The Grothendieck-Teichmüller group of a finite group and G-dessins d’enfants

Abstract

For each finite group G, we define the Grothendieck-Teichmüller group of G, denoted GT(G), and explore its properties. The theory of dessins d’enfants shows that the inverse limit of GT(G) as G varies can be identified with a group defined by Drinfeld and containing Gal(Formula Presented). We give, in particular, an identification of GT(G), in the case when G is simple and non-abelian, with a certain very explicit group of permutations that can be analyzed easily. With the help of a computer, we obtain precise information for G = PSL2(Fq) when q ∈ {4, 7, 8, 9, 11, 13, 16, 17, 19}, and we treat A7, PSL3(F3) and M11. In the rest of the paper we give a conceptual explanation for the technique which we use in our calculations. It turns out that the classical action of the Grothendieck-Teichmüller group on dessins d’enfants can be refined to an action on “G-dessins”, which we define, and this elucidates much of the first part.