## Journal of Differential Geometry

### Rigidity for quasi-Möbius actions on fractal metric spaces

Kyle Kinneberg

#### 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.

#### Article information

Source
J. Differential Geom., Volume 100, Number 2 (2015), 349-388.

Dates
First available in Project Euclid: 4 May 2015

https://projecteuclid.org/euclid.jdg/1430744124

Digital Object Identifier
doi:10.4310/jdg/1430744124

Mathematical Reviews number (MathSciNet)
MR3343835

Zentralblatt MATH identifier
1328.53052

#### Citation

Kinneberg, Kyle. Rigidity for quasi-Möbius actions on fractal metric spaces. J. Differential Geom. 100 (2015), no. 2, 349--388. doi:10.4310/jdg/1430744124. https://projecteuclid.org/euclid.jdg/1430744124