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

Masanori Adachi, Yoshifumi Matsuda, Hiraku Nozawa

TL;DR

The paper generalizes Milnor–Wood-type rigidity results for surface group actions on $S^1$ to general lattices in $\mathrm{PSU}(1,1)$ by using harmonic measures on suspension foliations and Thurston's $S^1$-connection. It proves a Gauss–Bonnet formula for the associated connection, showing $e(\rho)=\frac{1}{2\pi}\int_{\Sigma}K(z)\operatorname{vol}(dz)$ with $|K|\le 1$, and uses equality cases to deduce semiconjugacy to the Fuchsian action via an equivariant map $\mathfrak m:S^1\to S^1$. The work extends the Euler-number framework of BIW to orbifold settings with torsion, derives a cusp/Seifert-invariant refinement (EHN inequalities), and establishes rigidity for foliations with maximal Euler number, connecting to Matsumoto–Burger–Iozzi–Wienhard-type results. These results have implications for transverse foliations on Seifert $3$-orbifolds and provide a robust, geometric approach to rigidity phenomena in group actions on the circle.

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.

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

TL;DR

The paper generalizes Milnor–Wood-type rigidity results for surface group actions on to general lattices in by using harmonic measures on suspension foliations and Thurston's -connection. It proves a Gauss–Bonnet formula for the associated connection, showing with , and uses equality cases to deduce semiconjugacy to the Fuchsian action via an equivariant map . The work extends the Euler-number framework of BIW to orbifold settings with torsion, derives a cusp/Seifert-invariant refinement (EHN inequalities), and establishes rigidity for foliations with maximal Euler number, connecting to Matsumoto–Burger–Iozzi–Wienhard-type results. These results have implications for transverse foliations on Seifert -orbifolds and provide a robust, geometric approach to rigidity phenomena in group actions on the circle.

Abstract

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

Paper Structure

This paper contains 13 sections, 13 theorems, 54 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgments.
  3. Convention.
  4. Harmonic measures on the suspension foliation
  5. The Euler number of actions of lattices on $S^1$
  6. Definition of the Euler number
  7. The Seifert invariant and the translation number
  8. The Euler number and the translation number
  9. The Gauss--Bonnet formula for the connection associated with a harmonic measure
  10. Construction of the $S^1$-connection
  11. Proof of the Gauss--Bonnet formula
  12. Applications to rigidity
  13. Eisenbud--Hirsch--Neumann's inequality

Key Result

Theorem 1.1

Let $\Gamma$ be a lattice in $\operatorname{PSU}(1,1)$ with corresponding orbifold $\Sigma := \Gamma \backslash \mathbb D$ and $\rho : \Gamma \to \operatorname{Homeo}_{+}(S^1)$ a homomorphism. Assume that $\rho (\Gamma)$ has no finite orbit in $S^1$. Take a harmonic measure $\mu$ on the suspension f

Figures (1)

  • Figure 1: Subsurface $\Sigma^s$

Theorems & Definitions (26)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Definition 2.1
  • Lemma 2.2
  • Lemma 3.1
  • ...and 16 more