A characterization of Fuchsian actions by topological rigidity

We prove that any rigid representation of ${\pi }_1{\mathrm{\Sigma }}_g$ in ${\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}}_{+}(S^1)$ with Euler number at least $g$ is necessarily semi-conjugate to a discrete, faithful representation into $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{R})$. 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.