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