Towards a proof of the Classical Schottky Uniformization Conjecture

Abstract: As a consequence of 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 which are not classical ones, that is, they cannot be defined by a suitable collection of circles. This opened the question to if every closed Riemann surface can be uniformized by a classical Schottky group. Recently, Hou has observed this question has an affirmative answer by first noting that every closed Riemann surface can be uniformized by a Schottky group with limit set of Hausdorff dimension $<1$ and then by proving that such a Schottky group is necessarily a classical one. In this paper, we provide another argument, based on the density of Belyi curves on the the moduli space ${\mathcal M}{g}$ and the fact that 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. We show that every Belyi curve can be uniformized by a classical Schottky group, so ${\mathcal M}{g}^{cs}={\mathcal M}_{g}$.