Ideal triangles, hyperbolic surfaces and the Thurston metric on Teichm{\"u}ller space

TL;DR

This paper surveys the hyperbolic geometry of surfaces, Teichmüller spaces, and Thurston's metric, focusing on stretch lines and their properties, and explores analogies with Teichmüller's metric, highlighting open questions in the field.

Contribution

It provides a comprehensive overview of Thurston's metric on Teichmüller space, emphasizing the behavior of stretch lines and their relation to classical metrics, with new insights and open problems.

Findings

Analysis of stretch lines as geodesics for Thurston's metric

Identification of analogies between Thurston's and Teichmüller's metrics

Presentation of open questions in hyperbolic surface geometry

Abstract

These are notes on the hyperbolic geometry of surfaces, Teichm{\"u}ller spaces and Thurston's metric on these spaces. They are associated with lectures I gave at the Morningside Center of Mathematics of the Chinese Academy of Sciences in March 2019 and at the Chebyshev Laboratory of the Saint Petersburg State University in May 2019. In particular, I survey several results on the behavior of stretch lines, a distinguished class of geodesics for Thurston's metric and I point out several analogies between this metric and Teichm{\"u}ller's metric. Several open questions are addressed. The final version of these notes will appear in the book "Moduli Spaces and Locally Symmetric Spaces", edited by L. Ji and S.-T. Yau, International Press, 2021.