1
ON THE KTHEORY OF THE CLASSIFYING SPACE
OF A DISCRETE GROUP
Alejandro Adem*
Mathematics Department
University of Wisconsin
Madison, WI 53706
Dedicated to the memory of Jose Adem (19211991).
x0. INTRODUCTION
Let G be a finite group. By considering the action of G on itself by conju*
*gation,
one arrives at the class equation for G,
X 1
1  _1_G= ______ (0:*
*1)
(g)g6=1C(g)
where g ranges over conjugacy classes of elements in G, with centralizer C(g).
Now assume that is a discrete group of finite virtual cohomological dimen*
*sion
with centralizers of elements of finite order in homologically finite (for exa*
*mple
SLn(Z)). Brown [B2] proved the generalization of (0.1) to :
X
"O()  O() = O(C(fl)) (0:*
*2)
(fl)fl6=1
where fl ranges over conjugacy classes of elements of finite order in , "Odeno*
*tes
the topological Euler characteristic and O the usual grouptheoretic version (i*
*f 0 is
0)
a torsionfree subgroup of finite index in then by definition O() = O"(__[:0])*
*. The
___________________________
* Partially supported by an NSF grant.
2
point of (0.2) is that the difference between "O() and O() is determined by the
torsion in , in particular if is torsionfree they coincide.
Now let us recall a wellknown result in algebra: let R(G) denote the char*
*acter
ring of a finite group G. Then
No. of conjugacy
dim CR(G) C  1 = classes of nontrivial (0:*
*3)
elements in G :
The term on the left can be identified with dim CK*G(*) C, the dimension of the
equivariant Ktheory of a point. In this paper we will use this interpretation*
* to
outline an approach for generalizing this formula to infinite groups of finite*
* virtual
cohomological dimension (with suitable finiteness conditions), motivated by (0.*
*2).
By a result due to Serre [S], will act on a finite dimensional, contracti*
*ble
space X, with finite isotropy, and such that XH ' * for all H finite. Fix an
extension 1 ! 0 ! !ssG ! 1 with 0 torsionfree, G finite. Then 0 acts freely
on X, G acts on X=0 such that the finite subgroups in correspond under ss to
the isotropy subgroups in G.
We have
Theorem 3.1
M
K*G(X=0) C ~=K*(B0)G C K*(B(C(fl) \ 0))Hfl C
fl2C{1}
where C is the set of conjugacy classes of elements of finite order in , and Hf*
*l=
C(fl)=C(fl) \ 0, a finite group. *
* 
An immediate corollary of this is the formula
X
O(K*G(X=0) C)  "O() = "O(C(fl)) : (0:*
*4)
fl2C{1}
where O(K*G(X=0) C) = dim CK0G(X=0) C  dimC K1G(X=0) C.
3
The ingredient which makes (0.3) useful is the fact that R(G)^ ~= K*(BG) (*
*the
completion theorem). Let act on X x EG diagonally and through !ssG; then
B ' X=0xG EG, whence the completion theorem implies K*G(X=0)^ ~= K*(B),
and so, up to IGadic completion, the preceding results provide information on
K*(B). For example, if G is a pgroup completion is padic completion, and hence
(calculating over the padics Cp),
X
O(K*(B) Cp)  "O() = "O(C(fl)) : (0:*
*5)
fl2C{1}
We point out that will always contain subgroups of this form (G a pgroup) with
finite index.
The key technical device which we use is a result of N. Kuhn [K] expressing
K*G(Y ) C as a sum of K*(Y =C(g)) C, as g ranges over conjugacy classes of
elements in G.
The paper is organized as follows: in x1 we describe the complex X; in x2*
* we
outline the main properties of equivariant Ktheory, in x3 we prove our result *
*and
in x4 we provide examples, as well as a plocal version of the results in x3: u*
*sing
padic Ktheory we obtain an exact formula (4.2).
