Calculating and Drawing Belyi Pairs

Abstract

The paper contains a survey of the current state of a constructive part in dessin d’enfants theory. Namely, it is devoted to actual establishing the correspondence between the Belyi pairs and their combinatorial-topological representations. This correspondence is established in terms of equivalence of categories, and all relevant categories are introduced. Several connections with arithmetic are discussed. One of the sections presents a possible generalization of the theory, in which three branch points, allowed for the Belyi functions, are replaced by four. Several directions for further research are presented. Bibliography: 80 titles.