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29 pages, 1994, appeared in "The Grothendieck Theory of Dessins d 'Enfant", London Math. Soc. Lecture Notes 200, Cambridge Univ.

Abstract:

Dessins d'enfants are combinatorial objects, namely drawings with vertices and edges on topological surfaces. Their interest lies in their relation with the set of algebraic curves defined over the closure of the rationals, and the corresponding action of the absolute Galois group on them....A dessin d'enfant is defined as a graph \Gamma embedded in a connected compact oriented surface X in such a way that the complement X \Gamma is homeomeomorphic to a disjoint union of open discs). It is not art but profound mathematics, that someone might argue that resembles modern abstract art. Grothendieck's Theory of Dessins d'Enfants is however much more than a child's, or any children's drawings! Thus, it has impact on Theoretical Physics in the field of Topological Quantum Field Theories (TQFT) as well as in the high-interest mathematical field of Galois Theory. Topics covered include: the explicity association of algebraic curves to dessins, the study of the action of the Galois group on the dessins, computation and combinatorics, relations with modular forms, geometry, generating functions, Teichm\"uller and moduli spaces. Researchers in number theory, algebraic geometry or related areas of group theory will find much of interest in this paper by L. Schneps on "Desssins d'Enfants on the Riemann sphere. pp. 47-78." Other papers from the same book in which the available (PDF) article appeared are: J. M. Couvienne and L. Granboulan. Dessins from a Geometric Point of view, pp. 79-114 Y. Ihara : "Embedding of $Gal(Q_bar/Q)$ into $GT$" pp.289-306 P. Lochak and L. Schneps: "The Grothendieck-Teichm\"uller group and automorphisms of braid groups", pp. 323-358 M. Bauer and C. Itzykson. Triangulations p.179-236. G. Jones and D. Singerman. Maps, Hypermaps and Triangle Groups, pp. 115- 142. P. Degiovanni: TQFT and Galois Theory. p.359-376. Keywords: groupe cartographique, clean dessins, valency lists, lecture notes,oriented hypermaps, two dessins, tangential base points, given dessin, valency classes, noncongruence subgroups, horizontal divisors, maximal cyclotomic field, moduli field, rational conformal field theories, fundamental groupoid, valeurs critiques, profinite completions, topological field theories, conformal blocks, ramification orders, braid groups, arithmetic surfaces, aux sommets, profinite group, triangle group, ramification indices, New York, Leila Schneps, Kyoji Saito, London Math, Academic Press, Ecole Normale, Jean-Marc Couveignes, Pierre Lochak, Alexander Zvonkin, Linda Keen, Tuomas Sorvali, Yoshihide Okumura, Annals of Math, Grothendieck's Esquisse, Konstruktive Galoistheorie, Leningrad Math, Nikolai Adrianov, Scott Wolpert. http://people.math.jussieu.fr/~leila/Fschneps.pdf

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Open access: free download for individual study http://people.math.jussieu.fr/~leila/Fschneps.pdf Original Documents Copyright@1988 by Alexander Grothendieck http://www.ams.org/tran/2000-352-07/S0002-9947-00-02347-3/S0002-9947-00-02347-3.pdf Cambridge Uni

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