Real Schottky parametrizations

Abstract

A real algebraic curve of genus g is a pair (S,τ), where S is a closed Riemann surface of genus g and τ :S → S is an anticonformal involution. It was already known to Koebe that each real algebraic curve for which τ is a reflection can be uniformized by a real Schottky group, that is, a Schottky group that keeps invariant the unit circle. In the case that τ is an imaginary reflection, we produce uniformizations by either (i) real noded Klein–Schottky groups (once we have chosen some points on S as phantom nodes) or (ii) Klein–Schottky groups. We also give explicit descriptions of the real algebraic curves of genus 2 in terms of these types of uniformizing groups.