On representations of $mathrm{Gal}(overline{mathbb{Q}}/mathbb{Q})$, $widehat{GT}$ and $mathrm{Aut}(hat{F}_2)$

By work of Belyi, the absolute Galois group $G_{mathbb{Q}}=mathrm{Gal}(overline{mathbb{Q}}/mathbb{Q})$ of the field $mathbb{Q}$ of rational numbers can be embedded into $A=mathrm{Aut}(widehat{F_2})$, the automorphism group of the free profinite group $widehat{F_2}$ on two generators. The image of $G_{mathbb{Q}}$ lies inside $widehat{GT}$, the Grothendieck-Teichmuller group. While it is known that every abelian representation of $G_{mathbb{Q}}$ can be extended to $widehat{GT}$, Lochak and Schneps put forward the challenge of constructing irreducible non-abelian representations of $widehat{GT}$. We do this virtually, namely by showing that a rich class of arithmetically defined representations of $G_{mathbb{Q}}$ can be extended to finite index subgroups of $widehat{GT}$. This is achieved, in fact, by extending these representations all the way to finite index subgroups of $A=mathrm{Aut}(widehat{F_2})$. We do this by developing a profinite version of the work of Grunewald and Lubotzky, which provided a rich collection of representations for the discrete group $mathrm{Aut}(F_d)$.