Symmetric Riemann Surfaces, Torsion Subgroups and Schottky Coverings

Abstract

We consider a torsion-free Fuchsian group $G$ acting on $H$ which admits an orientation reversing involution $j$. That is, $jGj = G$. Let $T$ be the orientation preserving half of the torsion subgroup of the extended group $\left \langle {G,j} \right \rangle$. By considering invariant homology basis elements of the surface $H/G$, we show that the surface $H/T$ is planar, and that the group $G/T$ acts on $H/T$ as a Schottky group.