Infinite-type Schottky groups and group actions on infinite-type surfaces

Abstract

We introduce a certain class of purely loxodromic free Kleinian groups, called infinite-type Schottky groups, which are defined by a suitable collection of simple loops on the Riemann sphere, in a similar way as in the case of Schottky groups of finite rank. An infinite-type Schottky group $\hat{I}“$ admits a $\hat{I}“$-invariant connected component $\hat{I}{\text{c◯}}_{\hat{I}}“$ of its region of discontinuity $\hat{I}\text{c◯}(\hat{I}“)$, such that every other connected component of $\hat{I}\text{c◯}(\hat{I}“)\setminus \hat{I}{\text{c◯}}_{\hat{I}}“$ is a topological disc with trivial $\hat{I}“$-stabilizer, and $\hat{I}{\text{c◯}}_{\hat{I}}“/\hat{I}“$ is an infinite-type Riemann surface without planar ends. Let $F$ be a torsion-free purely hyperbolic Fuchsian group of the first kind such that $\hat{I}{\pounds }_F={\mathbb{H}}^2/F$ is an infinite-type Riemann surface with no planar ends. Then there exists an infinite-type Schottky group $\hat{I}“$ such that $\hat{I}{\pounds }_F$ is isomorphic to $\hat{I}{\text{c◯}}_{\hat{I}}“/F$ (retrosection theorem). If $G<\mathrm{Aut}(\hat{I}{\pounds }_F)$ acts freely and $\hat{I}{\pounds }_F/G$ is of finite-type, then we observe that (i) the existence of some infinite Schottky $\hat{I}“$ such that $\hat{I}{\text{c◯}}_{\hat{I}}“/\hat{I}“$ and $\hat{I}{\pounds }_F$ are conformally equivalent and for which $G$ lifts to a group of automorphisms of $\hat{I}{\text{c◯}}_{\hat{I}}“$, is equivalent to (ii) the existence of a $G$-invariant collection $\mathcal{F}$ of pairwise disjoint essential simple loops on $\hat{I}{\pounds }_F$ such that each connected component of $\hat{I}{\pounds }_F\setminus \mathcal{F}$ is a finite planar surface. This generalizes the situation for the case of closed Riemann surfaces and Schottky groups of finite rank.