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. Historical and introductory background.- Graph embeddings.- Dessins and triangle groups.- Galois actions.- Quasiplatonic surfaces, and automorphisms.- Regular maps.- Regular embeddings of complete graphs.- Wilson operations.- Further examples.- Arithmetic aspects.- Beauville surfaces.
"The book under review provides a very useful insight into the theory of dessins, and on its deep relationship with a number of mathematical distant fields. It is excellently written, and the Chapters are somewhat independent, in the sense that they have their own abstract, keywords and references. Of course, it is worth noting that the authors have contributed themselves to many of the results included in the book, most of which are really recent." (José Javier Etayo, zbMATH, August, 2017)
"This book allows the reader familiar with the analytic aspects of the theory of Riemann surfaces to develop a feeling for the arithmetic aspects of these objects. ... The reader interested in diving deeper into any of these aspects will most certainly find this well-written book to be of much help." (Ariyan Javanpeykar, Mathematical Reviews, January, 2017)
"The book is intended for properly trained mathematics graduate students or researchers. ... The text is also regularly interrupted by formulations of exercises which ask to prove some intermediate result or apply a definition in a particular situation (some brief hints are added at the end of the book). The book can thus be used as a textbook for a course on this fascinating topic." (Adhemar Bultheel, European Mathematical Society, euro-math-soc.eu, April, 2016)
