~ /~d : d ~ ott3 = b - l ~ o t c - t d k"
>
Section
7.3 4 - d i m .
AB-groups, H2(Q,Z)
with 3-step
Fitting
223
subgroup
= Z @ Z~ = Z 4 / A ,
A = {(kl, k~, ka, k4) II k~ = 0, k3 ~ 2Z, k~, k~ ~ Z) AB-groups: Vm>0, m~0mod2,
<(m, 0,1,0)>
= < (2l + ~, 0, ~, o) > E: < a,b,c,d,o~,~ll
[b, a] = c4Z+2d(4z+2)(2k'+k~) [c, a] = 1 [c, b] = d 4k~
[d, a] = 1 [d, b] = 1 [d, c] = 1
aa ab ac jfa
a 2 = cd k~ ad = d-la
-= a-loE = b - t a d ~k~ = c a d -21~ = a ~ c - = t - l d (21+l)(k~-k~)
>
/3b = b-tfld2(~-k~+k~-k')
~2 = dk~ fld = d ~
~ c = c - ~fld ~( ~ - ~
a f t = b - ~f l a c - ~d ~"
)
H 2 ( Q , Z ) = Z @ Z2 = Z4/A,
A = {(kl, k2, k3, k~)II kl = O, ]~3 C 2Z, k2, k4 e Z} AB-groups: Vm>O, < (m, 0, 1,0) >
s. Q = < (k,o,o, 1) > r
= d -1 and/3d = d
E : < a,b,c,d,a,1311
[b, a] = c4~d 2k(~+2k~) [c, a] = 1
[d, a] = 1 > [d, b] = 1
[c, b] = d TM
[d, c] = 1
o~a -= a-J-ac2kd 2kk~
a 2 = cd k~
ab ac /3a fib aft ~c
ad = d-la
= = = = = =
b-lac-2kd (1-2~)~-2k~ c a d -2k~ aft b - l f l c - 2 ~ d k~(l+2k)-2k~(l+ak)+2(ks-k') a - l b - l ~ a c - O + 2 k ) d ~" c - l Z d ~'~-2k~
132 = ad k~ fld = dj3
H 2 ( Q , 2 5 ) = Z = Z 4 / A , A = { ( k l , k2, k3, k4) II kl -- 0, k2, k3, k4 E Z}
AB-groups: V m ) 0, < (m, 0 , 0 , 0 ) )
224
Chapter
7: C l a s s i f i c a t i o n s u r v e y
~ d = d -1 a n d ~ d = d -1 E: < a,b,c,d,a,~[I
[b, a] = c4kd -2kkl+4kk* [c, a] = d -2k' [c,b]= 1
[d, a] = 1 [d, b] = 1 [d,c]= 1
a a = a - l a c k e d -(l+2k)kl+2kk2
a 2 = cd I'~
ab = ac = [3a = fib =
ad = d-la
b - l a c - 2 k d -2kk2 c a d -2k~ a/3d -2k~ b - l f l c - 2~ d - 2kk~
~2 = adk~ /3d = d-1/3
Z c = c - l Z d k~
a~ = a-lb-l~ac-(l+2~)d~* H2(Q,Z)
= E = E 4 / A , A = {(kl,k2, k3, k4) ]] kl = 0, ks, k3, k4 e Z}
AB-groups: All groups are isomorphic to one of the previous case.
Some comments 1. This group leads to all r a n k 4, 3-step n i l p o t e n t groups. 2. Using the results of the previous section we have to require t h a t a acts n o n - t r i v i a l on Z. This means t h a t we only have to consider those 3-dimensional A C - g r o u p s of category 2, which are torsion free. In fact, there's only one such a group, n a m e l y Q = < (k, o, o, 1) >, for k = 0 rood 2. 3. For c a t e g o r y 3, b o t h actions on 2~ are allowed. However, since a n y A C - g r o u p Q of this category has torsion, we only have to investigate the trivial action. We notice t h a t for kl ~ 0 m o d 2 < (k, 0) > ~ < (k, 1) >, b u t for k = 0 m o d 2 < ( k , 0 ) > ~ < (k, 1 ) > 4. For this category, b o t h the trivial a n d the non-trivial action are to be considered. 5. Since all groups Q have torsion, we only investigate the trivial action. 6. In Q we can c o m p u t e (aft) 2 = 1. This forces the fact t h a t for the only allowable action of Q on ~ , there is a torsion element which acts non-trivially. 7. Because f12 = 1 we have to require t h a t fl acts trivially. T h e only allowable a c t i o n for a is a non-trivial one, so we have to investigate those groups w i t h o u t torsion elements "involving" a. Therefore we m u s t consider k3 = 1 (else a 2 = 1) a n d k2 = 0 (else ( a c - l a ) 2 = 1). Conclusion: the only group to be considered is Q = < (k, 0, 1, 0) >. 8. a has to act n o n - t r i v i a l l y (So consider only the case k4 = 1), b u t for fl there are two choices. 9. T h e a c t i o n of a is a non-trivial one. Since (f12) = 1 we are only inter-
Section 7.3 4-dim. AB-groups, with 3-step Fitting subgroup
225
ested in a trivial action for/3. But in this case (aft) 2 = 1, which implies that a torsion element does act non-trivially. Conclusion: this category does not lead to AB-groups. Now, we also indicate an affine representation for any of these groups. We again write down this represention A : E ~ A f f ( R 4) by giving the images of the generators a, b, c, d, a and if necessary/3. In each case we use
A(d)--
1 0 0 0 0
0 1 0 0 0
0 0 0 0
0 0 0 1 0
1 0 0 0 1
The numbers of each representation described below, correspond to the numbers of groups in the table above:
. i A(a)--
3
-2% 1
00
0 0 0
1 0 0
2 - - 3
2
- - k -~~k 0 1 0
)~(c) -
i) 1 0 0 0 0
0 1 0 0 0
~(b)--
o
2k13 0 0
~3 0 1 0 0
M 3 0 0 1 0
0\ 1 0 0 1
3
~ _ 2~kk % 1 0 0
00 0 1 0
00 0 1 1
)
.
~(a)=/
1400Oo oO 13 000 1 3Oo 1 101/b/i 400 103 23kko 01 3 0000 1 i)
~(c)--
Oo o~000 1 0Oo 1 000 1 Oo 1/
/100 000 0~ 000Oo o)
Chapter 7: Classification survey
226
.
(i00 --6 i)
Q =<(2l+1,1)>
1 0 0 0
,~(a) :
0 1 0 0
~ 0 1 0
l
1 0 0 ~3 0 o
A(c) =
I o
A(b)=
(i -2._"oo013 2131)OoOl Oo1 -Z
-k2\
0 1
0 o
I o
0 0 0 0 0 0
1 0
0 1
I 0 0 -I 0 0 0 0 0 0
~(~) =
0 -1 1 0 0
0 0 0 -1 0
~2 \ 0 0 0 1
0
0
o 0 1
Q =< (21,1) >
J
1 0 0 k-~--k21 O) 3 1 0 -I 0
0
),(a)=
~(b)=(
I 0
A(~)= Q :<
~(a)=
0 0
1 0 0
0 1 0
1 0 1
1 -:tk13 k 2 l - ~3 -2k~13 -k3
0
1
l
0
0
0 0 0
0 0 0
1 0 0
0 1 0
0 1 1
-~ 0
0 I
o o 1 o o
0 0
0 0 0
0
0 0 1 0
I
0 0 0
0 0
1 0
~(a) =
0 1
0
-1
-1
o 0 0
o
1
o
0 0
0 0
-1 0
(21, o) > 1 0
0 1
0 0
~2 -I
0 0
o 0 0
o 0 0
1 0 0
o 1 0
1 0 1
~(b) :
1 0 0 0 0
-2~.~ _hA _~ 3 3 1 l 0 1 0 0 0 0
2
-k2 0 0 1 0
0"~ 0 0 1 1
J
227
Section 7.3 4-dim. A B - g r o u p s , with 3-step F i t t i n g subgroup
~(~)=
1
0
0
0
1
o 0 0
o 0 0
k~ 3
0
0
0
1
I 0 0
o 1 0
o 0 1
A(o0 =
,
Trivial action
~(a) =
1
0
0
0
1
0
o 0 0
o 0 0
1 0 0
~2 -k o 1 0
Good!)3
010 0 A(c)= 010 001 O00
-k4 0 1 0 1 l!
