Rigidity for quasi-Möbius actions on fractal metric spaces

Abstract

In Rigidity for quasi-Möbius group actions, M. Bonk and B. Kleiner proved a rigidity theorem for expanding quasi-Möbius group actions on Ahlfors $n$-regular metric spaces with topological dimension $n$. This led naturally to a rigidity result for quasi-convex geometric actions on $\textrm{CAT}(-1)$-spaces that can be seen as a metric analog to the “entropy rigidity” theorems of U. Hamenstädt and M. Bourdon. Building on the ideas developed in Rigidity for quasi-Möbius group actions, we establish a rigidity theorem for certain expanding quasi-Möbius group actions on spaces with different metric and topological dimensions. This is motivated by a corresponding entropy rigidity result in the coarse geometric setting.