caa a22 4500 44588469X CHVBK 20180317145555.0 cr unu---uuuuu 170323e20111201xx s 000 0 eng 10.1007/s10711-011-9579-y doi (NATIONALLICENCE)springer-10.1007/s10711-011-9579-y Genus 2 Belyĭ surfaces with a unicellular uniform dessin [Elektronische Daten] [Ernesto Girondo, David Torres-Teigell] 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. Springer Science+Business Media B.V., 2011 Riemann surfaces nationallicence Dessins d'enfant nationallicence Belyi surfaces nationallicence Genus 2 nationallicence Girondo Ernesto Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain aut Torres-Teigell David Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain aut Geometriae Dedicata Springer Netherlands 155/1(2011-12-01), 81-103 0046-5755 155:1<81 2011 155 10711 https://doi.org/10.1007/s10711-011-9579-y text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10711-011-9579-y text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Girondo Ernesto Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain aut NATIONALLICENCE 700 1- Torres-Teigell David Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain aut NATIONALLICENCE 773 0- Geometriae Dedicata Springer Netherlands 155/1(2011-12-01), 81-103 0046-5755 155:1<81 2011 155 10711 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer