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][1] 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!


  [1]: https://arxiv.org/abs/math/0608325