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I would like to know what is the recent progress about the group homomorphism $$ \mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\rightarrow \mathrm{Out}(\hat{F_{2}})$$

  • $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ is the absolute Galois group of $\mathbf{Q}$.
  • $\mathrm{Out}(\hat{F_{2}})$ is the group of outer automorphisms of the procompletion of the free group generated by two elements.

It is know to be injective homomorphism, what can we say about the image ?

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    $\begingroup$ I strongly recommend this paper by Guillot: arxiv.org/pdf/1309.1968v2.pdf. It develops the theory of dessins from scratch; of course, the faithful action of ${\rm Gal}(\overline{\mathbb Q})/{\mathbb Q})$ on the set of dessins yields the homomorphism you mention. Guillot also proves the remarkable fact that the action of the absolute Galois group on the subset of regular dessins is also faithful, a result that was also proved by Jaikin-Zapirain and Gonzalez-Diaz. Guillot has a follow-up article dealing with explicit computations pertaining to this homomorphism. $\endgroup$ – Nick Gill Dec 2 '15 at 9:29
  • $\begingroup$ @NickGill That is mean that the homomoprhism $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\rightarrow \mathrm{Out}(\hat{F_{2}})$ can not be surjective ? $\endgroup$ – Ofra Dec 2 '15 at 10:02
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    $\begingroup$ Yes! It certainly can't be surjective. It must lie in the Grothendieck-Teichmuller group, $\widehat{GT}$, a proper subgroup of ${\rm Out}(\hat{F_2})$. Guillot discusses this group at the end of his monograph. $\endgroup$ – Nick Gill Dec 2 '15 at 12:20
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For the sake of getting this off the unanswered stack...

Please refer to the following reference:

Guillot, P. An elementary approach to dessins d'enfants and the Grothendieck-Teichmüller group, Enseign. Math. vol 60, 2014. [arXiv version | journal version]

The article just cited develops the theory of dessins d'enfants from scratch. The final section discusses the image of the homomorphism mentioned in the OP's question.

In particular,

  1. Guillot includes a proof that the image must lie in the group $\widehat{GT_0}$, a proper subgroup of ${\rm Out}(\hat{F_2})$, first defined (I believe) by Drinfeld;
  2. Guillot discusses (but does not prove) a stronger result due to Ihara that the image must lie in the Grothendieck-Teichmuller group, $\widehat{GT}$, a proper subgroup of $\widehat{GT_0}$.

Please consult the bibliography of the given reference to find the relevant works of Drinfeld and Ihara.

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