Representations and K-theory of Discrete Groups

Alejandro Adem

ABSTRACT: Let Γ be a discrete group of finite virtual cohomological dimension with certain finiteness conditions of the type satisfied by arithmetic groups. We define a representation ring for Γ determined on its elements of finite order which is of finite type. Then we determine the contribution of this ring to the topological K-theory K^{∗}(BΓ), obtaining an exact formula for the difference in terms of the cohomology of the centralizers of elements of finite order in Γ.

§0. Introduction:

Let Γ denote a discrete group of finite virtual cohomological dimension. Examples 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 BΓ for such groups is a central problem in algebraic topology. Unfortunately, the cohomology H^{∗}(BΓ, ) is a very intractable object; consequently there are few available calculations (e.g. [So]). In sufficiently high dimensions the cohomology is known to depend only on the lattice F of finite subgroups in Γ ([B], [F]) but in general this yields a complicated spectral sequence involving the cohomology of the normalizers N(S), S ∈ F.

In this note we outline an approach to understanding the rôle of representations in the topology of BΓ as was done in the case of finite groups by Atiyah [At]. We define a representation ring determined on the elements of finite order in Γ, which for a large class of groups (including arithmetic groups) is of finite rank. Then we indicate to what degree the topological K-theory K^{∗}(BΓ) is determined by these representations. In fact we provide a precise description of the discrepancy in terms 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 Γ

From now on we will assume that Γ has a finite number of distinct conjugacy classes of elements of finite order and that their centralizers are homologically finite. These hypotheses are known to hold in particular for arithmetic groups.

1991 Mathematics Subject Classification 55R35 Research partially supported by an NSF grant.

Definition 1.1: Let V, W be two finite dimensional Γ-modules. We say that V is F-isomorphic to W if V |_{S} ≅ W |_{S} for all S ∈ F.

Definition 1.2: R_{F}(Γ) is the Grothendieck group on F-isomorphism classes of finite dimensional Γ-modules.

We can of course also describe R_{F}(Γ) as a quotient of the usual representation ring R(Γ). Let n(Γ) denote the number of distinct conjugacy classes of elements of finite order in Γ. Using character theory arguments, we prove

Proposition 1.3: R_{F}(Γ) is a commutative, unitary ring which as an abelian group is free of rank n(Γ). In particular Γ is torsion–free if and only if R_{F}(Γ) ≅ .

Similarly if F(p) denotes the family of all finite p-subgroups of Γ, and n_{p}(Γ) the number of distinct conjugacy classes of elements of order a power of p (p prime), then R_{F(p)}(Γ) can be defined, and it will be of rank n_{p}(Γ). The following examples illustrate that these rings are readily computable from subgroup data, unlike the cohomology.

Example 1.4: Γ = SL_{2}( ), n(Γ) = 8 and

R_{F}(SL_{2}( )) ≅ [w] / w^{8} + w^{6} − w^{2} − 1 = 0

Example 1.5 Γ = SL_{3}( ), n_{2}(Γ) = 5 and

R_{F(2)}(SL_{3}( )) ≅ [α_{1}, α_{2}, β_{1}, β_{2}] / \begin{matrix}α_{1}^{2} = α_{2}^{2} = 1, α_{1}β_{1} = β_{1} \\ β_{1}^{2} = 2(1 + α_{1}), β_{2}^{2} = 2(1 + α_{2}) \\ α_{2}β_{2} = β_{2}, α_{1}α_{2} = α_{1} + α_{2} − 1 \\ α_{1}β_{2} = 2α_{1} + β_{2} − 2, β_{1}β_{2} = 2β_{1} + 2β_{2} − 4 \\ α_{2}β_{1} = 2α_{2} + β_{1} − 2 . \end{matrix}

§2. Contribution to K-Theory

For the sake of clarity of exposition, we work at a fixed prime p; let K_{p}^{∗}( ) denote p-adic K-theory and _{p} the completion of the algebraic closure of _{p}. We choose a fixed normal subgroup Γ′ ⊆ Γ such that Γ′ is torsion-free, and G = Γ/Γ′ is finite. If γ ∈ Γ let C(γ) denote its centralizer; then it can be expressed as an extension

1 → C(γ) ∩ Γ′ → C(γ) → H_{γ} → 1

where |H_{γ}| < ∞. Our main result is the following.

