The Grothendieck-Teichm\"uller group of a finite group and $G$-dessins d'enfants

TL;DR

This paper introduces the Grothendieck-Teichmüller group associated with finite groups, explores its properties, and connects it to dessins d'enfants and the absolute Galois group, providing explicit calculations for specific groups.

Contribution

It defines GT(G) for finite groups, relates it to dessins d'enfants, and offers explicit computational and conceptual insights, especially for simple non-abelian groups and groups like PSL(2, q).

Findings

GT(G) is explicitly identified for simple non-abelian G.

Precise computational data obtained for PSL(2, q) with various q.

Refined action on equivariant dessins elucidates the theory.

Abstract

For each finite group G, we define the Grothendieck-Teichm\"uller 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-Teichm\"uller 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.