Metric weak proper discontinuity for pseudo-Anosov maps

TL;DR

This paper introduces a metric version of weak proper discontinuity for pseudo-Anosov maps and demonstrates the existence of many unbounded quasi-morphisms on certain homeomorphism groups, revealing new algebraic properties.

Contribution

It develops a metric framework for WPD in the context of pseudo-Anosov maps and applies it to construct unbounded quasi-morphisms on homeomorphism groups.

Findings

Existence of unbounded quasi-morphisms on homeomorphism groups.

Development of a metric version of WPD for pseudo-Anosov maps.

Application to groups of homeomorphisms of surfaces.

Abstract

We present a metric version of weak proper discontinuity (WPD) for pseudo-Anosov maps on surfaces. As an application, we show that there are plenty of quasi-morphisms on the homeomorphism group of a torus or a hyperbolic surface that are unbounded on certain homeomorphisms with large fixed set.