HYPERBOLIC STRUCTURES ON SURFACES

We give a brief introduction to hyperbolic structures on surfaces. Using the concepts of developing map and holonomy, we sketch a proof that every surface equipped with a complete hyperbolic metric is isometric to a quotient of ℍ by a Fuchsian group. We then define Teichmüller spaces and explain Fenchel-Nielsen coordinates. Finally, we introduce mapping class groups and show that they act properly discontinuously on Teichmüller space.