Coarse and fine geometry of the Thurston metric

Abstract

We study the geometry of the Thurston metric on the Teichmüller space $\mathcal{T}(S)$ 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, $S_{1,1}$. 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.

50 pages, 14 figures. v6: Minor change in introduction. v5: Remark 5.5 adds info about Figure 0. v4: Minor correction in Thm 3.10. v3: Revised according to referee report. v2: Minor corrections

References in corpus (3)

• Maximally stretched laminations on geometrically finite hyperbolic manifolds
• The shadow of a Thurston geodesic to the curve graph
• The Myers-Steenrod theorem for Finsler manifolds of low regularity

Cited by in corpus (5)

• Generalized stretch lines for surfaces with boundary
• Local Rigidity of Teichmüller space with Thurston metric
• Optimal Lipschitz Maps on One-holed Tori and the Thurston Metric Theory of Teichmueller Space
• Ideal triangles, hyperbolic surfaces and the Thurston metric on Teichm{ü}ller space
• Minimal stretch maps between Euclidean triangle?