Description of Origamis by Schottky groups

Abstract

Let $(S,\eta)$ be an origami pair, that is, $S$ is a closed Riemann surface of genus $g \geq1$ and $\eta:S \to E$ is a holomorphic branched covering, with at most one branch value, where $E$ is a genus one Riemann surface. As the lowest uniformizations of $S$ are provided by Schottky groups, we are interested in describing origami pairs in terms of virtual Schottky groups. In other words, we are interested in those Kleinian groups $K$ which contain, as a finite index subgroup, a Schottky group $\Gamma$ such that $S=\Omega/\Gamma$ and such that $\eta$ is induced by the inclusion $\Gamma \leq K$. We say that $K$ is an origami-Schottky group. We provide a geometrical structural picture, in terms of the Klein-Maskit combination theorems, of these origami-Schottky groups.