Rigidity for Quasi-Möbius Actions on Fractal Metric Spaces

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In 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 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.