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

Given a group $G$ and an integer $n\ge 0$ 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 $n\ge 2$ 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 $n\ge 2$. The main tools that we use are the Mayer-Vietoris sequence for Bredon cohomology, Bass-Serre theory, and the Lück-Weiermann construction.