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A NEW CHARACTERIZATION OF WIMAN SURFACES

Part of: Graph theory

Published online by Cambridge University Press:  27 July 2009

ADNAN MELEKOĞLU
ADNAN MELEKOĞLU*
Affiliation:
Department of Mathematics, Faculty of Arts and Sciences, Adnan Menderes University, 09010 Aydın, Turkey (email: amelekoglu@hotmail.com)
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Abstract

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Let ℳ be a regular map of genus g>1 and X be the underlying Riemann surface. A reflection of ℳ fixes some simple closed curves on X, which we call mirrors. Each mirror passes through at least two of the geometric points (vertices, face-centers and edge-centers) of ℳ. In this paper we study the surfaces which contain mirrors passing through just two geometric points, and show that only Wiman surfaces have this property.

Keywords

Riemann surfaceWiman surfaceregular mapgeometric pointreflectionpattern

MSC classification

Secondary: 05C10: Planar graphs; geometric and topological aspects of graph theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 80 , Issue 3 , December 2009 , pp. 360 - 368
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

References

[1]Bujalance, E. and Singerman, D., ‘The symmetry type of a Riemann surface’, Proc. London Math. Soc. (3) 51 (1985), 501–519.CrossRefGoogle Scholar
[2]Coxeter, H. S. M., Regular Polytopes (Dover Publications, New York, 1973).Google Scholar
[3]Harvey, W. J., ‘Cyclic groups of automorphisms of compact Riemann surfaces’, Quart. J. Math. Oxford 17 (1966), 86–97.CrossRefGoogle Scholar
[4]Jones, G. A. and Singerman, D., ‘Theory of maps on orientable surfaces’, Proc. London Math. Soc. (3) 37 (1978), 273–307.CrossRefGoogle Scholar
[5]Kulkarni, R. S., ‘A note on Wiman and Accola–Maclachlan surfaces’, Ann. Acad. Sci. Fenn. Ser. A. 1. Math. 16 (1991), 83–94.CrossRefGoogle Scholar
[6]Kulkarni, R. S., Riemann Surfaces Admitting Large Automorphism Groups, Contemporary Mathematics, 201 (American Mathematical Society, Providence, RI, 1997), pp. 63–79.Google Scholar
[7]Macbeath, A. M., ‘The classification of non-Euclidean plane crystallographic groups’, Canad. J. Math. 19 (1967), 1192–1205.Google Scholar
[8]Melekoğlu, A., ‘A geometric approach to the reflections of regular maps’, Ars Combin. 89 (2008), 355–367.Google Scholar
[9]Singerman, D., ‘Finitely maximal Fuchsian groups’, J. London Math. Soc. 6 (1972), 29–38.Google Scholar
[10]Singerman, D., ‘On the structure of non-Euclidean crystallographic groups’, Proc. Cambridge Philos. Soc. 76 (1974), 233–240.CrossRefGoogle Scholar
[11]Singerman, D., ‘Symmetries of Riemann surfaces with large automorphism group’, Math. Ann. 210 (1974), 17–32.CrossRefGoogle Scholar
[12]Singerman, D., ‘Unicellular dessins and a uniqueness theorem for Klein’s Riemann surface of genus 3’, Bull. London Math. Soc. 33 (2001), 701–710.CrossRefGoogle Scholar
[13]Wiman, A., Über die hyperelliptischen Curven und diejenigen vom geschlechte p=3 welche eindeutigen Transformationen in sich zulassen. Bihang Till Kongl. Svenska Veienskaps-Akademiens Handlingar, (Stockholm 1895–6), Vol. 21, pp. 1–23.Google Scholar