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Zbl 1076.14040
Oesterle, Joseph
Dessins d'enfants. (Dessins d'enfants.) (French)
Bourbaki seminar. Volume 2001/2002. Exposes 894–908. Paris: Societe Mathematique de France (ISBN 2-85629-149-X/pbk). Astérisque 290, 285-305, Exp. No. 907 (2003).
Grothendieck's dessins d'enfants are closely connected to the study of coverings of the three times punctured sphere, and such coverings can be considered from many different points of view. In this survey it is shown how all of them are equivalent, and how the absolute Galois group acts on these objects.
Reviewer: Ernesto Girondo (Madrid]
MR2074061 (2006c:14031) Oesterle, Joseph(F-PARIS6-IMJ) Dessins d'enfants. (French. French summary) Seminaire Bourbaki. Vol. 2001/2002. Asterisque No. 290 (2003), Exp. No. 907, ix, 285–305. 14G32 (14E20 14H30)
From the text (translated from the French): "In 1984, A. Grothendieck presented a research program,
entitled Esquisse d'un programme' (published in 1997 [in Geometric Galois actions, 1, 5--48, Cambridge Univ. Press, Cambridge, 1997; MR1483107 (99c:14034)]), as part of his application for a position at the CNRS (a position that he would hold until his retirement in 1988). In his program Grothendieck used the term dessin d'enfant' (in its ordinary sense) as a visual analogue of certain cell maps;
he explained that every finite oriented map is realized canonically over a complex algebraic curve' and that the Galois group of $\overline{\bold Q}$ over $\bold Q$ acts on the category of these maps in a natural way':
one derives this by comparing various approaches to the study of coverings of $\bold P_1 - \{0,1,\infty\}$.
Since then, the term `dessin d'enfant' has been used often, by various authors in various mathematical senses,
to denote objects (or isomorphism classes of objects) arising in those approaches.
In this paper we do not try to define the term; we content ourselves with using it to denote the theory as a whole.
"Here are some reasons why one should pay particular attention to finite coverings of the curve $\bold P_1 - \{0,1,\infty\}$: "(a) It is the simplest algebraic curve whose fundamental group is not commutative. "(b) It has many coverings over $\overline{\bold Q}$: according to a theorem of Belyi(, every integral algebraic curve over $\overline{\bold Q}$ has an open Zariski set that is realized as such a covering. "(c) It is identified with the moduli space $M_{0,4}$ of genus-0 curves equipped with four marked points. The study of the action of ${\rm Gal}(\overline{\bold Q}/\bold Q)$ on its $\pi_1$ is the starting point for the study of the Grothendieck-Teichmüller tower (consisting of the fundamental groupoids of all the moduli spaces $M_{g,n}$ on which ${\rm Gal}(\overline{\bold Q}/\bold Q)$ acts).''