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Abstract The set of equivalence classes of cobounded actions of a group on different hyperbolic metric spaces carries a natural partial order. Following Abbott–Balasubramanya–Osin, the group is ‐accessibleif the resulting poset has a largest element. In this paper, we prove that every nongeometric 3‐manifold has a finite cover with ‐inaccessible fundamental group and give conditions under which the fundamental group of the original manifold is ‐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 ‐inaccessible. 

Abstract Actions on hyperbolic metric spaces are an important tool for studying groups, and so it is natural, but difficult, to attempt to classify all such actions of a fixed group. In this paper, we build strong connections between hyperbolic geometry and commutative algebra in order to classify the cobounded hyperbolic actions of numerous metabelian groups up to a coarse equivalence. In particular, we turn this classification problem into the problems of classifying ideals in the completions of certain rings and calculating invariant subspaces of matrices. We use this framework to classify the cobounded hyperbolic actions of many abelian‐by‐cyclic groups associated to expanding integer matrices. Each such action is equivalent to an action on a tree or on a Heintze group (a classically studied class of negatively curved Lie groups). Our investigations incorporate number systems, factorization in formal power series rings, completions, and valuations. 

We prove several topological and dynamical properties of the boundary of a hierarchically hyperbolic group are independent of the specific hierarchically hyperbolic structure. This is accomplished by proving that the boundary is invariant under a “maximization” procedure introduced by the first two authors and Durham.