Pop's proof that $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})=\mathrm{Aut}\underline\pi^{alg}_{\overline{\mathbb{Q}}}$

Question

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 In there, it is referenced as unpublished at the time (2002):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?

Answer 1

Florian Pop has a manuscript on his web page, On I/OM, 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.