Rigidity for Quasi-Möbius Actions on Fractal Metric Spaces

Abstract

In \cite{BK02}, 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 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 \cite{BK02}, 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.

arXiv comment33 pages; v2: published version