Harmonic measures and rigidity for surface group actions on the circle

2022/07/18 by Masanori Adachi, Adachi, Masanori, Yoshifumi Matsuda +3
Mathematics · #Complex Variables (math.CV) #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · doi:10.48550/arxiv.2207.08411

openalex publication_date 2022/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study rigidity properties of actions of a torsion-free lattice of PSU(1,1) on the circle S¹. 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 S¹ 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.

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