Hostname: page-component-7c8c6479df-ph5wq Total loading time: 0 Render date: 2024-03-28T13:25:59.930Z Has data issue: false hasContentIssue false

Cyclic p-groups of symmetries of surfaces

Published online by Cambridge University Press:  18 May 2009

Ravi S. Kulkarni
Affiliation:
Inst. Mittag-Leffler, Auravägen 17, 5-182 62 Djursholm, Sweden
Colin Maclachlan
Affiliation:
Department of Mathematical Sciences, University of Aberdeen, Aberdeen, Scotland
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let Σg denote a compact orientable surface of genus g ≥ 2. We consider finite groups G acting effectively on Σg and preserving the orientation—for short, G acts on Σg or Gis a symmetry group of Σg. Each surface Σg admits only finitely many symmetry groups G and the orders of these groups are bounded by Wiman's bound of 84(g – 1). This bound is attained for infinitely many values of g [12], see also [9], and all values of g ≤ 104 for which it is attained are known [4].

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 1991

References

REFERENCES

1.Accola, R. D. M., On the number of automorphisms of a closed Riemann surface. Trans. Amer. Math. Soc. 131 (1968), 398408.CrossRefGoogle Scholar
2.Chetiya, B. P., On genuses of compact Riemann surfaces admitting solvable automorphism groups. Indian J. Pure appl. Math. 12 (1981), 13121318.Google Scholar
3.Conder, M. D. E., Generators for alternating and symmetric groups. J. London Math. Soc. 22 (1980), 7586.CrossRefGoogle Scholar
4.Conder, M. D. E., The genus of Compact Riemann surfaces with Maximal Automorphism Group. J. of Algebra 108 (1987), 204247.CrossRefGoogle Scholar
5.Glover, H. and Mislin, G., Torsion in the mapping class group and its cohomology. J. of Pure & Appl. Algebra 44 (1987), 177189.CrossRefGoogle Scholar
6.Glover, H. and Sjerve, D., The genus of PSL2(q) J. Reine und Angew Math. 380 (1987), 5986.Google Scholar
7.Harvey, W. J., Cyclic groups of automorphisms of a compact Riemann surface. Quart. J. Math. 17 (1966), 8697.CrossRefGoogle Scholar
8.Kulkarni, R. S., Symmetries of surfaces. Topology 26 (1987), 195203.CrossRefGoogle Scholar
9.Kulkarni, R. S., Normal subgroups of fuchsian groups. Quart. J. Math. Oxford 36 (1985), 325344.CrossRefGoogle Scholar
10.Kuribayashi, A. and Kimura, H., On Automorphism Groups of Compact Riemann surfaces of Genus 5. Proc. Japan Academy 63 Ser A (1987), 126130.Google Scholar
11.Kuribayashi, I. and Kuribayashi, A., On Automorphism groups of a Compact Riemann Surface of genus 4 as subgroups in GL(4, C). Bull. Facul. Sc. Eng. Chuo Univ. 28 (1985), 1128.Google Scholar
12.Macbeath, A. M., On a Theorem of Hurwitz Proc. Glasgow Math. Assoc. 5 (1961), 9096.CrossRefGoogle Scholar
13.Maclachlan, C., Groups of Automorphisms of compact Riemann surfaces Ph.D. Thesis Birmingham Univ. 1966.Google Scholar
14.Maclachlan, C., Abelian groups of automorphisms of compact Riemann surfaces. Proc. London Math. Soc. 15 (1965), 699712.CrossRefGoogle Scholar
15.Maclachlan, C., A bound for the number of automorphisms of a compact Riemann surface. J. London Math. Soc. 44 (1969), 265272.CrossRefGoogle Scholar
16.McCullough, D., Miller, A. and Zimmermann, B., Group actions on non-closed 2-manifolds (to appear).Google Scholar
17.Wiman, A., Uber die hyperelliptischen Kurven und diejenigen vom Geschlechte p = 3, welche eindeutigen Transformationen in sich Zulassen Bihang. Till Kongl. Svenska Veienskaps—Akademiems Hadlingar (Stockholm 1895–6), 21, 123.Google Scholar
18.Zomorrodian, R., Nilpotent automorphism groups of Riemann surfaces, Trans. Amer. Math. Soc. 288 (1985), 241255.Google Scholar