How to draw a group?

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. 1 Maps and cartographic groups The same graph can be drawn in diierent topological ways, as is shown in Figure 1 (we consider these graphs as drawn on the sphere). s &% '$ s s &% '$ s s s Q Q Figure 1: One graph, but two maps The combinatorial structure that reeects not only the graph properties but also those of its embedding is called a map.