An elementary approach to dessins d'enfants and the Grothendieck-Teichm\"uller group

TL;DR

This paper provides an elementary, self-contained overview of dessins d'enfants, demonstrating the faithfulness of Galois actions and embedding the absolute Galois group into the Grothendieck-Teichmüller group, with explicit finite approximations.

Contribution

It introduces an accessible approach to dessins d'enfants, proves the faithfulness of Galois actions on regular dessins, and embeds the Galois group into the Grothendieck-Teichmüller group with explicit finite approximations.

Findings

Galois action on dessins d'enfants is faithful.

The absolute Galois group embeds into the Grothendieck-Teichmüller group.

Explicit finite group approximations of GT_0 are provided.

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 this absolute Galois group embeds into the Grothendieck-Teichm\"uller group $GT_0$ introduced by Drinfel'd. There are explicit approximations of $GT_0$ 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 Galois action on the subset of regular dessins - that is, those exhibiting maximal symmetry -- is also faithful.