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 $\Gamma$ admits a $\Gamma$-invariant connected component $\Omega_{\Gamma}$ of its region of discontinuity $\Omega(\Gamma)$, such that every other connected component of $\Omega(\Gamma) \setminus \Omega_{\Gamma}$ is a topological disc with trivial $\Gamma$-stabilizer, and $\Omega_{\Gamma}/\Gamma$ 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 $\Sigma_{F}={\mathbb H}^{2}/F$ is an infinite-type Riemann surface with no planar ends. Then there exists an infinite-type Schottky group $\Gamma$ such that $\Sigma_{F}$ is isomorphic to $\Omega_{\Gamma}/F$ (retrosection theorem). If $G < {\rm Aut}(\Sigma_{F})$ acts freely and $\Sigma_{F}/G$ is of finite-type, then we observe that (i) the existence of some infinite Schottky $\Gamma$ such that $\Omega_{\Gamma}/\Gamma$ and $\Sigma_{F}$ are conformally equivalent and for which $G$ lifts to a group of automorphisms of $\Omega_{\Gamma}$, is equivalent to (ii) the existence of 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. This generalizes the situation for the case of closed Riemann surfaces and Schottky groups of finite rank.