zbMATH — the first resource for mathematics

Hyperelliptic compact non-orientable Klein surfaces without boundary. (English) Zbl 0688.14024
Let X be a compact non-orientable Klein surface (KS). X may be represented as \(D/\Gamma\), D being the hyperbolic plane and \(\Gamma\) a non-Euclidean crystallographic group (NEC group), i.e. a discrete subgroup of the group G of all isometries of the hyperbolic plane, including orientation-reversing ones. X is q-hyperelliptic if there exists an involution \(\Phi\) of X such that \(X/<\Phi >\) has algebraic genus q. In the theorem 2.2 a characterisation of those compact non- orientable KS’s without boundary is given, which are q-hyperelliptic. If such a q-hyperelliptic KS has the algebraic genus \(p>4q+1,\) then: (1) the automorphism \(\Phi\) (the automorphism of q-hyperellipticity) is central and unique (see proposition 2.3); (2) if additionally \(X/<\Phi >\) has boundary, then \(|\text{Aut}(X)| \leq 12(p-1)\). The equality is possible only for \(p=2\) and in this case Aut\((X)\) is the dihedron group \(D_ 6\) with 12 elements (s. theorem 3.1).
Reviewer: A.Duma

14H05 Algebraic functions and function fields in algebraic geometry
20H15 Other geometric groups, including crystallographic groups
30F10 Compact Riemann surfaces and uniformization
Full Text: DOI
[1] ACCOLA, R. D. M., Riemann Surfaces with Automorphism Groups admitting parti-tions, Proceedings of American Math. Society, Vol. 21 (1969), 477-482. · Zbl 0174.37401 · doi:10.2307/2037029
[2] ACCOLA, R. D. M., Strongly branched coverings of closed Riemann Surfaces, Pro ceedings of American Math. Society, Vol. 26 (1970), 315-322. · Zbl 0212.42501 · doi:10.2307/2036396
[3] ALLING, N. L. AND GREENLEAF, N., Foundations of the Theory of Klein Surfaces, Lect. Notes in Math., 219 (1971), Springer · Zbl 0225.30001
[4] BUJALANCE, E., Proper periods of normal NEC Subgroups with even index, Rev Mat. Hisp.-Amer. (4) 41 (1981), 121-127. · Zbl 0513.20035 · eudml:39785
[5] BUJALANCE, J. A., Normal subgroups of even index in an NECgroup, Arch. Math 49 (1987), 470-478. · Zbl 0617.20026 · doi:10.1007/BF01194293
[6] BUJALANCE, E., ETAYO, J. J. AND GAMBOA, J. M., Hyperelliptic Klein Surfaces, Quart. J. Math. Oxford, 36 (1985), 141-157 · Zbl 0573.14010 · doi:10.1093/qmath/36.2.141
[7] BUJALANCE, E., ETAYO, J. J. AND GAMBOA, J. M., Superficies de Klein Elipticas Hiperelipticas Memorias de R. Acad. Ciencias, tomo X, 1985.
[8] BUJALANCE, E. AND ETAYO, J. J., A characterization of O-Hyperelliptic compac planar Klein Surface, Abh. Math. Sem. Univ. Hamburg,
[9] HOARE, A. M. H. AND SINGERMAN, D., The orientability of Subgroups of plan Groups, London Math. Soc.: Lect. Notes Ser. 71 (1982), 221-227. · Zbl 0489.20036
[10] MACBEATH, A. M., The classification of non-Euclidean plane crystallographi groups, Canad. J. Math. 6, 1967. · Zbl 0183.03402 · doi:10.4153/CJM-1967-108-5
[11] MACLACHLAN, C., Smooth covering of Hyperelliptic Surfaces, Quart. J. Math Oxford, 22 (1971), 117-123. · Zbl 0208.10101 · doi:10.1093/qmath/22.1.117
[12] MAY, C. L., Large Automorphism Groups of Compact Klein Surfaces with bound ary, Glasgow Math. J. 18, 1977. · Zbl 0363.14008 · doi:10.1017/S0017089500002950
[13] SINGERMAN, D., On the structure of non Euclidean Crystallographic Groups, Proc Cambridge Phil. Soc. 76 (1974), 233-240. · Zbl 0284.20053
[14] WILKIE, M. C., On non Euclidean Crystallographic groups, Math. Zeit. 91 (1966), 87-102 · Zbl 0166.02602 · doi:10.1007/BF01110157 · eudml:170507
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.