Yang-Hui He and James Read have a new paper on the ArXiv entitled “Hecke Groups, Dessins d’Enfants and the Archimedean Solids”.
Abstract.We show that every clean dessin d’enfant can be associated with a conjugacy class of subgroups of a certain Hecke group, or free product of Hecke groups. This dessin is naturally viewed as the Schreier coset graph for that class of subgroups, while the Belyi map associated to the dessin is the map between the associated algebraic curve and . With these points in mind, we consider the well-studied Archimedean solids, finding a representative the associated class of Hecke subgroups in each case by specifying a generating set for that representative, before finally discussing the congruence properties of these subgroups.
You can download the paper here.