Formula (3.1) indicates that the Ktheory of B can be calculated given eno*
*ugh
information about its elements of finite order and their centralizers. It is in*
*teresting
to compare this with results obtained by Brown (see [B3]) for computing the (hi*
*gh
dimensional) cohomology of B. In the general situation there is a spectral sequ*
*ence
involving the cohomology of the normalizers of the finite subgroups, converging*
* to
the cohomology of in sufficiently high dimensions (or in any dimension if Farr*
*ell
Cohomology is used). This can be difficult to deal with, except in the rank one
situation, where the spectral sequence only has one line. In contrast, K*(B) C
seems to be much more accessible in terms of subgroup data, a fact which is of *
*course
evident for finite groups. One may expect that by using the AtiyahHirzebruch
4
spectral sequence or other techniques, this can (in some cases) yield informati*
*on on
the cohomology.
x1. GROUPS OF FINITE VCD
Definition 1.1: A discrete group is said to have finite virtual cohomological
dimension (vcd < 1) if there is a subgroup of 0 of finite index such that 0
has finite cohomological dimension. *
* 
Examples of this type of group include arithmetic groups (such as SLn(Z)) *
*and
mapping class groups.
The key geometric ingredient in the analysis of these groups is a result d*
*ue to
Serre [S]:
Theorem 1.2 (Serre)
If is a group with vcd < 1, then there exists a finite dimensional, proper
contractible complex X with the following additional property: XH is contract*
*ible
for all finite subgroups H . 
This can be easily summarized as follows. Fix 0 a normal subgroup of
finite cohomological dimension. Then by the wellknown result of Eilenberg and
Ganea [EG] we can find a finite dimensional K(0; 1) complex, whose universal
cover X0 is a contractible 0complex. If r = [ : 0], then acts faithfully as a
group of automorphisms of the principal 0bundle ! =0 and hence embeds in
the full automorphism group r xT (0)r. As the latter group acts on (X0)r, we
obtain a action on X = (X0)r. One checks that this satisfies all the condition*
*s.
In particular [B] if H is finite, XH ~=(X0)k, where k = [ : 0]=H, hence it*
* is
contractible.
In a more abstract language, X can be regarded as an "E (0)space." We
recall what that means. Given H / G, let FG (H) be the family of all subgroups
K G such that K \ H = {e}. Then a Gspace X is said to be an EG (H)space if
5
(1) XK = ; for all K 62 FG (H)
(2) XK is contractible for all K 2 FG (H).
The orbit space EG (H)=H is called a BG H space, and as a G=Hspace is unique
up to weak G=Hequivalence.
In our situation F (0) consists of all finite subgroups in , whence Serre's
construction is a particular finite dimensional model for the generalized class*
*ifying
space B 0; (see [tD] for more on this).
Let us now fix an extension
1 ! 0! ! G ! 1
where cd0 < 1, G < 1. We will now make a detailed examination of the fixed
point sets X, where fl 2 is an element of finite order, the subgroup *
*it
generates, and X a complex as in 1.2.
First we have
Lemma 1.3: Let g 2 G, then
a a
(X=0)= XH =0\ N(H) ' B(0\ N(H))
H2C H2C
where C ranges over all 0conjugacy classes of finite subgroups H mapping
onto .
Proof: This is explicitly described in [B], pg. 267, and follows from looking *
*at in
verse images under ss : ! G and the additional fact that XH is a free, contra*
*ctible
0\ N(H)space. 
Now consider the C(g)action on this fixedpoint set. We describe its orb*
*it
space as follows
Lemma 1.4:
a a
(X=0) =C(g) = X=C(fl)
(g) (fl)
6
where the elements range over conjugacy classes of elements of finite order in *
*G and
respectively.
`
Proof: Let Cg(g) = ss1 (C(g)), then this group acts on fl2C"Xfl, where C"is *
*the
collection of all elements of finite order in mapping onto g. If x 2 Xfl, c 2 *
*Cg(g),
take cx; clearly cx 2 Xcflc1, and ss(cflc1 ) = g.
We now take the orbit space
0 1
a ffi a
@ XflA gC(g)= Xfl=Cg(g)\ C(fl)
fl2C" fl2C
where now
Cg(g)conjugacy classes of
C =
fl 2 of finite order, ss(fl) =:g
Clearly C(fl) Cg(g), hence Cg(g)\ C(fl) = C(fl).
Next we claim that if fl 2 of finite order, ss(fl) = g, and if fl0 = fl1f*
*lfl11 also
satisfying ss(fl0) = g, then fl1 2 Cg(g). For we have
ss(fl1)gss(fl1)1 = g; hence ss(fl1) 2 C(g)
and so fl1 2 Cg(g).
As elements of finite order in are mapped injectively under ss, as we ran*
*ge
over conjugacy classes of elements of G, we obtain the asserted disjoint union.*
* 
We have the following cohomological corollary of 1.4.