A(b)=
1 0 0 0 0
k3 -1 0 0 0
0 0 1 0 0
0 0 0 -1 0
k42 0 0 0 1
J
/ /lk3ooo/ ),(,~)=
0
-I
0
~
0
o 0 0
o 0 0
1 0 0
o -1 0
1
-2k~ _k_xk_k 3 3 _ 1 k 0 1 0 0 0 0
0 1
3kk 4 --O0 012 i)
2
Non-trivial action i
~(~)=
0 o
0 0
~,(c)=
4~k 0 _kk_~ --k3 3
3
i
0
-k
0
o 0 0
1 0 0
o
1 0 1
1 0 0 1 o o 0 0 0 0
1 0
_2k_k 3 0 1 0 0
0 0 o 1 0
/ /
O) 1 o 0 1
A(a) --
.
Q = < (2t, o) >
A(a) =
I 0 0 0 0
-2k~ 3 1 0 0 0
0 0 1 0 0
k3l -l 0 1 0
k2 0 1 0 1
0
4klk - 2k2
0
1
0
0
0 0 0 0
I
-1 0 0 0 0
k
0 0
0 0
1
0
0
0 0
1 0
1 1
3
-2k~ 3 -I 0 0 0
0 0 1 0 0
0
0 0
0 -1 0
1
0 1
) (1 3ooo) A(b)=
0
1
I
0 0 0
0 0 0
1 0 0 0 1 1 0 0 1
0
0
/ /
Chapter7:Classificationsurvey
228
A(c) =
Q :<
1
0
~3
0
1
0
_k~_ O\ 3 0
1
0
0
1
0
0
0 0
0 0
0 0
1 0
0 1
A(a) =
1 0 0
k3 -1 0
0 0 0
0 0 1
k-t2 "~ 0 0
0
0
1
0
0
0
0
0
0
1
J
( 2 / + 1,0) >
A(a)=
1 0 0 0 0
.X(c) =
-2kz 3 1 0 0 0
0 0 1 0 0
1 0 0 0 0
kl 3 0 1 0 0
0 1 0 0 0
k3+2k31 1 l 2 0 1 0 _ k3 0 0 1 0
~(b)--
0 1 0 0 1
o
o
1
o
~Oo
o 0
o l 0
0
0 0 1 0 0
k_~ 2 0 o 0 1
/ 1 0 o 0 0
~(o,)=
2k 3 -1 o 0 0
0 0 o 1 0
.
Q =< (21,o, 1,o) >
I 01 01 00 -2k2I -2l 00 I ~(a)=
o 0 0
o 0 0
1 0 0
o
1 0 1
I
0
I 1 -4ki -4k~t+2k210 ~-k2-k4 ) 3
X(b)=
l1
~(~)=
0 o 0 0
0 1 o 0 0
0 0 1 0 0
2
0
1
2l
0
0
0 0
0 0
1 0
0 1
0 1
0
0
0
0
1
2kl 3 0 o 1 0
0 1 o 0 1
A(a) =
-1 0 0 0 0
2k2 1 0 0 0
0 O -1 0 0
-kl 3 0 0 -1 0
0 1
0 0 1
J
Section 7.3 4-dim. AB-groups, with 3-step Fitting subgroup
)
Q =<(2/+1,
=
~+ 3 k21 l 1 0 0
2k3 0 0 -I
0 0 0 1
0
1
O,l,O)>
1 0 0 1 0 0 0 0 0 0
:~(b) =
~3- 2 k 2 -1 0 0 0
1 0 0 0 0
229
0 0 1 0 0
-k2-2k21 - 1 - 21 0 1 0
1
-sk~
0 0 0 0
1 0 0 0
0 0
I -1 0
i
0 1
0
I 0
0 0
0 0
-4k13 + k 2 8kl/3 + 2k21 1+2l 1 0 0
3
A(c)=
4k~ _ 2 k 2 3
lO04k~ 010 001 000 000 ~+ 3
3
0 0 1 0
2k2
0 0 0 1 0
0 0
-2kz 3 0
0
-1
0
0
0 0
-1 0
0 1
kl - k2 - k4 0 0 1 1
1
J
0 1 0 0 1
2k~l+k21 2k3 0
~-
21
3
-1
5+/
0
0
0 0
I 0
0 -1
0
0
0
0
1
1
J
~
a d = d -1, ~ d = d 1 0
A(a) =
0 0 0
0 1 0 0 0
0 0 1 0 0
2kk~ _ 2kk2 3 -2k 0 1 0
k2 + 2kkl
_ k2 _ 2kk2
0 1 0 1
_ k4
)
Chapter 7: Classification survey
230
A(b)= (
1
a(c)=
~(Z) =
0
0
o 1 o 0 0 1 0 0 0
o 0 0
'~d=d -1, Zd=d
)~(a) =
(
0 0 0 0
2kkl
+ 2kk2 2k 1 0 0
0
3 0 0 1 0
0 0 1 1
A(a)=
kl + ~ 3-
2 k 2 - 3kk2 + 2 k 3 - 2k4 k
0 0 1
0 -1 0
3
3
1
0
-2k
0
o 0
1 0
o 1
1 0
0
0
0
1
0 1 0 0 0
-1 0 ~
:~(a) =
0 1 0 0 1
3
!2 1
-1
1 0 0 0 0
:~(b) =
-8kkl
3 1 0 0 0
3 - kk2 -k 1 0 0
-1
~
-4kl
1 0 0 0 0
1 2k 0-1 0 0 0 0
~3 2k 1 0 0
2
0 0 0 1 0
0 -2k 0 -1 0
)~(c)=
o
o
i~ ~ 0
1
o
~ 0
0
I
- ~ 2 - ~2 + k2 + 2kk2 + k3 + k4 "~ 0 0 1 1
O) 1
0 0 1
:~(~) =
0 -k
J
-1 0
-2k~3 -1
kkl k
0
0
1
0
0 0
0 0
0 0
-I 0
0 "~ 0 i 1
Appendix The
use
of Mathematica
~
W e a r e g o i n g to i l l u s t r a t e t h e use of M a t h e m a t i c a | w i t h a c o n c r e t e exa m p l e o f t h e c o n s t r u c t i o n o f a 4 - d i m e n s i o n a l A B - g r o u p . We will e x e c u t e all t h e c o m p u t a t i o n s n e e d e d in case n u m b e r 3 in t h e t a b l e o f s e c t i o n 7.2.
A.1
C h o o s e a crystallographic group Q
T h e g r o u p E to b e c o n s t r u c t e d fits in a s h o r t e x a c t sequence
1---+Z---~E---+Q---+I w h e r e Q is t h e 3 - d i m e n s i o n a l c r y s t a l l o g r a p h i c g r o u p listed in [10] on p a g e 62 as follows ( t h e n u m b e r i n g o f t h e lines is a d d e d ) : 1 2
3
FAMILY I I :
...
