Bredon cohomological dimensions for groups acting on $\mathrm{CAT}(0)$-spaces

Abstract

Let $G$ be a group acting isometrically with discrete orbits on a separable complete $\mathrm{CAT}(0)$-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 $g\ge 2$ 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.