Largest hyperbolic actions 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 non-geometric 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 3–dimensional graph manifolds) has a finite index subgroup that is H–inaccessible.