Harmonic measures and rigidity for surface group actions on the circle

We study rigidity properties of actions of a torsion-free lattice of 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 and Wienhard.22D50, 53C12; 31A15, 57M60, 57R20, 58D19where vol denotes the hyperbolic volume form on WD n D.In the case where is a uniform lattice, the Gauss-Bonnet formula is classical and is not new.The main point of this result is the estimate of the curvature, already stated in [23], and the behavior of the connection !near the cusps of , which is proved based on Harnack's inequality and the isoperimetric inequality.We remark that there is a similar curvature estimate in [5, 6] by Besson, Courtois and Gallot.As a corollary, we obtain an alternative proof of the generalization of the Milnor-Wood inequality and Matsumoto's rigidity theorem to torsion-free lattices due to Burger, Iozzi and Wienhard [7].