Lowest uniformizations of compact Klein surfaces

Abstract

A Schottky group is a purely loxodromic Kleinian group, with non-empty region of discontinuity, isomorphic to a free group of finite rank. An extended Schottky group is an extended Kleinian group whose orientation-preserving half is a Schottky group. The collection of uniformizations of either a closed Riemann surface or a compact Klein surface is partially ordered. In the case of closed Riemann surfaces, the lowest uniformizations are provided by Schottky groups. In this paper we provide simple arguments to see that the lowest uniformizations of compact Klein surfaces are exactly those produced by extended Schottky groups.