2014/07/11 by Pierre Guillot, Guillot, Pierre
Arts and Humanities · Mathematics · #FOS: Mathematics #Group Theory (math.GR) #Historical Studies and Socio-cultural Analysis #History and Theory of Mathematics #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.1407.3112
openalex publication_date 2014/07/11 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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.