The Riley slice: a concrete moduli space of hyperbolic manifolds

It has been known since at least the time of Poincaré that isometries of 3-dimensional hyperbolic space $ \mathbb{H}^3 $ can be represented by $ 2 \times 2$ matrices over the complex numbers: the matrices represent fractional linear transformations on the sphere at infinity, and hyperbolic space is rigid enough that every hyperbolic motion is determined by such an action at infinity. A discrete subgroup of $ \PSL(2,\mathbb{C}) $ is called a Kleinian group; the quotient of $ \mathbb{H}^3 $ by the action of such a group is an orbifold; and its boundary at infinity is a Riemann surface. The Riley slice is arguably the simplest example of a moduli space of Kleinian groups; it is naturally embedded in $ \mathbb{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 (joint with Gaven Martin and Jeroen Schillewaert) which extend the work of Keen and Series.