Closed quasi-Fuchsian surfaces in hyperbolic knot complements

Masters, Joseph D.; Zhang, Xingru

doi:10.2140/gt.2008.12.2095

Cited by 10 publications

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References 16 publications

(29 reference statements)

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“…Masters and Zhang prove and apply a special case of a version of this result. In [11] they introduced a refined version of subgroup separability for a surface with boundary. We found a new proof [3] of a slight generalization of this theorem, and this result is used heavily in this paper.…”

Section: Comparison With the Proof Of Masters And Zhangmentioning

confidence: 99%

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The paper contains a new proof that a complete, non-compact hyperbolic 3-manifold M with finite volume contains an immersed, closed, quasi-Fuchsian surface.A complete finite-volume hyperbolic 3-manifold with cusps is a non-compact hyperbolic 3-manifold with finite volume and universal cover hyperbolic space. We give a new proof of the following result of Masters and Zhang [11], [12].Theorem 0.1. Suppose M is a complete finite-volume hyperbolic 3-manifold with cusps.

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“…A complete finite-volume hyperbolic 3-manifold with cusps is a non-compact hyperbolic 3-manifold with finite volume and universal cover hyperbolic space. We give a new proof of the following result of Masters and Zhang [11], [12].…”

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Finite-volume hyperbolic 3–manifolds contain immersed quasi-Fuchsian surfaces

Baker

Cooper

2015

Algebr. Geom. Topol.

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Abstract. The paper contains a new proof that a complete, non-compact hyperbolic 3-manifold M with finite volume contains an immersed, closed, quasi-Fuchsian surface.A complete finite-volume hyperbolic 3-manifold with cusps is a non-compact hyperbolic 3-manifold with finite volume and universal cover hyperbolic space. We give a new proof of the following result of Masters and Zhang [11], [12].Theorem 0.1. Suppose M is a complete finite-volume hyperbolic 3-manifold with cusps. Then there is a π 1 -injective immersion f : S −→ M of a closed, orientable surface S with genus at least 2 and f * (π 1 S) is a quasi-Fuchsian subgroup of π 1 M .Cooper, Long and Reid [6] showed that such manifolds contain geometrically finite closed surface groups but there might be accidental parabolics. Kahn and Markovic [9] have shown that a closed hyperbolic 3-manifold contains an immersed QF (quasi-Fuchsian) surface.A prefabricated 3-manifold, Z, is the union of a finite number of convex pieces, each of which is either a rank-2 cusp or a QF manifold Q i with rank-1 cusps. We require simple combinatorics: there are exactly two rank-1 cusps, with slopes that intersect once, inside each rank-2 cusp of Z. See (1.1) for the precise definition. The convex combination theorem [2] is used to ensure Z has a convex thickening, CH(Z). In this case ∂Z consists of closed incompressible surfaces without parabolics. The main theorem follows from (1.2) which says there is a covering space of M that has a convex core which is a prefabricated manifold. This construction of QF surfaces is similar to the method used in [2] and [5].The crucial step is to control how the QF manifold pieces of Z intersect. In section (3) we study the intersection Q 1 ∩ Q 2 of two QF manifolds with cusps. This is governed by a finite collection of convex subsurfaces immersed by local isometries into ∂Q i . A compact core of Q i is homeomorphic to an interval times a compact surface F . A spider is a compact subsurface X ⊂ F , satisfying certain conditions. After taking finite covers, each component of Q 1 ∩ Q 2 is described by a spider.The crucial step relies on a result about surfaces: the spider theorem (2.5). Each component of X ∩ ∂F is an arc called a foot of the spider X. We show that if every component of ∂F contains at least one spider foot then, after replacing F by a suitable finite cover of F , and choosing certain lifts of the spiders, every boundary component of F contains exactly one spider foot. This ensures the above mentioned simple combinatorics for Z.In section (5) we discuss the relation between our proof and that of Masters and Zhang. Acknowledgements: The authors thank IHP and Université de Rennes 1 for hospitality during completion of this work.

