The Grothendieck Theory of Dessins d'Enfants

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The focus of this book is on combinatorial objects, dessins d'enfants, which are 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 articles contained here unite basic elements of the subject with recent advances. 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, Teichmüller and moduli spaces. Researchers in number theory, algebraic geometry or related areas of group theory will find much of interest in this book.

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Table of 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.

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