Hyperbolicity in Teichm\"uller space

TL;DR

This paper reviews key properties of Teichmüller geodesics, demonstrating their non-backtracking, fellow traveling in the thick part, and triangle edge proximity, highlighting hyperbolic-like behavior in Teichmüller space.

Contribution

It organizes existing results on Teichmüller geodesics and establishes new insights into their geometric properties, especially regarding their hyperbolic-like behavior.

Findings

Teichmüller geodesics do not backtrack.

Geodesic segments in the thick part exhibit fellow traveling.

Edges passing through the thick part are close to other edges in geodesic triangles.

Abstract

We review and organize some results describing the behavior of a Teichm\"uller geodesic and draw several applications: 1) We show that Teichm\"uller geodesics do not back track. 2) We show that a Teichm\"uller geodesic segment whose endpoints are in the thick part has the fellow travelling property. This fails when the endpoints are not necessarily in the thick part. 3) We show that if an edge of a Teichm\"uller geodesic triangle passes through the thick part, then it is close to one of the other egdes.