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Section: Comparison With the Proof Of Masters And Zhangmentioning

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Finite-volume hyperbolic 3–manifolds contain immersed quasi-Fuchsian surfaces

Baker

Cooper

2015

Algebr. Geom. Topol.

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“…Assume that no element of H is peripheral. Given a (possibly empty) finite subset B ⊂ π 1 (F ) \ H, there exists a finite-sheeted cover p :iii)F \ S is connected and incl * :iv) The covering is conservative.This theorem, without (iii), is due to Masters and Zhang [4] and is a key ingredient in their proof that cusped hyperbolic 3-manifolds contain quasi-Fuchsian surface groups [4], [5]. Without (iii), (iv) the theorem is a special case of well-known theorems on subgroup separability of free groups [1] and surface groups [6], [7].…”

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“…This theorem, without (iii), is due to Masters and Zhang [4] and is a key ingredient in their proof that cusped hyperbolic 3-manifolds contain quasi-Fuchsian surface groups [4], [5]. Without (iii), (iv) the theorem is a special case of well-known theorems on subgroup separability of free groups [1] and surface groups [6], [7].…”

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Conservative subgroup separability for surfaces with boundary

Baker

Cooper

2013

Algebr. Geom. Topol.

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Abstract. If F is a surface with boundary, then a finitely generated subgroup without peripheral elements of G = π 1 F can be separated from finitely many other elements of G by a finite index subgroup of G corresponding to a finite coverF with the same number of boundary components as F .Suppose F is a compact, orientable surface with nonempty boundary. A non-trivial element of π 1 (F ) is peripheral if it is represented by a loop freely homotopic into ∂F . A covering space p :F −→ F is called conservative if F andF have the same number of boundary components:Theorem 0.1 (Main theorem). Let F be a compact, connected, orientable surface with ∂F = φ and H ⊂ π 1 (F ) a finitely generated subgroup. Assume that no element of H is peripheral. Given a (possibly empty) finite subset B ⊂ π 1 (F ) \ H, there exists a finite-sheeted cover p :iii)F \ S is connected and incl * :iv) The covering is conservative.This theorem, without (iii), is due to Masters and Zhang [4] and is a key ingredient in their proof that cusped hyperbolic 3-manifolds contain quasi-Fuchsian surface groups [4], [5]. Without (iii), (iv) the theorem is a special case of well-known theorems on subgroup separability of free groups [1] and surface groups [6], [7]. For a discussion of subgroup separability and 3-manifolds, see [3].The proof in [4] uses the folded graph techniques due to Stallings, see [2]. The shorter proof below uses cut and cross-join of surfaces. A cover is called good if properties (i)-(ii) hold and very good if (i)-(iii) hold. The idea is to start with a good cover and then pass to a second cover which is very good. Then cross-join operations (defined below) are used to reduce the number of boundary components of a very good cover until it is conservative.

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Quasifuchsian state surfaces

Futer

Kalfagianni

Purcell

2014

Trans. Amer. Math. Soc.

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This paper continues our study, initiated in [arXiv:1108.3370], of essential state surfaces in link complements that satisfy a mild diagrammatic hypothesis (homogeneously adequate). For hyperbolic links, we show that the geometric type of these surfaces in the Thurston trichotomy is completely determined by a simple graph--theoretic criterion in terms of a certain spine of the surfaces. For links with A- or B-adequate diagrams, the geometric type of the surface is also completely determined by a coefficient of the colored Jones polynomial of the link.Comment: 21 pages, 9 figures: To appear in the Transactions of the AM

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Closed quasi-Fuchsian surfaces in hyperbolic knot complements

Abstract: We show that every hyperbolic knot complement contains a closed quasi-Fuchsian surface.57N35; 57M25

Cited by 10 publications

References 16 publications

Finite-volume hyperbolic 3–manifolds contain immersed quasi-Fuchsian surfaces

Finite-volume hyperbolic 3–manifolds contain immersed quasi-Fuchsian surfaces

Conservative subgroup separability for surfaces with boundary

Quasifuchsian state surfaces

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