Towards a proof of the Classical Schottky Uniformization Conjecture

2017/09/27 by Hidalgo, Rubén A.
#30F10 #30F40 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1709.09515

Abstract

By Koebe's retrosection theorem, every closed Riemann surface of genus g ≥ 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 ⊂ \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.

Related

• Schottky uniformizations of Automorphisms of Riemann surfaces
• Crossing the transcendental divide: from Schottky groups to algebraic curves
• Uniformization By Classical Schottky Groups
• Schottky spaces and universal Mumford curves over ℤ
• On smooth moduli space of Riemann surfaces