A geometric criterion to be pseudo-Anosov

TL;DR

This paper introduces a geometric criterion linking the properties of hyperbolic 3-manifolds and surface group extensions to identify when a surface homeomorphism is pseudo-Anosov, enhancing understanding of surface dynamics.

Contribution

It provides a new geometric criterion based on hyperbolic geometry and Gromov-hyperbolic extensions to determine pseudo-Anosov mapping classes.

Findings

A class is pseudo-Anosov if its geodesic representative is 'wide' in a hyperbolic 3-manifold.

The criterion connects surface group geometry with surface homeomorphism dynamics.

Applicable to mapping classes arising from fundamental group elements of surfaces.

Abstract

We establish a criterion for certain mapping classes of a surface homeomorphisms to be pseudo-Anosov in terms of the geometry of hyperbolic 3-manifolds and Gromov-hyperbolic surface group extensions. Specifically, any element of the fundamental group of a surface S gives rise to a mapping class on the punctured surface, and we show that such a class is pseudo-Anosov if its geodesic representative is "wide" in some hyperbolic 3-manifold homeomorphic to the trivial interval bundle over S.