Largest hyperbolic action of 3-manifold groups

The set of equivalence classes of cobounded actions of a group $G$ on different hyperbolic metric spaces carries a natural partial order. Following Abbott–Balasubramanya–Osin, the group $G$ is $H$-accessible if the resulting poset has a largest element. In this paper, we prove that every nongeometric 3-manifold has a finite cover with $H$-inaccessible fundamental group and give conditions under which the fundamental group of the original manifold is $H$-inaccessible. We also prove that every Croke–Kleiner admissible group (a class of graphs of groups that generalizes fundamental groups of three-dimensional graph manifolds) has a finite index subgroup that is $H$-inaccessible.