Schottky uniformizations of Automorphisms of Riemann surfaces

TL;DR

This paper studies how automorphisms of finite order on Riemann surfaces lift to Schottky uniformizations, characterizing the structure of associated Kleinian groups and counting their topologically distinct types.

Contribution

It provides a structural description of Kleinian groups arising from automorphisms of Riemann surfaces within Schottky uniformizations, including classification and counting results.

Findings

Determines the number of topologically different Kleinian groups for fixed automorphism order and Schottky rank.

Establishes a structural framework for these groups using Klein-Maskit's combination theorems.

Counts the number of conjugacy classes of normal Schottky subgroups for prime automorphism orders.

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.