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ON FIXED POINTS OF DOUBLY SYMMETRIC RIEMANN SURFACES

Published online by Cambridge University Press:  01 September 2008

GRZEGORZ GROMADZKI and
EWA KOZŁOWSKA-WALANIA
GRZEGORZ GROMADZKI
Affiliation:
Institute of Mathematics, University of Gdańsk, Wita Stwosza 57, Gdańsk 80-952, Poland e-mail: greggrom@math.univ.gda.pl
EWA KOZŁOWSKA-WALANIA
Affiliation:
Institute of Mathematics, University of Gdańsk, Wita Stwosza 57, Gdańsk 80-952, Poland e-mail: retrakt@math.univ.gda.pl
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Abstract

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In this paper, we study ovals of symmetries and the fixed points of their products on Riemann surfaces of genus g ≥ 2. We show how the number of these points affects the total number of ovals of symmetries. We give a generalisation of Bujalance, Costa and Singerman's theorems in which we show upper bounds for the total number of ovals of two symmetries in terms of g, the order n and the number m of the fixed points of their product, and we show their attainments for n holding some divisibility conditions. Finally, we give an upper bound for m in terms of n and g, and we study conditions under which it has given parity.

Keywords

Primary 30FSecondary 14H

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 50 , Issue 3 , September 2008, pp. 371 - 378
Copyright
Copyright © Glasgow Mathematical Journal Trust 2008

References

REFERENCES

1.Bujalance, E., Costa, A. F. and Singerman, D., Application of Hoare's theorem to symmetries of Riemann surfaces, Ann. Acad. Sci. Fenn. 18 (1993), 307–322.Google Scholar
2.Izquierdo, M. and Singerman, D., Pairs of symmetries of Riemann surfaces, Ann. Acad. Sci. Fenn. 23 (1998), 3–24.Google Scholar
3.Izquierdo, M. and Singerman, D., On the fixed point set of automorphisms of non-orientable surfaces without boundary, Geometry Topology Monographs, 1 (the Epstein birthday schrift) (1998) 295–301.CrossRefGoogle Scholar
4.Macbeath, A. M., Action of automorphisms of a compact Riemann surface on the first homology group, Bull. London Math. Soc. 5 (1973), 103–108.Google Scholar
5.Natanzon, S. M., Finite groups of homeomorphisms of surfaces and real forms of complex algebraic curves, Trans. Moscow Math. Soc. 51 (1989), 1–51.Google Scholar
6.Gromadzki, G., On a Harnack–Natanzon theorem for the family of real forms of Riemann surfaces, J. Pure Appl. Algebra 121 (1997), 253–269.CrossRefGoogle Scholar
7.Singerman, D., On the structure of non-Euclidean crystallographic groups, Proc. Camb. Phil. Soc. 76 (1974), 233–240.CrossRefGoogle Scholar