2022/04/25 by Alex Elzenaar, Elzenaar, Alex, Gaven Martin +3
Computer Science · Mathematics · #20H10 #30F40 #30F60 #57K32 #57R18 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2204.11422
openalex publication_date 2022/04/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ℤp*ℤq acting on the Riemann sphere, 2≤ p,q ≤ ∞. The Riley slice is the case p=q=∞ (i.e. two parabolic generators).