Teichmüller Spaces of Compact Riemann Surfaces.

Abstract

In this thesis, we study the Teichmüller space of compact Riemann surfaces — the space of marked compact Riemann surfaces. First, we study a topological structure on the Teichmüller space of higher genus surfaces S g (g ≥ 2) through its description separately through Fenchel-Nielsen coordinates and as the space of discrete faithful representation of π 1 (S g ) into Isom(H 2 ) ∼ = PSL(2, R). Later we introduce the descrip- tion of the Teichmüller space using quasiconformal maps, which yields the Teichmüller metric on the Teichmüller space. Lastly, we briefly have a look at the Thurston com- pactification of the Teichmüller space.