CRYSTALSYSTEM 2:
Q-CLASS 2/I: ORDER 2; ISDM TYPE 2.1; 2 Z-CLASSES REL : A2=I
4
5
...
Z-CLASS 2 / 1 / 1 :
Z(P2);
...
6 7 8
GEN:
9
SPGR: Ol A [0,0,0] FF 02 A [0,1,0]/2
I0
0 0 0 -I
A -1
0
0
1
0
IT 3; OBT 1 IT 4; OBT 1
Mathematica| is a registered trademark of Wolfram Research, Inc.
...
232
Appendix:
11
Z-CLASS
T h e use o f M a t h e m a t i c a
...
We extract the information we need in the following way: On line 3 we see that the holonomy group F of Q is of order 2 and so F is isomorphic to ~2 (= Isomorphism Type 2.1 in [10, Table 6B]). Line 4 describes the holonomy group F . There is one generator A and one relation A 2 = 1. We will use (2 (or a i f a ) to denote this generator in the sequel. The group Q has an affme representation (seen in G l ( 4 , R)) as follow S : First there are the three translations~ which are always the same and which we denote by a, b, c:
(1001) (1000
a=
0 0 0
1 0 0
0 1 0
0 0 1
b=
0 0 0
1 0 0
0 1 0
1 0 1
C z
1
0
0
0
0
1
0
0
0 0
0 0
1 0
1 1
Further, the affine transformation corresponding to a is indicated in lines 6,7,8 (the rotational part) and in line 9 (the translational part). So -1 0 0 0 / 0 1 0 0 ( 2 ---0 0 -1 0 " 0 0 0 1 * A presentation for Q can be written down now very easily following section 5.2. One should keep in mind that the action of F on Z 3 is given by the rotational part of (2. For this group, we can see, since the translational part of (2 equals zero, that Q = Z 3 >~Z2. For other groups, one will have to use the matrix representation of Q to complete its presentation. Conclusion:
Q =< a,b,c,(2
II
[b,a]
= i
[c,a]
= i
[c,b]= 1
(22 = 1
(2a = a - i a
(2b = b(2
(2C ~
C--1(2
>.
A . 2 D e t e r m i n a t i o n o f c o m p u t a t i o n a l consistent p r e s e n t a t i o n s
233
9 Line 10 denotes the next crystallographic group (number 4) and the symbol FF in front of its indicates that this group is torsion free. (Fixedpoint Free space group). Continuing as in section 5.2 we now describe any extension of Q by Z compatible with the trivial action of Q on Z (which factors through Z2). (Since a 2 = ls we cannot hope that extensions compatible with the non trivial action s will be torsion free). Such a group E has a presentation as follows: E:
all
[b,a]=d h
[d,a 1 = 1
[c,a] = d z2
[d,b] = 1
>.
[c,b] = d [d,c] = 1 a a = a - l a d 14 a 2 = d l~
ab = bad t5 ac = c-lad
ad = da 16
for some integers ll, 12~...s 17. We use M a t h e m a t i c a ~ to find the computational consistent ones, which is described in the following section.
A.2
Determination of computational consistent presentations
The following little p r o g r a m contains a function "matrixmacht" which computes formal powers of a unitriangular matrix, based on lemma 4.4.5. This p r o g r a m is funned automatically each time we start a M a t h e m a t i c a ~ session.
(* Personal Commands Used in the Infra-nil computations
*)
(, ....................................................
,)
(* Remark: All variables are written with 3 identical symbols to *) (* prevent confusion with variables used within a *) (* Mathematica session. *)
(* Definition of the binomial coefficients (* xxx : is a formal parameter *) (* nnn : is an integer *) b i n [ x x x _ , n n n ] :=(xxx-nnn+i)/nnn bin [xxx_, 0 ] :=1
*)
bin[xxx,nnn-1]
Appendix: The use of Mathematica
234
(* matrixmacht (* (* (*
: Power of an uppertringularmatrix "aaa" with *) formal power "xxx". *) Based on the formula: *) $A'x=((A-l)+I)'x=\sum_x \bn{x}{l} (A-l)'x$ *)
matrixmacht[aaa_
, xxx
] :=Sum[ (bin[xxx , 111]* MatrixPower [aaa-ldent ityMatrix [ Length [aaa] ] ,Iii] ), {iii, 0, Length [aaa] }]
(* Some convenient abbreviations *) mat [ a a a _ ] : = T a b l e F o r m [Expand [ a a a ] ] com [ a a a _ , b b b _ ] :=Expand [ I n v e r s e [ a a a ] . I n v e r s e [bbb] . a a a . b b b ]
Now we are r eady to start the p r o g r a m used do determine the comp u t a t i o n a l consistent presentations for E . T h e p r o g r a m is stored in a file with the n a m e g r o e p . 3 and looks like
(* Determining the computational consistent presentations and affine *) (* representations for this class of AC-groups. *) (* A general canonical type representation built up from the data (* i n t h e b o o k o f N e u b u e s e r e . a . a={{1,A1,A2,A3,A4},
{0,1,0,0,1}, {o,o,l,O,O5, {o,o,o,l,O}, {o,o,o,o,155
b={{1,B1,B2,B3,B4}, {0,1,0,0,05, {0,0,1,0,15, {0,0,0,1,05, {o,o,o,o,1}} c={{1,C1,C2,C3,C45, {O,l,O,O,O5, {0,0,1,0,0}, {0,0,0,1,15, {o,o,o,o,1}} d={{1, o,o,o ,1}, {o,1 ,o,o,o}, {o,o,l,O,O5,
*) *)
A.2 Determination of computational consistent presentations
235
{0,0,0,1,0},
{o,o,o,o,1}} alfa={{l,alfl,alf2,alf3,alf4},(* The "I" on this line indicates that we*) {0,-I, O, O, 0}, (* are dealing with a trivial action of *) {0, O, i, O, 0}, (* alia on d *) {0, O, 0,-I, 0},
{o, o, o, o, i } } (*The conditions, which should be satisfied by the unknowns Al,...,alf4*) *) (*And the conditions on ii,12,...,17 for computational consistency. *) ( * A l l matrices printed should be zero. Print Print Print Print Print Print Print
[ [ [ [ [ [ [
Expand [ Expand [ Expand[ Expand [ Expand [ Expand[ Expand[
com[b ,a] -matrixmacht [d ,II] ] ] comic, a] -matrixmacht [d ,12] ] ]
