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Fuchsian groups generated by half-turns and geometrical characterization of hyperelliptic and symmetric Riemann surfaces

Etayo Gordejuela, J. Javier and Martínez García, Ernesto (2004) Fuchsian groups generated by half-turns and geometrical characterization of hyperelliptic and symmetric Riemann surfaces. Mathematica Scandinavica, 95 (2). pp. 226-244. ISSN 0025-5521

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Abstract

We construct a special type of fundamental regions for any Fuchsian group $F$ generated, by an even number of half-turns, and for certain non-Euclidean crystallographic groups (NEC groups in short). By comparing these regions we give geometrical conditions for F to be the canonical Fuchsian subgroup of one of those NEC groups. Precisely speaking, we deal with NEC groups of algebraic genus 0 having all periods in the signature equal to 2. By means of these conditions we give a characterization of hyperelliptic and symmetric Riemann surfaces.

Item Type:Article
Uncontrolled Keywords:Fuchsian groups and automorphic functions; Fuchsian groups and their generalizations
Subjects:Sciences > Mathematics > Group Theory
ID Code:15787
References:

Bujalance, E., Etayo, J. J., Characterization of q -Hyperelliptic compact planar Klein surfaces, Abh. Math. Sem. Univ. Hamburg 58 (1988), 95–104.

Bujalance, E., Etayo, J. J., Gamboa, J. M., Hyperelliptic Klein surfaces, Quart. J. Math. Oxford (2) 36 (1985), 141–157.

Bujalance, E., Etayo, J. J., Gamboa, J. M., Superficies de Klein elípticas hiperelípticas, Memorias de la Real Academia de Ciencias, Tomo XIX (1985).

Bujalance, E., Singerman, D., The symmetry type of a Riemann surface, Proc. London Math. Soc. (3) 51 (1985), 501–519.

Bujalance, J. A., Hyperelliptic compact non-orientable Klein surfaces without boundary, Kodai Math. J. 12 (1989), 1–8.

Maskit, B., A New Characterization of Hyperellipticity, Michigan Math. J. 47 (2000), 3–14.

Macbeath, A. M., The classification of non-euclidean crystallographic groups, Canad. J. Math. 19 (1967), 1192–1205.

Schmutz Schaller, P., Geometric characterization of hyperelliptic Riemann surfaces, Ann. Acad. Sci. Fenn. Math. 25 (2000), 85–90

Singerman, D., On the structure of non-euclidean crystallographic groups, Proc. Cambridge Philos. Soc. 76 (1974), 233–240.

Wilkie, H. C., On non-Euclidean crystallographic groups, Math. Z. 91 (1966), 87–102.

Wiman, A., Über die hyperelliptischen Kurven und diejenigen vom Geschlecht p=3 , welche eindeutigen Transformationen in sich zulassen, Bihang Till. Kongl. Svenska Vetenskaps-Akademiens Handlingar 21,1 n.1 (1895), 23 pp.

Deposited On:27 Jun 2012 11:31
Last Modified:27 Jun 2012 11:31

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