Higher Dessins d'Enfants

Dessins d'Enfants are seemingly simple combinatorial objects that have been introduced by Grothendieck in order to encode coverings of the two-sphere ramified at three points. Such a covering defines an algebraic curve defined over a number field, and by a celebrated theorem of Belyi, every algebraic curve defined over a number field occurs in this way. As a consequence there is a faithful action of the absolute Galois group of the rationals on dessins d'enfants.

In this talk we present an extension of this theory to an arbitrary finite number of points on the sphere. Several new phenomena occur: a faithful braid group action as well as the dependence of the Galois action upon the chosen complex structure on the punctured sphere. We link this to Galois actions on étale fundamental groups and derive an anabelian theorem for higher dessins.