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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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  • $\begingroup$ facebook.com/pages/Dessin-denfant/107153052654583 $\endgroup$ May 2, 2011 at 14:31
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    $\begingroup$ 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? $\endgroup$
    – Dr Shello
    May 2, 2011 at 22:47
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    $\begingroup$ @GeraldEdgar Why? $\endgroup$
    – 54321user
    Sep 29, 2016 at 23:26

10 Answers 10

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Given a compact Riemann surface $X$, every holomorphic function on $X$ is constant. This is obvious if you think about holomorphic functions as locally conformal mappings, that is, transformations of $X$ into a plane which locally preserve angles ("infinitesimal similarities"). Such a transformation is continuous, thus has a compact image, but the image is also open, so it must be constant: collapses $X$ into a point. In other words, the ring $\mathcal{O}(X)$ of holomorphic functions on $X$ reduces to the constants $\mathcal{O}(X) = \mathbb{C}$.

That's why, in the theory of compact Riemann surfaces, one is interested in meromorphic functions, i. e., those with at least one pole. These functions constitute a field denoted by $\mathcal{M}(X)$, and can be seen as branched coverings $X \rightarrow \mathbb{S}^2$ of the Riemann sphere/complex projective line.

One can show that there always exist non-constant meromorphic functions on $X$ (Riemann's existence theorem). This means that every compact Riemann surface is a branched covering of the sphere. This can be used to show that the field $\mathcal{M}(X)$ is a finite field extension of $\mathcal{M}(\mathbb{S}^2) =\mathbb{C}(z)$ (this last field is the field of rational functions = polynomial fractions).

Moreover, one can show that each finite extension of $\mathbb{C}(z)$ gives rise to a unique compact Riemann surface, up to isomorphisms, with the given extension as its field of meromorphic functions (Dedekind-Weber theory of algebraic function fields in one variable). This compact Riemann surface can always be realized as an algebraic curve in the complex projective space $\mathbf{P}^3(\mathbb{C})$ (without singularities, naturally). This means that it is the set of zeroes of some homogeneous polynomials with complex coefficients, in projective space.

One interesting question is to ask when a compact Riemann surface $X$ can be given by an equation with coefficients in $\overline{\mathbb{Q}}$ (the algebraic closure of rationals). It is known that this is the case if and only if there is a meromorphic function $X \rightarrow \mathbb{S}^2$ with at most three critical values (Belyi's theorem). A function of this kind is a covering of the sphere which is branched over three points (or less).

Thus, the study of compact Riemann surfaces/smooth plane algebraic curves over the algebraic numbers $\overline{\mathbb{Q}}$ is reduced to the study of coverings of the sphere branched over three points, which we can assume to be the points $0, 1, \infty$.

These branched coverings $f: X \rightarrow \mathbb{S}^2$ can be given a geometric representation. The fiber of $0$, is a finite set of points in $X$, which can be marked as black points. Points in the fiber of $1$ are usually colored white. The preimage of the interval $[0,1]$ (as a curve joining $0$ and $1$) is given by a set of curves joining black and white points, alternatively. This graph on $X$, formed by black points, white points and curves, is the dessin d'enfant associated to the branched covering $f: X \rightarrow \mathbb{S}^2$.

It is a remarkable fact that the branched covering $f: X\rightarrow \mathbb{S}^2$ determines the Riemann surface structure of $X$ (by pullback of the same structure in $\mathbb{S}^2$), and so, compact Riemann surfaces over algebraic numbers (with a specified meromorphic function branched over at most three points) are just orientable compact surfaces with certain graphs in them (the graphs associated with branched coverings).

Now, Grothendieck was interested in the Galois group of $\overline{\mathbb{Q}}$ over $\mathbb{Q}$, which acts on the coefficients of the equation of an algebraic curve over $\overline{\mathbb{Q}}$, giving another curve of the same type. Considering the dessins associated to those curves, that group then transforms a dessin into another dessin, and so, dessins can be used to obtain a geometric interpretation of the absolute Galois group.

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    $\begingroup$ This is a terrific answer. $\endgroup$ Apr 5 at 0:24
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    $\begingroup$ Not every Riemann surface can be realized as a smooth plane curve (e.g. by the degree-genus formula). Also, in the deduction "branched cover determines complex structure so curves over $\bar{\mathbb{Q}}$ are just compact top. surfaces + a dessin d'enfants", is hidden the fact that a dessin d'enfants actually determines a branched cover, which is not immediately clear $\endgroup$
    – Chris
    Apr 5 at 12:13
  • $\begingroup$ True. I corrected the mistake about embedding curves in the plane and remarked that a dessin specifies the branched cover. $\endgroup$
    – Compacto
    Apr 5 at 22:44
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You can find a nice introduction to them in The best rejected proposal ever (updated link, 2016), followed up by some discussions about the cartographer's groups and more.

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    $\begingroup$ In the two years since this answers was posted the link-address has changed. It is now neverendingbooks.org/the-best-rejected-proposal-ever $\endgroup$
    – Nick Gill
    Jan 22, 2013 at 10:39
  • $\begingroup$ Wayback Machine link $\endgroup$ Apr 5 at 14:02
  • $\begingroup$ @TimothyChow, it's still up at the link that @‍NickGill posted, now edited into the main post by @‍YCor. $\endgroup$
    – LSpice
    Apr 5 at 19:10
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    $\begingroup$ @LSpice Yes, but sometimes I like to pre-emptively put in a Wayback Machine link for references that are not permanent (like a DOI). $\endgroup$ Apr 5 at 22:50
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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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  • $\begingroup$ That's good to know! They really found a lot about algebraic curves in 19th century. $\endgroup$ Oct 22, 2009 at 21:24
  • $\begingroup$ Your link doesn't work. Can you update it? $\endgroup$
    – draks ...
    Sep 29, 2016 at 17:56
  • $\begingroup$ Done. Thanks for catching the broken link. $\endgroup$ Sep 29, 2016 at 19:19
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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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    $\begingroup$ PDF available here... $\endgroup$
    – draks ...
    Mar 15, 2018 at 12:30
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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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    $\begingroup$ What about Graphs on Surfaces and Their Applications by Lando and Zvonkin? BTW, the book edited by Schneps contains a lot of French ... $\endgroup$
    – Junyan Xu
    Dec 20, 2011 at 3:06
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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. Exposés 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 $\overline{\mathbf Q}$ over $\mathbf Q$ acts on the category of these maps in a natural way’: one derives this by comparing various approaches to the study of coverings of $\mathbf P_1 - \{0,1,\infty\}$. 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 $\mathbf P_1 - \{0,1,\infty\}$:

“(a) It is the simplest algebraic curve whose fundamental group is not commutative.

“(b) It has many coverings over $\overline{\mathbf Q}$: according to a theorem of Belyi, every integral algebraic curve over $\overline{\mathbf Q}$ 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 $\operatorname{Gal}(\overline{\mathbf Q}/\mathbf Q)$ 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 $\operatorname{Gal}(\overline{\mathbf Q}/\mathbf Q)$ acts).”

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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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There is a french talk Cartes et dessins d'enfants 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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