Geometric description of Virtual Schottky groups

TL;DR

This paper provides a geometric structural decomposition of virtual Schottky groups, specifically when the quotient group is abelian, enhancing understanding of their automorphism groups of Riemann surfaces.

Contribution

It introduces a new geometric decomposition of virtual Schottky groups with abelian quotients, linking group structure to Riemann surface automorphisms.

Findings

Decomposition of virtual Schottky groups when the quotient is abelian

Connection between group structure and Riemann surface automorphisms

Enhanced understanding of Schottky uniformizations

Abstract

A virtual Schottky group is a Kleinian group $K$ containing a Schottky group $G$ as a finite index normal subgroup. These groups correspond to those groups of automorphisms of closed Riemann surfaces which can be realized at the level of their Schottky uniformizations. In this paper we provides a geometrical structural decomposition of $K$ in the particular case when $K/G$ is an abelian group.