Concrete One Complex Dimensional Moduli Spaces of Hyperbolic Manifolds and Orbifolds: Kleinian Groups and the Generalised Riley Slices

The Riley slice is arguably the simplest example of a moduli space of Kleinian groups; it is naturally embedded in ℂ, and has a natural coordinate system (introduced by Linda Keen and Caroline Series in the early 1990s) which reflects the geometry of the underlying 3-manifold deformations. The Riley slice arises in the study of arithmetic Kleinian groups, the theory of two-bridge knots, the theory of Schottky groups, and the theory of hyperbolic 3-manifolds; because of its simplicity it provides an easy source of examples and deep questions related to these subjects. We give an introduction for the non-expert to the Riley slice and much of the related background material, assuming only graduate level complex analysis and topology; we review the history of and literature surrounding the Riley slice; and we announce some results of our own, extending the work of Keen and Series to the one complex dimensional moduli spaces of Kleinian groups isomorphic to ℤ/aℤ*ℤ/bℤ acting on the Riemann sphere, 2 ≤ a,b ≤ ∞. The Riley slice is the case of two parabolic generators where a = b = ∞.