  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.
