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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Extensions of finite cyclic group actions on non-orientable surfaces


Authors: E. Bujalance, F.J. Cirre and M.D.E. Conder
Journal: Trans. Amer. Math. Soc. 365 (2013), 4209-4227
MSC (2010): Primary 57M60; Secondary 14H37, 20H15, 30F50
Published electronically: February 18, 2013
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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.


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Additional Information

E. Bujalance
Affiliation: Departamento de Matemáticas Fundamentales, Facultad de Ciencias, UNED, c/ Senda del Rey s/n, 28040 Madrid, Spain
Email: eb@mat.uned.es

F.J. Cirre
Affiliation: Departamento de Matemáticas Fundamentales, Facultad de Ciencias, UNED, c/ Senda del Rey s/n, 28040 Madrid, Spain
Email: jcirre@mat.uned.es

M.D.E. Conder
Affiliation: Department of Mathematics, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand
Email: m.conder@auckland.ac.nz

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05776-5
PII: S 0002-9947(2013)05776-5
Received by editor(s): May 17, 2011
Received by editor(s) in revised form: November 19, 2011
Published electronically: February 18, 2013
Additional Notes: The first author was partially supported by MTM2011-23092
The second author was partially supported by MTM2011-23092
The third author was partially supported by the N.Z. Marsden Fund UOA-1015
Article copyright: © Copyright 2013 American Mathematical Society