Harmonic measures and rigidity for surface group actions on the circle

Masanori Adachi, Yoshifumi Matsuda, Hiraku Nozawa

Published 2022-07-18Version 1

Let $\Gamma$ be a torsion-free lattice of $\operatorname{PSU}(1,1)$. We study rigidity properties of $\Gamma$-actions on the circle $S^1$ via foliated harmonic measures on the suspension bundles. We will show a curvature estimate and a Gauss-Bonnet formula for an $S^1$-connection obtained by taking the average of the flat connection on the suspension bundle 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.