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Rigidity Properties for Hyperbolic Generalizations

Published online by Cambridge University Press:  18 November 2019

Brendan Burns Healy*
Department of Mathematical Sciences, University of Wisconsin-Milwaukee, 3200 N Cramer St, Milwaukee, WI 53211, USA Email:


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.

MSC classification

© Canadian Mathematical Society 2019

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