Cyclic and dihedral actions on Klein surfaces with 2 boundary components

Abstract

In this paper, we obtain the cyclic and dihedral groups which occur as a group of automorphisms of Klein surfaces of topological genus g g with 2 boundary components, either orientable or non-orientable. For each of those groups G G we determine the values of g g such that G G acts on a surface of genus g g both in orientable or non-orientable cases. As a noteworthy result we obtain that whenever the cyclic group C N C_N acts on a surface of genus g g and given orientability character, the dihedral group D N D_N of order 2 N 2N also acts on a surface of the same topological type. Besides, we give a topological action of the group on the surface, by using the uniformization by means of NEC groups. The results are computable, and we exhibit explicitly, distinguishing according to the orientability, all groups C N C_N and D N D_N acting on surfaces of each genus g g for 1 ≤ g ≤ 5 1\leq g\leq 5 .