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Abelian actions on compact nonorientable Riemann surfaces

Part of: Riemann surfaces Other groups of matrices Low-dimensional topology Special aspects of infinite or finite groups

Published online by Cambridge University Press:  02 December 2021

Jesús Rodríguez Open the ORCID record for Jesús Rodríguez [Opens in a new window]
Jesús Rodríguez*
Affiliation:
Departamento de Matemáticas Fundamentales, Facultad de Ciencias Universidad Nacional de Educación a Distancia, 28040 Madrid, Spain
*
Email: jesus.rms@madrid.uned.es
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Abstract

Given an integer $g>2$ , we state necessary and sufficient conditions for a finite Abelian group to act as a group of automorphisms of some compact nonorientable Riemann surface of genus g. This result provides a new method to obtain the symmetric cross-cap number of Abelian groups. We also compute the least symmetric cross-cap number of Abelian groups of a given order and solve the maximum order problem for Abelian groups acting on nonorientable Riemann surfaces.

Keywords

Nonorientable surfaceKlein surfaceAutomorphism groupSymmetric cross-cap number

MSC classification

Primary: 57M60: Group actions in low dimensions
Secondary: 20F05: Generators, relations, and presentations 20H10: Fuchsian groups and their generalizations 30F50: Klein surfaces
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 64 , Issue 3 , September 2022 , pp. 634 - 648
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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