# What do dessins tell us about the absolute Galois group?

I have sometimes seen it asserted that one manifestiation of how complicated the absolute Galois group $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ is is that one can not "pin down" any single element of this group other than complex conjugation; the only thing you can do is to study it indirectly via its representations, like the cyclotomic characters and the $\ell$-adic representations attached to modular forms.

Similarly one of the motivations for studying dessin d'enfants that I've heard stated in seminar talks etc. is that since the absolute Galois group acts faithfully on the set of all dessins, we can learn things about the absolute Galois group by studying its action on this set. Just as an example I quote:

Fortunately, Belyi's Theorem provides us with an explicit realisation of $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ in terms of bipartite maps, which is beginning to add to our rather meagre knowledge of this complicated group.

My question is: just what kind of Galois-theoretical or number-theoretical statements do people hope to prove by studying dessins? Are there at this point any theorems proven via dessins which can be stated without using the language of dessins?

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Have you looked at the book "Galois groups over $Q$"? It has articles by Ihara and Deligne and others along these lines. Also work of Goncharov might fit what you're looking for. Maybe results on multiple-zeta values satisfy your last request? I don't know if they directly relate to specific dessin, but they certainly relate to $\pi_1$ of the thrice-punctured projective line. – Marty Jan 26 '11 at 16:36
One reason it's hard to "pin down" any elements is that, in some precise sense, the group itself is not defined up to unique isomorphism. The point is that an algebraic closure of the rationals is only well-defined up to isomorphism, not up to unique isomorphism, so the Galois group is only defined up to inner automorphism. This means that there is almost no hope of pinning down any element---one can only really hope to pin down conjugacy classes, or invariants which are constant on conjugacy classes---and this is precisely why the representation theory of Galois groups works so well. – Kevin Buzzard Jan 26 '11 at 19:11
@Kevin: Even though I agree that, given a filed $K$, the algebraic closure of $K$ is only well defined up to non-canonical isomorphism, there does exist a preferred algebraic closure of $\mathbb Q$. That is: the subset of $\mathbb C$ consisting of algebraic numbers. In that sense, the absolute Galois group of $\mathbb Q$ does have a canonical model. – André Henriques Jan 26 '11 at 23:00
Grothendieck conjectured that $G_{\mathbb{Q}}$ is the automorphism group of the tower formed by the $\pi_1(M_{g,n})$ together with the morphism coming from deleting points, gluing curves on marked points etc... Nakamura and Schneps gave an explicit finite set relations for this automorphism group (Inv. Math. 141) as a subgroup of $Aut(\pi_1(M_{0,4}))$. It contains $G_{\mathbb{Q}}$ but of course we have no idea how to decide whether the inclusion is strict of how to construct a non trivial element of this group but it is still pretty amazing conjecture in my opinion. – AFK Jan 27 '11 at 0:13
This is Grothendieck-Teichmuller theory. For Dessins d'enfants per se the standard reference is "Geometric Galois Actions" I and II ed. by Lochak and Schneps. – AFK Jan 27 '11 at 0:17