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Minimum topological genus of compact bordered Klein surfaces admitting a prime-power automorphism

Published online by Cambridge University Press:  18 May 2009

E. Bujalance
Affiliation:
Departamento de Matemáticas Fundamentales, Fac. Ciencias, U. N. E. D., 28040 Madrid, Spain
J. M. Gamboa
Affiliation:
Departamento de Algebra, Fac. C. Matemáticas, Universidad Complutense, 28040 Madrid, Spain
C. Maclachlan
Affiliation:
Department of Mathematics, University of Aberdeen, Aberdeen AB9 2TY, Scotland
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In the nineteenth century, Hurwitz [8] and Wiman [14] obtained bounds for the order of the automorphism group and the order of each automorphism of an orientable and unbordered compact Klein surface (i. e., a compact Riemann surface) of topological genus g s 2. The corresponding results of bordered surfaces are due to May, [11], [12]. These may be considered as particular cases of the general problem of finding the minimum topological genus of a surface for which a given finite group G is a group of automorphisms. This problem was solved for cyclic and abelian G by Harvey [7] and Maclachlan [10], respectively, in the case of Riemann surfaces and by Bujalance [2], Hall [6] and Gromadzki [5] in the case of non-orientable and unbordered Klein surfaces. In dealing with bordered Klein surfaces, the algebraic genus—i. e., the topological genus of the canonical double covering, (see Alling-Greenleaf [1])—was minimized by Bujalance- Etayo-Gamboa-Martens [3] in the case where G is cyclic and by McCullough [13] in the abelian case.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 1995

References

1.Ailing, N. L. and Greenleaf, N., Foundations of the theory of Klein surfaces. Lecture Notes in Mathematics No 219 (Springer-Verlag, 1971).CrossRefGoogle Scholar
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