Thurston's metric on Teichm\"uller space and isomorphisms between Fuchsian groups

TL;DR

This paper explores the relationship between Thurston's metric on Teichmüller space and Fuchsian group isomorphisms, providing new formulas and updating existing results on boundary isomorphisms.

Contribution

It introduces a new formula for Thurston's asymmetric metric on punctured surfaces and refines results on boundary isomorphisms of Fuchsian groups.

Findings

Derived a new formula for Thurston's metric on punctured surfaces

Updated results on boundary isomorphisms of Fuchsian groups

Linked Thurston's metric with Sorvali's ideas on Fuchsian group isomorphisms

Abstract

The aim of this paper is to relate Thurston's metric on Teichm\"uller space to several ideas initiated by T. Sorvali on isomorphisms between Fuchsian groups. In particular, this will give a new formula for Thurston's asymmetric metric for surfaces with punctures. We also update some results of Sorvali on boundary isomorphisms of Fuchsian groups.