I've heard of this result in a paper on which Yves André proves the p-adic analogue (that is, $\mathrm{Gal}(\overline{\mathbb{Q}}\_p/\mathbb{Q}\_p)=\mathrm{Aut}\underline\pi^{temp}\_{{\mathbb{C}}\_p}$), conditional on Pop's theorem. In there, it is referenced as unpublished at the time (2002):


* F. Pop, A combinatorial description of $G\_\mathbb{Q}$, talk at Oberwolfach, July 1998 (cf. Tagungsbericht 28), manuscript in preparation.


A report of the 1998 Oberwolfach conference survives. The relevant section reads:



> 
> Let $\mathcal{Var}\_\mathbb{Q}$ be the category of all
>  $\mathbb{Q}$-varieties and morphisms of such varieties. Taking the
>  fundamental group functor we get
> 
> 
> $$\overline{\pi}\_1: \mathcal{Var}\_\mathbb{Q} \to \mathcal{G}$$ $$X
>  \mapsto \pi\_1(\bar{X})$$
> 
> 
> into the category $\mathcal{G}$ of all profinite groups and outer
>  homomorphisms. Let $\mathcal{G}\_\mathbb{Q}$ be the image of
>  $\mathcal{Var}\_\mathbb{Q}$ under $\overline{\pi}\_1$. For every $X$
>  there exists a canonical representation $$\rho\_X:G\_\mathbb{Q} \to
>  \mathrm{Out}(\pi\_1(\bar{X}))=\mathrm{Aut}\_\mathcal{G}(\pi\_1(\bar{X}))$$
>  which behaves functorially. Then we get a homomorphism
> 
> 
> $$\imath\_\mathbb{Q} : G\_\mathbb{Q} \to \mathrm{Aut}(G\_\mathbb{Q})$$
>  $$\sigma \mapsto (\rho\_X(\sigma))\_X$$
> 
> 
> We gave a positive answer to the question of Oda-Matsumoto, asking
>  whether $\imath\_\mathbb{Q}$ is an isomorphism.
> 
> 
> 


It seems like Pop didn't publish anything else about this result.


Is the proof (or a sketch of it) available anywhere?



Pop's proof that $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})=\mathrm{Aut}\underline\pi^{alg}_{\overline{\mathbb{Q}}}$
Florian Pop has a manuscript on his web page, [On I/OM](https://www.math.upenn.edu/~pop/Research/files-Res/IOM_29May14.pdf), dated 29 May 2014, where he proves a pro-$\ell$-abelian by central version of the Ihara/Oda-Matsumoto conjecture over any field, and explains how it implies the initial conjecture over the rational numbers.
I did not read the paper, though, and I don't know its status with regard to publication.


