Harmonic measures and rigidity for surface group actions on the circle

Masanori Adachi; Yoshifumi Matsuda; Hiraku Nozawa

doi:10.2140/agt.2025.25.2391

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arXiv · 2207.08411

Harmonic measures and rigidity for surface group actions on the circle

Masanori Adachi·Yoshifumi Matsuda·Hiraku Nozawa

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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.

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Masanori Adachi, Yoshifumi Matsuda, Hiraku Nozawa. 2022-07-18. Harmonic measures and rigidity for surface group actions on the circle. https://doi.org/10.2140/agt.2025.25.2391

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