# A complex hyperbolic Riley slice

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### Cites background from "A complex hyperbolic Riley slice"

...spherical CR structure on a closed hyperbolic 3 manifold. Recently, in [15] Deraux and Falbel have described a spherical CR structure on the complement of the ﬁgure eight knot. In the article to come [87], Parker and Will produce an example of a spherical CR structure on the complement of the Whitehead link that is not conjugate to Schwartz’s one. The question of knowing which hyperbolic 3 manifolds a...

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### "A complex hyperbolic Riley slice" refers background in this paper

...6 of [2] in the context of the Poincaré disc....

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### "A complex hyperbolic Riley slice" refers background in this paper

...Whitehead link complement from an ideal octahedron equipped with face identiﬁcations. The Whitehead link is depicted in Figure 2. We refer to Section 10.3 of Ratcliﬀe [29] and Section 3.3 of Thurston [35] for classical information about the topology of the Whitehead link complement and its hyperbolic structure. 1.6 Further remarks Other discrete groups appearing in U. As well as the ideal triangle gro...

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...e obtain a true combinatorial octahedron. The face identiﬁcations given above make the quotient manifold homeomorphic to the complement of the Whitehead link (compare for instance with Section 3.3 of [35]). 7 Technicalities. 7.1 The triple intersections: proofs of Proposition 4.5 and Lemma 6.2. In this section we ﬁrst prove Proposition 4.5, which states that the triple intersection must contain a poin...

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### "A complex hyperbolic Riley slice" refers background or methods in this paper

...This is the limit of Dirichlet polyhedra as the centre point approaches ∂H2 C ; see Section 9.3 of Goldman [15]....

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...Theorem 1.2 (Goldman, Parker [16], Schwartz [31, 33])....

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...Élie Cartan defined an invariant of triples of pairwise distinct points p1, p2, p3 in ∂H 2 C ; see Section 7.1 of Goldman [15]....

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...In fact, there is a 1-parameter family of parabolic conjugacy classes, see for instance Chapter 6 of [15]....

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...Lemma 2.3 (Section 7.1.3 of Goldman [15])....

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### "A complex hyperbolic Riley slice" refers background in this paper

...Groups generated by complex reflections of higher order are also interesting, see [22] for example, but we do not consider them here....

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...A generalisation to the case of H2 C has already appeared in Mostow [22] and Deraux, Parker, Paupert [9]....

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...We can now state the version of the Poincaré polyhedron theorem that we need (compare [22] or [9])....

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