Bounded combinatorics and the Lipschitz metric on Teichm\"uller space

TL;DR

This paper investigates geodesics in Teichmüller space with Thurston's Lipschitz metric, demonstrating their stability and contraction properties when endpoints have bounded combinatorics, using length ratios of finitely many curves for estimates.

Contribution

It establishes coboundedness and strong contraction of geodesics with bounded combinatorics in Lipschitz metric, advancing understanding of Teichmüller space geometry.

Findings

Geodesics with bounded combinatorics are cobounded.

Closest-point projections to these geodesics are strongly contracting.

Lipschitz distance can be estimated via finitely many curve length ratios.

Abstract

Considering the Teichm\"uller space of a surface equipped with Thurston's Lipschitz metric, we study geodesic segments whose endpoints have bounded combinatorics. We show that these geodesics are cobounded, and that the closest-point projection to these geodesics is strongly contracting. Consequently, these geodesics are stable. Our main tool is to show that one can get a good estimate for the Lipschitz distance by considering the length ratio of finitely many curves.