com[c ,hi -matrixmacht [d ,13] ] ] alla. a-Inverse [a] .alia .matrixmacht [d, 14] ] ] alfa.b-b.alfa.matrixmacht [d,15] ] ] alla. c-Inverse [c] .alia .matrixmacht [d, 16] ] ] alia. alfa-matrixmacht [d, 17] ] ]
We now list a recorded version of a Mathematica e session to show how we use this program:
eulerT, math Mathematica 2.1 for SPARC Copyright 1988-92 Wolfram Research, Inc. In[l] := <
>
{o, o, o, o, o}, {o, o, o, o, o}}
{{0, O, O, O, -A3 + CI - 12}, {0, O, O, O, 0}, {0, O, O, O, 0},
>
{o, o, o, o, o}, {o, o, o, o, o}}
{ { 0 , O, O, O, -B3 + C2 - 13}, {0, O, O, O, 0}, {0, O, O, O, 0},
> {{0
> {{0
>
{o, o, o, o, o}, {o, o, o, o, o}} O, 2 A2, O,alfl - Ai + 2 A4 - 14}, {0, O, O, O, 0}, {0, O, O, O, 0},
{o, o, o, o, o}, {o, o, o, o, o}} 2 BI, O, 2 B3, a l f 2 - 15}, {0, O, O, O, 0}, {0, O, O, O, 0},
{o, o, o, o, o}, {o, o, o, o, o}}
{{0, O, 2 C2, O,alI3 - C3 + 2 C4 - 16}, {0, O, O, O, 0}, {0, O, O, O, 0},
>
{o, o, o, o, o}, {o, o, o, o, o}}
Appendix: The use of Mathematica
236
{{o, o, 2 all2, o, 2 all4 - 17}, {o, O, O, o, o}, {o, o, o, o, o}, >
{o, o, o, O, 0}, {0, o, o, o, o}}
In[2]:= A2=BI-II;A3=CI-12;B3=C2-13;alf2=I5;alf4=IT/2; In[3]:= < {o, o, o, o, o}} {{o o, o, o, o}, {o , o, o, o, o}, {o, o, o, o, o}, {o, o, o, o, o}, > {o, o, o, o, o}} {{o o, o, o, o}, {o , o, o, o, o}, {o, o, o, o, o}, {o, o, o, o, o}, > {o, o, o, o, o}} {{o o , 2 Bi - 2 11, O, all1 - A1 + 2 A4 - 14}, {0, O, O, O, 0},
> {{0
{o, o, o, o, o}, {0, O, O, O, 0}, {0, O, O, O, 0}} 2 B I , O, 2 C2 - 2 13, 0}, {0, O, O, O, 0}, {0, O, O, O, 0},
> {0, O, O, O, 0}, {0, O, O, O, 0}} {{0 O, 2 C2, O,alf3 - C3 + 2 C4 - 1 6 } ,
{0,
O, O, O, 0 } ,
{0,
O, O, O, 0 } ,
{{o
{o, o, o, o, o}, {o, o, o, o, o}} O, 2 15, O, 0}, {o, o, o, o, o}, {o, o, o, o, o}, {o, o, o, o, o},
>
{o, o, o, o, o}}
>
In[4]:= B1=O;I1=O;C2=O;13=O;15=O;alf1=A1-2
A4 + 14;
In[5]:= < {o, o, o, o, o}} {{o o, o, o, o}, {o, o, o, o, o}, {o, o, o, o, o}, {o, o, o, o, o}, > {0, O, O, O, 0}} {{0 o, o, o, o}, {o, o, o, o, o}, {o, o, o, o, o}, {o, o, o, o, o}, > {0, o, o, o, o}} {{0 o, o, o, o}, {o, o, o, o, o}, {o, o, o, o, o}, {o, o, o, o, o}, > {0, o, o, o, o}} {{0 O, o, o, o}, {o, o, o, o, o}, (o, o, o, o, o}, {o, o, o, o, o}, > {0, O, O, O, 0}} {{o o, O, O , a l f 3 - C3 + 2 C4 - 1 6 } , { 0 , O, O, O, 0 } , { 0 , O, O, O, 0 } ,
A.2 Determination of computational consistent presentations > {o, o, o, o, o}, {o, o, o, o, o}} {{o, o, o, o, o}, {o, o, o, o, o}, {o, o, o, o, o}, {o, o, o, o, o}, >
{0, O, O, O, 0}}
In[6]:=
alf3=C3
-9 C4 + 16;
In[7] := < < g r o e p 3 . 1
{{o, o, o, o, o}, {o , O, O, O, 0}, {0, O, O, O, 0}, {0, O, O, O, 0}, > {o, o, o, o, o}} {{o, o, o, o, o}, {o , o, o, o, o}, {o, o, o, o, o}, {o, o, o, o, o}, > {o, o, o, o, o}} {{o, o, o, o, o}, {o , o, o, o, o}, {o, o, o, o, o}, {o, o, o, o, o}, > {o, o, o, o, o}} {{o, o, o, o, o}, {o , o, o, o, o}, {o, o, o, o, o}, {o, o, o, o, o}, > {o, o, o, o, o}} {{o, o, o, o, o}, {o, o, o, o, o}, {o, o, o, o, o}, {o, o, o, o, o}, > {o, o, o, o, o}} {{o, o, o, o, o}, {o, o, o, o, o}, {o, o, o, o, o}, {o, o, o, o, o},
> {o, o, o, o, o}} {{o, o, o, o, o}, {o, o, o, o, o}, {o, o, o, o, o}, {o, o, o, o, o}, >
{o, o, o, o, o}}
In[B] := P r i n t [11," ",12, .... ,13," ",14," 0 12 0 14 0 16 17
",15,"
In [9] := 12=kl ; 14=k2 ;16=k3 ;17=k4 ; In[lO] := mat [a]
AI
0
Cl
0
1
0
0
1
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
Out [ I 0 ] / / T a b l o F o r m = 1
In[11] := A 1 = O ; C 1 = k 1 / 2 ; A 4 = O ;
-
kl
A4
",16,"
",17]
237
Appendix: The use of Mathematiea
238 In[12] := mat [b] Out [12]//TableForm= 1
0
B2
0
B4
0
1
0
0
0
0
0
1
0
1
0
0
0
1
0
0
0
0
0
1
In[13] := B2=O;B4=O; In[14] := mat [c] Out [14]//TableForm=
kl 1
2
0
C3
C4
0
1
0
0
0
0
0
1
0
0
0
0
0
1
1
0
0
0
0
1
In[15] := C3=0;C4=0;
In[16]
:= mat [alfa]
k4
Out [16]//TableForm=
In[17] := mat
1
k2
0
k3
2
0
-1
0
0
0
0
0
1
0
0
0
0
0
-1
0
0
0
0
0
1
[a]
Out [17]//TableForm=
-kl 1
0
0
2
0
A.2 Determination of computational consistent presentations
In[18]
1
0
0
1
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
1
0
0
0
0
0
1
0
0
0
0
0
1
0
1
0
0
0
1
0
0
0
0
0
1
:= m a t [ b ]
Out [ 1 8 ] / / T a b l e F o r m =
In[19]
0
239
:= m a t [ c ]
kl
Out [19]//TableForm= 1
2
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
1
0
0
0
0
1
In[20] := Save ["representatie3. i" ,a,b,c ,d,alfa] In[21] := Quit eulerT,
The program specifies the computational consistent groups E, which
240
Appendix:
T h e use o f M a t h e m a t i c a
can all be presented by means of the four parameters kl, k2, k3 and k4: E:
< a,b,c,d,a[[
[b,a] : 1 [c,a] = d ~1 [c,b]= 1 a a = a - l o r d k2
[d,a]= 1 [d,b] = 1 [d,c]= 1 ot 2 : d k4
ab = ba
ad = da
> .
OLC = c-loLd k3
A.3
C o m p u t a t i o n of
H2(Q,Z)
The previous section also shows t h a t the set of standard cocycles SZ2(Q,,~)
~- 2 4
and t h a t a canonical representation for each group E is saved in a file with the name r e p r e s e n t a r 3.1. The following step in the process is to determine which s t a n d a r d cocycles are cohomologous to zero. This is done with a program having the name c o c y k 3 . 1 :
(* Which standard cocycles are cohomologous (* By theorem 5.2.2 we may suppose that kl=0 <
to zero?
,) ,)
(* The action of alfa on d can be read of as the first entry of the *) (* matrix which represents alfa *) actalfa=alfa[[1,1]]
For [kk=0, kk<=l ,kk++, For [mm=O ,r~n<=1 ,Irm++, {product--mat rixmacht [a, x 1] .matrixmacht [b, x2] . matrixmacht [c, x3] .MatrixPower [all a,kk] . matrixmacht [a, yl] .matrixmacht [b, y2] . matrixmacht [c, y3] .MatrixPower [alfa ,mm] , (* Product is a general product of two elements. *) (* Product can be written in the form: *) (* a~acoef b'bcoef c'ccoef d'f(x,y) alfa'alfacoef *) (* The following commands determine all these *) (* coefficients. ,) alfacoef=Mod[kk+rmn,2], (* Computation in Z-modulo 2 *) product=product. MatrixPower [alfa ,-alfacoef] , acoef=product[[2,5]], (, This is a direct consequence *)
A.3 Computation of H 2(Q, Z)
241
bcoef=product[[3,S]], (* of t h e c a n o n i c a l t y p e *) ccoef=product[[4,S]], (* representation *) product--matrixmacht [c,-ccoef] .matrixmacht [b,-bcoef] . matrixmacht [a ,-acoef] .product, (* The power of d equals the cocycle ~) f=Simplify [product [ [1, S] ] ], (* A check for exactness of the program: *) product [[1,5]]=0, If [product==IdentityMatrix [S] ,Print [] ,Print ["Something' s wrong :"] ] , Print ["Cocykel: ~ , Print [f] , term=(actalfa)~(kk), (* term= the action of x on d *) (* The general form of delta gamma for x and y *) deltaganuna=( (term yl + xl -acoef) gammaa + (term y2 + x2 -bcoef) ga~nab + (term y3 + x3 -ccoef) garmnac + (term mm + kk -alfacoef )gammaalfa ), dif f erence=Expand [f-delt agarmma], P r i n t [] , Print ["f (x,y) -deltagamma(x ,y) :"], Print [difference] , Print ["_ ..................................... ", ,t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
I,]
)]] Print [.......................................... , ~t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
i,]
The recorded M a t h e m a t i c ae session is as follows:
euler~ math Mathematica 2.1 for SPARC Copyright 1988-92 Wolfram Research, Inc. In[l]:= <
Cocykel: 0 f(x,y)-deltagarmua(x,y): 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cocykel: k2 yl + k3 y3 f(x,y)-deltagarmua(x,y):
242
Appendix: The use of Mathematica
- 2 gamraaa y l + k2 y l - 2 ganunac y3 + k3 y3 .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Cocykel: k4 + k2 y l + k3 y3 f(x,y)-deltagarmna(x,y): - 2 g a m m a a l f a + k4 - 2 gammaa y l + k2 y l - 2 gammac y3 + k3 y3 .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
gammaa=k2/2;gan~ac=k3/2;ganunaalfa=k4/2;
In[2]:=
I n [ 3 ] := < < c o c y k 3 . 1 Cocykel: 0 f (x , y ) - d e l t aganma (x ,y) : 0 .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Cocykel: 0 f (x ,y) -delt agalmua (x ,y) : 0 .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Cocykel: k2 yl + k3 y3 f (x ,y) -deltagamma (x ,y) : 0 .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Cocykel : k4 + k2 yl + k3 y3 f (x, y) -de it agan~a (x ,y) : 0 .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
In[4] := Quit eulerT.
This shows that a cocycle f depending on the four variables kl, k2, k3 and k4 is cohomologous to 0 if and only if kl
= 0,
This allows us to conclude that
H (Q,
k2, k3, k4 E 27,
A.4 Investigation of the torsion
A.4
243
Investigation of the torsion
The following problem is to discover in which groups E there is torsion. To achieve this, we first look for all torsion elements in the crystallographic group Q. Since the order of the holonomy group is 2, the order of a torsion element will also be equal to two. The program used to find torsion in Q is called t o r s i o n and is the following:
(* Searching the torsion elements of the crystallographic group Q *) (* Loading the representation for E and so for Q *) <
(* L o o k i n g a t t h e e l e m e n t i n q r a t h e r t h a n i n E *) testcutoff=IdentityMatrix[4] For[i=l,i<=4,i++, For[j=l,j<=4,j++, testcutoff[[i,j]]=testpower[[i+l,j+l]] ]] ( * What a r e t h e p o s s i b l e t o r s i o n s ? * ) Print[mat[testcutoff]] Print[ ....] Print[Solve[testcutoff==IdentityMatrix[4],{ml,m2,m3}] Print[ .... ]
The result of this program is: eulerZ math Mathematica 2.1 for SPARC Copyright 1988-92 Wolfram Research, Inc. I n [ l ] := < < t o r s i o n 1
0
0
0
0
1
0
2 m2
]
Appendix: The use of Mathematica
244 0
0
1
0
0
0
0
1
{{m2 -> 055
I n [2] := Quit euler7~
B e f o r e we continue, we have to notice t h a t the file r e p r e s e n t a t i e 3 . 1 is in a d a n g e r o u s f o r m , since it contains a lot of r e d u n d a n c y , which m a y cause errors. T h e r e f o r e we do the following: euler~, c a t representatie3.1 a = {{I, AI, O; C1 - 12, A4},
{0, I, O, O, 1}, {0, O, I, O, 0},
{o, o, o, 1, o5, {o, o, o, o, 1}} AI=O C1 = kl/2 12 = kl A4=
0
b = {{I,
O, B2, O,B4}, ( 0 ,
I,
O, O, 0 } , {0, O, I ,
O, 1}, ( 0 ,
O, O, I ,
0},
{0, o, o, o, 1}5 B2=O B4=O c = {{I,CI, 0,C3, C4}, {0, I, O, O, 05, {0, O, I, O, 05, {0, O, O, l, I},
{o, o, o, o, 155 C3=0 C4=0 d = {{I, O, O, O, I}, {0, 1, O, O, 0}, {0, O, I, O, 0}, {0, O, O, I, 05,
{o, o, o, o, 15} alfa = {{I, A1 - 2.A4 + 14, O, C3 - 2.C4 + 16, 17/25, {0, -I, O, O, 05,
{o, o, i, o, o}, {o, o, o, -i, o5, {o, o, o, o, t55 14 = k2
A.4 Investigation of the torsion
245
16 = k3 17 = k4 euler% math Mathematica 2.1 for SPARC Copyright 1988-92 Wolfram Research, Inc. In[l]:= <>representatie3.1 In[4]:= b>>>representatie3.1 In [5] := c>>>representatie3.1 In[6]:= d>>>representatie3.1 In[7]:= alfa>>>representatie3.1 In [8] := Quit euler~, cat representatie3.1 {{1, o, 0 , - k l / 2 , 0}, {0, 1, O, O, i } , {o, o, l, o, o}, {o, o, o, 1, o}, {o, o, o, o, i } } -[-[1, O, O, O, 0}, {0, 1, O, O, 0}, {0, O, 1, O, 1 } , {0, O, O, 1, 0}, {o, o, o, o, i } } {{1, kl/2, O, O, 0}, {0, 1, O, O, 0}, {0, O, 1, O, 0}, {0, O, O, 1, 1}, {0, O, O, O, 1}} -[-[1, O, O, O, 1}, {0, 1, O, O, 0}, {0, O, 1, O, 0}, {0, O, O, 1, 0}, {o, o, o, o, i } } {{1,k2, O,k3, k4/2}, {0, -1, O, O, 0}, {0, O, I, O, 0}, {0, O, O, -1, 0}, {0, O, O, O, 1}} eulerY, vi representatie3.1 (--> a little bit of file editing <--) euler~, cat representatie3.1 a={{1, O, 0,-kl/2, 0}, {0, 1, O, O, 1}, {0, O, 1, O, 0}, {0, O, O, I, 0}, {0, O, O, O, I}} b={{l, O, O, O, 0}, {0, I, O, O, 0}, {0, O, 1, O, 1}, {0, O, O, 1, 0}, {o, o, o, o, i } } c={{l, kl/2, O, O, 0}, {0, I, O, O, 0}, {0, O, I, O, 0}, {0, O, O, I, I}, {0, O, O, O, I}} d={{1, O, O, O, 1}, {0, 1, O, O, 0}, {0, O, 1, O, 0}, {0, O, O, 1, 0}, {0, O, O, O, 1}} alfa={{1, k2, O, k3, k4/2}, {0, -1, O, O, 0}, {0, O, i, O, 0}, {o, o, o, -I, o} , {o, o, o, o, I}}
euler~
Appendix: The use of Mathematica
246
Now we are really ready to start the computations concerning torsion. We know from corollary 6.1.2 that we only have to deal with kl,k2, k3, k4 E {0,1}. Let q = a'~lb'~2cm3a be a torsion element in Q (of order 2), then for the lift ~ = amlb'~2c"~3a of q in E we have t h a t
q2 = (am1bm2cm5 Ct)2 = dP(ml ,,n2,,~3) where P ( m l , m2, m3) is some polynomial function in the variables rnl, m2 and m3. If q acts non trivially on d, we know that ~ is indeed a torsion element. So suppose that the action of q on d is trivial. Now it's easy to see t h a t there exists a lift ~ of q which is a torsion element of order 2 if and only if P(ml,m2, m3) :- 0 m o d 2. To investigate this, the following lemma is very helpful: L e m m a A . 4 . 1 Let P(x) be an integer valued polynomial function (i.e.
