Belyi functions: Examples, properties and applications

Abstract

Let $X$ be a Riemann surface, and $f : X \to \Cbar$ a non-constant meromorphic function on $X$ (here $\Cbar = \C \cup \{\infty\}$ is the complex Riemann sphere). The function $f$ is called a {\em Belyi function}, and the pair $(X,f)$, a {\em Belyi pair}, if $f$ is unramified outside $\{0,1,\infty\}$. The study of Belyi functions, otherwise called the theory of {\em dessins d'enfants}, provides a link between many important theories. First of all, it is related to Riemann surfaces, as follows from the definition. Then, to Galois theory since, according to the Belyi theorem, a Belyi function on $X$ exists if and only if $X$ is defined over the field $\Qbar$ of algebraic numbers. It is also related to combinatorics of maps, otherwise called embedded graphs, since $f^{-1}([0,1])$ is a graph drawn on the two-dimensional manifold underlying $X$. Therefore, certain Galois invariants can be expressed in purely combinatorial terms. More generally, many properties of functions, surfaces, fields, and groups in question may be ``read from'' the corresponding pictures, or sometimes constructed in a ``picture form''. Group theory is related to all the above subjects and therefore plays a central role in this theory.