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Full Description
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. The study of this group via such realted combinatorial methods as its action on the Dessins and on certain fundamental groups of moduli spaces was initiated by Alexander Grothendieck in his unpublished Esquisse d'un Programme, and developed by many of the mathematicians who have contributed to this volume. The various articles here unite all of the basics of the subject as well as the most recent advances. Researchers in number theory, algebraic geometry or related areas of group theory will find much of interest in this book.
Contents
1. Noncongruence subgroups, covers, and drawings B. Birch; 2. Dessins d'enfant on the Riemann sphere L. Schneps; 3. Dessins from a geometric point of view J-M. Couveignes and L. Granboulan; 4. Maps, hypermaps and triangle groups G. Jones and D. Singerman; 5. Fields of definition of some three point ramified field extensions G. Malle; 6. On the classification of plane trees by their Galois orbit G. Shabat; 7. Triangulations M. Bauer and C. Itzykson; 8. Dessins d'enfant and Shimura varieties P. Cohen; 9. Horizontal divisors on arithmetic surfaces associated with Belyi uniformizations Y. Ihara; 10. Algebraic representation of the Teichmüller spaces K. Saito; 11. On the embedding of Gal(Q/Q) into GT Y. Ihara; Appendix M. Emsalem and P. Lochak; 12. The Grothendieck-Teichmüller group and automorphisms of braid groups P. Lochak and L. Schneps; 13. Moore and Seiberg equations, topological field theories and Galois theory P. Degiovanni.
