Rigidity Properties for Hyperbolic Generalizations

TL;DR

This paper explores the limitations of geometric and topological rigidity in acylindrically hyperbolic and relatively hyperbolic groups, highlighting the absence of well-defined limit sets and the decay of rigidity under relaxed conditions.

Contribution

It provides new insights into the lack of rigidity properties in certain hyperbolic groups and summarizes how these properties diminish when conditions are loosened.

Findings

No well-defined limit set for acylindrical actions on hyperbolic spaces.

Relatively hyperbolic groups exhibit specific quasi-isometry behaviors.

Rigidity properties decay as action conditions are relaxed.

Abstract

We make a few observations on the absence of geometric and topological rigidity for acylindrically hyperbolic and relatively hyperbolic groups. In particular, we demonstrate the lack of a well-defined limit set for acylindrical actions on hyperbolic spaces, even under the assumption of universality. We also prove a statement about relatively hyperbolic groups inspired by a remark by Groves, Manning, and Sisto about the quasi-isometry type of combinatorial cusps. Finally, we summarize these results in a table in order to assert a meta-statement about the decay of metric rigidity as the conditions on actions on hyperbolic spaces are loosened.