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If I am not mistaken, Alexander Grothendieck introduced dessins d'enfants as they are known today, in order to better understand the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$.

(The group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acts faithfully on the set of dessins d'enfants.)

Has one really developed a better understanding of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ through the dessins ?

If so, what are some of the properties / observations in this better understanding ?

(I think I have read, in a paper of Deligne, that the study of dessins d'enfants in the context of the absolute Galois group has not lead to a better understanding of the latter.)

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    $\begingroup$ Couldn't the fact that the absolute Galois group acts faithfully on a set of some kind of "discrete" objects be itself something we've learned about this group through the use of dessins d'enfants? $\endgroup$
    – Sam Hopkins
    Commented May 31, 2022 at 15:27
  • $\begingroup$ @SamHopkins: isn't it pretty easy to let the absolute Galois group act faithfully on a set of interesting combinatorial objects ? Or does the structure of the dessins add much value ? $\endgroup$
    – THC
    Commented May 31, 2022 at 17:11
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    $\begingroup$ This seems like a good question to me. Any time one defines object X to study object Y, it’s always reasonable to ask whether X can be used to prove any theorems about Y that don’t mention X. Certainly, the mere fact of acting on a set of discrete objects isn’t enough — by definition, the absolute Galois group is an inverse limit of finite groups, and so is determined by a sequence of finite actions. One important fact in this area is that the absolute Galois groups embeds into the outer automorphism group of the free profinite group of rank 2. Can this be seen using dessins d’enfants? $\endgroup$
    – HJRW
    Commented Jun 1, 2022 at 12:31
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    $\begingroup$ @HJRW: I think this prior MO question addresses your last question- mathoverflow.net/questions/225011/… $\endgroup$
    – Sam Hopkins
    Commented Jun 1, 2022 at 14:11
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    $\begingroup$ @HJRW: yes, dessins are one way to "see" the embedding you mention. See my paper P. Guillot An elementary approach to dessins d'enfants and the Grothendieck-Teichmüller group, on the arxiv. $\endgroup$
    – Pierre
    Commented Jun 10, 2022 at 8:55

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