Classifying spaces for families of abelian subgroups of braid groups, RAAGs and graphs of abelian groups

Abstract

Given a group $G$ and an integer $n\geq 0$, we consider the family ${\mathcal F}_n$ of all virtually abelian subgroups of $G$ of $\textrm{rank}$ at most $n$. In this article, we prove that for each $n\ge 2$ the Bredon cohomology, with respect to the family ${\mathcal F}_n$, of a free abelian group with $\textrm{rank}$ $k \gt n$ is nontrivial in dimension $k+n$; this answers a question of Corob Cook et al. (Homology Homotopy Appl. 19(2) (2017), 83–87, Question 2.7). As an application, we compute the minimal dimension of a classifying space for the family ${\mathcal F}_n$ for braid groups, right-angled Artin groups, and graphs of groups whose vertex groups are infinite finitely generated virtually abelian groups, for all $n\ge 2$. The main tools that we use are the Mayer–Vietoris sequence for Bredon cohomology, Bass–Serre theory, and the Lück–Weiermann construction.