Published online by Cambridge University Press: 04 August 2010
Introduction. Belyi's Theorem and the associated theory of dessins d'enfants has recently played an important rôle in Galois theory, combinatorics and Riemann surfaces. In this mainly expository article we describe some consequences for Riemann surface theory. It is organised as follows: in §1 we describe the ideas of critical points and critical values which leads in §2 to the definition of a Belyi function. In §3 we state Belyi's Theorem and define a Belyi surface. In §4 the close connection with triangle groups is described and this leads in §5 to an account of maps and hypermaps (or dessins d'enfants) on a surface. These are closely related to Belyi surfaces but in order to investigate this connection we introduce the idea of a smooth Belyi surface and a platonic surface in §6. The only new result in this article is Theorem 7.1 which describes the connection between regular maps and thir underlying Riemann sufaces.
The definition of a Riemann surface is given in Beardon's lecture in this volume and the important ideas of uniformization are also discussed there. In the definition of a Riemann surface the transition functions are complex analytic and this allows us to describe most of the important ideas of complex analysis on a Riemann surface.