Coarse and fine geometry of the Thurston metric

TL;DR

This paper explores the geometric properties of the Thurston metric on Teichmüller space, providing detailed insights especially for once-punctured tori, including geodesic behavior and an analogue of Royden's theorem.

Contribution

It offers a comprehensive analysis of the Thurston metric's geometry, focusing on both coarse and fine structures, with new results for the once-punctured torus case.

Findings

Detailed description of geodesic behavior in the Thurston metric

Results on the coarse geometry applicable to all finite type surfaces

An analogue of Royden's theorem for the Thurston metric

Abstract

We study the geometry of the Thurston metric on the Teichm\"uller space $\mathcal{T}(S)$ of hyperbolic structures on a surface $S$. Some of our results on the coarse geometry of this metric apply to arbitrary surfaces $S$ of finite type; however, we focus particular attention on the case where the surface is a once-punctured torus, $S_{1,1}$. In that case, our results provide a detailed picture of the infinitesimal, local, and global behavior of the geodesics of the Thurston metric, as well as an analogue of Royden's theorem.