Corollary 1.5:
0 1
a M
H* @ (X=0)=C(g); QA ~= H*(BC(fl); Q) :
(g) (fl)
finite order
Proof: We have extensions
1 ! 0\ C(fl) ! C(fl) ! Hfl! 1
7
where Hflis finite. Hence we obtain
H*(BC(fl); Q) ~=H*(B0\ C(fl); Q)Hfl~= H*(X=C(fl); Q);
the last equality because X is a contractible C(fl)space with a free C(fl)*
* \ 0
action. *
* 
x2. EQUIVARIANT KTHEORY
We recall some wellknown facts about Gequivariant Ktheory, for G a fini*
*te
group. Our main reference is [AS1]. In this section we assume Y is a compact
Gspace.
K*G(Y ) is defined by using Gvector bundles over Y . Using the natural m*
*ap
Y ! *, we have a homomorphism
K*G(*) ! K*G(Y )
with which K*G(Y ) is a (we will assume finitely generated) K*G(*)module. Reca*
*ll
that K*G(*) = R(G), the complex character ring of the finite group G. Denote by
YG = (Y x EG)=G the usual Borel construction. If F is a Gvector bundle on Y ,
then (F x EG)=G is a vector bundle on YG ; the assignment F 7! (F x EG)=G is
additive hence it induces a homomorphism
ff : K*G(Y ) ! K*(Y xG EG) :
Here the term on the right is defined as
limK*G(Y x EG(n))
for a suitable filtration of EG, and the map above can be alternatively constru*
*cted
as follows: the natural projection Y x EGn ! Y induces
K*G(Y )ffn!K*(Y xG EGn) :
8
In particular, if IGn denotes the kernel of R(G) = K*G(*) ! K*G(EGn), then ffn
factors
K*G(Y )=IGnK*G(Y )ffn!K*(Y xG EG) :
The main result due to Atiyah and Segal concerning this map is
Theorem 2.1 (Completion Theorem)
Let Y be a compact Gspace such that K*G(Y ) is finite over R(G). Then the
homomorphisms
ffn : K*G(Y )=IGn ! K*G(Y x EGn)
induce an isomorphism of prorings. *
* 
In particular if IG R(G) is the augmentation ideal, and K*G(Y ) is endowed
with the IGadic topology, then 2.1 can be rephrased as saying that
ff : K*G(Y ) ! K*(Y xG EG)
induces an isomorphism of the IGadic completion of K*G(Y ) with K*(Y xG EG).
The principle then is that K*G(Y ) can be approximated by K*(Y xG EG).
To conclude this section we describe a more recent result due to N. Kuhn [*
*K]
relating K*G(Y ) to the Ktheory of Y =C(g). Namely, he proves that there i*
*s an
isomorphism
M
K*G(Y ) C ~= K*(Y =C(g)) C (2:*
*2)
(g)
where the sum is taken over all conjugacy classes of g 2 G (note the case Y = *
**,
we recover the formula for dim CR(G) C). This correspondence can be outlined
fi
as follows. Let F be a Gvector bundle on Y ; then on F fiY the element g s*
*till
fi
acts, leaving points in the base fixed. Therefore F fiY will split as a dire*
*ct sum of
vector bundles corresponding to the eigenspaces of g; putting the eigenvalue in*
* the
second factor gives an element in K(Y ) C. C(g) acts on Y , hence we *
*can
9
take invariants to obtain an element in K(Y )C(g) C = K(Y =C(g)) C;
the same holds for K1G(Y ). This description is also given in [HH] , [AS2], w*
*here it
is pointed out that this is related to work in string theory concerning orbifol*
*d Euler
characteristics.
x3. APPLICATION TO K*(B)
The goal of this section will be to use the preceding results to obtain an*
* approx
imation to K*(B), for a discrete group of finite v.c.d., with suitable finiten*
*ess
assumptions.
As before, fix an extension
1 ! 0! ! G ! 1
where cd 0 < 1, G < 1. Let X be an admissible complex as before, with
suitable (e.g. compactness) finiteness assumptions on X=0. Then, using ! G,
acts diagonally on X x EG without any nontrivial isotropy. As X x EG ~ *, this
means
B ' (X x EG)= = X=0xG EG
and hence
K*(B) ~=K*(X=0xG EG) :
It is now evident that the completion theorem provides a method for approa*
*ch
ing K*(B), namely via the map
K*G(X=0)ff!K*(B)
which will induce an isomorphism K*G(X=0)^ ~= K*(B).
