$\mathrm{SL}_2(\mathbb{R})$-dynamics on the moduli space of one-holed tori

TL;DR

This paper investigates the dynamics of the $ ext{SL}_2( ext{R})$ action on the moduli space of dilation tori with one boundary, revealing that orbits are either closed or dense, and Teichmüller flow orbits escape to infinity.

Contribution

It characterizes the orbit structure of the $ ext{SL}_2( ext{R})$-action on a specific moduli space of dilation tori, including orbit closure properties and flow behavior.

Findings

Every orbit is either closed or dense.

Teichmüller flow orbits escape to infinity.

Provides a classification of orbit types in this moduli space.

Abstract

We study the $\mathrm{SL}_2(\mathbb{R})$-action on the moduli space of (triangulable) dilation tori with one boundary component. We prove that every orbit is either closed or dense, and that every orbit of the Teichmuller flow escapes to infinity.