Infinite Schottky groups and group actions on infinite type surfaces

TL;DR

This paper introduces infinite Schottky groups, a class of purely loxodromic free Kleinian groups, and explores their actions on infinite type Riemann surfaces, establishing a correspondence with certain group actions and surface decompositions.

Contribution

It defines infinite Schottky groups via simple loops, proves a retrosection theorem for infinite type surfaces, and characterizes automorphism group actions in terms of invariant simple loop collections.

Findings

Infinite Schottky groups are associated with infinite type Riemann surfaces without planar ends.

Every such surface can be obtained via an infinite Schottky uniformization.

Conditions are given for lifting automorphism group actions to the universal cover.

Abstract

In this paper, we introduce a collection of purely loxodromic free Kleinian groups, called infinite Schottky group, which are defined by a suitable collection of simple loops in a similar way as in the case for Schottky groups of finite rank. An infinite Schottky group $\Gamma$ admits a $\Gamma$-invariant connected component $\Omega$ of its region of discontinuity, such that every other component is a topological disc and has trivial $\Gamma$-stabilizer, and $\Omega/\Gamma$ is an infinite type Riemann surface without planar ends. Every infinite type Riemann surface $\Sigma_{F}$ without planar ends can be so obtained (retrosection theorem). If $G < {\rm Aut}(\Sigma_{F})$ acts freely and $\Sigma_{F}/G$ is of finite type, then we observe that it lifts to a group of automorphisms of $\Omega$, for a suitable infinite Schottky uniformization of it by a infinite Schottky group $\Gamma$, if and only if there is a $G$-invariant collection ${\mathcal F}$ of pairwise disjoint essential simple loops on $\Sigma_{F}$ such that each connected component of $\Sigma_{F} \setminus {\mathcal F}$ is a finite planar surface, generalizing the situation for the case of Schottky groups of finite rank.