Genus 2 Belyĭ surfaces with a unicellular uniform dessin
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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.
KeywordsRiemann surfaces Dessins d’enfant Belyi surfaces Genus 2
Mathematics Subject Classification (2000)30F10 11G99 30F35
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- 1.Adrianov, N.M., Shabat, G.B.: Belyĭ functions of dessins d’enfants of genus 2 with four edges. Uspekhi Mat. Nauk 60(6), (366), 229–230 (2005) (in Russian); Russ. Math. Surv. 60(6), 1237–1239 (2005) (in English)Google Scholar
- 3.Beardon A.F.: The Geometry of Discrete Groups, Graduate Texts in Mathematics 91. Springer, New York (1983)Google Scholar
- 4.Belyĭ, G.V.: On Galois extensions of a maximal cyclotomic field. Izv. Akad. Nauk SSSR Ser. Mat. 43, 269–276 (1979) (in Russian); Math. USSR Izv. 14, 247–256 (1980) (in English)Google Scholar
- 8.Girondo, E., Torres-Teigell, D., Wolfart, J.: Shimura curves with many uniform dessins (2009)Google Scholar
- 10.Haataja, J.: HTessellate, version 1.3.0. Mathematica® package for hyperbolic geometry computations, freely downloadable at the webpage http://www.funet.fi/pub/sci/math/riemann/mathematica/.
- 14.Wolfram Research, Inc: Mathematica, Version 7.0, Champaign (2008)Google Scholar