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The action of the groups Dm × Dn on unbordered Klein surfaces

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
Uncontrolled Keywords:Klein surfaces; Strong symmetric genus; Symmetric crosscap number
Subjects:Sciences > Mathematics > Group Theory
ID Code:15815
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