Genus 2 Belyĭ surfaces with a unicellular uniform dessin

Abstract

A bicoloured graph embedded in a compact oriented surface and dividing it into a union of simply connected components (faces) is known as a dessin d’enfant. It is well known that such a graph determines a complex structure on the underlying topological surface, but a given compact Riemann surface may correspond to different dessins. In this paper we deal with all unicellular (one-faced) uniform dessins of genus 2 and their underlying Riemann surfaces. A dessin is called uniform if white vertices, black vertices and faces have constant degree, say p, q and r respectively. A uniform dessin d’enfant of type (p, q, r) on a given surface S corresponds to the inclusion of the torsion-free Fuchsian group K uniformizing S inside a triangle group Δ(p, q, r). Hence the existence of different uniform dessins on S is related to the possible inclusion of K in different triangle groups. The main result of the paper states that two unicellular uniform dessins belonging to the same genus 2 surface must necessarily be isomorphic or obtained by renormalisation. The problem is approached through the study of the face-centers of the dessins. The displacement of such a point by the elements of K must belong to a prescribed discrete set of (hyperbolic) distances determined by the signature (p, q, r). Therefore looking for face-centers amounts to finding points correctly displaced by every element of K.