On representations of [formula omitted], [formula omitted] and [formula omitted

Abstract

By work of Belyĭ [2], the absolute Galois group GQ=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 GQ lies inside GTˆ, the Grothendieck-Teichmüller group. While it is known that every abelian representation of GQ 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 GQ 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(Fd).