Matsumoto dichotomy on foliated $S^1$-bundles

TL;DR

This paper investigates the Matsumoto dichotomy for ergodic harmonic measures on foliated circle bundles over hyperbolic manifolds, showing restrictions on actions with non-discrete images and fixed points.

Contribution

It extends the Matsumoto dichotomy to actions of hyperbolic manifold groups, demonstrating that certain actions cannot admit Type I Matsumoto maps under specific conditions.

Findings

Actions with non-discrete images cannot admit Type I Matsumoto maps.

The study addresses a question posed by Matsumoto regarding harmonic measures.

Provides new insights into the structure of group actions on the circle in the context of foliated bundles.

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.