This question is about a link between an open question in low-dimensional topology and a conjecture of Grothendieck, proved by Mochizuki. Let's start by stating them.

Recall that a subgroup $K$ of a group $G$ is characteristic if it is preserved by the action of $\operatorname{Aut}(G)$. In this case, we get an induced homomorphism of outer automorphism groups, $\operatorname{Out}(G)\to \operatorname{Out}(G/K)$. When $G/K$ is finite, the kernel of such a homomorphism is called a congruence subgroup.

Here's the topological question. It makes use of the fact that $\operatorname{Mod}(S)$ is naturally a subgroup of $\operatorname{Out}(\pi_1(S))$, so congruence subgroups make sense there too.

CSP conjecture: Let $S$ be a hyperbolic surface of finite type. Every finite-index subgroup of the mapping class group $\operatorname{Mod}(S)$ contains a congruence subgroup.

Here's the algebro-geometric result. This is far from my area of expertise, so apologies in advance for any errors.

Theorem (Mochizuki, conjectured by Grothendieck): Every smooth algebraic curve over $\mathbb{Q}$ is determined up to isomorphism by its étale fundamental group, equipped with the natural action of $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$.

In his famous problem list about mapping class groups (posted to the arXiv in 2006), Ivanov writes that he was told by Voevodsky that the above conjecture implies the above theorem. He goes on:

I am not aware of any publication where this conjecture of Grothendieck is deduced from the solution of the congruence subgroup problem for the mapping class groups.

Hence my question.

Has anyone written down a proof of this implication since 2006? If not, where would be a good place to get started learning about this? Best of all would be if someone knows the proof and could explain it!

This question is about a link between an open question in low-dimensional topology and a conjecture of Grothendieck, proved by Mochizuki. Let's start by stating them.

Recall that a subgroup $K$ of a group $G$ is characteristic if it is preserved by the action of $\operatorname{Aut}(G)$. In this case, we get an induced homomorphism of outer automorphism groups, $\operatorname{Out}(G)\to \operatorname{Out}(G/K)$. When $G/K$ is finite, the kernel of such a homomorphism is called a congruence subgroup.

Here's the topological question. It makes use of the fact that $\operatorname{Mod}(S)$ is naturally a subgroup of $\operatorname{Out}(\pi_1(S))$, so congruence subgroups make sense there too.

CSP conjecture: Let $S$ be a hyperbolic surface of finite type. Every finite-index subgroup of the mapping class group $\operatorname{Mod}(S)$ contains a congruence subgroup.

Here's the algebro-geometric result. This is far from my area of expertise, so apologies in advance for any errors.

Theorem (Mochizuki, conjectured by Grothendieck): Every smooth algebraic curve over $\mathbb{Q}$ is determined up to isomorphism by its étale fundamental group, equipped with the natural action of $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$.

In his famous problem list about mapping class groups (posted to the arXiv in 2006), Ivanov writes that he was told by Voevodsky that the above conjecture implies the above theorem. He goes on:

I am not aware of any publication where this conjecture of Grothendieck is deduced from the solution of the congruence subgroup problem for the mapping class groups.

Hence my question.

Has anyone written down a proof of this implication since 2006? If not, where would be a good place to get started learning about this? Best of all would be if someone knows the proof and could explain it!

UPDATE 1 October 2024:

I think, with the aid of @coLaideronette's answer, the question can be sharpened. Here, Voevodsky states a certain "combinatorial conjecture", and from it deduces Grothendieck's conjecture in the genus-zero case. I haven't found time to completely understand Voevodsky's paper, but his "combinatorial conjecture" is plausibly equivalent to, or implied by, the CSP.

[His combinatorial conjecture states that a certain homomorphism is injective. The CSP can be rephrased as: the natural map $\widehat{\operatorname{Mod}(S)}\to \operatorname{Out}(\widehat{\pi_1(S)})$ is injective, where $\widehat{-}$ denotes profinite completion.]

Interestingly, the CSP for genus-zero curves was proved by Diaz--Donager--Harbater in 1989, before Voevodsky's paper, so if this interpretation is correct then perhaps Voevodsky's result can be upgraded to an unconditional proof in that case.

Hence, my question can be rephrased as:

1. Is my interpretation of Voevodsky's theorem correct?
2. If so, has anyone generalised Voevodsky's result, that CSP implies Grothendieck's conjecture for genus-zero curves, to curves of higher genus?

As stated above, I am aware that Grothendieck's conjectures have been proved by Mochizuki and others, but my interest is in applications of the CSP. It may be that the answer to 2 is buried in one of the proofs of Grothendieck's conjecture -- if so, I'd appreciate a reference to a specific result, rather than just to the whole paper.