Harmonic measures and rigidity for surface group actions on the circle
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866911110505955328 |
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| author | Adachi, Masanori Matsuda, Yoshifumi Nozawa, Hiraku |
| author_facet | Adachi, Masanori Matsuda, Yoshifumi Nozawa, Hiraku |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_08411 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Harmonic measures and rigidity for surface group actions on the circle Adachi, Masanori Matsuda, Yoshifumi Nozawa, Hiraku Geometric Topology Complex Variables Differential Geometry Dynamical Systems 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. |
| title | Harmonic measures and rigidity for surface group actions on the circle |
| topic | Geometric Topology Complex Variables Differential Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2207.08411 |