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APPEARED IN BULLETIN OF THE

arXiv:math/9301211v1 [math.KT] 1 Jan 1993APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 28, Number 1, January 1993, Pages 95-98REPRESENTATIONS AND K-THEORYOF DISCRETE GROUPSAlejandro AdemAbstract. Let Γ be a discrete group of finite virtual cohomological dimension withcertain finiteness conditions of the type satisfied by arithmetic groups.

We define arepresentation ring for Γ, determined on its elements of finite order, which is of finitetype. Then we determine the contribution of this ring to the topological K-theoryK∗(BΓ), obtaining an exact formula for the difference in terms of the cohomologyof the centralizers of elements of finite order in Γ.0.

IntroductionLet Γ denote a discrete group of finite virtual cohomological dimension. Exam-ples of this type of group include finite groups, arithmetic groups, and mappingclass groups, making them an important class of objects in both topology and al-gebra.

In particular, understanding the classifying space BΓ for such groups is acentral problem in algebraic topology. Unfortunately, the cohomology H∗(BΓ, Z) isa very intractable object; consequently, there are few available calculations (e.g., see[So]).

In sufficiently high dimensions the cohomology is known to depend only onthe lattice F of finite subgroups in Γ [B, F], but in general this yields a complicatedspectral sequence involving the cohomology of the normalizers N(S), S ∈F.In this note we outline an approach to understanding the rˆole of representationsin the topology of BΓ as was done in the case of finite groups by Atiyah [At]. Wedefine a representation ring determined on the elements of finite order in Γ, whichfor a large class of groups (including arithmetic groups) is of finite rank.

Then weindicate to what degree the topological K-theory K∗(BΓ) is determined by theserepresentations.In fact, we provide a precise description of the discrepancy interms of the rational cohomology of the centralizers of elements of finite order in Γ.Complete details will appear elsewhere.1. A reduced representation ring for ΓFrom now on we will assume that Γ has a finite number of distinct conjugacyclasses of elements of finite order and that their centralizers are homologically finite.These hypotheses are known to hold, in particular, for arithmetic groups.Definition 1.1.

Let V, W be two finite-dimensional CΓ-modules. We say that Vis F-isomorphic to W if VS∼= WS for all S ∈F.1991 Mathematics Subject Classification.

Primary 55R35.Research partially supported by an NSF grantReceived by the editors May 20, 1992c⃝1993 American Mathematical Society0273-0979/93 $1.00 + $.25 per page1

2ALEJANDRO ADEMDefinition 1.2. RF(Γ) is the Grothendieck group on F-isomorphism classes offinite-dimensional CΓ-modules.We can, of course, also describe RF(Γ) as a quotient of the usual representationring R(Γ).

Let n(Γ) denote the number of distinct conjugacy classes of elements offinite order in Γ. Using character theory arguments, we proveProposition 1.3.

RF(Γ) is a commutative, unitary ring, which as an abelian groupis free of rank n(Γ). In particular, Γ is torsion-free if and only if RF(Γ) ∼= Z.Similarly, if F(p) denotes the family of all finite p-subgroups of Γ and np(Γ) thenumber of distinct conjugacy classes of elements of order a power of p (p prime),then RF(p)(Γ) can be defined and will be of rank np(Γ).

The following examplesillustrate that these rings are readily computable from subgroup data, unlike thecohomology.Example 1.4. Γ = SL2(Z), n(Γ) = 8, andRF(SL2(Z)) ∼= Z[w],w8 + w6 −w2 −1 = 0.Example 1.5.

Γ = SL3(Z), n2(Γ) = 5, andRF(2)(SL3(Z)) ∼= Z[α1, α2, β1, β2],α21 = α22 = 1, α1β1 = β1,β21 = 2(1 + α1), β22 = 2(1 + α2),α2β2 = β2, α1α2 = α1 + α2 −1,α1β2 = 2α1 + β2 −2,β1β2 = 2β1 + 2β2 −4,α2β1 = 2α2 + β1 −2.2. Contribution to K-TheoryFor the sake of clarity of exposition, we work at a fixed prime p; let K∗p( ) denotep-adic K-theory and Cp the completion of the algebraic closure of Qp.

We choosea fixed normal subgroup Γ′ ⊆Γ such that Γ′ is torsion-free, so G = Γ/Γ′ is finite.If γ ∈Γ, let C(γ) denote its centralizer; then it can be expressed as an extension1 →C(γ) ∩Γ′ →C(γ) →Hγ →1where |Hγ| < ∞. Our main result is the following.Theorem 2.1.

