Hyperbolic Structures on Surfaces and Geodesic Currents

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 Teichmüller 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 Teichmüller 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 Teichmüller 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.