Rigidity for Quasi-M\"obius Actions on Fractal Metric Spaces

TL;DR

This paper extends rigidity theorems for expanding quasi-M"obius actions from Ahlfors regular spaces to more general fractal metric spaces, linking geometric group actions to entropy rigidity concepts.

Contribution

It introduces a new rigidity theorem for expanding quasi-M"obius actions on fractal metric spaces with varying dimensions, broadening previous results.

Findings

Established a rigidity theorem for new classes of fractal metric spaces.

Connected quasi-M"obius group actions with entropy rigidity in coarse geometry.

Extended prior rigidity results to spaces with different metric and topological dimensions.

Abstract

In \cite{BK02}, M. Bonk and B. Kleiner proved a rigidity theorem for expanding quasi-M\"obius 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. Hamenst\"adt and M. Bourdon. Building on the ideas developed in \cite{BK02}, we establish a rigidity theorem for certain expanding quasi-M\"obius group actions on spaces with different metric and topological dimensions. This is motivated by a corresponding entropy rigidity result in the coarse geometric setting.