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There was an observation that any algebraic curve over Q can be rationally mapped to P^1 without three points and this led Grothendieck to define a special class of these mappings, called the Children's Drawings, or, in French, Dessins d'Enfants (his quote was something like "things as simple as the drawings...").

I'm not an expert in this field, so could somebody please write more about those dessins, and what things they are related to? What's their importance? How does the cartographic group act on these?

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Is there any way to see in any geometric way the shuffle relations (satisfied by multi-zeta values) or the "automorphisms of the associator" (arising from Drienfeld's work on quasi-Hopf algebras) that appear in the larger body of work related to dessins? –  Dr Shello May 2 '11 at 22:47

9 Answers 9

You can find a nice introduction to them in The best rejected proposal ever, followed up by some discussions about the cartographer's groups and more.

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In the two years since this answers was posted the link-address has changed. It is now neverendingbooks.org/the-best-rejected-proposal-ever –  Nick Gill Jan 22 '13 at 10:39

This is not my area at all, but the Notices published a piece a few years ago called "What is a dessin d'enfant?

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There's also a Wikipedia article that attempts to answer this question.

By the way, the original invention of these things was much earlier than Grothendieck. See Klein’s dessins d’enfant and the buckyball on lieven le bruyn's blog.

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That's good to know! They really found a lot about algebraic curves in 19th century. –  Ilya Nikokoshev Oct 22 '09 at 21:24

In Leila Schneps - Dessins d'enfants on the Riemann Sphere you can find a definition of dessins and the Grothendieck correspondence between Belyi morphisms and dessins. It also has pictures of how the cartographic group acts on the flag set of a dessin.
Grothendieck correspondence means that there is a bijection between isomorphism classes of dessins and isomorphism classes of Belyi morphisms (morphisms f:X->P1C which are ramified only over three points).

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Historically, one of the first papers on th subject is Drawing curves over number fields, by G.B. Shabat and V.A. Voevodsky (The Grothendieck Festschrift, Vol. III, 199–227, Progr. Math., 88), which I strongly recommend. Another nice and historical paper is Triangulations, by M. Bauer and C. Itzykson (many references: R.C.P. 25, Vol. 44 (1992), Discrete Math. 156 (1996), no. 1-3 or L. Schneps's book below). Both papers are also concerned with the (combinatorial) cellular decomposition of moduli spaces of curves. They appeared before L. Schneps book The Grothendieck theory of dessins d'enfants (London Mathematical Society Lecture Note Series, 200), which is now the main reference on the subject.

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What about Graphs on Surfaces and Their Applications by Lando and Zvonkin? BTW, the book edited by Schneps contains a lot of French ... –  Junyan Xu Dec 20 '11 at 3:06

There are many good answers to this question already. However it seems important to me that the contribution of Jones and Singerman to this subject is noted. These two British mathematicians from the University of Southampton wrote an important paper on this subject some time before Grothendieck wrote his Esquisse.

The paper in question is:

MR0505721 Zbl0391.05024 Jones, Gareth A.; Singerman, David Theory of maps on orientable surfaces. Proc. London Math. Soc. (3) 37 (1978), no. 2, 273–307.

The paper is beautifully written, and outlines the correspondence between maps on topological surfaces, maps on Riemann surfaces, and groups with certain distinguished generators. They do not consider the Galois action, this being the aspect of the area that so excited Grothendieck. Their notion of a map is a particular instance of a dessin d'enfant (these days a map is also known as a clean dessin), the more general notion of hypermap which was considered subsequently corresponds to the general dessin d'enfant.

A later paper, by Bryant and Singerman, extended the treatment to surfaces with boundary.

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Also might be interesting:

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 termdessin 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 $\overline{\bf Q}$ over $\bf Q$ acts on the category of these maps in a natural way': one derives this by comparing various approaches to the study of coverings of $\bf P_1 - \{0,1,\infty\}$. 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 $\bf P_1 - \{0,1,\infty\}$: "(a) It is the simplest algebraic curve whose fundamental group is not commutative. "(b) It has many coverings over $\overline{\bf Q}$: according to a theorem of Belyi(, every integral algebraic curve over $\overline{\bf Q}$ 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 ${\rm Gal}(\overline{\bf Q}/\bf Q)$ 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 ${\rm Gal}(\overline{\bf Q}/\bf Q)$ acts).''

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There is a french talk by Alexander Zvonkin which can be a good introduction to this subject as well.

If readers are interested I can translate parts of it in english.

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A very modern compendium of thoughts on this topic can be found here:

Theory of motives, homotopy theory of varieties, and dessins d'enfants: http://www.aimath.org/WWN/motivesdessins/motivesdessins.pdf

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