Uniformization by Classical Schottky Groups

Koebe’s Retrosection Theorem [8] states that every closed Riemann surface can be uniformized by a Schottky group. In [10] Marden showed that non-classical Schottky groups exist, and a first explicit example of a non-classical Schottky group was given by Yamamoto in [14]. Work on Schottky uniformizations of surfaces with certain symmetry has been done by people such as Hidalgo [7]. The natural question to ask is whether Koebe’s Theorem holds if we restrict to classical Schottky groups, that is Schottky groups where some set of defining curves can be taken to be circles. In this work we look at lifts of collars on a closed Riemann surface to the domain of discontinuity for its uniformizing Schottky group. We get a bound on the lengths of the core curves of the collars, and hence find surfaces which can be uniformized by classical Schottky groups. We have the following main theorem and corollary: