JOURNAL OF ALGEBRA
ARTICLE NO.
182, 721]737 Ž1996.
0197
Automorphisms and Cohomology of Discrete Groups
Alejandro AdemU
Department of Mathematics, Uni¨ ersity of Wisconsin, Madison, Wisconsin 53706
Communicated by Richard G. Swan
Received March 27, 1995
INTRODUCTION
Let G denote a discrete group of finite cohomological dimension.
Calculating the cohomology of such groups is notoriously difficult, involving complicated geometric information attached to the group. For example, the cohomology of torsion-free arithmetic groups involves delicate
questions about symmetric spaces and number theory. Perhaps the key
difficulty lies in that there is no practical method for building up the
cohomology of G from that of its subgroups Žsuch as one can do for finite
groups..
In this paper we outline a method for constructing non-trivial classes in
H U Ž G, Fp . Žwhere Fp denotes a field with p elements. based on the use of
finite automorphism groups of G. Given an explicit presentation for G,
AutŽ G . can often be approached, and in particular its finite subgroups are
sometimes accessible. Furthermore, it is an elementary observation that
many interesting classes of groups admit numerous finite symmetries. An
obvious but very important example is given by a normal torsion-free
subgroup G in an arithmetic group U; any finite subgroup K : U will act
on it via conjugation.
Given G : AutŽ G ., we can form the semidirect product G s G =w G.
Let H : G denote a finite subgroup mapping onto G under the natural
projection map. Then C G Ž H . s C G Ž H . l G : G and in the particular case
when H s Ž1, x .< x g G .4 , we have that
C G Ž H . s G G s g g G < g ? g s g for all g g G 4 .
U
Partially supported by an NSF Young Investigator Award. E-mail address: adem@math.
wisc.edu.
721
0021-8693r96 $18.00
Copyright Q 1996 by Academic Press, Inc.
All rights of reproduction in any form reserved.
722
ALEJANDRO ADEM
After recalling that there is a 1 y 1 correspondence between the G-conjugacy classes of such subgroups H : G and the non-abelian cohomology
H 1 Ž G, G . we prove the following
THEOREM 3.3.
If P : AutŽ G . is a finite p-group, then
dim F p H U Ž G, Fp . G
Ý
HgH
1Ž
dim F p H U C G Ž H . , Fp .
ž
/
P , G.
Here dim F p H U Ž , Fp . denotes the total dimension of the mod p cohomology. If M is a set, we denote its cardinality by aw M x; then as a
corollary we obtain that
dim F p H U Ž G, Fp . G max a H 1 Ž P , G . , dim F p H U Ž G P , Fp . .
½
5
The results above clearly indicate that each of the subgroups C G Ž H . will
contribute to the mod p cohomology of G, in particular the cohomology of
the fixed subgroup G P will produce cohomology for G, in what can be
thought of as a group-theoretic version of a classical result due to P.
Smith. This has interesting consequences, some of which we describe in
Section 3.
To illustrate this, we apply it to the case of a level q congruence
subgroup Ž q an odd prime. GnŽ q . : SL nŽZ. and P s ² A:, where A is the
involution defined by
Ai j
¡
¢
0,
s~y1,
1,
i/j
i s j, 1 F i F n y 1
i s j s n.
Given B g SL nŽZ., define an action via conjugation,
B̂i j s
½
yBi j
Bi j
i s n or j s n but not both
otherwise.
Then the classes in H 1 Ž P, GnŽ q .. are equal to
B g Gn Ž q . Bˆ s By1 4 r; ,
where B1 ; B2 if there exists C g G Ž q . with Cy1 B1Cˆ s B2 . For each class
w B x g H 1 Ž P, GnŽ q .. we have that the corresponding centralizer is the
subgroup
ˆ s DB 4 .
C Ž w B x . s D g Gn Ž q . BD
AUTOMORPHISMS AND COHOMOLOGY
723
Then our result indicates that
dim F 2 H U Ž Gn Ž q . , F 2 . G
Ý dim F
w Bx
2
H U Ž C w B x , F2 . .
In particular C Žw I x. ( Gny 1Ž q . from which we derive the rather interesting fact that
dim F 2 H U Ž Gn Ž q . , F 2 . G dim F 2 H U Ž Gny1 Ž q . , F 2 . .
Using an automorphism of order p, an odd prime, we show in Application 3.8 that if n s k Ž p y 1. q t, with 0 F t - p y 1, then
dim F p H U Ž Gn Ž q . , Fp . G 2 k?ŽŽ py3.r2. ? dim F p H U Ž Gt Ž q . , Fp . .
Our theorem seems to be a basic result for demonstrating the existence
of non-trivial cohomology for G. However, it can also be used conversely to
prove the finiteness of H 1 Ž P, G . and H U Ž C G Ž H ., Fp ..
In Section 5 we describe a formula for computing the number of
conjugacy classes of elements of finite order in a semidirect product. More
precisely, we prove
THEOREM 5.1. If G is a torsion-free discrete group and G : AutŽ G . is a
finite automorphism group, then G s G =w G has the following number of
conjugacy classes of elements of finite order:
Ý
a H 1 Ž ² g : , G . rCG Ž g . .
Žg.
conjugacy
classes in G
Here CG Ž g . s centralizer of g in G, which acts on H 1 Ž² g :, G . Žsee
Section 5 for details..
The proof of Theorem 3.3 is an application of Smith theory to a
construction of Serre for the group G. More precisely, we construct an
admissible G-complex X and analyze the G-action on XrG Žsee Section 1
for definitions.. The sets H 1 Ž K, G . arise naturally from fixed-point data, as
do the subgroups C G Ž H .. Although Theorem 5.1 is purely algebraic, it is
closely related to the calculation of K GU Ž XrG ., as described in wAx.
This work is motivated by the paper of Rohlfs and Schwermer wRSx,
where they use finite automorphisms to construct non-trivial cohomology
via intersection theory. Our main contribution is the introduction of
cohomological ideas which produce non-zero classes in a more general
context, without using products. From our point of view the ring structure
of the H U Ž C G Ž H .. and possible intersections are best expressed homologi-
724
ALEJANDRO ADEM
cally, as we do in Section 4. We think that the results in this paper may
help put their results in perspective, showing how specific information
about arithmetic groups is required in cohomology calculations. The subgroups C G Ž H . yield ‘‘special cycles’’ which are in fact simply arising from
BC G Ž H . : BG, and they contribute non-trivial cohomology without the
need of any intersection arguments.
The author is grateful to J. Schwermer, J. Robbin, and P. Kropholler for
useful comments.
1. PRELIMINARIES
In this section we provide the necessary algebraic and topological
background.
To begin we will assume that G is a discrete group of finite cohomological dimension and that G ¨ AutŽ G . is a finite automorphism group. We
w
will need the following algebraic concept.
DEFINITION 1.1. The non-abelian cohomology H 1 Ž G, G . is the set of
equivalence classes of functions u : G ª G satisfying u Ž g 1 g 2 . s
u Ž g 1 . w Ž g 1 .w u Ž g 2 .x for all g 1 , g 2 g G, where u is said to be equivalent to u X
if there exists g g G with
u Ž g . s gu X Ž g . w Ž g . gy1
for all g g G.
Note in particular the distinguished element 1 g H 1 Ž G, G ., corresponding to the trivial homomorphism.
Using G and G, we can construct the semidirect product G =w G,
consisting of the set of pairs Žg , g . g G = G with product
Ž g 1 , g1 . Ž g 2 , g2 . s Žg 1 w Ž g1 . w g 2 x , g1 g2 . .
By construction G 1 G =w G with quotient G, hence G =w G has finite
¨ irtual cohomological dimension Ži.e., it has a subgroup of finite index which
has finite cohomological dimension..
Next we recall a construction due to Serre Žsee wB, p. 190x for details..
THEOREM 1.2. If Q is a discrete group of finite ¨ .c.d., then there exists a
finite dimensional G-complex X with the following properties
Ž1. X H / B m H : G is finite.
Ž2. For all H : G finite, X H is contractible.
AUTOMORPHISMS AND COHOMOLOGY
725
From now on we denote by X the complex associated to G =w G. Note
that G acts freely on X, a contractible space and hence XrG , BG, the
classifying space of G. We recall a basic fixed-point formula due to K.
Brown wBx applied to this particular case: if K : G is any subgroup, then
@ X HrG l NG=G Ž H . ,
K
Ž XrG . s
w
Hg C
Ž 1.3.
where C is a set of representatives for the G-conjugacy classes of finite
subgroups in G =w G whose image in G is K. Note that by the defining
properties of X, we have
K
Ž XrG . ,
@ B Ž G l NG=G Ž H . . .
w
Hg C
Ž 1.4.
Finally, we recall a basic result from Smith theory Žsee wAP, p. 210x for
details.:
THEOREM 1.5. Let Y be a finite dimensional complex with an action of a
finite p-group P; then
dim Y P
dim Y
Ý
dim F p H Ž Y , Fp . G
i
is0
Ý
dim F p H i Ž Y P , Fp . .
is0
Note in particular that if H U Ž Y, Fp . has finite total dimension, then Y P
has a finite number of components, each of which has finite total dimension. The proof is based on using a central subgroup in P of order p and
applying induction on < P < to reduce it to the case of Zrp. In that case
equality occurs only under rather restrictive conditions.
2. SEMI-DIRECT PRODUCTS AND NON-ABELIAN
COHOMOLOGY
To simplify notation, let G s G =w G. We now state the main result in
this section, which is a basic fact Žsee wSx. which we include for completeness.
THEOREM 2.1. Let p : G ª G be the natural projection map. There is a
one-to-one correspondence between G-conjugacy classes of finite subgroups of
G mapping onto K : G and H 1 Ž K, G ..
Proof. Let K : G be a finite subgroup such that p Ž K . s K; note that
p maps K isomorphically onto K, as ker p s G is torsion-free. We can
describe K as
Ks
½ Žg
K
x ,
x . x g K , gxK g G
5
726
ALEJANDRO ADEM
for some function g K : K ª G. Note that
Ž gxK , x . ž gyK , y / s ž gxKw Ž x .
gyK , xy
/
and hence
gxKy s gxKw Ž x . gyK ,
which simply means that g K is a cocycle.
Hence we can define a function
¡K : G finite¦ cocycles
¥ª ½
KªG 5
¢p Ž Kwith
§
. sK
r :~
via r Ž K . s g K . Conversely, given any cocycle j : K ª G, we consider
K Ž j . s Ž jx , x . x g K 4 .
We claim that K Ž j . is a finite subgroup in G, isomorphic to K. Indeed, if
Ž j x , x ., Ž j y , y . g K Ž j ., then
y1
, xy1 .
Ž j x , x . s Ž w Ž xy1 . jy1
x
s Ž j xy1 , xy1 .
Žhere we use the fact that 1 s j 1 s j x xy1 s j x w Ž x .w j xy 1 x, from which
Ž .w
x
Ž y1 .w jy1
x s j xy 1 . and
jy1
x s w x j xy1 and w x
x
Ž j x , x . ? Ž j y , y . s Ž j x w Ž x . j y , xy .
s Ž j x y , xy .
as j is a cocycle. It remains to show that this association induces a
bijection modulo G-conjugacy and equivalence of cocycles, respectively.
Assume first that K, K 9 are G-conjugate, i.e., there exists Ž j , 1. g G
such that Ž j , 1. K Ž j , 1.y1 s K 9. Hence we have
y1
Ž j , 1 . Ž gxK , x . Ž j , 1 . s Ž gxK 9 , x .
for all Ž gxK , x . g K .
« Ž jgxKw Ž x . jy1 , x . s Ž gxK 9 , x .
X
« gxK s jgxKw Ž x . jy1
X
for all x g K .
This means precisely that g K s g K as elements in H 1 Ž K, G .. The converse can be proved similarly and our proof is complete.
AUTOMORPHISMS AND COHOMOLOGY
727
Remark. Under this correspondence, the trivial cocycle 1 g H 1 Ž K, G .
corresponds to the subgroup K s Ž1, x .< x g K 4 : G. This will be a distinguished class in our considerations.
Given K : G mapping onto K, we define its centralizer in G
CG Ž K . s Ž g , y . g G Ž g , y . m Ž g , y .
y1
s m ;m g K 4 .
Note that this simplifies to yield
y1
Ž g , y . Ž gxK , x . Ž g , y . s Ž gxK , x .
ž gw Ž y .
gxK w Ž yxyy1 . gy1 , yxyy1 s Ž gxK , x .
/
gw Ž y . gxK w Ž x . gy1 s gxK ,
yxyy1 s x
for all Žgxk , x . g K. Note the special case when y s 1, i.e., Žg , y . g C G Ž K .
l G s C G Ž K ., then the second condition is superfluous and we have
ggxKw Ž x . gy1 s gxK .
For later use we note that if NG Ž K . denotes the normalizer of K in G,
then NG Ž K . l G s C G Ž K .; indeed if ŽggxKw Ž x .w gy1 x, x . g K, then necessarily we recover the identity above.
Note that if j K s 1, then
CG Ž K . s g g G < w Ž x . w g x s g ; x g K 4 ,
the subgroup of ‘‘fixed points’’ in G under the action of K. We will denote
this group by G K .
3. FIXED POINTS, CENTRALIZERS, AND COHOMOLOGY
Let X be the finite dimensional G-complex described in Section 1.
Combining Ž1.4. with the observation at the end of Section 2 and Theorem
2.1, we obtain
PROPOSITION 3.1.
Let K : G be any subgroup, then
K
Ž XrG . ,
@
KgH 1Ž K , G .
BC G Ž K . .
728
ALEJANDRO ADEM
Note that we have established a 1 y 1 correspondence
p 0 Ž Ž XrG .
K
. l H 1Ž K , G. .
This result appears in a very special context in wRSx. The explanation for
this is that in fact there is an underlying action of a semidirect product on
the symmetric spaces which they consider.
In addition we have
COROLLARY 3.2. If XrG is compact, then for e¨ ery G : AutŽ G . finite,
H 1 Ž G, G . is a finite set and BC G Ž G . admits a compact model for all
G g H 1 Ž G, G ..
We can now prove one of our main results. For a space X, let
dim F p H U Ž X , Fp . s
dim X
Ý
dim F p H i Ž X , Fp . .
is0
THEOREM 3.3. Let G be a discrete group of finite cohomological dimension and P ; AutŽ G . a finite p-group. Then we ha¨ e that
dim F p H U Ž G, Fp . G
dim F p H U C G Ž H . , Fp ,
ž
Ý
/
HgH 1Ž P , G .
and in particular
dim F p H U Ž G, Fp . G max aH 1 Ž P , G . , dim F p H U Ž G P , Fp . .
½
5
Proof. The proof is a straightforward application of Theorem 1.5 and
Proposition 3.1.
To make this meaningful, we assume from now on that H U Ž G, Fp . is
totally finite. This will imply that H 1 Ž P, G . is finite for all P : AutŽ G . and
that each C G Ž H . is homologically finite mod p. Of course the right hand
side of Theorem 3.3 achieves its maximum possible value Žas we vary the
automorphism group. when < P < s p. We now give a few applications.
Application 3.4. G s F Ž n., free group on n generators. In this case we
obtain that for any P : AutŽ G .,
nq1G
Ý
HgH 1Ž P , G .
Ž rank C G Ž H . q 1 .
AUTOMORPHISMS AND COHOMOLOGY
729
and in particular
n q 1 G max aH 1 Ž P , G . , rank G P q 1 4 .
Application 3.5. Suppose that G is a discrete group with the mod p
homology of a point. Then H 1 Ž P, G . s 14 , and G P is also mod p homologous to a point. There are examples of groups G Žof finite c.d.. satisfying
such a condition. Perhaps the simplest one is Higman’s group, for which a
presentation can be given as
2
G s² x 0 , x 1 , x 2 , x 2 < x iy1 x i xy1
iy1 s x i , i s 0, 1, 2, 3 mod Ž 4 .: .
This group evidently has an automorphism of order 4 Žsimply rotating
the generators. and we deduce that its fixed point group must be mod 2
acyclic.
Application 3.6. Suppose that G has the same mod p cohomology as a
sphere. Then, for any P : AutŽ G ., we have either H 1 Ž P, G . s 14 and G P
has the mod p homology of a sphere, or aw H 1 Ž P, G .x s 2 and for both
classes w H x g H 1 Ž P, G . we have that C G Ž H . is mod p acyclic. Examples of
such groups G can be easily provided. Let r, s, t be positive, pairwise
relatively prime integers satisfying 1rr q 1rs q 1rt - 1. Denote by T the
group generated by elements g 1 , g 2 , g 3 subject to the relations
g 1r s g 2s s g 3t s g 1 g 2 g 3 .
Then if G s w T, T x : T, it is a well-known fact wMix that G has the integral
homology of a 3-sphere and in fact has cohomological dimension three.
Application 3.7. Let GnŽ p . : SL nŽZ. denote the level p congruence
subgroup, for p an odd prime. It is known that the GnŽ p . have finite
cohomological dimension. Now if A g GL nŽZ. is a finite subgroup then it
acts on GnŽ p . via conjugation.
Let
y1
As
y1
..
.
y1
1
0
,
an element of order two which clearly acts non-trivially on GnŽ p .. It is
730
ALEJANDRO ADEM
direct to verify that GnŽ p . ² A: ( Gny1Ž p . and so Theorem 3.3 yields
dim F 2 H U Ž Gn Ž p . , F 2 . G dim F 2 H U Ž Gny1 Ž p . , F 2 .
for all n G 2.
One should note however that H 1 Ž² A:, GnŽ p .. may have more than one
element. The cocycles can be described as u : ² A: ª GnŽ p . such that
Au Ž A . A s u Ž A .
y1
or more precisely if for B g GL nŽZ. we denote
B̂i j s
½
yBi j ,
Bi j ,
i s n or j s n but not both
otherwise,
then
Z 1 Ž ² A: , Gn Ž p . . s B g Gn Ž p . Bˆ s By1 4 .
Two elements B1 , B2 are equivalent if there exists C g GnŽ p . with
Cy1 B1Cˆ s B2 .
The cocycle B corresponds to the subgroup Ž B, A., Ž1, 1.4 : G, with
ˆ s DB 4 .
C G Ž B, A . s D g Gn Ž p . BD
It should be possible to use the above to obtain explicit numerical lower
bounds on dim H U Ž GnŽ p ., F 2 ..
Application 3.8. Consider G g GL py 1ŽZ. an element of order p. Then,
for p odd G 5,
CGL Ž T . ( Z ŽŽ py3.r2. [ Zr2 [ Zrp.
Let G s Gpy 1Ž q ., q any odd prime. Then, from the extension
p
1 ª G ª GL py 1 Ž Z . ª GL py1 Ž Fq . ª 1
and the action of H s ²T : by conjugation, we infer that there is an exact
sequence
1 ª G H ª Z ŽŽ py3.r2. [ Zrp [ Zr2 ª CG Ž p Ž H . . ,
731
AUTOMORPHISMS AND COHOMOLOGY
where G s GL py 1ŽFq .. We deduce that
G H ( Z ŽŽ py3.r2.
and hence that, for all odd q,
dim F p H U Ž Gpy1 Ž q . , Fp . G 2 ŽŽ py3.r2. .
Now let n be any integer larger than p y 1, and write n s k Ž p y 1. q t,
where 0 F t - p y 1. Consider the n = n block matrix S with T in the
upper left hand corner, and then n y p q 1 = n y p q 1 identity matrix
in the bottom right. It is not hard to see that the centralizer of this matrix
in GL nŽZ. is precisely the diagonal product CGL py 1 ŽZ.ŽT . = GL nypq1ŽZ..
Using the previous result we can deduce that
dim F p H U Ž Gn Ž q . , F 2 . G 2 k?ŽŽ py3.r2. ? dim F p H U Ž Gt Ž q . , Fp . .
Given the cohomological applications in this section, it is quite natural
to expect some consequences from the well-known localization methods
Žsee wAPx.. For example, we have:
THEOREM 3.9.
Let P s ŽZrp . r ¨ AutŽ G ., where G is a discrete group of
w
finite cohomological dimension. Then there exists a class C g H U Ž P, Fp . such
that the localized map induced by inclusion
Cy1 H U Ž G =T P , Fp . ª Cy1
[
HgH 1Ž P , G .
H U C G Ž H . , Fp
ž
/
m H U Ž P , Fp .
is an isomorphism. In particular if H 1 Ž P, G . s 14 , the inclusion G P ¨ G
induces an isomorphism H U Ž G =w P, Fp . ª H U Ž G P = P, Fp . after localizing.
It is evident that in many interesting situations this formula can be used
to relate the structure of H *Ž G, Fp . to that of the H *Ž C G Ž H ., Fp . Žand in
particular H *Ž G P, Fp ... For example, if BG is a Poincare
´ Duality space, it
follows from Proposition 4.1 that if H *Ž G, Fp . is totally non-homologous to
zero in H *Ž G =T P, Fp ., then each BC G Ž H . will be a Poincare
´ Duality
space. One may conjecture more generally that G P must be a Poincaré
Duality group if G is.
More delicate comparisons between the cohomology of G and that of
G P Žas well as the other components. are possible in several cases, in
addition one can consider the action of the Steenrod algebra.
732
ALEJANDRO ADEM
4. INTERSECTIONS
In this section we will summarize how our methods can be used together
with intersection theory to produce non-trivial classes in the cohomology
of a discrete group. This is based on and motivated by the work of Rohlfs
and Schwermer wRSx. We present a simplified account of this in a purely
topological setting, which has the advantage of wider applicability, although the arithmetic case is of preponderant interest.
To begin we need a little background in homology of manifolds. Let us
assume that Y N is a connected N-manifold, X m 1 ¨ Y N, Z m 2 ¨ Y N compact submanifolds such that B / X l Z is a compact L-manifold, and
m1 q m 2 s N. Assume in addition that they are all R-oriented Ž R a
coefficient field, which we now suppress.. We have Gysin]Thom isomorphisms Žsee wSpx.
u X : H U Ž X . ª H m 2q) Ž Y , Y y X .
u Z : H U Ž Z . ª H m 1q) Ž Y , Y y Z . .
In particular we have distinguished classes u X Ž1. g H m 2 Ž Y, Y y X .,
u Z Ž1. g H m 1 Ž Y, Y y Z . and their product
u X Ž 1. j u Z Ž 1. g H N Ž Y , Y y X j Y y Z . s H N Ž Y , Y y X l Z . .
LŽ
Ž Ž .
Ž ..
Hence uy1
X l Z .. Let Ž X l Z . i denote a
X l Z u X 1 j uZ 1 g H
component in X l Z with fundamental class m i g HLŽŽ X l Z . i .. Hence
Ž Ž .
Ž ..w m i x g R; then we
we can evaluate, to obtain r i s uy1
X l Z u X 1 j uZ 1
have that the inclusions Ž Y, B. ª Ž Y, Y y X . and Ž Y, B. ª Ž Y, Y y Z .
induce homomorphisms
U
j X : H U Ž Y , Y y X . ª Hcomp
ŽY .,
U
j Y : H U Ž Y , Y y Z . ª Hcomp
ŽY .,
and that from the commutativity of
j Xmj Z
6
HU Ž Y , Y y X . m HU Ž Y , Y y Z .
U
U
Hcomp
Ž Y . m Hcomp
ŽY .
j
6
U
Hcomp
ŽY .
6
6
HU Ž Y , Y y X l Z .
j
we can conclude that
Ž j X u X Ž 1 . j jZ u Z Ž 1 . . w m Y x s Ý r i .
i
AUTOMORPHISMS AND COHOMOLOGY
733
We denote this number by r Ž X, Z .; this can evidently be applied to
show
PROPOSITION 4.1. Under the abo¨ e conditions, if r Ž X, Z . / 0 then there
m2 Ž .
m1 Ž .
exist classes ¨ X g Hcomp
Y , ¨ Z g Hcomp
Y such that ¨ X j ¨ Z / 0.
Remarks. Ž1. Note that by Poincare
´ Duality we obtain non-trivial dual
classes in Hm 1Ž Y ., Hm 2Ž Y ..
Ž2. In case X, Z intersect transversally, the invariant above is simply
the elementary intersection number X ? Z.
Ž3. More generally assume Y is a C` -manifold, X, Z closed immersed
submanifolds. Then the intersection is said to be clean if the components
of X l Z are immersed submanifolds of V and if for all such components
W of X l Z one has TW s TX < W l TZ < W . In this case the number above
can be interpreted as an intersection number X ? Z, which in turn can be
computed from the Euler number of the excess bundle N of the intersection.
We now specialize to our type of situation. We assume G s H = K is a
finite automorphism group of G, construct X as before, and consider the
intersection diagram
6
XrG H
6
XrG G
6
XrG
6
XrG K
assuming the relevant additional hypotheses. First we observe that if
H0 , K 0 correspond to the trivial elements in H 1 Ž H, G ., H 1 Ž K, G ., respectively, then
X H 0rC G Ž H0 . l X K 0rC G Ž K 0 . s
@ X GrC G Ž G . ,
GgL
where L : H 1 Ž G, G . is defined as ker res H = res K , and
res H : H 1 Ž G, G . ª H 1 Ž H , G .
res K : H 1 Ž G, G . ª H 1 Ž K , G .
are the restriction maps. This condition simply arises from the fact that G
must restrict trivially on each factor Žgiven our choice of H0 and K 0 ..
Using a subscript to denote the distinguished components corresponding
734
ALEJANDRO ADEM
to 1, we have that
H
K
Ž XrG . 0 l Ž XrG . 0 ,
@ BC G Ž G . .
GgL
It is now possible to use Proposition 4.1 to produce non-trivial classes in
H U Ž G, R ., which will be arising from the subgroups C G Ž H ., C G Ž K .. Specifically, we have that if
r Ž X H 0rC G Ž H0 . , X K 0rC G Ž K 0 . . / 0
Žan intersection of classifying spaces. then there exist classes x H , x K g
H U Ž G, R . such that
res GC G Ž H . x H s X H 0rC G Ž H0 .
res GC G Ž K . x K s X K 0rC G Ž K 0 .
U
U
/0
/ 0.
In certain situations the number r Ž , . can be computed in terms of
intrinsic information associated to the group G. In wRSx the important case
of an invariant arithmetic subgroup in an algebraic group having two
commuting automorphisms of finite order is discussed. The associated
symmetric space and its quotient will inherit an action of the finite group
they generate, and the respective fixed-point sets are called ‘‘special
cycles.’’ Understanding their intersection is a critical element in their
method for producing non-trivial classes in H U Ž G, C.. In fact they obtain
an impressive general formula for r in purely arithmetic terms, using
Euler characteristics. The key technical point is that the ‘‘clean intersection formula’’ can be applied under certain rather general assumptions.
This important result illustrates how the cohomology of certain discrete
groups arises from underlying arithmetic data. However, from our point of
view it is simply an example of how the presence of symmetries on a
topological space forces the existence of non-trivial cohomology. This is of
course one of the guiding principles in fixed-point theory.
By using a cohomological approach we are able to extend this method to
a much broader context. In fact the specific geometric conditions can be
sufficiently weakened so that we can see how the group theoretic nature of
G produces non-trivial cohomology}the new consideration here is the
nature of its finite automorphisms. One would expect our more general
format to have additional specific geometric applications to other classes
of groups, and that other more sophisticated methods from equivariant
topology can be usefully applied in analyzing the cohomology of discrete
groups.
735
AUTOMORPHISMS AND COHOMOLOGY
5. CONJUGACY CLASSES IN A SEMI-DIRECT PRODUCT
In this section we will apply our methods to study a purely algebraic
problem, namely, how many conjugacy classes of elements of finite order
are there in G s G =w G? This has some relevance to the calculation of
the complex K-theory of the classifying space of this group Žsee wAx., but
we will not elaborate on this here.
Let g g G, and denote its centralizer in G by C Ž g .. Consider an
element u g Z 1 Ž² g :, G .; if h g C Ž g ., define hŽ u .Ž x . s w Ž h.w u Ž x .x. Then
h Ž u . Ž x1 x 2 . s w Ž h . u Ž x1 x 2 . s w Ž h . u Ž x1 . w Ž x1 . u Ž x 2 .
s w Ž h . u Ž x 1 . w Ž hx 1 . u Ž x 2 .
s w Ž h . u Ž x1 . w Ž x1 . w Ž h . u Ž x 2 .
s h Ž u . Ž x1 . w Ž x1 . h Ž u . Ž x 2 . .
This means that hŽ u . is also a cocycle, and this clearly induces a C Ž g .action on the set H 1 Ž² g :, G .. We can now state
THEOREM 5.1.
Let G denote a torsion-free discrete group and G ¨ AutŽ G .
w
a finite automorphism group. Then the number of conjugacy classes of
elements of finite order in G =w G is precisely equal to
Ýa
H 1 Ž ² g : , G . rC Ž g . ,
Žg.
where the sum ranges o¨ er all conjugacy classes of elements in G.
Proof. Consider Žg , g . g G s G =w G of finite order. If p : G ª G is
the natural projection then p Ž²Žg , g .:. s ² g :. Also ²Žg , g .: is a finite
subgroup mapping isomorphically onto ² g : : G. If TorsŽ G . is the set of all
elements of finite order in G, define
u : Tors Ž G . ª
@ g 4 = H 1 Ž ² g :, G .
ggG
by u ŽŽ g , g .. s Ž g, w²Ž g , g .:x. g g 4 = H 1 Ž² g :, G .. If ²Ž g 1 , g 1 .: ;
G
²Žg 2 , g 2 .:, then g 1 s g 2 , and we have that w²Žg 1 , g 1 .:x s w²Žg 2 , g 2 .:x in
H 1 Ž² g :, G .. Furthermore, if u Žg 1 , g 1 . s u Žg 2 , g 2 ., then g 1 s g 2 and there
exists an n ) 0 such that Žg 1 , g 1 . is G conjugate to Žg 2 , g 2 . n, but then
736
ALEJANDRO ADEM
g 1 s g 2n s g 2 , and so Žg 1 , g 1 . ; Žg 2 , g 2 .. It is also clear that u is onto,
hence it establishes a bijection
G
u
6
Tors Ž G . rG
@ g 4 = H 1 Ž ² g :, G . .
ggG
Assume now that j : ² g : ª G is a cocycle. We define h j : ² hghy1 : ª G
by h j Ž hxhy1 . s w Ž h.w j Ž x .x, this will again be a cocycle and in fact h
induces a bijection H 1 Ž² g :, G . ª H 1 Ž² hghy1 :, G .. Hence Ž g, j . ¬
Ž hghy1 , h j . defines a G-action on the set @ g g G g 4 = H 1 Ž² g :, G .. Using
the natural G-action on TorsŽ G .rG induced by conjugation, it is direct to
verify that u is G-equivariant, as is its inverse. We deduce that there is a
bijection
@ g 4 = H 1 Ž ² g :, G .
Tors Ž G . rG (
G
ggG
(
@ g 4 = H 1 Ž ² g : , G . rC Ž g . .
Žg.
From this we infer that the number of conjugacy classes of elements of
finite order in G is precisely
Ýa
H 1 Ž ² g : , G . rC Ž g . .
Žg.
The following is a simple application of this formula.
EXAMPLE 5.2. Let A be a free abelian group, and G : GLŽ A. a finite
subgroup. Then for G s A =w G the number of conjugacy classes of
elements of finite order can be computed from the standard cohomology
invariants H 1 Ž² g :, A. CŽ g .. For example, let A s Z py1, with an action of
Zrp represented by the Ž p y 1. = Ž p y 1. matrix
0
1
0
Ms ?
?
?
0
0
0
1
0
?
?
0
???
0
0
?
?
?
0
1
y1
y1
?
?
?
?
y1
0
Note that this matrix could be taken as a representative for the element
T discussed in Application 3.8. In this case the number of conjugacy
AUTOMORPHISMS AND COHOMOLOGY
737
classes is precisely
py1
Ý
aH 1 Ž Zrp, A . q 1 s p 2 y p q 1.
1
More generally, it is well known wCRx that any integral representation of L
of Zrp decomposes as
s
L ( Zr [
t
[P [ [A ,
1
i
1
i
where
Pi m Z p ( Z p w Zrp x ,
Ai m Z p ( A m Z p
Ž A as above.. Furthermore H 1 ŽZrp, L. s ŽZrp . t. Hence we obtain that G
has exactly Ž p y 1. p t q 1 conjugacy classes of elements of finite order.
REFERENCES
wAx
wAPx
wBx
wCRx
wMix
wRSx
wSx
wSpx
A. Adem, On the K-theory of the classifying space of a discrete group, Math. Ann. 292
Ž1992., 319]327.
C. Allday and V. Puppe, ‘‘Cohomological Methods in Transformation Groups,’’
Cambridge Univ. Press, Cambridge, UK, 1993.
K. Brown, ‘‘Cohomology of Groups,’’ Graduate Texts in Math., Vol. 87, SpringerVerlag, New YorkrBerlin, 1982.
C. Curtis and I. Reiner, ‘‘Representation Theory of Finite Groups and Associative
Algebras,’’ Interscience, New York, 1962.
J. Milnor, On the 3-dimensional Brieskorn Manifolds M Ž p, q, r ., in ‘‘ Knots, Groups
and 3-Manifolds,’’ pp. 175]225, Ann. of Math. Stud., Vol. 84, Princeton Univ. Press,
Princeton, NJ, 1975.
J. Rohlfs and J. Schwermer, Intersection numbers of special cycles, J. Amer. Math.
Soc. 6, No. 3 Ž1993..
J.-P. Serre, ‘‘Galois Cohomology,’’ 5th ed., Lecture Notes in Math., Vol. 5, SpringerVerlag, New YorkrBerlin, 1994.
E. Spanier, ‘‘Algebraic Topology,’’ Springer-Verlag, New YorkrBerlin, 1989.