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 the absolute Galois group of the rational field. 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= PSL(2, q) when q= 4, 7, 8, 9, 11, 13, 16, 17, 19, and we treat A7, PSL(3, 3) 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 equivariant dessins, which we define, and this elucidates much of the first part.

The title has changed. Many little improvements and changes in terminology. Identical to the version which is to appear in the SIGMAP proceedings