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 rank at most $n$. In this article we prove that for each $n\ge2$ the Bredon cohomology, with respect to the family $\mathcal{F}_n$, of a free abelian group with rank $k > n$ is nontrivial in dimension $k+n$; this answers a question of Corob Cook, Moreno, Nucinkis and Pasini. 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 $n\ge2$. The main tools that we use are the Mayer-Vietoris sequence for Bredon cohomology, Bass-Serre theory, and the L\"uck-Weiermann construction.