By work of Belyĭ [2], the absolute Galois group $G_{\mathbb{Q}}=\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ of the field $\mathbb{Q}$ of rational numbers can be embedded into $A=\mathrm{Aut}({\hat{F}}_2)$, the automorphism group of the free profinite group ${\hat{F}}_2$ on two generators. The image of $G_{\mathbb{Q}}$ lies inside $\widehat{GT}$, the Grothendieck-Teichmüller group. While it is known that every abelian representation of $G_{\mathbb{Q}}$ can be extended to $\widehat{GT}$, Lochak and Schneps [13] 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 $A=\mathrm{Aut}({\hat{F}}_2)$. We do this by developing a profinite version of the work of Grunewald and Lubotzky [7], which provided a rich collection of representations for the discrete group $\mathrm{Aut}(F_d)$.

通过 Belyĭ [2] 的工作，绝对的伽罗瓦群$G_{\mathbb{问}}=\mathrm{加尔}(\overline{\mathbb{问}}/\mathbb{问})$领域的$\mathbb{问}$的有理数可以嵌入到$\mathrm{拥有}=\mathrm{其他}({\widehat{F}}_2)$, 自由有限群的自同构群${\widehat{F}}_2$在两台发电机上。的形象$G_{\mathbb{问}}$躺在里面$\widehat{G吨}$, Grothendieck-Teichmüller 群。众所周知，每一个阿贝尔表示$G_{\mathbb{问}}$可以扩展到$\widehat{G吨}$, Lochak 和 Schneps [13] 提出了构建不可约非阿贝尔表示的挑战$\widehat{G吨}$. 我们实际上是这样做的，即通过展示丰富的算术定义表示$G_{\mathbb{问}}$可以扩展到有限索引子群$\widehat{G吨}$. 事实上，这是通过将这些表示一直扩展到有限索引子群来实现的$\mathrm{拥有}=\mathrm{其他}({\widehat{F}}_2)$. 我们通过开发 Grunewald 和 Lubotzky [7] 工作的一个有限版本来做到这一点，它为离散组提供了丰富的表示集合$\mathrm{其他}(F_d)$.