Classifying spaces for the family of virtually abelian subgroups of orientable $3$-manifold groups

Abstract

For a group $G$, let ${\mathcal{F}}_n$ be the family of all the subgroups of $G$ containing a subgroup isomorphic to ${\mathbb{Z}}^r$ for some $r=0,1,2,\dots ,n$ of finite index. Joecken, Lafont and Sánchez Saldaña computed the ${\mathcal{F}}_1$-dimension of 3-manifold groups. The goal of this article is to compute the ${\mathcal{F}}_n$-geometric dimension of 3-manifold groups for all $n\ge 2$.

22 pages, 1 table. Proofs of Theorem 3.1, Theorem 4.5 c), and Proposition 5.1 were expanded for the sake of clarity. Title was changed