On the $K$-theory of linear groups

Abstract

We prove that for a finitely generated linear group over a field of positive characteristic the family of quotients by finite subgroups has finite asymptotic dimension. We use this to show that the $K$-theoretic assembly map for the family of finite subgroups is split injective for every finitely generated linear group $G$ over a commutative ring with unit under the assumption that $G$ admits a finite-dimensional model for the classifying space for the family of finite subgroups. Furthermore, we prove that this is the case if and only if an upper bound on the rank of the solvable subgroups of $G$ exists.