Harmonic measures and rigidity for surface group actions on the circle
Abstract
We study rigidity properties of actions of a torsion-free lattice of on the circle . 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 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.
Cite
@article{arxiv.2207.08411,
title = {Harmonic measures and rigidity for surface group actions on the circle},
author = {Masanori Adachi and Yoshifumi Matsuda and Hiraku Nozawa},
journal= {arXiv preprint arXiv:2207.08411},
year = {2025}
}
Comments
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