The action of the groups D m × D n on unbordered Klein surfaces

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 \({\widetilde{\sigma}(G)}\). 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, σ0(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 Dm × Dn. 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.