Theorem 2.1: Let Γ be a discrete group of finite v.c.d. satisfying our finiteness assumptions. Then there is an exact sequence

0 → I_{p} → K_{p}^{∗}(BΓ) ⊗ _{p}⟶^{φ_{p}}R_{F(p)}(Γ) ⊗ _{p} → 0

where φ_{p} is a surjection of rings and we have an additive decomposition

I_{p} ≅ ⨁_{(γ)} K̃_{p}^{∗}(B(C(γ) ∩ Γ′))^{H_{γ}} ⊗ _{p}

where the sum is taken over conjugacy classes of elements of order a finite power of p.

Corollary 2.2:

K_{p}^{∗}(BΓ) ⊗ _{p} ≅ R_{F(p)}(Γ) ⊗ _{p}

if and only if H̃^{∗}(BC(γ), ) ≡ 0 for every element γ ∈ Γ of order a power of p.

The corollary follows from the fact that I_{p} is determined by the cohomology of BC(γ); it is of course independent of the choice of the extension.

Example 2.3: Γ = G_{1}∗_{H}G_{2}, an amalgamated product of finite groups. Then

\begin{matrix}K_{p}^{0}(BΓ) ⊗ _{p} ≅ R_{F(p)}(Γ) ⊗ _{p} , \\ K_{p}^{1}(BΓ) ⊗ _{p} ≅ ( _{p})^{v_{p}(Γ)} \end{matrix}

where

v_{p}(Γ) = n_{p}(Γ) − n_{p}(G_{1}) − n_{p}(G_{2}) + n_{p}(H)

represents the total sum of dim_{}H^{1}(BC(γ), ) as γ ∈ Γ ranges over conjugacy classes of elements of order a power of p.

Example 2.4: Γ = SL_{3}( )

K_{2}^{∗}(BΓ) ⊗ _{2} ≅ R_{F(2)}(Γ) ⊗ _{2} ,

whence we can use 1.5 to determine this (compare with [So], [TY]).

Example 2.5: Γ = GL_{p−1}( ), p odd prime. If Cℓ(p) = class number of p, then R_{F(p)}(Γ) can be computed from the extension

0 → R_{F(p)}(Γ) → ( ⨁_{Cℓ(p)} R( /p))^{Δ} → ^{t(p)−1} → 0

where Δ = Galois group, t(p) = number of Δ-orbits in the set of ideal classes. Hence rk_{}R_{F(p)}(Γ) = 1 + Cℓ(p), and in this case

I_{p} ≅ K̃_{p}^{∗}(BΓ′)^{G} ⊗ _{p} ⊕ ( ⨁_{Cℓ(p)} K̃_{p}^{∗} ((S^{1})^{\frac{p−3}{2}} ) ⊗ _{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 Γ-complex X with finite isotropy, contractible fixed point sets and X/Γ of finite type. By essentially identifying bundles which agree on finite subgroups, we construct a surjection of rings

K_{Γ}^{∗}(X) R_{F}(Γ) .

Next we identify K_{Γ}^{∗}(X) ≅ K_{G}^{∗}(X/Γ′) (Γ′, G as before) and use an additive decomposition for K_{G}^{∗}(X/Γ′) 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 Γ.

The final technical step is to complete this map, as by the Atiyah-Segal completion theorem, K_{G}^{∗}(X/Γ′)^{∧} ≅ K^{∗}(BΓ) (at IG ⊆ R(G)). Doing this locally leads to the statement in 2.1.

REFERENCES

[A] A. Adem, “On the K-theory of the classifying space of a discrete group,” Math. Ann. 292, 319–327 (1992).

[At] M. F. Atiyah, “Characters and cohomology of finite groups,” Publ. Math. IHES (Paris) 9 (1961), 23–64.

[B] K. Brown, “High-dimensional cohomology of discrete groups,” Proc. Nat. Acad. Sci. USA 73 (No. 6), 1795–97 (1976).

[F] F. T. Farrell, “An extension of Tate cohomology to a class of infinite groups,” J.Pure Applied Algebra 10 (1977), 153–161.

[S] J-P. Serre, “Cohomologie des groupes discretes,” Ann. Math. Studies 70, 77–169 (1971).

[So] C. Soulé, “The cohomology of SL_{3}( ),” Topology 17, 1–22 (1978).

[TY] M. Tezuka and N. Yagita, “Complex K-theory of BSL_{3}( ),” preprint 1992.

Department of Mathematics, University of Wisconsin, Madison WI 53706