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110
THE COLORED JONES POLYNOMIALS AND THE SIMPLICIAL VOLUME OF A Knot
, 1999
"... We show that the set of colored Jones polynomials and the set of generalized Alexander polynomials defined by Akutsu, Deguchi and Ohtsuki intersect nontrivially. Moreover it is shown that the intersection is (at least includes) the set of Kashaev’s quantum dilogarithm invariants for links. Theref ..."
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Cited by 96 (10 self)
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We show that the set of colored Jones polynomials and the set of generalized Alexander polynomials defined by Akutsu, Deguchi and Ohtsuki intersect nontrivially. Moreover it is shown that the intersection is (at least includes) the set of Kashaev’s quantum dilogarithm invariants for links. Therefore Kashaev’s conjecture can be restated as follows: The colored Jones polynomials determine the hyperbolic volume for a hyperbolic knot. Modifying this, we propose a stronger conjecture: The colored Jones polynomials determine the simplicial volume for any knot. If our conjecture is true, then we can prove that a knot is trivial if and only if all of its Vassiliev invariants are trivial.
Tameness of hyperbolic 3–manifolds
"... Marden conjectured that a hyperbolic 3manifold M with finitely generated fundamental group is tame, i.e. it is homeomorphic to the interior of a compact manifold with boundary [42]. Since then, many consequences of this conjecture have been developed by Kleinian group theorists and 3manifold topol ..."
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Cited by 65 (5 self)
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Marden conjectured that a hyperbolic 3manifold M with finitely generated fundamental group is tame, i.e. it is homeomorphic to the interior of a compact manifold with boundary [42]. Since then, many consequences of this conjecture have been developed by Kleinian group theorists and 3manifold topologists. We prove this
The classification of Kleinian surface groups II: The Ending Lamination Conjecture
, 2004
"... Thurston’s Ending Lamination Conjecture states that a hyperbolic 3manifold with finitely generated fundamental group is uniquely determined by its topological type and its end invariants. In this paper we prove this conjecture for Kleinian surface groups. The main ingredient is the establishment o ..."
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Cited by 58 (15 self)
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Thurston’s Ending Lamination Conjecture states that a hyperbolic 3manifold with finitely generated fundamental group is uniquely determined by its topological type and its end invariants. In this paper we prove this conjecture for Kleinian surface groups. The main ingredient is the establishment of a uniformly bilipschitz model for a Kleinian surface group. The first half of the proof appeared in [47], and a subsequent paper [15] will establish the Ending Lamination Conjecture in general.
Knot Floer homology detects fibred knots
"... Abstract Ozsváth and Szabó conjectured that knot Floer homology detects fibred knots in S 3. We will prove this conjecture for nullhomologous knots in arbitrary closed 3–manifolds. Namely, if K is a knot in a closed 3–manifold Y, Y − K is irreducible, and ̂HFK(Y, K) is monic, then K is fibred. The ..."
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Cited by 43 (4 self)
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Abstract Ozsváth and Szabó conjectured that knot Floer homology detects fibred knots in S 3. We will prove this conjecture for nullhomologous knots in arbitrary closed 3–manifolds. Namely, if K is a knot in a closed 3–manifold Y, Y − K is irreducible, and ̂HFK(Y, K) is monic, then K is fibred. The proof relies on previous works due to Gabai, Ozsváth–Szabó, Ghiggini and the author. A corollary is that if a knot in S 3 admits a lens space surgery, then the knot is fibred. AMS Classification 57R58, 57M27; 57R30.
Iteration on Teichmüller space
 Invent. Math
, 1994
"... this paper we use Riemann surface techniques to study the third iteration, and provide a new proof of a fundamental step in the geometrization of 3manifolds. (An expository account appears in [Mc2].) ..."
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Cited by 35 (11 self)
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this paper we use Riemann surface techniques to study the third iteration, and provide a new proof of a fundamental step in the geometrization of 3manifolds. (An expository account appears in [Mc2].)
