An elementary approach to dessins d’enfants and the Grothendieck–Teichmüller group

Abstract

We give an account of the theory of dessins d'enfants which is both elementary and self-contained. We describe the equivalence of many categories (graphs embedded nicely on surfaces, finite sets with certain permutations, certain field extensions, and some classes of algebraic curves), some of which are naturally endowed with an action of the absolute Galois group of the rational field. We prove that the action is faithful. Eventually we prove that Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$ embeds into the Grothendieck-Teichmüller group $\widehat{{\mathcal{G}\mathcal{T}}_0}$ introduced by Drinfeld. There are explicit approximations of $\widehat{{\mathcal{G}\mathcal{T}}_0}$ by finite groups, and we hope to encourage computations in this area.

Our treatment includes a result which has not appeared in the literature yet: the action of Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$ on the subset of regular dessins – that is, those exhibiting maximal symmetry – is also faithful.