The Teichmüller Space

Abstract

We introduce the Teichmüller space ℐ_gbased on marked Riemann surfaces. Our main goal is to construct certain rather simple (6g - 6)-parameter families of compact Riemann surfaces of genusgand to show that they are models of ℐ_g in a natural way. The construction has already been outlined in Section 1.7 and gives rise to the Fenchel-Nielsen parameters. In the subsequent chapters we work with these models rather than ℐ_g. The various models are real analytically equivalent and we shall use them to define the real analytic structure of ℐ_g. In Section 6.4 we briefly discuss two distance functions on ℐ_g. In Sections 6.5 and 6.6 we study the Teichmüller modular group. In the final two sections we compare the Fenchel-Nielsen parameters with other parameters which are frequently used in the literature.