P(z) E Z, Vz E Z) of total degree < M. Then Yz, k,n e Z : P(z) = P(z + kn(M!))mod n. Proof: Every integer valued polynomial in the variable x of degree < M , is a Z - l i n e a r combination of polynomials of the form
(o <
i<
(0) enot t 0 o. t nt z + kn(M!)~
(z + M!kn)((z - 1) + M l k n ) . . . ( ( z - i + 1) + Mlkn)
i
1.2.3...i
] =_ z ( z - 1 ) ( z - 2 ) . . . ( z - i +
l) m o d n .
1.2.3...i This finishes the proof.
The lelmna tells us that in checking if P(ml,m2, m3) =- 0 m o d 2, we m a y restrict ourselves to mi E {0, 1 , . . . , 2(degree of P in m~)! - 1}, for i = 1, 2, 3. We used all of this in writing the following p r o g r a m ( l ; o r s i o n 3 . 1 ) to find the AB-groups:
A.4 Investigation of the torsion
247
(~ maxki= the maximum value of ki < the order of the torsion we are *) (* looking for. *) maxkl=1 maxk2=l maxk3=l maxk4=l order=2 For[kl=O,kl<--maxkl,kl++, For[k2=O,k2<--maxk2,k2++, For[k3=O,k3<=maxk3,k3++, For[k4=O,k4<--maxk4,k4++, {<O,Print["torsion"], Print["These kind of e l e m e n t s have infinite order!"] ], Print[ ....], ml=., (* Cleaning up *) m2=., m3=., f=., a=o,
c=. , d=o ,
alfa=.}
248
Appendix: The use of Mathematica ]]]] The o u t p u t with M a t h e m a t i c a~ looks Hke:
euler~ math Mathematica 2.1 for SPARC Copyright 1988-92 Wolfram Research, Inc. In[1]:= <
249
A.5 Summary
group: torsion
(kl=l,k2=l,k3=l,k4=O)
group: torsion
(kl=l,k2=l,k3=l,k4=l)
In [2] := quit eulerT.
We have to interpret this as follows. A group E determined by a 4-tuple ( k l , k2, k3, k4) is an AB-group if and only if kl = k2 = k3 k4 -
0mod2 0 mod 2 0 mod 2 1 mod 2.
We denote (the cohomology class of) the group E (which is determined by kl, k2, k3, k4) as < (kl, k2, k3, k4) >. Since < (kl, k2, k3, k4) >-~< ( - k l , - k 2 , - k 3 , - k 4 ) >, we m a y always suppose that kl is positive. (We exclude kl = 0, since in this case Fitt (E) would be abelian). Of course, the relations on the cohomology-level show that we m a y restrict ourselves to k2, k3, k4 C {0, 1}. All this allows us to conclude that for a fixed kl (i.e. a fixed nilmanifold), there is at most one AB-group of this kind, namely if kl is even, we m a y take E = < (kl, 0, 0, 1) >.
A.5
Summary
The previous sections show that any extension of Q by Z has a presentation of the form E: < a,b,c,d,a[I
[b,a]= 1 [c,a] = d kl [c,b]= l a a = a - l a d k2 ab = ba ac = c - l a d k3
[d,a]= 1 [d,b] = 1 [d,c]= 1 a 2 = d k4 ad = da
>.
250
Appendix: The use of Mathematica
1000) 00)0
A canonical type affme representation A : E given by
~(a)=
0
1
0
0
0
0
1
0
0 ~0
0 0
0 0
1 0
0
1
0 0
0 0
0 0
I
A(c)=
1 0
1 k2 o k~
;~(~) =
AtT(R4) for such an E is
0
0 0 1 0
~(b)=
and
0
-~
0 0 0 0
1 0 0 0
0 1 0 0
0 0 1 0
0
~ )
0
-1
0
0
0
o
o
1
o
o
0 0
0 0
0 0
-1 0
0 1
.
We also showed that the group H2(Q, ~) =~ Z | (Z2) a. Finally, we indicated that for a fixed nilpotent group, there is maximal one AB-group of this kind, containing this nilpotent group as its maximal nilpotent normal subgroup. More precise, for kl > 0 even, there is one AB-group, determined by the parameters (kl, k2, k3, k4) = (]cl, 0, 0, 1), containing N : < a,b,c, dll [c,a] = dk' > as its maximal nilpotent normal subgroup.
Bibliography [1] Auslander, L. Bieberbach's Theorem on Space Groups and Discrete Uniform Subgroups of Lie Groups. Ann. of Math. (2), 1960, 71 (3), pp. 579-590. [2] Auslander, L. The structure of complete locally a]:fine manifolds. Topology, 1964, 3 Suppl. 1., pp. 131-139. [3] Auslander, L. and Markus, L. Holonomy of Flat A]:finely Connected Manifolds. Ann. of Math., 1955, 62 (1), pp. 139-151. [4] Benoist, Y. Une nilvaridtd non affine. C. R. Acad. Sci. Paris S6r. I Math., 1992, 315 pp. 983-986. [5] Benoist, Y. Une nilvaridtd non affine. J. Differential Geom., 1995, 41 pp. 21-52. [6] Bieberbach, L. Uber die Bewegungsgruppen der Euklidisehen Raume L Math. Ann., 1911, 70, pp. 297-336. [7] Bieberbach, L. {lber die Bewegungsgruppen der Euklidischen Raume II. Math. Ann., 1912, 72, pp. 400-412. [8] Borel, A. and Harish-Chandra. Arithmetic subgroups of algebraic groups. Ann. of Math. (2), 1962, 75, pp. 485-535. [9] Boyom, N. B. The lifting problem for affine structures in nilpotent Lie groups. Trans. Amer. Math. Soc., 1989, 313, pp. 347-379. [10] Brown, H., Billow, R., Neubiiser, J., Wondratscheck, H., and Zassenhaus, H. Crystallographic groups of four-dimensional Space. Wiley New York, 1978. [11] Brown, K. S. Cohomology of groups, volume 87 of Grad. Texts in Math. Springer-Verlag New York Inc., 1982.