We have
10
Theorem 3.1:
M
K*G(X=0) C ~= K*(B(C(fl) \ 0))Hfl C
(fl)
where fl ranges over conjugacy classes of elements of finite order in , and Hfl=
C(fl)=C(fl) \ 0, a finite group.
Proof: Simply combine (2.2) with (1.4) and identify
K*(X=C(fl)) C = K*([X=C(fl) \ 0]=Hfl) C
= K*(B(C(fl) \ 0))Hfl C :

From this we derive (if each C(fl) is homologically finite):
Corollary 3.2.
X
O(K*G(X=0)) = O"(C(fl)) :
(fl)
finite order
Proof: Use that fact that the Euler characteristic of K*(Y ) equals O(Y ). *
* 
What we have is an expression for K*G(X=0) involving only centralizers of
elements of finite order in , which after IGadic completion determines K*(B).
In some cases this completion process can be straightforward, in particular if *
*G is
a pgroup, it is just padic completion. Computing ranks over Cp leads to
Theorem 3.3: Let be a discrete group of finite v.c.d. such that (a) the cen
tralizers C(fl) are of finite homological type for all fl 2 of finite order an*
*d (b)
there is a torsionfree subgroup 0, normal in and such that =0 = G is a finite
pgroup. Then
X
O(K*(B) Cp)  "O() = O"(C(fl)) :
(fl)
fl of finite
order 6= 1
11

Remark: An elementary case of the above results occurs when is torsionfree;
then X= ' B and hence O(K*(B)) = "O(); on the righthand side it vanishes
because there is no torsion.
Recall now the usual Euler characteristic for discrete groups:
O() = "O(0)=[ : 0]
(we assume it is welldefined). Then the results above should be compared with a
theorem due to K. Brown [B2]:
X
"O()  O() = O(C(fl)) :
(fl)6=(1)
fl finite order
Brown's result can be considered as the generalization of the "class equat*
*ion"
from finite group theory. Our result is an extension of the formula for dimC R(*
*G)C.
x4. EXAMPLES AND A LOCAL VERSION
Assume we have a group fitting into an extension 1 ! 0 ! ! G ! 1 as
described before. Then will contain a subgroup (p)of finite index normalizing 0
and with (p)=0 ~=Sylp(G). Hence the class of groups to which the exact formula
3.3 can be applied is a large one. Here we concentrate on familiar examples.
Example 4.1: Let = K *N H, the amalgamated product of two finite groups
over a common subgroup. In this case the group is virtually free [S] and we may
take 0 to be a free group of finite index. Here X can be taken to be a tree, X=0
a finite graph on which G = =0 acts with orbit space
N
o o
K H
12
i.e. there are two orbits of vertices, with stabilizers K, H respectively, and *
*one orbit
of edges, with stabilizer N. In this case we use a spectral sequence due to Seg*
*al [Se]
for computing equivariant cohomology (indeed it is a very simple MayerViaetoris
sequence in this case) to obtain the exact sequence:
0 ! K0G(X=0) ! R(K) R(H) ! R(N) ! K1G(X=0) ! 0:
This may be used to compute K*G(X=0). Compare this to the formula [B]
O() = _1__K+ _1__H _1__N:
This applies to = SL2(Z) = Z=6 *Z=2 Z=4 to yield
ae
dimC K*G(X=0) C = 0;8;ifi*fodd;* even.
A similar formula can obviously be proved for any virtually free group of *
*finite
vcd, taking into account the edge and vertex stabilizers of the corresponding t*
*ree.
