Volumes of end-periodic mapping tori

TL;DR

This paper provides an overview connecting the dynamics of end-periodic homeomorphisms on infinite-type surfaces to the geometry of their associated mapping tori, extending Brock's finite-type theorem to this setting.

Contribution

It introduces two recent results that relate the dynamics of end-periodic homeomorphisms to the geometry of their mapping tori, generalizing known finite-type theorems.

Findings

Relation between end-periodic homeomorphisms and mapping torus geometry

Extension of Brock's theorem to infinite-type surfaces

Illustrative overview of recent results in the field

Abstract

In this expository paper, we provide an intuition and illustration-driven overview of two recent results that tie the dynamics of certain homeomorphisms of infinite-type surfaces, called end-periodic homeomorphisms, to the geometry of their associated (compactified) mapping tori. These results are analogues of a theorem of Brock in the finite-type setting for mapping tori of pseudo-Anosov homeomorphisms.