This paper looks into the relationship between dessins d’enfants and their associated Riemann surfaces. A dessin on a topological surface is an embedding of a bipartite graph whose complement has simply connected components. Associated to such a dessin there is a triangle group , whose periods are determined by the valencies of the graph, and a finite index subgroup of , uniquely determined up to conjugacy, such that the quotient of the upper half plane by is a Riemann surface . This is referred to as the Riemann surface underlying the dessin. The surface is smooth Belyi if is torsion-free. The surface is quasiplatonic if the dessin is regular, meaning in addition that is normal in . If moreover the dessin happens to have no free edges (i.e. it is clean), then becomes platonic. One first result in this paper asserts that quasiplatonic surfaces of genus are platonic, and then gives an example in genus 3 where this is no longer true. The authors also observe that every smooth dessin of genus 2 is arithmetic (i.e. the associated triangle group is an arithmetic Fuchsian group).
The main focus of this paper is the study of the injectivity of the map . The authors specialize to the family of dessins which are uniform maps and hence ask whether two non-isomorphic such dessins can have the same underlying Riemann surface. A dessin is uniform if all black vertices (in the bipartite description) have equal valency , all white vertices have equal valency and all cells have equal valency . The associated triangle group in this case is of the form . If then the dessin becomes a map. The main theorem of this paper now states that if and are two non-arithmetic maps of maximal type, then and are conformally equivalent if and only if and are isomorphic maps. The maximal type of a uniform map translates into a condition on and for which the associated triangle group is not contained in a larger triangle group. The proof of the main theorem is based on a fundamental theorem of Margulis on commensurators of Fuchsian groups [R. J. Zimmer, Ergodic theory and semisimple groups (1984; Zbl 0571.58015)].
The paper ends with a list of examples where distinct maps and dessins may lie on the same Riemann surface, and also gives a comparison between the automorphism group of a dessin and the one for its underlying Riemann surface.