Harmonic measures and rigidity for surface group actions on the circle

TL;DR

This paper investigates the rigidity of surface group actions on the circle using harmonic measures, providing curvature estimates, a Gauss-Bonnet formula, and new proofs of existing rigidity theorems.

Contribution

It introduces a curvature estimate and a Gauss-Bonnet formula for circle actions, offering new insights and alternative proofs for known rigidity results.

Findings

Curvature estimate for circle actions

Gauss-Bonnet formula for the associated connection

Characterization of harmonic measures on suspension foliations

Abstract

We study rigidity properties of actions of a torsion-free lattice of $\operatorname{PSU}(1,1)$ on the circle $S^1$. We follow the approaches of Frankel and Thurston proposed in preprints via foliated harmonic measures on the suspension bundles. Our main results are a curvature estimate and a Gauss--Bonnet formula for the $S^1$ connection obtained by taking the average of the flat connection with respect to a harmonic measure. As consequences, we give a precise description of the harmonic measure on suspension foliations with maximal Euler number and an alternative proof of rigidity theorems of Matsumoto and Burger--Iozzi--Wienhard.