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Matsumoto dichotomy on foliated $S^1$-bundles

KyeongRo Kim, Hongjun Lee

TL;DR

The work investigates Matsumoto maps associated with circle actions arising from suspensions of hyperbolic group actions, focusing on the Matsumoto dichotomy that classifies ergodic harmonic measures as Type I (singleton) or Type II (full boundary). It proves a non-existence result: a non-discrete action with uniformly bounded fixed points cannot admit a Type I Matsumoto map, forcing Type II, and uses this to classify ergodic harmonic measures for foliated $S^1$-bundles over closed hyperbolic surfaces. This yields a complete dichotomy in the surface-bundle setting: discrete faithful actions produce a unique Type I measure, while non-discrete or non-faithful actions yield Type II, aligning with known rigidity phenomena. The results address questions posed by Matsumoto, connect universal circle actions to boundary dynamics, and have implications for the amenability of foliations and rigidity of group actions on $S^1$.

Abstract

Given an ergodic harmonic measure on a foliated circle bundle over a closed hyperbolic manifold, Matsumoto constructed a map from the fiber circle to the space of nonempty closed subsets of the boundary sphere of the universal cover of the base manifold. This map is well-defined at almost every point. Also, the map is equivariant under two actions of the fundamental group of the base manifold: the holonomy action on the fiber and the action on the space of closed subsets induced by the boundary sphere action. Matsumoto established a dichotomy for these maps, which corresponds to a dichotomy of ergodic harmonic measures. (Indeed, the Matsumoto dichotomy also concerns ergodic harmonic measures on compact hyperbolic laminations.) The dichotomy says that a Matsumoto map either maps each point to a singleton (Type I) or to the entire sphere (Type II). In this paper, we study actions of closed hyperbolic manifold groups on the circle in the context of the Matsumoto dichotomy. We essentially show that the suspension of any action with a non-discrete image cannot admit a Matsumoto map of type I under the condition of a uniformly bounded number of fixed points. As a consequence, we address a question posed in Matsumoto's paper.

Matsumoto dichotomy on foliated $S^1$-bundles

TL;DR

The work investigates Matsumoto maps associated with circle actions arising from suspensions of hyperbolic group actions, focusing on the Matsumoto dichotomy that classifies ergodic harmonic measures as Type I (singleton) or Type II (full boundary). It proves a non-existence result: a non-discrete action with uniformly bounded fixed points cannot admit a Type I Matsumoto map, forcing Type II, and uses this to classify ergodic harmonic measures for foliated -bundles over closed hyperbolic surfaces. This yields a complete dichotomy in the surface-bundle setting: discrete faithful actions produce a unique Type I measure, while non-discrete or non-faithful actions yield Type II, aligning with known rigidity phenomena. The results address questions posed by Matsumoto, connect universal circle actions to boundary dynamics, and have implications for the amenability of foliations and rigidity of group actions on .

Abstract

Given an ergodic harmonic measure on a foliated circle bundle over a closed hyperbolic manifold, Matsumoto constructed a map from the fiber circle to the space of nonempty closed subsets of the boundary sphere of the universal cover of the base manifold. This map is well-defined at almost every point. Also, the map is equivariant under two actions of the fundamental group of the base manifold: the holonomy action on the fiber and the action on the space of closed subsets induced by the boundary sphere action. Matsumoto established a dichotomy for these maps, which corresponds to a dichotomy of ergodic harmonic measures. (Indeed, the Matsumoto dichotomy also concerns ergodic harmonic measures on compact hyperbolic laminations.) The dichotomy says that a Matsumoto map either maps each point to a singleton (Type I) or to the entire sphere (Type II). In this paper, we study actions of closed hyperbolic manifold groups on the circle in the context of the Matsumoto dichotomy. We essentially show that the suspension of any action with a non-discrete image cannot admit a Matsumoto map of type I under the condition of a uniformly bounded number of fixed points. As a consequence, we address a question posed in Matsumoto's paper.

Paper Structure

This paper contains 9 sections, 16 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Matsumoto Dichotomy
  3. Compact hyperbolic $C^2$ foliations
  4. Harmonic measures
  5. Matsumoto Dichotomy
  6. The Matsumoto Map
  7. Discontinuity of Matsumoto maps
  8. Application: Matsumoto dichotomy on foliated $S^1$-bundles over closed hyperbolic surfaces
  9. Application: some remarks on Amenable foliations

Key Result

Theorem 1.1

Let $\Gamma$ be a subgroup of $\operatorname{Isom_+}(\mathbb{H}^n)$ such that $\mathbb{H}^n/\Gamma$ is a closed manifold, and $\rho:\Gamma\to \operatorname{Homeo}_+(S^1)$ an action of $\Gamma$ on $S^1$. Assume that $\rho$ satisfies one of the following conditions: Then, any ergodic harmonic measure of the suspension foliation of $\rho$ is of Type II.

Theorems & Definitions (28)

  • Theorem 1.1
  • Corollary 1.3
  • Theorem 2.1: Local Structure of Harmonic Measure
  • Theorem 2.2: Matsumoto12
  • Corollary 2.3: Matsumoto12
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Theorem 2.8: Matsumoto12
  • ...and 18 more