ABSTRACT
ABSTRACT: Let X be a Riemann surface, and f : X → C a non-constant meromorphic function on X (here C = C∪{∞} is the complex Riemann sphere). The function f is called a Belyi function, and the pair (X , f ), a Belyi pair, if f is unramified outside {0, 1, ∞}. The study of Belyi functions, otherwise called the theory of 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 Q 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.
