Hyperbolic Structures on Surfaces and Geodesic Currents
Hyperbolic Structures on Surfaces and Geodesic Currents
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表面上的双曲结构和测地线流
DOI：
10.1007/978-3-319-60940-9_3
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发表时间：
2017
期刊：
影响因子：
--
通讯作者：
C. Leininger
中科院分区：
文献类型：
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作者：
J. Aramayona;C. Leininger
本章包含课程“曲面上的双曲结构和测地线流”的讲义，作者在2012年9月在CRM（巴塞罗那）举行的自由群自同构暑期学校：几何，拓扑和动力学。注记的主要目的是用曲面上的测地流来说明Bonahon对Thurston的Teichmuller空间紧化的描述[4]。本章的计划如下。第3.2节讨论曲面上的双曲结构，解释了为什么一个具有完全双曲结构的曲面与H2的商等距于一个Fuchsian群。在第3.3节中，我们将回顾Teichmuller空间和测地线叠层的一些基本特征，并以关于Thurston紧化的“经典”构造的一些话结束。在3.4节中，我们将介绍测地流，并解释Bonahon对Teichmuller空间紧化的解释。最后，在3.5节中，我们将把测地线流的概念推广到其他场合，如曲面上的负曲度量、曲面上的平坦度量和自由群。
This chapter contains the lecture notes from the course “Hyperbolic structures on surfaces and geodesic currents”, given by the authors during the summer school on Automorphisms of Free Groups: Geometry, Topology, and Dynamics, held at the CRM (Barcelona) in September 2012. The main objective of the notes is to give an account of Bonahon’s description [4] of Thurston’s compactification of Teichmuller space in terms of geodesic currents on surfaces. The plan of the chapter is as follows. Section 3.2 deals with hyperbolic structures on surfaces, explaining why a surface equipped with a complete hyperbolic structure is isometric to the quotient of H2 by a Fuchsian group. In Section 3.3 we will review some basic features of Teichmuller spaces and measured geodesic laminations, ending with some words about the “classical” construction of Thurston’s compactification. In Section 3.4, we will introduce geodesic currents, and explain Bonahon’s interpretation of the compactification of Teichmuller space. Finally, in Section 3.5 we will present some generalizations of the notion of geodesic currents to other settings, such as negatively curved metrics on surfaces, flat metrics on surfaces, and free groups.