A characterization of Fuchsian actions by topological rigidity

TL;DR

This paper characterizes Fuchsian actions of surface groups on the circle through topological rigidity, showing that rigid representations with high Euler number are semi-conjugate to discrete, faithful hyperbolic actions.

Contribution

It provides a topological rigidity criterion that uniquely characterizes Fuchsian actions among surface group actions on the circle.

Findings

Rigid representations with Euler number ≥ g are semi-conjugate to Fuchsian groups.

The work extends and complements Matsumoto's results on surface group actions.

It offers a new topological perspective on the classification of surface group actions.

Abstract

We prove that any rigid representation of $\pi_1\Sigma_g$ in $\mathrm{Homeo}_+(S^1)$ with Euler number at least $g$ is necessarily semi-conjugate to a discrete, faithful representation into $\mathrm{PSL}(2,\mathbb{R})$. Combined with earlier work of Matsumoto, this precisely characterizes Fuchsian actions by a topological rigidity property. Though independent, this work can be read as an introduction to the companion paper {\em rigidity and geometricity for surface group actions on the circle}, by the same authors.