Classical Two-parabolic T-Schottky groups

Abstract

A $T$-Schottky group is a discrete group of M\"obius transformations whose generators identify pairs of, possibly-tangent, Jordan curves on the complex sphere, ${\hat{\IC}}$. If the curves are Euclidean circles then the group is termed classical $T$-Schottky. We describe the boundary of the space of classical $T$-Schottky groups affording two parabolic generators within the larger parameter space of all $T$-Schottky groups with two parabolic generators. This boundary is surprisingly different from that of the larger space. It is analytic while the boundary of the larger space appears to be fractal. Approaching the boundary of the smaller space does not correspond to pinching, circles necessarily become tangent but extra parabolics need not develop. As an application we construct an explicit one parameter family of two parabolic generator non-classical $T$-Schottky groups.