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The Riley slice is arguably the simplest example of a moduli space ofKleinian groups; it is naturally embedded in $ \mathbb{C} $, and has a naturalcoordinate system (introduced by Linda Keen and Caroline Series in the early1990s) which reflects the geometry of the underlying 3-manifold deformations.The Riley slice arises in the study of arithmetic Kleinian groups, the theoryof two-bridge knots, the theory of Schottky groups, and the theory ofhyperbolic 3-manifolds; because of its simplicity it provides an easy source ofexamples and deep questions related to these subjects. We give an introductionfor the non-expert to the Riley slice and much of the related backgroundmaterial, assuming only graduate level complex analysis and topology; we reviewthe history of and literature surrounding the Riley slice; and we announce someresults of our own, extending the work of Keen and Series to the one complexdimensional moduli spaces of Kleinian groups isomorphic to$\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 parabolicgenerators).

Concrete one complex dimensional moduli spaces of hyperbolic manifolds and orbifolds