How to draw a group?
Abstract A map is at the same time a group. To represent a map (that is, a graph drawn on the sphere or on another surface) we usually use a pair of permutations on the set of the ‘ends’ of edges. These permutations generate a group which we call a cartographic group . The main motivation for the study of the cartographic group is the so-called theory of “dessins d'enfants’ of Grothendieck, which relates the theory of maps to Galois theory [24]. In the present paper we address the questions of identifying the cartographic group for a given map, and of constructing the maps with a given cartographic group.