There is a local way of analyzing Ktheory, by using padic Ktheory, Kp (*
*see
[H]). The field C is replaced by Cp, the completion of the algebraic closure of*
* Qp,
and instead of characters on G we specialize to class functions on
T orsp(G) = {g 2 G  g is of order pn, for some n 0} :
Then Atiyah's result can be reformulated as
Kp(BG) Cp ~=Cpvalued class functions on T orsp(G):
In this setting we obtain a local version of our main result:
Theorem 4.2:
M
(i) K*p(B) Cp ~= K*p(B0\ C(fl))Hfl Cp
fl2T(fl)ors
p()
X
(ii) O(K*p(B) Cp) = "O(C(fl))
fl2(fl)Tors
p()
13

Example 4.3 : Now let = SL3(Z), 0= (3), G = SL3(F3). Note G = 24.33.13
but that has no 13torsion. Hence we deduce
M
K*p(B) C ~= K*(BC(fl) \ (3))Hfl Cp for p = 2; 3:
fl2T(fl)ors
p()
In particular for p = 3, we see that up to conjugacy,
0 1 0 1
0 0 1 0 1 0
@ 1 0 0A and @ 1 1 0 A
0 1 0 0 0 1
are the only elements of finite order in which are in T orsp(). Their centrali*
*zers
are cyclic of order 3 and cyclic of order 6 respectively. We obtain
K*3(B) C3 ~=K*3(X=) C3 [K*3(*) C3]2 :
However, from the work of Soule [So], X= is homotopically trivial, hence
K*3(BSL 3(Z)) C3 ~=[K*3(*) C3]3 :
Similarly, we have that the centralizers of elements of order 2 or 4 in SL*
* 3(Z)
are rationally acyclic, hence K*2(BSL (Z)) C2 is of rank equal to the number of
distinct conjugacy classes of elements in SL 3(Z) of order 2 or 4.
Example 4.4 Let k be a totally real number field with ring of integers O, and
let ik denote the Dedekind zeta function associated to k. The centralizer of ev*
*ery
finite subgroup in = SL 2(O) is finite, except for 1. Let [T orsp()] denote t*
*he
number of conjugacy classes of nontrivial elements in T orsp(). Then, for any
p 6= 2
K*p(BSL 2(O)) Cp ~=K*p(X=) Cp [Cp][Torsp()]
*
*14
and
X 2
O(K*p(BSL 2(O)) Cp) = 2ik(1) + 1  ____ + [T orsp()]
(H) H
where H ranges over conjugacy classes of maximal finite subgroups. For this
formula we use an identity due to K. Brown [B1] for "O(SL 2(O)).
Example 4.5: Let = GLp1 (Z), p an odd prime. It is wellknown that this group
has no subgroups of order p2, and furthermore the number of conjugacy classes *
*of
elements of order p in is equal to the class number of p, Cl(p). The centrali*
*zer of
any such element fl will be isomorphic to the group of units U in Z[i], where *
*i is a
primitive pth root of unity. It is also well known that U splits as a direct *
*product
< fl > xZ(p3)=2x Z=2. Hence we obtain
M
K*p(BGLp1 (Z)) Cp ~=K*p(B0)G Cp K*p((S1)(p3)=2) Cp
Cl(p)
where 0 is a normal torsionfree subgroup of GLp1 (Z) with finite factor grou*
*p G.
In addition, we obtain that
O(K*p(BGLp1 (Z)) Cp) = "O(GLp1 (Z)):
We point out that using results due to Ash [A], formulae of this type can be c*
*alcu
lated for GLn(Z), provided that p  1 n 2p  3:
REFERENCES
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Mathematics.
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Journal of Differential Geometry 3 (1969) 118.
[AS2] M. F. Atiyah and G. B. Segal, unpublished.
[B] K. Brown, "Cohomology of Groups," SpringerVerlag GTM 87 (1982).
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Math. 27 (1974) 229264.
15
[B2] K. Brown, "Complete Euler Characteristics and FixedPoint Theory," J.
Pure & Applied Algebra 24 (1982) 103121.
[B3] K. Brown, "Highdimensional Cohomology of Discrete Groups," Proc. Natl.
Acad. Sci. (USA) Vol. 73, No. 6 17951797 (1976).
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Abstract Groups," Ann. Math. 65 (1957) 517518.
[H] M. Hopkins, "Characters and Elliptic Cohomology," Advances in Homotopy
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[HH] F. Hirzebruch and T. H"ofer, "On the Euler Number of an Orbifold," Math
ematische Annalen 286 (1990), 255260.
[Se] G.B. Segal, "Equivariant KTheory," Pub. Math. Inst. des Hautes Etudes
Scient. (Paris), 34 (1968).
[S] JP. Serre, "Cohomologie des Groupes Discrets," Ann. Math. Studies 70
(1971) 77169.
[So] C. Soule, "The Cohomology of SL 3(Z)," Topology 17 (1978) 122.
[tD] T. Tom Dieck, "Transformation Groups and Representation Theory," Lec
ture Notes in Mathematics 766, SpringerVerlag, 1979.