Chebyshev polynomials and inequalities for Kleinian groups

Alaqad, Hala; Gong, Jianhua; Martin, Gaven

doi:10.1142/s0219199721501029

Cited by 3 publications

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“…Lemma 1]; • The Riley slice is invariant under a particular dynamical system [79, Lemma 2], c.f. [90], [92], [13], [44, Section 3], and our upcoming preprint [46];…”

Section: The Riley Slice As a Quasiconformal Deformation Spacementioning

confidence: 99%

Concrete one complex dimensional moduli spaces of hyperbolic manifolds and orbifolds

Elzenaar¹,

Martin²,

Schillewaert³

2022

Preprint

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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 Z p * Z q acting on the Riemann sphere, 2 ≤ p, q ≤ ∞. The Riley slice is the case p = q = ∞ (i.e. two parabolic generators).

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“…Lemma 1]; • The Riley slice is invariant under a particular dynamical system [79, Lemma 2], c.f. [90], [92], [13], [44, Section 3], and our upcoming preprint [46];…”

Section: The Riley Slice As a Quasiconformal Deformation Spacementioning

confidence: 99%

Concrete one complex dimensional moduli spaces of hyperbolic manifolds and orbifolds

Elzenaar¹,

Martin²,

Schillewaert³

2022

Preprint

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“…Then, D 2 is the projection of the three complex dimensional moduli space D to ℂ 2 . Applications of this set being closed are in the generalizations of Jørgensen's inequality quantifying the isolated nature of the elementary groups in the moduli space of discrete groups [4].…”

Section: Introductionmentioning

confidence: 99%

Projections in Moduli Spaces of the Kleinian Groups

Alaqad

Gong

Martin

2022

Abstract and Applied Analysis

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A two-generator Kleinian group f , g can be naturally associated with a discrete group f , ϕ with the generator ϕ of order two and where f , ϕ f ϕ − 1 = f , g f g − 1 ⊂ f , g , f , ϕ : f , g f g − 1 = 2 . This is useful in studying the geometry of the Kleinian groups since f , g will be discrete only if f , ϕ is, and the moduli space of groups f , ϕ is one complex dimension less. This gives a necessary condition in a simpler space to determine the discreteness of f , g . The dimension reduction here is realised by a projection of principal characters of the two-generator Kleinian groups. In applications, it is important to know that the image of the moduli space of Kleinian groups under this projection is closed and, among other results, we show how this follows from Jørgensen’s results on algebraic convergence.

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