Schottky uniformizations of Automorphisms of Riemann surfaces

Abstract

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 $(\Omega,\Gamma,P:\Omega \to S)$, where $\Gamma$ is a Schottky group with region of discontinuity $\Omega$ and $P:\Omega \to S$ is a regular holomorphic cover map with $\Gamma$ as its deck group. Let $\tau:S \to S$ be a conformal (respectively, anticonformal) automorphism of $S$ of finite order $n$, and let $(\Omega,\Gamma,P:\Omega \to S)$ be a Schottky uniformization of $S$. Assume that $\tau$ lifts with respect to the previous Schottky uniformization, that is, there exists a M\"obius (respectively, extended M\"obius) transformation $\kappa$, keeping $\Omega$ invariant, with $P \circ \kappa=\tau \circ P$. The Kleinian (respectively, extended Kleinian) group $K=< \Gamma, \kappa >$ contains $\Gamma$ as a finite index normal subgroup and $K/\Gamma \cong {\mathbb Z}_{n}$. 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.