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On q-n-gonal Klein Surfaces

On q-n-gonal Klein Surfaces
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摘要 We consider proper Klein surfaces X of algebraic genus p ≥ 2, having an automorphism φ of prime order n with quotient space X/（φ） of algebraic genus q. These Klein surfaces axe called q-n-gonal surfaces and they are n-sheeted covers of surfaces of algebraic genus q. In this paper we extend the results of the already studied cases n ≤ 3 to this more general situation. Given p ≥ 2, we obtain, for each prime n, the （admissible） values q for which there exists a q-n-gonal surface of algebraic genus p. Furthermore, for each p and for each admissible q, it is possible to check all topological types of q-n-gonal surfaces with algebraic genus p. Several examples are given： q-pentagonal surfaces and q-n-gonal bordered surfaces with topological genus g = 0, 1. We consider proper Klein surfaces X of algebraic genus p ≥ 2, having an automorphism φ of prime order n with quotient space X/（φ） of algebraic genus q. These Klein surfaces axe called q-n-gonal surfaces and they are n-sheeted covers of surfaces of algebraic genus q. In this paper we extend the results of the already studied cases n ≤ 3 to this more general situation. Given p ≥ 2, we obtain, for each prime n, the （admissible） values q for which there exists a q-n-gonal surface of algebraic genus p. Furthermore, for each p and for each admissible q, it is possible to check all topological types of q-n-gonal surfaces with algebraic genus p. Several examples are given： q-pentagonal surfaces and q-n-gonal bordered surfaces with topological genus g = 0, 1.
作者 B.ESTRADA R.A.HIDALGO E.MARTINEZ
机构地区 Departamento de Matemáticas Fundamentales Departamento de Matemática
出处 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2007年第10期1833-1844,共12页 数学学报（英文版）
基金 The first and third authors are partially supported by Project MTM2005-01637 the second is partially supported by Projects Fondecyt 1030252,1030373 UTFSM 12.05.21
关键词 Klein surfaces Riemann surfaces automorphism groups non-Euclidean crystallographic groups Klein surfaces, Riemann surfaces, automorphism groups, non-Euclidean crystallographic groups
分类号 O153 [理学—基础数学]
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参考文献14

  • 1Alling, N. L., Greenleaf, N.: Foundations of the Theory of Klein Surfaces, Lect. Notes in Math., 219, Springer-Verlag, 1971.
  • 2Wilkie, H. C.: On non-Euclidean crystallographic groups. Math. Z., 91, 87-102 (1966).
  • 3Macbeath, A. M.: The classification of non-euclidean crystallographic groups. Can. J. Math., 19, 1192-1205 (1967).
  • 4Preston, R.: Projective Structures and Fundamental Domains on Compact Klein Surfaces, Ph.D. Thesis, University of Texas, 1975.
  • 5May, C. L.: Large automorphism group of compact Klein surfaces with boundary. Glasgow Math. J., 18, 1-10 (1977).
  • 6Bujalance, E., Etayo, J. J., Gamboa, J. M., Gromadzki, G.: Automorphism Groups of Compact Bordered Klein Surfaces, A Combinatorial Approach, Lect. Notes in Math., 1439, Springer-Verlag, 1990.
  • 7Bujalance, E., Etayo, J. J., Gamboa, J. M.: Hyperelliptic Klein surfaces. Quart. J. Math. Oxford (2), 36, 141-157 (1985).
  • 8Bujalance, E., Etayo, J. J., Gamboa, J. M.: Superficies de Klein elipticas hiperelipticas, Memorias de la Real Academia de Ciencias, XIX, 1985.
  • 9Bujalance, E., Etayo, J. J.: Characterization of q-Hyperelliptic compact planar Klein surfaces. Abh. Math. Sere Univ. Hamburg, 58, 95-104 (1988).
  • 10Estrada, B., Martinez, E.: On q-Hyperelliptic k-bordered tori. Glasgow Math. J., 43, 343-357 (2001).
Acta Mathematica Sinica,English Series
2007年 第10期

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