Let Γ be a discrete group of finite v.c.d. satisfying our finitenessassumptions.

Then there is an exact sequence0 →Ip →K∗p(BΓ) ⊗Cpϕp−→RF(p)(Γ) ⊗Cp →0where ϕp is a surjection of rings, and we have an additive decompositionIp ∼=M(γ)˜K∗p(B(C(γ) ∩Γ′))Hγ ⊗Cpwhere the sum is taken over conjugacy classes of elements of order a finite powerof p.

REPRESENTATIONS AND K-THEORY OF DISCRETE GROUPS3Corollary 2.2.K∗p(BΓ) ⊗Cp ∼= RF(p)(Γ) ⊗Cpif and only if ˜H∗(BC(γ), Q) ≡0 for every element γ ∈Γ of order a power of p.The corollary follows from the fact that Ip is determined by the cohomology ofBC(γ); it is, of course, independent of the choice of the extension.Example 2.3. Γ = G1∗HG2 is an amalgamated product of finite groups.

ThenK0p(BΓ) ⊗Cp ∼= RF(p)(Γ) ⊗Cp ,K1p(BΓ) ⊗Cp ∼= (Cp)vp(Γ)wherevp(Γ) = np(Γ) −np(G1) −np(G2) + np(H)represents the total sum of dimQ H1(BC(γ), Q) as γ ∈Γ ranges over conjugacyclasses of elements of order a power of p.Example 2.4. Γ = SL3(Z) andK∗2(BΓ) ⊗C2 ∼= RF(2)(Γ) ⊗C2 ,whence we can use Example 1.5 to determine this ring (compare with [So, TY]).Example 2.5.

Γ = GLp−1(Z), where p is odd prime. If Cl(p) = class number ofp, then RF(p)(Γ) can be computed from the extension0 →RF(p)(Γ) →MCl(p)R(Z/p)∆→Zt(p)−1 →0where ∆= Galois group and t(p) = number of ∆-orbits in the set of ideal classes.Hence rkZRF(p)(Γ) = 1 + Cl(p), and in this caseIp ∼= ˜K∗p(BΓ′)G ⊗Cp ⊕MCl(p)˜K∗p(S1)(p−3)/2⊗Cp.Sketch of Proof of 2.1.

From a theorem of Serre [S] for the class of groups weconsider that there exists a finite-dimensional Γ-complex X with finite isotropy,contractible fixed point sets, and X/Γ of finite type.By essentially identifyingbundles that agree on finite subgroups, we construct a surjection of ringsK∗Γ(X) ։ RF(Γ) .Next we identify K∗Γ(X) ∼= K∗G(X/Γ′) (Γ′, G as before) and use an additive decom-position for K∗G(X/Γ′) ⊗C obtained previously by the author [A] to estimate thekernel of this ring map in terms of the centralizers of elements of finite order in Γ.The final technical step is to complete this map, as by the Atiyah-Segal comple-tion theorem, K∗G(X/Γ′)∧∼= K∗(BΓ) (at IG ⊆R(G)). Doing this locally leads tothe statement in Theorem 2.1.□

4ALEJANDRO ADEMReferences[A]A. Adem, On the K-theory of the classifying space of a discrete group, Math. Ann.

292(1992), 319–327.[At]M. F. Atiyah, Characters and cohomology of finite groups, Inst.

Hautes ´Etudes Sci. Publ.Math., vol.

9, Presses Univ. France, Paris, 1961, pp.

23–64.[B]K. Brown, High-dimensional cohomology of discrete groups, Proc.

Nat. Acad.

Sci. U.S.A.73 (1976), 1795–97.[F]F.

T. Farrell, An extension of Tate cohomology to a class of infinite groups, J. Pure Appl.Algebra 10 (1977), 153–161.[S]J.-P.

Serre, Cohomologie des groupes discretes, Ann. of Math.

Stud., vol. 70, PrincetonUniv.

Press, Princeton, NJ, 1971, pp. 77–169 v.[So]C. Soul´e, The cohomology of SL3(Z), Topology 17 (1978), 1–22.[TY]M.

Tezuka and N. Yagita, Complex K-theory of BSL3(Z), preprint 1992.Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706E-mail address: adem@math.wisc.edu

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