Infinite Schottky groups and group actions on infinite type surfaces

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.