Towards a proof of the Classical Schottky Uniformization Conjecture

TL;DR

This paper investigates the classical Schottky uniformization conjecture, showing that Belyi curves can be uniformized by classical Schottky groups, implying the density of such uniformizations in the moduli space of Riemann surfaces.

Contribution

It proves that all Belyi curves can be uniformized by classical Schottky groups, advancing the understanding of the classical Schottky uniformization conjecture.

Findings

Belyi curves can be uniformized by classical Schottky groups

The locus of Riemann surfaces uniformized by classical Schottky groups is dense in moduli space

The set of classical Schottky uniformized surfaces forms a non-empty open subset

Abstract

By Koebe's retrosection theorem, every closed Riemann surface of genus $g \geq 2$ is uniformized by a Schottky group. Marden observed that there are Schottky groups that are not classical ones, that is, they cannot be defined by a suitable collection of circles. This opened the question of whether every closed Riemann surface can be uniformized by a classical Schottky group. In this paper, we observe that every Belyi curve can be uniformized by a classical Schottky group. Since Belyi curves form a dense locus in the moduli space ${\mathcal M}_{g}$ and the locus ${\mathcal M}_{g}^{cs} \subset {\mathcal M}_{g}$ of those Riemann surfaces uniformized by classical Schottky groups is a non-empty open set, this ensures that ${\mathcal M}_{g}^{cs}$ is open and dense in ${\mathcal M}_{g}$.