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GENUS 2 SEMI-REGULAR COVERINGS WITH LIFTING SYMMETRIES

Published online by Cambridge University Press:  01 September 2008

YOLANDA FUERTES  and
ALEXANDER MEDNYKH
YOLANDA FUERTES
Affiliation:
Departamento de Matemáticas, U. Autonoma de Madrid, 28049 Madrid, Spain e-mail: [email protected]
ALEXANDER MEDNYKH
Affiliation:
Sobolev Institute of Mathematics, Novosibirsk State University, 630090, Novosibirsk, Russia e-mail: [email protected]
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Abstract

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In this paper, we obtain algebraic equations for all genus 2 compact Riemann surfaces that admit a semi-regular (or uniform) covering of the Riemann sphere with more than two lifting symmetries. By a lifting symmetry, we mean an automorphism of the target surface which can be lifted to the covering. We restrict ourselves to the genus 2 surfaces in order to make computations easier and to make possible to find their algebraic equations as well. At the same time, the main ingredient (Main Proposition) depends neither on the genus, nor on the order of the group of lifting symmetries. Because of this, the paper can be thought as a generalisation for the non-normal case to the question of lifting automorphisms of a compact Riemann surface to a normal covering, treated, for instance, by E. Bujalance and M. Conder in a joint paper, or by P. Turbek solely.

Keywords

30F1014H3014H45
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 50 , Issue 3 , September 2008 , pp. 379 - 394
Copyright
Copyright © Glasgow Mathematical Journal Trust 2008

References

REFERENCES

1.Abu-Osman, M. T. and Rosenberger, G., Embedding property of surface groups, Bull. Malaysian Math. Soc. 3 (2) (1980), 21–27.Google Scholar
2.Bétréma, J. and Zvonkin, A., Plane trees and Shabat polynomials. Proceedings of the 5th conference on formal power series and algebraic combinatorics (Florence, 1993). Discrete Math. 153(1–3) (1996), 47–58.CrossRefGoogle Scholar
3.Boalch, P., Some explicit solutions to the Riemann–Hilbert problem. http://arxiv.org/pdf/math.DG/0501464.pdf.Google Scholar
4.Bolza, O., On binary sextics with linear transformations into themselves, Amer. J. Math. 10 (1888), 47–60.CrossRefGoogle Scholar
5.Broughton, S. A., Classifying finite group actions on surfaces of low genus, J. Pure Appl. Algebra 69 (1990), 233–270.CrossRefGoogle Scholar
6.Bujalance, E. and Conder, M., On cyclic groups of automorphisms of Riemann surfaces, J. London Math. Soc. 59 (2) (1999), 573–584.CrossRefGoogle Scholar
7.Farkas, H. and Kra, I., Riemann surfaces, Graduate Texts in Math. 71 (Springer-Verlag, New York, 1980).Google Scholar
8.Fuertes, Y. and Streit, M., Genus 3 normal coverings of the Riemann sphere branched over 4 points, Rev. Mat. Iberoamericana 22 (2) (2006), 413–454.CrossRefGoogle Scholar
9.González-Diez, G., Loci of curves which are prime Galois coverings of 1, Proc. London Math. Soc. 62 (3) (1991), 469–489.CrossRefGoogle Scholar
10.Harvey, W. H., On branch loci in Teichmüller space, Trans. Amer. Math. Soc. 153 (1971), 387–399.Google Scholar
11.Hulpke, A., Kuusalo, T., Naatanen, M. and Rosenberger, G., On orbifold coverings by genus 2 surfaces, Scientia, Series A: Math. Sci. 11 (2005), 45–55.Google Scholar
12.Hurwitz, A., Über Riemannăsche Flachen mit gegebenen Verzweigungpunkten, Math. Ann. 39 (1891), 1–61.CrossRefGoogle Scholar
13.Jones, G. A., Characters and surfaces: A survey. The atlas of finite groups: Ten years on (Birmingham, 1995), 90–118, London Math. Soc., Lecture Note Series 249 (Cambridge Univ. Press, Cambridge, UK, 1998).Google Scholar
14.Jones, G. and Singerman, D., Theory of maps on orientable surfaces, Proc. London Math. Soc. 37 (3) (1978), 273–307.CrossRefGoogle Scholar
15.Jones, G. and Singerman, D., Belyi functions, hypermaps and Galois groups, Bull. London Math. Soc. 28 (1996), 561–590.CrossRefGoogle Scholar
16.Kruglov, V. E., Structure of the partial indices of the Riemann problem with permutation-type matrices, Math. Notes 35 (2) (1984), 89–93.CrossRefGoogle Scholar
17.Kulkarni, R. S., Riemann surfaces admitting large automorphism groups. Extremal Riemann surfaces (San Francisco, CA, 1995), Contemporary Math. 201, Amer. Math. Soc., Providence, RI (1997), 63–79.CrossRefGoogle Scholar
18.Kuribayashi, A. and Kimura, H., Automorphism groups of compact Rieman surfaces of genus five, J. Algebra 134 (1990), 80–103.CrossRefGoogle Scholar
19.Kuribayashi, A. and Komiya, K., On Weierstrass points and automorphisms of curves of genus three, in Algebraic Geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), 253–299, Lecture Notes Math. 732 (Springer-Verlag, Berlin, Germany, 1979).CrossRefGoogle Scholar
20.Kuribayashi, I. and Kuribayashi, A., Automorphism groups of compact Riemann surfaces of genera three and four, J. Pure Appl. Algebra 65 (1990), 277–292.CrossRefGoogle Scholar
21.Kwak, J. H. and Mednykh, A., Enumeration of branched coverings of closed orientable surfaces whose branch orders coincide with multiplicity, Studia Sci. Math. Hungarica 44 (2) (2007), 215–223.Google Scholar
22.Maclachahn, C. and Harvey, W. H., On mapping-class groups and Teichmüller space, Proc. London Math. Soc. 30 (3) (1975), 496–512.CrossRefGoogle Scholar
23.Maclachlan, C. and Rosenberger, G., Commensurability classes of two-generators Fuchsian groups. Discrete groups and geometry (Birmingham, 1991), 171–189, London Math. Soc. Lecture Note Series 173, (Cambridge Univ. Press, Cambridge, UK, 1992).Google Scholar
24.Magaard, K., Shaska, T., Shpectorov, S. and Völklein, H., The locus of curves with prescribed automorphism group. Communications in arithmetic fundamental groups (Kyoto, 1999/2001), Sürikaisekikenkyüsho Kökyüroku No. 1267 (2002), 112–141.Google Scholar
25.Mednykh, A. D., On branched coverings of Riemann surfaces with the trivial group of covering transformations, Sov. Math. Dokl. 18 (1977), 1156–1158 [translation from Dokl. Akad. Nauk SSSR 235 (1977), 1267–1269].Google Scholar
26.Singerman, D., Subgroups of Fuchsian groups and finite permutation groups, Bull. London Math. Soc. 2 (1970), 319–323.CrossRefGoogle Scholar
27.Singerman, D. and Syddall, R. I., Belyi uniformization of elliptic curves, Bull. London Math. Soc. 29 (1997), 443–451.CrossRefGoogle Scholar
28.Turbek, P., A necessary and sufficient condition for lifting the hyperelliptic involution, Proc. Amer. Math. Soc. 125 (9) (1997), 2615–2625.CrossRefGoogle Scholar