Harmonic measures and rigidity for transverse foliations on Seifert $3$-manifolds
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arXiv
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| Format: | Preprint |
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2025
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| author | Adachi, Masanori Matsuda, Yoshifumi Nozawa, Hiraku |
| author_facet | Adachi, Masanori Matsuda, Yoshifumi Nozawa, Hiraku |
| contents | Thurston proposed, in part of an unfinished manuscript, to study surface group actions on $S^1$ by using an $S^1$-connection on the suspension bundle obtained from a harmonic measure. Following the approach and previous work of the authors, we study the actions of general lattices of $\mathrm{PSU}(1,1)$ on $S^1$. We prove the Gauss--Bonnet formula for the $S^1$-connection associated with a harmonic measure, and show that a harmonic measure on the suspension bundle of the action with maximal Euler number has rigidity, having a form closely related to the Poisson kernel. As an application, we prove a semiconjugacy rigidity for foliations with maximal Euler number, which is analogous to theorems due to Matsumoto, Minakawa and Burger--Iozzi--Wienhard. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_16336 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Harmonic measures and rigidity for transverse foliations on Seifert $3$-manifolds Adachi, Masanori Matsuda, Yoshifumi Nozawa, Hiraku Geometric Topology 22D50, 53C12, 57M60, 58D19, 57R20, 31A15 Thurston proposed, in part of an unfinished manuscript, to study surface group actions on $S^1$ by using an $S^1$-connection on the suspension bundle obtained from a harmonic measure. Following the approach and previous work of the authors, we study the actions of general lattices of $\mathrm{PSU}(1,1)$ on $S^1$. We prove the Gauss--Bonnet formula for the $S^1$-connection associated with a harmonic measure, and show that a harmonic measure on the suspension bundle of the action with maximal Euler number has rigidity, having a form closely related to the Poisson kernel. As an application, we prove a semiconjugacy rigidity for foliations with maximal Euler number, which is analogous to theorems due to Matsumoto, Minakawa and Burger--Iozzi--Wienhard. |
| title | Harmonic measures and rigidity for transverse foliations on Seifert $3$-manifolds |
| topic | Geometric Topology 22D50, 53C12, 57M60, 58D19, 57R20, 31A15 |
| url | https://arxiv.org/abs/2504.16336 |