### Impacto

Etayo Gordejuela, J. Javier and Martínez García, Ernesto
(2011)
*The action of the groups Dm × Dn on unbordered Klein surfaces.*
Revista de la Real Academia de Ciencias Exactas Fisicas y Naturales Serie A: Matemáticas, 105
(1).
pp. 95-106.
ISSN 1578-7303

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## Abstract

Every finite group G may act as an automorphism group of Klein surfaces either bordered or unbordered either orientable or non-orientable. For each group the minimum genus receives different names according to the topological features of the surface X on which it acts. If X is a bordered surface the genus is called the real genus ρ(G). If X is a non-orientable unbordered surface the genus is called the symmetric crosscap number of G and it is denoted by [(s)\tilde](G)(G). Finally if X is a Riemann surface it has two related parameters. If G only contains orientation-preserving automorphisms we have the strong symmetric genus, σ 0(G). If we allow orientation-reversing automorphisms we have the symmetric genus σ(G). In this work we obtain the strong symmetric genus and the symmetric crosscap number of the groups D m × D n . The symmetric genus of these groups is 1. However we introduce and obtain a new parameter, denoted by τ as the least genus g ≥ 2 of Riemann surfaces on which these groups act disregarding orientation

Item Type: | Article |
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Uncontrolled Keywords: | Klein surfaces; Strong symmetric genus; Symmetric crosscap number |

Subjects: | Sciences > Mathematics > Group Theory |

ID Code: | 15815 |

References: | Alling, N.L., Greenleaf, N.: Foundations of the Theory of Klein Surfaces. Lect. Not. in Math., vol. 219. Springer, Berlin (1971) Bujalance E.: Cyclic groups of automorphisms of compact non-orientable Klein surfaces without boundary. Pac. J. Math. 109, 279–289 (1983) Etayo, J.J.: Sobre grupos de automorfismos de superficies de Klein, Tesis Doctoral. Universidad Complutense (1983) Etayo J.J., Martínez E.: The real genus of cyclic by dihedral and dihedral by dihedral groups. J. Algebra. 296, 145–156 (2006) Etayo J.J., Martínez E.: The symmetric crosscap number of the groups C m × D n . Proc. R. Soc. Edinburgh A 138, 1197–1213 (2008) Gromadzki G.: Abelian groups of automorphisms of compact non-orientable Klein surfaces without boundary. Comment. Math. Prace Mat. 28, 197–217 (1989) Gross J.L., Tucker T.W.: Topological Graph Theory. Wiley, New York (1987) Macbeath A.M.: The classification of non-Euclidean crystallographic groups. Can. J. Math. 19, 1192–1205 (1967) May C.L.: The symmetric crosscap number of a group. Glasgow Math. J. 41, 399–410 (2001) May C.L., Zimmerman J.: There is a group of every strong symmetric genus. Bull. Lond. Math. Soc. 35, 433–439 (2003) Preston, R.: Projective Structures and fundamental domains on compact Klein surfaces, Thesis. Univ. of Texas (1975) Singerman D.: Automorphisms of compact non-orientable Riemann surfaces. Glasgow Math. J. 12, 50–59 (1971) Tucker, T.W.: Symmetric embeddings of Cayley graphs in non-orientable surfaces. In: Alavy, I., et al. (eds.) Graph Theory, Combinatorics and Applications, pp. 1105–1120 (1991) Wilkie H.C.: On non-Euclidean crystallographic groups. Math. Z. 91, 87–102 (1966) |

Deposited On: | 03 Jul 2012 09:41 |

Last Modified: | 06 Feb 2014 10:32 |

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