  For a topological group $G$ let $E_{\textsf{com}}(G)$ 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 $E_{\textsf{com}}(G)$ 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
$E_{\textsf{com}}(G)$.
