Concrete one complex dimensional moduli spaces of hyperbolic manifolds and orbifolds

TL;DR

This paper introduces the Riley slice, a fundamental moduli space of Kleinian groups, reviews its background, and extends existing work to one complex dimensional moduli spaces associated with groups isomorphic to , broadening understanding of hyperbolic 3-manifolds.

Contribution

It provides an accessible introduction to the Riley slice and extends the analysis to new moduli spaces of Kleinian groups with specific algebraic structures.

Findings

Extended the work of Keen and Series to new moduli spaces

Connected the Riley slice to broader classes of Kleinian groups

Provided background and survey for non-experts in the field

Abstract

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, extending the work of Keen and Series to the one complex dimensional moduli spaces of Kleinian groups isomorphic to $\mathbb{Z}_p*\mathbb{Z}_q$ acting on the Riemann sphere, $2\leq p,q \leq \infty$. The Riley slice is the case $p=q=\infty$ (i.e. two parabolic generators).