Extensions of finite cyclic group actions on non-orientable surfaces

Bujalance, E.; Cirre, F.J.; Conder, M.D.E.

doi:10.1090/S0002-9947-2013-05776-5

Extensions of finite cyclic group actions on non-orientable surfaces
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by E. Bujalance, F.J. Cirre and M.D.E. Conder PDF

Trans. Amer. Math. Soc. 365 (2013), 4209-4227 Request permission

Abstract:

Conditions are derived for the extension of an action of a cyclic group on a compact non-orientable surface to the faithful action of some larger group on the same surface. It is shown that if such a cyclic action is realised by means of a non-maximal NEC signature, then the action always extends. The special case where the full automorphism group is cyclic of the largest possible order (for given genus) is also considered. This extends previous work by the authors for group actions on orientable surfaces. In addition, the smallest algebraic genus of a non-orientable surface on which a given cyclic group acts as the full automorphism group is determined.

References

Norman L. Alling and Newcomb Greenleaf, Foundations of the theory of Klein surfaces, Lecture Notes in Mathematics, Vol. 219, Springer-Verlag, Berlin-New York, 1971. MR 0333163
Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
E. Bujalance, Normal N.E.C. signatures, Illinois J. Math. 26 (1982), no. 3, 519–530. MR 658461
Emilio Bujalance, Cyclic groups of automorphisms of compact nonorientable Klein surfaces without boundary, Pacific J. Math. 109 (1983), no. 2, 279–289. MR 721920
Emilio Bujalance and Marston Conder, On cyclic groups of automorphisms of Riemann surfaces, J. London Math. Soc. (2) 59 (1999), no. 2, 573–584. MR 1709666, DOI 10.1112/S0024610799007115
Emilio Bujalance, José J. Etayo, José M. Gamboa, and Grzegorz Gromadzki, Automorphism groups of compact bordered Klein surfaces, Lecture Notes in Mathematics, vol. 1439, Springer-Verlag, Berlin, 1990. A combinatorial approach. MR 1075411, DOI 10.1007/BFb0084977
Emilio Bujalance, Grzegorz Gromadzki, and Peter Turbek, On non-orientable Riemann surfaces with $2p$ or $2p+2$ automorphisms, Pacific J. Math. 201 (2001), no. 2, 267–288. MR 1875894, DOI 10.2140/pjm.2001.201.267
José Luis Estévez and Milagros Izquierdo, Non-normal pairs of non-Euclidean crystallographic groups, Bull. London Math. Soc. 38 (2006), no. 1, 113–123. MR 2201610, DOI 10.1112/S0024609305017984
Leon Greenberg, Maximal Fuchsian groups, Bull. Amer. Math. Soc. 69 (1963), 569–573. MR 148620, DOI 10.1090/S0002-9904-1963-11001-0
W. Hall: Automorphisms and coverings of Klein surfaces, Ph.D. Thesis, University of Southampton, UK, 1978.
Ravi S. Kulkarni, Riemann surfaces admitting large automorphism groups, Extremal Riemann surfaces (San Francisco, CA, 1995) Contemp. Math., vol. 201, Amer. Math. Soc., Providence, RI, 1997, pp. 63–79. MR 1429195, DOI 10.1090/conm/201/02610
A. M. Macbeath, The classification of non-euclidean plane crystallographic groups, Canadian J. Math. 19 (1967), 1192–1205. MR 220838, DOI 10.4153/CJM-1967-108-5
Coy L. May, The symmetric crosscap number of a group, Glasg. Math. J. 43 (2001), no. 3, 399–410. MR 1878584, DOI 10.1017/S0017089501030038
D. Singerman, Automorphisms of compact non-orientable Riemann surfaces, Glasgow Math. J. 12 (1971), 50–59. MR 296286, DOI 10.1017/S0017089500001142
David Singerman, Finitely maximal Fuchsian groups, J. London Math. Soc. (2) 6 (1972), 29–38. MR 322165, DOI 10.1112/jlms/s2-6.1.29
Thomas W. Tucker, Symmetric embeddings of Cayley graphs in nonorientable surfaces, Graph theory, combinatorics, and applications. Vol. 2 (Kalamazoo, MI, 1988) Wiley-Intersci. Publ., Wiley, New York, 1991, pp. 1105–1120. MR 1170849