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

2025/04/23 by Adachi, Masanori, Matsuda, Yoshifumi, Nozawa, Hiraku
#22D50 #31A15 #53C12 #57M60 #57R20 #58D19 #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2504.16336

Abstract

Thurston proposed, in part of an unfinished manuscript, to study surface group actions on S¹ by using an S¹-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 PSU(1,1) on S¹. We prove the Gauss--Bonnet formula for the S¹-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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