  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 $n\geq 2$.
