Bredon cohomological dimension for virtually abelian stabilisers for CAT(0) groups

TL;DR

This paper establishes an upper bound on the Bredon cohomological dimension for groups acting on CAT(0) spaces with virtually abelian stabilizers, covering classes like Coxeter and right-angled Artin groups.

Contribution

It provides a new upper bound for the Bredon cohomological dimension for a broad class of CAT(0) groups with virtually abelian stabilizers, partially answering Lafont's question.

Findings

Upper bound for Bredon cohomological dimension established

Applicable to Coxeter, Right-angled Artin, and related groups

Advances understanding of group actions on CAT(0) spaces

Abstract

Given a discrete group $G$, for any integer $r\geqslant0$ we consider the family of all virtually abelian subgroups of $G$ of rank at most $r$. We give an upper bound for the Bredon cohomological dimension of $G$ for this family for a certain class of groups acting on $\mathrm{CAT}(0)$ spaces. This covers the case of Coxeter groups, Right-angled Artin groups, fundamental groups of special cube complexes and graph products of finite groups. Our construction partially answers a question of J.-F. Lafont.