Representations and Ktheory
of Discrete Groups
Alejandro Adem
ABSTRACT: Let be a discrete group of finite virtual cohomological dim*
*en
sion with certain finiteness conditions of the type satisfied by arith*
*metic groups.
We define a representation ring for determined on its elements of fin*
*ite order
which is of finite type. Then we determine the contribution of this r*
*ing to the
topological Ktheory K*(B), obtaining an exact formula for the differe*
*nce in
terms of the cohomology of the centralizers of elements of finite orde*
*r in .
x0. Introduction:
Let denote a discrete group of finite virtual cohomological dimension. E*
*xam
ples of this type of group include finite groups, arithmetic groups and mapping*
* class
groups, making them an important class of objects in both topology and algebra.*
* In
particular, understanding the classifying space B for such groups is a central *
*problem
in algebraic topology. Unfortunately, the cohomology H*(B; Z) is a very intract*
*able
object; consequently there are few available calculations (e.g. [So]). In suffi*
*ciently high
dimensions the cohomology is known to depend only on the lattice F of finite su*
*bgroups
in ([B], [F]) but in general this yields a complicated spectral sequence invol*
*ving the
cohomology of the normalizers N(S), S 2 F.
In this note we outline an approach to understanding the r^ole of represen*
*tations in
the topology of B as was done in the case of finite groups by Atiyah [At]. We d*
*efine
a representation ring determined on the elements of finite order in , which for*
* a large
class of groups (including arithmetic groups) is of finite rank. Then we indic*
*ate to
what degree the topological Ktheory K*(B) is determined by these representatio*
*ns.
In fact we provide a precise description of the discrepancy in terms of the rat*
*ional
cohomology of the centralizers of elements of finite order in . Complete detai*
*ls will
appear elsewhere.
x1. A Reduced Representation Ring for
From now on we will assume that has a finite number of distinct conjugacy
classes of elements of finite order and that their centralizers are homological*
*ly finite.
These hypotheses are known to hold in particular for arithmetic groups.
__________________________
1991 Mathematics Subject Classification 55R35
Research partially supported by an NSF grant.
1
Definition 1.1: Let V; Wfbeitwoffiniteidimensional Cmodules. We say that V is
Fisomorphic to W if V fifi~=W fififor all S 2 F. 
S S
Definition 1.2: RF () is the Grothendieck group on Fisomorphism classes of fin*
*ite
dimensional Cmodules. 
We can of course also describe RF () as a quotient of the usual representa*
*tion ring
R(). Let n() denote the number of distinct conjugacy classes of elements of fi*
*nite
order in . Using character theory arguments, we prove
Proposition 1.3: RF () is a commutative, unitary ring which as an abelian group*
* is
free of rank n(). In particular is torsionfree if and only if RF () ~=Z. 
Similarly if F(p) denotes the family of all finite psubgroups of , and np*
*() the
number of distinct conjugacy classes of elements of order a power of p (p prime*
*), then
RF(p)() can be defined, and it will be of rank np(). The following examples ill*
*ustrate
that these rings are readily computable from subgroup data, unlike the cohomolo*
*gy.
Example 1.4: = SL2(Z), n() = 8 and
,
RF (SL 2(Z)) ~=Z[w] w8 + w6  w2  1 = 0 
Example 1.5 = SL3(Z), n2() = 5 and
ff21= ff22= 1; ff1fi1 = fi1
, fi2 2
1 = 2(1 + ff1); fi2 = 2(1 + ff2)
RF(2)(SL 3(Z)) ~=Z[ff1; ff2; fi1; fi2] ff2fi2 = fi2; ff1ff2 = ff1 + ff2  *
*1 
ff1fi2 = 2ff1 + fi2  2; fi1fi2 = 2f*
*i1 + 2fi2  4
ff2fi1 = 2ff2 + fi1  2 :
x2. Contribution to KTheory
For the sake of clarity of exposition, we work at a fixed prime p; let K*p*
*( ) denote
padic Ktheory and Cp the completion of the algebraic closure of Qp. We choos*
*e a
fixed normal subgroup 0 such that 0 is torsionfree, and G = =0 is finite. If
fl 2 let C(fl) denote its centralizer; then it can be expressed as an extension
1 ! C(fl) \ 0! C(fl) ! Hfl! 1
where Hfl < 1. Our main result is the following.
