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

TL;DR

This paper extends the understanding of the geometric dimensions of 3-manifold groups by computing the $ $-dimensional classifying spaces for families of virtually abelian subgroups, generalizing previous results.

Contribution

It provides a comprehensive calculation of the $ $-dimensional classifying spaces for all $n q 2$ for 3-manifold groups, expanding prior work on $ =1$.

Findings

Computed $ $-dimensional classifying spaces for all $n q 2$ for 3-manifold groups.

Extended previous results from $ =1$ to all $n q 2$.

Enhanced understanding of the subgroup structure of 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 S\'anchez Salda\~na 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 $n\geq 2$.