On the K-theory of linear groups

TL;DR

This paper investigates the K-theory of finitely generated linear groups over fields of positive characteristic, establishing finite asymptotic dimension for certain quotient families and analyzing the split injectivity of the K-theoretic assembly map.

Contribution

It proves the finite asymptotic dimension of quotients by finite subgroups and characterizes when the K-theoretic assembly map is split injective for these groups.

Findings

Finite asymptotic dimension for quotients by finite subgroups.

Split injectivity of the K-theoretic assembly map under certain conditions.

Characterization of when the classifying space exists based on solvable subgroup ranks.

Abstract

We prove that for a finitely generated linear group G 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.