Dessins d'Enfants on Riemann Surfaces

Provides basic material about maps and hypermaps on Riemann surfacesPresents many elementary and less elementary examples of Galois actions on dessins and their algebraic curves
Emphasises the role of group theory in the classification of regular maps, regular dessins, and quasiplatonic surfaces
Explains the links between the theory of dessins and other areas of arithmetic and geometry

This volume provides an introduction to dessins d'enfants and embeddings of bipartite graphs in compact Riemann surfaces. The first part of the book presents basic material, guiding the reader through the current field of research. A key point of the second part is the interplay between the automorphism groups of dessins and their Riemann surfaces, and the action of the absolute Galois group on dessins and their algebraic curves. It concludes by showing the links between the theory of dessins and other areas of arithmetic and geometry, such as the abc conjecture, complex multiplication and Beauville surfaces.

Dessins d'Enfants on Riemann Surfaces will appeal to graduate students and all mathematicians interested in maps, hypermaps, Riemann surfaces, geometric group actions, and arithmetic.