COARSE AND FINE GEOMETRY OF THE THURSTON METRIC

Jing Tao,Kasra Rafi,Anna Lenzhen,David Dumas

doi:10.1017/fms.2020.3

Abstract

We study the geometry of the Thurston metric on the Teichmüller space of hyperbolic structures on a surface$S$. Some of our results on the coarse geometry of this metric apply to arbitrary surfaces$S$of finite type; however, we focus particular attention on the case where the surface is a once-punctured torus. In that case, our results provide a detailed picture of the infinitesimal, local, and global behavior of the geodesics of the Thurston metric, as well as an analogue of Royden’s theorem.

Highlights

Let S be a surface of finite type, i.e. the complement of a finite set in a compact surface
We show that the geodesic from X to Y is unique when Λ(X, Y ) is of type (b) or (c), and when it has type (a), the envelope has a simple, explicit description
For curves that interact with the maximally stretched lamination Λ(X, Y ), meaning they belong to the lamination or intersect it essentially, we show that becoming short on a geodesic with endpoints in the thick part of T(S) is equivalent to the presence of large twisting

Summary

Introduction

Let S be a surface of finite type, i.e. the complement of a finite set in a compact surface. Note that in part (iv) of the theorem, a chain-recurrent lamination properly containing α has multiple completions, but they all give the same stretch path (see Corollary 2.3) This theorem highlights a distinction between two cases in which the dTh-geodesic from X to Y is unique—the cases (b) and (c) discussed above. Such in-envelopes are limiting cases of the envelopes of type (iv) where X is replaced by a lamination. Returning to the case of an arbitrary surface S of finite type, in Section 3, we establish results on the coarse geometry of Thurston metric geodesic segments. The notion of the maximally stretched lamination for a pair of hyperbolic surfaces has been generalized to higher-dimensional hyperbolic manifolds [Kas, GK17] and to vector fields on H2 equivariant for convex cocompact subgroups of PSL(2, R) [DGK16]

Background

Twisting parameter along a Thurston geodesic

Thurston norm and rigidity