On the K-theory of groups with Finite Decomposition Complexity

TL;DR

This paper proves the injectivity of the assembly map in algebraic K- and L-theory for groups with finite quotient finite decomposition complexity, under certain conditions, extending previous results to a broader class of groups.

Contribution

It establishes the injectivity of the assembly map for groups with finite quotient finite decomposition complexity, a significant generalization of earlier finite decomposition complexity results.

Findings

Assembly map is injective for groups with finite quotient finite decomposition complexity.

Applicable to finitely generated linear groups over fields with characteristic zero.

Includes groups with finite dimensional models for ar Emma and bounded finite subgroup order.

Abstract

It is proved that the assembly map in algebraic K- and L-theory with respect to the family of finite subgroups is injective for groups $\Gamma$ with finite quotient finite decomposition complexity (a strengthening of finite decomposition complexity introduced by Guentner, Tesser and Yu) that admit a finite dimensional model for $\underbar E\Gamma$ and have an upper bound on the order of their finite subgroups. In particular this applies to finitely generated linear groups over fields with characteristic zero with a finite dimensional model for $\underbar E\Gamma$.