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 ?

regulardessins 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. $\endgroup$