A characterization of Fuchsian actions by topological rigidity

Kathryn Mann and Maxime Wolff

Vol. 302 (2019), No. 1, 181–200

DOI: 10.2140/pjm.2019.302.181

Abstract

We give a simple proof that any rigid representation of ${\pi }_1\left({\Sigma }_g\right)$ in ${Homeo}^{+}\left(S_1\right)$ with Euler number at least $g$ is necessarily semiconjugate to a discrete, faithful representation into $PSL\left(2,ℝ\right)$. Combined with earlier work of Matsumoto, this precisely characterizes Fuchsian actions by a topological rigidity property. We have proved this result in greater generality, but with a much more involved proof, in arxiv:1710.04902.

Keywords

rigidity, geometricity, Euler class, surface group actions on the circle

Mathematical Subject Classification 2010

Primary: 20H10, 37E10, 37E45, 57S25, 58D29

Milestones

Received: 9 March 2018

Revised: 16 January 2019

Accepted: 21 January 2019

Published: 5 November 2019