252
Bibliography
[12] Burde, D. and Grunewald, F. Modules for certain Lie algebras of maximal class. 3. Pure Appl. Algebra, 1995, 99 pp. 239-254. [13] Charlap, L. S. Bieberbach Groups and Flat Manifolds. Universitext. Springer-Verlag, New York Inc., 1986. [14] Conner, P. E. and Raymond, F. Deforming Homotopy Equivalences to Homeomorphisms in Aspherical Manifolds. Bull. A.M.S., 1977, 83 (1), pp. 36-85. [15] Dekimpe, K. Almost Bieberbach Groups: cohomology, construction and classification. Doctoral Thesis, K.U.Leuven, 1993. [16] Dekimpe, K. Polynomial structures and the uniqueness of affinely fiat infra-nilmanifolds. 1994. preprint, to appear in Math. Zeitschrift. [17] Dekimpe, K. A note on the torsion elements in the centralizer of a finite index subgroup. 1995. To appear in the Bull. of the Belgian Math. Soc. - Simon Stevin. [18] Dekimpe, K. The construction of affine structures on virtually nilpotent groups. Manuscripta Math., 1995, 87 pp. 71 - 88. [19] Dekimpe, K. and Hartl, M. Affine Structures on 4-step Nilpotent Lie Algebras. 1994. To appear in Journal of Pure and Applied Algebra. [20] Dekimpe, K. and Igodt, P. Computational aspects of a]:fine representations for torsion free nilpotent groups via the Seifert construction. 3. Pure Applied Algebra, 1993, 84, pp. 165-190. [21] Dekimpe, K. and Igodt~ P. The structure and topological meaning of almost-torsion free groups. Comm. Algebra, 1994, (7), pp. 25472558. [22] Dekimpe, K. and Igodt, P. groups. 1995. Prepint.
Polynomial structures on polycyclic
[23] Dekimpe, K. and Igodt, P. Polycyclic-by-finite groups admit a bounded-degree polynomial structure. 1996. Prepint. [24] Dekimpe, K. and Igodt, P. Polynomial structures for iterated central extensions of abelian-by-nilpotent groups. Algebraic Topology: New Trends in Localization and Periodicitiy~ 1996~ pages pp. 155-166. Progress in Mathematics, Birkh~iuser.
Bibliography
253
[25] Dekimpe, K., Igodt, P., Kim, S., and Lee, K. B. A]:fine structures for closed 3-dimensional manifolds with NIL-geometry. Quart. J. Math. Oxford (2), 1995, 46, pp. 141-167. [26] Dekimpe, K., Igodt, P., and Lee, K. B. Polynomial structures for nilpotent groups. Trans. Amer. Math. Soc., 1996, 348, pp. 77-97. [27] Dekimpe, K., Igodt, P., and Malfait, W. On the Fitting subgroup of almost crystallographic groups. Tijdschrift van het Belgisch Wiskundig Genootschap, 1993, B 1, pp. 35-47. [28] Dekimpe, K., Igodt, P., and Malfait, W. There are only finitely many infra-nilmanifolds under each nilmanifold: a new proof. Indagationes Math., 1994, 5 (3), pp. 259-266. [29] Dekimpe, K. and Malfait, W. Affine structures on a class of virtually nilpotent groups. To appear in Topology and its Applications. [30] Fried, D., Goldman, W., and Hirsch, M. Affine manifolds with nilpotent holonomy. Comment. Math. Helv., 1981, 56 pp. 487-523. [31] Fried, D. and Goldman, W. M. Three-Dimensional A]:fine Crystallographic Groups. Adv. in Math., 1983, 47 1, pp. 1-49. [32] Frobenius, G. Uber die unzerlegbaren diskreten Bewegungsgruppen. Sitzungsber. Akad. Wiss. Berlin, 1911, 29 pp. 654-665. [33] Gorbacevi~, V. V. Discrete subgroups of solvable Lie groups of type (E). Math. USSR Sbornik, 1971, 14 N ~ 2, pp. 233-251. [34] Gorbacevi~, V. V. Lattices in solvable Lie groups and deformations of homogeneous spaces. Math. USSR Sbornik, 1973, 20 2, pp. 249-266. [35] Gromov, M. Almost fiat manifolds. J. Differential Geometry, 1978, 13, pp. 231-241. [36] Grunewald, F. and Margulis, G. Transitive and quasitransitive actions of a]:fine groups preserving a generalised Lorentz-structure. J. Geometry and Physics, 1988, 5 (4), pp. 493-531. [37] Grunewald, F. and Segal, D. On affine crystallographic groups. J. Differential Geom., 1994, 40 (3), pp. 563-594. [38] Igodt, P. and Lee, K. B. Applications of group cohomology to space constructions. Trans. A.M.S., 1987, 304 pp. 69-82.
254
Bibliography
[39]
Igodt~ P. and Lee, K. B. On the uniqueness of certain affinely flat infra-nilmanifoIds. Math. Zeitschrift, 1990, 204 pp. 605-613.
[4o]
Igodt, P. and Malfait, W. Extensions realising a faithful abstract kernel and their automorphisms. Manuscripta Math., 1994, 84, pp. 135-161.
[41] Igodt, P. and Malfait, W. Representing the automorphism group of an almost crystallographic group. Proc. Amer. Math. Soc., 1996, 124 (2), pp. 331-340. [42] Kamishima, Y., Lee, K. B., and Raymond, F. The Seifert construction and its applications to infra-nilmanifolds. Quarterly J. of Math. Oxford (2), 1983, 34, pp. 433-452. [43] Kim, H. Complete left-invariant a fjfine structures on nilpotent Lie groups. J. Differential Geom., 1986, 24, pp. 373-394. [44] Lee, K. B. Aspherical manifolds with virtually 3-step nilpotent fundamental group. Amer. J. Math., 1983, 105 pp. 1435-1453. [45] Lee, K. B. There are only finitely many infra-nilmanifolds under each nilmanifold. Quart. J. Math. Oxford (2), 1988, 39, pp. 61-66. [46] Lee, K. B. and Raymond, F. Topological, affine and isometric actions on flat Riemannian manifolds. J. Differential geometry, 1981, 16 pp. 255-269. [47] Lee, K. B. and Raymond, F. Geometric realization of group extensions by the Seifert construction. Contemporary Math. A. M. S., 1984, 33, pp. 353-411. [48] Lee, K. B. and Raymond, F. Rigidity of almost crystallographic groups. Contemporary Math. A. M. S., 1985, 44, pp. 73-78. [49] Lee, K. B. and Raymond, F. Examples of solvmanifolds without certain affine structure. Illinois J. Math., 1990, 37 pp. 69-77. [50] Mac Lane, S. Homology, volume 114 of Die Grundlehren der Math. Wissenschaften. Spriager-Verlag Berlin Heidelberg New York, 1975. [51] Mal'cev, A . I . On a class of homogeneous spaces. A.M.S., 1951, 39, pp. 1-33.
Translations
Bibliography
255
[52]
Milnor, J. On fundamental groups of complete afjfinely fiat manifolds. Adv. Math., 1977, 25 pp. 178-187.
