Harmonic measures and rigidity for surface group actions on the circle

Abstract

We study rigidity properties of actions of a torsion-free lattice of $\mathrm{PSU}(1,1)$ 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.

23 pages, to appear in Algebraic & Geometric Topology. v2, v3: minor corrections. v4: simplified the proof of Claim 4.1 & updated the bibliographic information on Thurston's paper. v5: final version incorporating referee's suggestions