Dessins d'Enfants on Riemann SurfacesThis 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. |
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abelian absolute Galois group acts transitively algebraic curve algebraic number AutM automorphism group Beauville Beauville group Beauville structure Beauville surfaces Bely˘ı function bipartite map black vertices coefficients commutator compact Riemann surface conjugacy classes conjugate coprime corresponding defined over Q dessin of type Dessins d’Enfants edges elements elliptic curve epimorphism equation equivalent Example Exercise Fermat curve field of definition finite group fixed points Fuchsian groups function ˇ fundamental region G acts G Š Galois Actions Galois conjugate generalised genus g group G D holomorphic Hurwitz hyperbolic hypermaps induced integer isomorphic Lemma Math Mathematical meromorphic functions moduli monodromy group normal subgroup number field orbits pairs permutation group polynomial prime power proof quasiplatonic curves quotient regular dessins regular embeddings regular maps Schneps Sect semidirect product shows Springer Theorem triangle group triple unique valency vertex Wilson operations Wolfart