Bredon cohomological dimensions for groups acting on CAT(0)-spaces

TL;DR

This paper establishes upper bounds for the Bredon cohomological dimension of groups acting on CAT(0)-spaces, with applications to mapping class groups, linear groups, and Baumslag-Solitar groups, advancing understanding of their classifying spaces.

Contribution

It provides new bounds for Bredon cohomological dimensions for groups acting on CAT(0)-spaces and applies these results to various important classes of groups.

Findings

Mapping class groups admit (9g-8)-dimensional classifying spaces.

Finitely generated linear groups of positive characteristic have finite-dimensional classifying spaces.

Generalized Baumslag-Solitar groups have 3-dimensional models for classifying spaces.

Abstract

Let G be a group acting isometrically with discrete orbits on a separable complete CAT(0)-space of bounded topological dimension. Under certain conditions, we give upper bounds for the Bredon cohomological dimension of G for the families of finite and virtually cyclic subgroups. As an application, we prove that the mapping class group of any closed, connected, and orientable surface of genus g greater than 1 admits a (9g-8)-dimensional classifying space with virtually cyclic stabilizers. In addition, our results apply to fundamental groups of graphs of groups and groups acting on Euclidean buildings. In particular, we show that all finitely generated linear groups of positive characteristic have a finite dimensional classifying space for proper actions and a finite dimensional classifying space for the family of virtually cyclic subgroups. We also show that every generalized Baumslag-Solitar group has a 3-dimensional model for the classifying space with virtually cyclic stabilizers.