Riemann Surfaces and Covering Spaces

Abstract

In this chapter, we shall discuss the automorphisms of Riemann surfaces as a preview of the higher-dimensional results to come later. In no sense are we going to try to survey completely the enormous collection of results on the subject obtained in the nineteenth century (cf. [Farkas/Kra 1992], and historically [Fricke/Klein 1897]) nor the continuing investigation of the subject up to our own time. Even less shall we explore the interaction of the theory of Riemann surface automorphisms with number theory, dynamical systems, and so on. Rather, we are going to focus concretely on the circle of ideas involving invariant metrics, since that subject will be one of our major themes in higher dimensions.