Free Kleinian groups and volumes of hyperbolic 3manifolds
 J. Differential Geom
, 1996
"... The central result of this paper, Theorem 6.1, gives a constraint that must be satisfied by the generators of any free, topologically tame Kleinian group without parabolic elements. The following result is case (a) of Theorem 6.1. ..."
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Cited by 32 (24 self)
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The central result of this paper, Theorem 6.1, gives a constraint that must be satisfied by the generators of any free, topologically tame Kleinian group without parabolic elements. The following result is case (a) of Theorem 6.1.
Scalar curvature and geometrization conjectures for 3manifolds
 in Comparison Geometry (Berkeley 1993–94), MSRI Publications
, 1997
"... Abstract. We first summarize very briefly the topology of 3manifolds and the approach of Thurston towards their geometrization. After discussing some general properties of curvature functionals on the space of metrics, we formulate and discuss three conjectures that imply Thurston’s Geometrization ..."
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Cited by 31 (9 self)
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Abstract. We first summarize very briefly the topology of 3manifolds and the approach of Thurston towards their geometrization. After discussing some general properties of curvature functionals on the space of metrics, we formulate and discuss three conjectures that imply Thurston’s Geometrization Conjecture for closed oriented 3manifolds. The final two sections present evidence for the validity of these conjectures and outline an approach toward their proof.
Paradoxical decompositions, 2generator Kleinian groups, and volumes of hyperbolic 3manifolds
 J. Amer. Math. Soc
, 1992
"... The ɛthin part of a hyperbolic manifold, for an arbitrary positive number ɛ, is defined to consist of all points through which there pass homotopically nontrivial curves of length at most ɛ. For small enough ɛ, the ɛthin part is geometrically very simple: it is a disjoint union of standard neighb ..."
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Cited by 28 (19 self)
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The ɛthin part of a hyperbolic manifold, for an arbitrary positive number ɛ, is defined to consist of all points through which there pass homotopically nontrivial curves of length at most ɛ. For small enough ɛ, the ɛthin part is geometrically very simple: it is a disjoint union of standard neighborhoods of closed geodesics and cusps. (Explicit descriptions of
The classification of puncturedtorus groups
 ANNALS OF MATH
, 1999
"... Thurston’s ending lamination conjecture proposes that a finitely generated Kleinian group is uniquely determined (up to isometry) by the topology of its quotient and a list of invariants that describe the asymptotic geometry of its ends. We present a proof of this conjecture for puncturedtorus grou ..."
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Cited by 28 (3 self)
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Thurston’s ending lamination conjecture proposes that a finitely generated Kleinian group is uniquely determined (up to isometry) by the topology of its quotient and a list of invariants that describe the asymptotic geometry of its ends. We present a proof of this conjecture for puncturedtorus groups. These are free twogenerator Kleinian groups with parabolic commutator, which should be thought of as representations of the fundamental group of a punctured torus. As a consequence we verify the conjectural topological description of the deformation space of puncturedtorus groups (including Bers ’ conjecture that the quasiFuchsian groups are dense in this space) and prove a rigidity theorem: two puncturedtorus groups are quasiconformally conjugate if and only if they are topologically conjugate.
Towards the Poincaré Conjecture and the Classification of 3Manifolds
, 2003
"... The Poincaré Conjecture was posed ninetynine years ago and may possibly have been proved in the last few months. This note will be an account of some of the major results over the past hundred years which have paved the way towards a proof and towards the even more ambitious project of classifying a ..."
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Cited by 27 (0 self)
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The Poincaré Conjecture was posed ninetynine years ago and may possibly have been proved in the last few months. This note will be an account of some of the major results over the past hundred years which have paved the way towards a proof and towards the even more ambitious project of classifying all compact 3dimensional manifolds. The final paragraph provides a brief description of the latest developments, due to Grigory Perelman. A more serious discussion of Perelman’s work will be provided in a subsequent note by Michael Anderson.