Dessins d’enfants and origami curves

Abstract

This chapter is a detailed exposition of Grothendieck’s theory of dessins d’enfants and its relation with Teichmüller theory. In the first part, the authors consider several aspects of dessins d’enfants, including their appearance through Belyi pairs and their Galois actions, ribbon graphs and their monodromy maps, and the relation to the theory of subgroups of the free group in two generators. In th second part, the authors give a detailed account of origami curves, also commonly known as translation surfaces. These are special Teichmüller curves. The authors study Galois actions on origamis, and they show in particular how such objects are related to boundary points of origami curves.