[53]
Moser, L. Elementary surgery along torus knots and solvable fundamental groups of closed 3-manifolds. Thesis, University of Wisconsin, 1970.
[54] Nisse, M. Structure affine des infranilvaridtds et infrasolvaridtds. C. R. Acad. Sci. Paris S~rie I, 1990, 310 pp. 667-670. [55] Orlik, P. Seifert Manifolds. Lect. Notes in Math. 291. SpringerVerlag, 1972. [56] Passman, D. S. The Algebraic Structure of Group Rings. Pure and Applied Math. John Wiley & Sons, Inc. New York, 1977. [57] Rub, E. A. Almost fiat manifolds. J. Differential Geometry, 1982, 17, pp. 1-14. [58] Scheuneman, J. AjCjfinestructures on three-step nilpotent Lie algebras. Proc. A.M.S., 1974, 46 (3), pp. 451-454. [59] Segal, D. Polycyclic Groups. Cambridge University Press, 1983. [60] Wolf, J. A. Spaces of constant curvature. Publish or Perish, Inc. Berkeley, 1977. [61] Wolfram, S. Mathematica. Wolfram Research, Inc., 1993. [62] Zassenhaus, H. [)ber einen Algorithmus zur Bestimmung der Raumgruppen. Commentarii Mathematici Helvetici, 1948, 21, pp. 117-141. [63] Zieschang, H. Finite Groups of Mapping Classes of Surfaces. Lect. Notes in Math. 875. Springer-Verlag, 1981.
Index . A ( - 2 , s, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~(-2,~,t) .......................................................... Aft(G) ............................................................. A(z)
96 95 13
.............................................................
BUT~
135
k,(R ) .......................................................
x(E)
61
..............................................................
134
d(E) ............................................................... exp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F(F) ............................................................... F i t t (P) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F i t t (F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~(N) ............................................................... rk ................................................................. 7-/(IRK) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81 7 39 20 41 2 19 49
log . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
#(a)
................................................................. K
2
k
. M ( R ,JR ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . or (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49 136
~N,E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P(•
K
21
) .............................................................
52 52
p ( R K , •k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S Z ~2 ( Q , g m ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ / H
* . . . . . . . . .
. . . . o . . o o . . . . . . . . . . . . . , . . . . .
. . . .
.~
. . . .
106 ..
. . . . .
U T K (_R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . z(c)
. . o . . o o o o
3
61
................................................................
Zi(N) ............................................................... 1-cohomology 3-dimensional 4-dimensional 4-dimensional
1
w i t h n o n - a b e l i a n coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 17 AC-groups ......................................... 159 A B - g r o u p s , w i t h 2 - s t e p n i l p o t e n t F i t t i n g s u b g r o u p .. 168 A B - g r o u p s , w i t h 3 - s t e p n i l p o t e n t F i t t i n g s u b g r o u p .. 219
AB-group .......................................................... abstract kernel ..................................................... AC-group .......................................................... a d a p t e d lower c e n t r a l series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . affine defect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15 28 15 3 81
Index
257
affme d i f f e o m o r p h i s m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . affme s t r u c t u r e of a g r o u p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13 80
affmely flat m a n i f o l d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
a l g e b r a i c c h a r a c t e r i z a t i o n of a l m o s t - c r y s t a l l o g a p h i c g r o u p s . . . . . . . . . . a l g e b r a i c c h a r a c t e r i z a t i o n of c r y s t a l l o g r a p h i c g r o u p s . . . . . . . . . . . . . . . . a l m o s t - B i e b e r b a c h group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34 15 15
almost-crystallographic group ...................................... a l m o s t t o r s i o n free g r o u p s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15 40
Auslander conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
Betti number ...................................................... Bieberbach group
..................................................
134 13
Bieberbach theorems ............................................... blocked D D D - m a t r i x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14 61
blocked upper triangular matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
Cam )bell-Baker-Hausdorff formula .................................. c a n o n i c a l t y p e affme r e p r e s e n t a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8 57
c a n o n i c a l t y p e affine r e p r e s e n t a t i o n of a Lie a l g e b r a . . . . . . . . . . . . . . . . . 65 canomcal type polynomial representation ........................ 53,81 canonical type representation ....................................... 49 canonical type smooth representation ............................... canomcal type structure ............................................ c e n t r a l series of a Lie a l g e b r a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . closure o f the F i t t i n g s u b g r o u p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . commutator ......................................................... commutator subgroup ............................................... c o m p a t i b l e basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c o m p a t i b l e set of g e n e r a t o r s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c o m p l e t e (affinely fiat m a n i f o l d ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c o m p l e t e affine s t r u c t u r e on a Lie a l g e b r a . . . . . . . . . . . . . . . . . . . . . . . . . . . computational consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c o m p u t a t i o n a l consistent p--tuple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . conjugation ..........................................................
52 48 64 41 1 2 65 2,48 78 80 105 106 1
c o o r d i n a t e s y s t e m o n a n i l p o t e n t Lie g r o u p G . . . . . . . . . . . . . . . . . . . . . . . crystallographic group ..............................................
72 13
DDD-matrix
.......................................................
61
.....................................................
61
DDD-property
Index
258
Diagonal Distance Degree property
.................................
essential covering ...................................................
61 19
essential extension ..................................................
20
Euler characteristic ................................................
134
exp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
exponential map
7
.....................................................
first B i e b e r b a c h t h e o r e m Fitting subgroup
...........................................
...................................................
14 20
flat R i e m a n n i a n m a n i f o l d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
group extension with non-abelian kernel ............................
28
g r o u p o f aiYme d i f f e o m o r p h i s m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
Heisenberg group ...................................................
19
holonomy group
16
....................................................
isolator .............................................................. Koszul-Vinberg (KV) structures
....................................
3 67
large manifold .....................................................
147
local Seifert i n v a r i a n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
146
log . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lorentz-flat manifold ............................................... l o w e r c e n t r a l series
7 78
..................................................
1
Mal~cev c o m p l e t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
matrix_macht ...................................................... Milnor's conjecture, problem~ question ..............................
233 79
n i l p o t e n t Lie g r o u p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
nilpotent group ......................................................
1
non-abelian
2 - c o h o m o l o g y sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
orientation module
................................................
28 135
polynomial diffeomorphism .........................................
52
polynomial manifold ................................................
81
Index
259
polynomial structure of a group r o o t set
....................................
.............................................................
81 3
second Bieberbach theorem .........................................
14
Seifert c o n s t r u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
Seifert F i b e r Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
146
Seifert i n v a r i a n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
144
small manifold ....................................................
147
s t a b l e affme r e p r e s e n t a t i o n of a n i l p o t e n t g r o u p . . . . . . . . . . . . . . . . . . . . .
74
s t a b l e afflne r e p r e s e n t a t i o n of a v i r t u a l l y n i l p o t e n t g r o u p . . . . . . . . . . . .
75
standard cocycle ..................................................
106
standard section ...................................................
106
third Bieberbach theorem
..........................................
14
t o r s i o n free c e n t r a l series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2,58
t o r s i o n free f i l t r a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
t o r s i o n free p s e u d o c e n t r a l series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
uniform lattice ...................................................... unique automorphism extension property (UAEP) unique isomorphism extension property (UIEP)
...................
......................
8 9 9
u p p e r c e n t r a l series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
virtually P
1
..........................................................
virtually nilpotent group ............................................. wallpaper group ...................................................
1 121
Printing: Weihert-Druck GmbH, Darmstadt Binding: Theo Gansert Buchbinderei GmbH, Weinheim
