Teichm¨uller Spaces of Compact Riemann Surfaces

Abstract

In this thesis, we study the Teichm¨uller space of compact Riemann surfaces — the space of marked compact Riemann surfaces. First, we study a topological structure on the Teichm¨uller space of higher genus surfaces Sg (g ≥ 2) through its description separately through Fenchel-Nielsen coordinates and as the space of discrete faithful representation of π1(Sg) into Isom(H2) ∼ =PSL(2,R). Later we introduce the description of the Teichm¨uller space using quasiconformal maps, which yields the Teichm¨uller metric on the Teichm¨uller space. Lastly, we briefly have a look at the Thurston compactification of the Teichm¨uller space.