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The Riley slice: a concrete moduli space of hyperbolic manifolds

  • Alexander Elzenaar (MPI MiS, Leipzig)
E1 05 (Leibniz-Saal)

Abstract

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.

Mirke Olschewski

MPI for Mathematics in the Sciences Contact via Mail

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