Hyperbolic Structures on Surfaces

In this thesis, we study hyperbolic structures on surfaces, which provide a way to endow a surface with a geometry that locally resembles the hyperbolic plane. We start with a brief review of hyperbolic geometry and a characterization of hyperbolic structures. Then, we study the space of all marked hyperbolic structures on a surface, called the Teichmüller space of the surface, and we see how a surface's mapping class group acts on its Teichmüller space. Finally, we generalize our study to a space of PSL(2,R)-representations of a surface's fundamental group. We will state a conjecture of Goldman on the dynamics of the mapping class group action on this representation space, and we will discuss Marché and Wolff's proof of Goldman's conjecture in the case of a closed surface of genus 2.