Representations and 𝐾-theory of discrete groups

Abstract

Let Γ \Gamma 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 \Gamma , 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 ∗ ( B Γ ) K^{\ast }(B\Gamma ) , obtaining an exact formula for the difference in terms of the cohomology of the centralizers of elements of finite order in Γ \Gamma .