The Absolute Galois Group of the Rationals, Grothendieck's Dessin D'Enfants, and Galois Invariants

Abstract

We review a theorem of G.V. Belyi which establishes an equivalence between the category of irreducible algebraic curves over the algebraic closure of the rationals and the category of finite covers of the Riemann sphere ramified at three points. We observe that the latter is also equivalent to the category of A. Grothendieck's dessins d'enfants: finite bipartite graphs embedded on smooth, oriented, compact topological surfaces. Through these categorical equivalences, one obtains a highly non-trivial action of the absolute Galois group of the rationals on a collection of relatively simple combinatorial objects. We then analyze recent work by Girondo, Gonzalez-Diez, Hidalgo, and Jones which provides two new Galois invariants for dessins called ``Zapponi orientability" and ``twist-orient type". We conclude with speculations for future work on ``Zapponi orientability" and the Grothendieck-Teichmuller group.