The classifying space for commutativity of geometric orientable 3-manifold groups

For a topological group $G$ let $E_{\mathsf{\text{com}}}(G)$ be the total space of the universal transitionally commutative principal $G$-bundle as defined by Adem–Cohen–Torres-Giese. So far this space has been most studied in the case of compact Lie groups; but in this paper we focus on the case of infinite discrete groups. For a discrete group $G$, the space $E_{\mathsf{\text{com}}}(G)$ is homotopy equivalent to the geometric realization of the order complex of the poset of cosets of abelian subgroups of $G$. We show that for fundamental groups of closed orientable geometric $3$-manifolds, this space is always homotopy equivalent to a wedge of circles. On our way to prove this result we also establish some structural results on the homotopy type of $E_{\mathsf{\text{com}}}(G)$.