Towards a proof of the Classical Schottky Uniformization Conjecture

TL;DR

This work addresses whether every closed Riemann surface can be uniformized by a classical Schottky group. It shows that all Belyi curves admit classical Schottky uniformizations by refining Belyi maps and manipulating annulus moduli, and then leverages the density of Belyi curves in the genus g moduli space to deduce that the locus of surfaces with classical uniformizations is open and dense. The key contribution is a complete construction that converts Belyi data into a circular Schottky presentation, bridging a gap toward the classical Schottky Uniformization Conjecture. The result significantly clarifies the geometric structure of uniformizations and informs the global topology of the moduli space of genus g surfaces.

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 ${\mathcal M}_{g}$.

Theorems & Definitions (7)

• Theorem 1
• Lemma 1: Grötzch inequality
• Lemma 2
• Lemma 3
• Lemma 4
• Theorem 2
• proof