Representations  and  K-theory of  Discrete  Groups
Alejandro Adem

ABSTRACT: Let  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 $\Gamma$ 
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 $K^*(B\Gamma)$,obtaining an exact formula for the 
difference in terms of the cohomology of the centralizers of 
elements of finite order in $\Gamma$.