Schottky uniformizations of Automorphisms of Riemann surfaces
It is well known that the collection of uniformizations of a closed Riemann surface $S$ is partially ordered; the lowest ones are the Schottky unformizations, that is, tuples $(Ω,Γ,P:Ω\to S)$, where $Γ$ is a Schottky group with region of discontinuity $Ω$ and $P:Ω\to S$ is a regular holomorphic cover map with $Γ$ as its deck group. Let $τ:S \to S$ be a conformal (respectively, anticonformal) automorphism of $S$ of finite order $n$, and let $(Ω,Γ,P:Ω\to S)$ be a Schottky uniformization of $S$. Assume that $τ$ lifts with respect to the previous Schottky uniformization, that is, there exists a Möbius (respectively, extended Möbius) transformation $κ$, keeping $Ω$ invariant, with $P \circ κ=τ\circ P$. The Kleinian (respectively, extended Kleinian) group $K=< Γ, κ>$ contains $Γ$ as a finite index normal subgroup and $K/Γ\cong {\mathbb Z}_{n}$. We provide a structural picture of $K$ in terms of the Klein-Maskit's combination theorems and some basic groups. Some consequences are (i) the determination of the number of topologically different types of such groups (fixed $n$ and the rank of the Schottky normal subgroup) and (ii) for $n$ prime, the number of normal Schottky normal subgroups, up to conjugacy, that $K$ has.