### Impacto

Etayo Gordejuela, J. Javier and Martínez García, Ernesto
(2006)
*The real genus of cyclic by dihedral and dihedral by dihedral groups.*
Journal of Algebra, 296
(1).
pp. 145-156.
ISSN 0021-8693

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Official URL: http://www.sciencedirect.com/science/article/pii/S0021869305002097

## Abstract

Every finite group G acts as an automorphism group of several bordered compact Klein surfaces. The minimal genus of these surfaces is called the real genus and denoted by ρ(G).

The systematical study was begun by C.L. May and continued by him in several other papers about the topic. As a consequence of these works, he and other authors obtained the groups such that 0⩽ρ(G)⩽8. The real genus of many families of groups has also been calculated. In this work we are interested in finding the real genus of each groupDr×Ds, where both factors are dihedral groups. Results depend on the real genus of groupsCm×Dn, where Cm is a cyclic group. The case m odd was studied by May and the authors have studied the case m even. The result of May needs to be slightly corrected.

In this work we complete the proof of May for the case m odd and we calculate the real genus of the groupsDr×Ds.

Item Type: | Article |
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Uncontrolled Keywords: | Group actions in low dimensions; Fuchsian groups and their generalizations |

Subjects: | Sciences > Mathematics > Algebraic geometry |

ID Code: | 15791 |

References: | N.L. Ailing, N. Greenleaf, Foundations of the Theory of Klein Surfaces, Lecture Notes in Math., vol. 219, Springer, 1971. E. Bujalance, E. Martínez, A remark on NEC groups of surfaces with boundary, Bull. London Math. Soc. 21 (1989) 263–266. J.J. Etayo, E. Martínez, El género real de los grupos C2m × Dn, in: Contribuciones Matemáticas. Libro Homenaje al Profesor Enrique Outerelo, Universidad Complutense, 2004, pp. 171–182. G. Gromadzki, B. Mockiewicz, The groups of real genus 6, 7 and 8, Houston J. Math. 28 (2002) 691–699. A.M. Macbeath, The classification of non-Euclidean crystallographic groups, Canad. J. Math. 19 (1967) 1192–1205. C.L. May, Large automorphism groups of compact Klein surfaces with boundary, Glasgow Math. J. 18 (1977) 1–10. C.L. May, The groups of real genus 4, Michigan Math. J. 39 (1992) 219–228. C.L. May, Finite groups acting on bordered surfaces and the real genus of a group, Rocky Mountain J. Math. 23 (1993) 707–724. C.L. May, Groups of small real genus, Houston J. Math. 20 (1994) 393–408. C.L. May, Finite metacyclic groups acting on bordered surfaces, Glasgow Math. J. 36 (1994) 23–240. D. McCullough, Minimal genus of abelian actions on Klein surfaces with boundary, Math. Z. 205 (1990) 421–436. R. Preston, Projective structures and fundamental domains on compact Klein surfaces, Thesis Univ. of Texas, 1975. H.C. Wilkie, On non-Euclidean crystallographic groups, Math. Z. 91 (1966) 87–102. |

Deposited On: | 28 Jun 2012 08:54 |

Last Modified: | 06 Feb 2014 10:31 |

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