The shadow of a Thurston geodesic to the curve graph

TL;DR

This paper investigates the geometry of Thurston geodesics in Teichmuller space, revealing that their shadows in the curve graph are quasi-geodesics, but with differences in short curve sets compared to Teichmuller geodesics.

Contribution

It demonstrates that the shadow of a Thurston geodesic to the curve graph is a reparametrized quasi-geodesic and compares it to Teichmuller geodesics, highlighting differences in short curves.

Findings

Shadow of Thurston geodesic is a reparametrized quasi-geodesic

Short curves differ between Thurston and Teichmuller geodesics

Provides geometric insights into Thurston metric and geodesic behavior

Abstract

We study the geometry of the Thurston metric on Teichmuller space by examining its geodesics and comparing them to Teichmuller geodesics. We show that, similar to a Teichmuller geodesic, the shadow of a Thurston geodesic to the curve graph is a reparametrized quasi-geodesic. However, we show that the set of short curves along the two geodesics are not identical.