Counting Geodesics, Teichmuller Space, and Random Hyperbolic Surfaces
Scott A. Wolpert · Notices of the American Mathematical Society · 2021
The study of geodesics provides a theme for understanding hyperbolic metrics on finite-area surfaces, as well as the geometry of the moduli space of Riemann surfaces. We begin with the fundamentals of the Teichmüller theory of hyperbolic surfaces. Then we describe how Thurston’s random geodesic metric generalizes to the pressure metric in higher Teichmüller theory and how Mirzakhani’s recursive integration scheme is a tool for understanding random finite-area hyperbolic surfaces.