Description of Origamis by Schottky groups

TL;DR

This paper characterizes origami pairs, which are special branched covers of elliptic curves, using Schottky and Kleinian groups, providing a geometric framework for understanding their structure.

Contribution

It introduces the concept of origami-Schottky groups and describes their structure using Klein-Maskit combination theorems, linking origami pairs to Kleinian group theory.

Findings

Defines origami-Schottky groups as Kleinian groups containing a Schottky subgroup

Provides a geometric description of these groups using Klein-Maskit theorems

Establishes a connection between origami pairs and Kleinian group structures

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.