Complex hyperbolic geometry of the figure-eight knot

Deraux, Martin; Falbel, Elisha

doi:10.2140/gt.2015.19.237

Cited by 37 publications

(92 citation statements)

References 21 publications

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“…4 (3,3,4)-group uniformising the figure eight knot complement. Obtained by Deraux and Falbel in [8]. 5 (3,3, n)-groups, proved to be discrete by in [26].…”

Section: Main Definitions and Discreteness Resultsmentioning

confidence: 88%

“…Remark 1. From (8), it is easy to see that projections of boundaries of Cygan spheres onto the z-factor are closed Euclidean discs in C. This correspond to the vertical projection onto C in the Heisenberg group. This fact is often useful to prove that two Cygan spheres are disjoint.…”

Section: Cygan Spheres and Geographical Coordinatesmentioning

confidence: 97%

“…We give them in (α 1 , α 2 ) coordinates and illustrate them in Figure 1 1. The groups corresponding to α 1 = 0 and α 2 = ± arccos 1/8 have been studied in great detail by Deraux and Falbel who proved that they give a spherical CR uniformisation of the figureeight knot complement [8]. This illustrates the fact that there is no statement for Theorem 1.3 analogous to the second part of Theorem 1.2: the group from [8] is contained in a discrete (non-faithful) (3, 3, ∞) triangle groups where I 2 I 1 I 3 I 1 is elliptic.…”

Section: Further Remarksmentioning

confidence: 99%

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A complex hyperbolic Riley slice

Parker

Will

2017

Geom. Topol.

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The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. We study subgroups of PU.2; 1/ generated by two noncommuting unipotent maps A and B whose product AB is also unipotent. We call U the set of conjugacy classes of such groups. We provide a set of coordinates on U that make it homeomorphic to R 2 .By considering the action on complex hyperbolic space H 2 C of groups in U , we describe a two-dimensional disc Z in U that parametrises a family of discrete groups. As a corollary, we give a proof of a conjecture of Schwartz for .3; 3; 1/-triangle groups. We also consider a particular group on the boundary of the disc Z where the commutator OEA; B is also unipotent. We show that the boundary of the quotient orbifold associated to the latter group gives a spherical CR uniformisation of the Whitehead link complement.20H10, 22E40, 51M10; 57M50

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“…4 (3,3,4)-group uniformising the figure eight knot complement. Obtained by Deraux and Falbel in [8]. 5 (3,3, n)-groups, proved to be discrete by in [26].…”

Section: Main Definitions and Discreteness Resultsmentioning

confidence: 88%

Section: Cygan Spheres and Geographical Coordinatesmentioning

confidence: 97%

Section: Further Remarksmentioning

confidence: 99%

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A complex hyperbolic Riley slice

Parker

Will

2017

Geom. Topol.

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“…] . For this manifold, one get a complete list of representation whose peripheral holonomy is unipotent (see [Fal08] and more recently [DE13], which shows that ρ 2 and ρ 3 are intimately related). Up to some Galois conjugations there are only 4 of them:…”

Section: Example : the 8-knot Complementmentioning

confidence: 99%

Deformation of hyperbolic manifolds in 𝑃𝐺𝐿(𝑛,𝐂) and discreteness of the peripheral representations

Guilloux¹

2014

Proc. Amer. Math. Soc.

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Abstract. Let M be a cusped hyperbolic 3-manifold, e.g. a knot complement. Thurston [Thu79] showed that the space of deformations of its fundamental group in PGL(2, C) (up to conjugation) is of complex dimension the number ν of cusps near the hyperbolic representation. It seems natural to ask whether some representations remain discrete after deformation. The answer is generically not. A simple reason for it lies inside the cusps: the degeneracy of the peripheral representation (i.e. representations of fundamental groups of the ν peripheral tori). They indeed generically become non discrete, except for a countable set. This last set corresponds to hyperbolic Dehn surgeries on M , for which the peripheral representation is no more faithful.We work here in the framework of PGL(n, C). The hyperbolic structure lifts, via the n-dimensional irreducible representation, to a representation ρ geom . We know from the work of that the space of deformations of ρ geom has complex dimension (n − 1)ν.We prove here that, unlike the PGL(2)-case, the generic behaviour becomes the discreteness (and faithfulness) of the peripheral representation: in a neighbourhood of the geometric representation, the non-discrete peripheral representations are contained in a real analytic subvariety of codimension ≥ 1.

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“…The case n = 4 combines work of Deraux, Falbel and Wang [3,6]. The cleanest statement may be found in Theorem 4.2 of Deraux [2], which also treats the case n = 5.…”

Section: Introductionmentioning

confidence: 97%

Complex hyperbolic (3,3,n) triangle groups

2016

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The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. Let p, q, r be positive integers. Complex hyperbolic (p, q, r) triangle groups are representations of the hyperbolic (p, q, r) reflection triangle group to the holomorphic isometry group of complex hyperbolic space H 2 C , where the generators fix complex lines. In this paper, we obtain all the discrete and faithful complex hyperbolic (3, 3, n) triangle groups for n ≥ 4. Our result solves a conjecture of Schwartz in the case when p = q = 3.

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Complex hyperbolic geometry of the figure-eight knot

Abstract: Abstract. We show that the figure eight knot complement admits a uniformizable spherical CR structure, i.e. it occurs as the manifold at infinity of a complex hyperbolic orbifold. The uniformization is unique provided we require the peripheral subgroups to have unipotent holonomy.

Cited by 37 publications

References 21 publications

A complex hyperbolic Riley slice

A complex hyperbolic Riley slice

Deformation of hyperbolic manifolds in 𝑃𝐺𝐿(𝑛,𝐂) and discreteness of the peripheral representations

Complex hyperbolic (3,3,n) triangle groups

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