On representations of Gal(Q‾/Q), GTˆ and Aut(Fˆ₂)

By work of Belyĭ [2], the absolute Galois group G_Q=Gal(Q‾/Q) of the field Q of rational numbers can be embedded into A=Aut(Fˆ₂), the automorphism group of the free profinite group Fˆ₂ on two generators. The image of G_Q lies inside GTˆ, the Grothendieck-Teichmüller group. While it is known that every abelian representation of G_Q can be extended to GTˆ, Lochak and Schneps [13] put forward the challenge of constructing irreducible non-abelian representations of GTˆ. We do this virtually, namely by showing that a rich class of arithmetically defined representations of G_Q can be extended to finite index subgroups of GTˆ. This is achieved, in fact, by extending these representations all the way to finite index subgroups of A=Aut(Fˆ₂). 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 Aut(F_d).