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.

中文翻译：

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瑟斯顿度量的粗细几何

我们研究了表面上双曲结构的 Teichmüller 空间上的 Thurston 度量的几何 $新元 . 我们关于该度量的粗几何的一些结果适用于任意表面 $新元 有限类型；但是，我们特别关注表面是曾经被刺穿的环面的情况。在这种情况下，我们的结果提供了瑟斯顿度量的测地线的无穷小、局部和全局行为的详细图片，以及罗伊登定理的类似物。