Harmonic measures and rigidity for transverse foliations on Seifert $3$-manifolds

Masanori Adachi; Yoshifumi Matsuda; Hiraku Nozawa

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arXiv · 2504.16336

Harmonic measures and rigidity for transverse foliations on Seifert $3$-manifolds

Masanori Adachi·Yoshifumi Matsuda·Hiraku Nozawa

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Abstract

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.

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BibTeX RIS

Masanori Adachi, Yoshifumi Matsuda, Hiraku Nozawa. 2025-04-23. Harmonic measures and rigidity for transverse foliations on Seifert $3$-manifolds. https://arxiv.org/abs/2504.16336

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