2
Theorem 2.1: Let be a discrete group of finite v.c.d. satisfying our finitenes*
*s as
sumptions. Then there is an exact sequence
0 ! Ip ! K*p(B) Cp'p!RF(p)() Cp ! 0
where 'p is a surjection of rings and we have an additive decomposition
M
Ip ~= "K*p(B(C(fl) \ 0))Hfl Cp
(fl)
where the sum is taken over conjugacy classes of elements of order a finite pow*
*er of p.

Corollary 2.2:
K*p(B) Cp ~=RF(p)() Cp
if and only if H"*(BC(fl); Q) 0 for every element fl 2 of order a power of p.*
* 
The corollary follows from the fact that Ip is determined by the cohomolog*
*y of
BC(fl); it is of course independent of the choice of the extension.
Example 2.3: = G1*H G2, an amalgamated product of finite groups. Then
K0p(B) Cp ~=RF(p)() Cp ;
K1p(B) Cp ~=(Cp)vp()
where
vp() = np()  np(G1)  np(G2) + np(H)
represents the total sum of dimQ H1(BC(fl); Q) as fl 2 ranges over conjugacy c*
*lasses
of elements of order a power of p. 
Example 2.4: = SL3(Z)
K*2(B) C2 ~=RF(2)() C2 ;
whence we can use 1.5 to determine this (compare with [So], [TY]). 
Example 2.5: = GL p1(Z), p odd prime. If C`(p) = class number of p, then
RF(p)() can be computed from the extension
0 1
M
0 ! RF(p)() ! @ R(Z=p)A ! Zt(p)1! 0
C`(p)
3
where = Galois group, t(p) = number of orbits in the set of ideal classes. He*
*nce
rkZRF(p)() = 1 + C`(p), and in this case
0 1
M i p3_j
Ip ~=K"*p(B0)G Cp @ K"*p (S1) 2 CpA : 
C`(p)
Sketch of Proof of 2.1: From a theorem of Serre [S] for the class of groups we
consider there exists a finite dimensional complex X with finite isotropy, con*
*tractible
fixed point sets and X= of finite type. By essentially identifying bundles whic*
*h agree
on finite subgroups, we construct a surjection of rings
K*(X) i RF () :
Next we identify K*(X) ~=K*G(X=0) (0, G as before) and use an additive decompos*
*i
tion for K*G(X=0) obtained previously by the author [A] to estimate the kernel *
*of this
ring map in terms of the centralizers of elements of finite order in .
The final technical step is to complete this map, as by the AtiyahSegal c*
*ompletion
theorem, K*G(X=0)^ ~=K*(B) (at IG R(G)). Doing this locally leads to the
statement in 2.1. 
REFERENCES
[A] A. Adem, "On the Ktheory of the classifying space of a discrete group," *
*Math.
Ann. 292, 319327 (1992).
[At] M. F. Atiyah, "Characters and cohomology of finite groups," Publ. Math. I*
*HES
(Paris) 9 (1961), 2364.
[B] K. Brown, "Highdimensional cohomology of discrete groups," Proc. Nat. Ac*
*ad.
Sci. USA 73 (No. 6), 179597 (1976).
[F] F. T. Farrell, "An extension of Tate cohomology to a class of infinite gr*
*oups,"
J.Pure Applied Algebra 10 (1977), 153161.
[S] JP. Serre, "Cohomologie des groupes discretes," Ann. Math. Studies 70, 7*
*7169
(1971).
[So] C. Soule, "The cohomology of SL3(Z)," Topology 17, 122 (1978).
[TY] M. Tezuka and N. Yagita, "Complex Ktheory of BSL 3(Z)," preprint 1992.
Department of Mathematics, University of Wisconsin, Madison WI 53706
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