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145
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 101 (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.
Shrinkwrapping and the taming of hyperbolic 3manifolds
 J. Amer. Math. Soc
"... Thurston and many others developed the theory of geometrically finite ends of hyperbolic 3–manifolds. It remained to understand those ends which are not geometrically finite; such ends are called geometrically infinite. ..."
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Cited by 83 (0 self)
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Thurston and many others developed the theory of geometrically finite ends of hyperbolic 3–manifolds. It remained to understand those ends which are not geometrically finite; such ends are called geometrically infinite.
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
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.
JSJSplittings for Finitely Presented Groups over Slender Groups
, 1998
"... We generalize the JSJsplitting of Rips and Sela to give decompositions of finitely presented groups which capture splittings over certain classes of small subgroups. Such classes include the class of all 2ended groups and the class of all virtually Z \Phi Z groups. The approach, called "track zipp ..."
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Cited by 42 (0 self)
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We generalize the JSJsplitting of Rips and Sela to give decompositions of finitely presented groups which capture splittings over certain classes of small subgroups. Such classes include the class of all 2ended groups and the class of all virtually Z \Phi Z groups. The approach, called "track zipping", is relatively elementary, and differs from the RipsSela approach in that it does not rely on the theory of R trees but rather on an understanding of certain embedded 1complexes (called patterns) in a presentation 2complex for the ambient group.
Involutory Hopf algebras and 3manifold invariants
 Intern. J. Math
, 1991
"... We establish a 3manifold invariant for each finitedimensional, involutory Hopf algebra. If the Hopf algebra is the group algebra of a group G, the invariant counts homomorphisms from the fundamental group of the manifold to G. The invariant can be viewed as a state model on a Heegaard diagram or a ..."
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Cited by 40 (4 self)
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We establish a 3manifold invariant for each finitedimensional, involutory Hopf algebra. If the Hopf algebra is the group algebra of a group G, the invariant counts homomorphisms from the fundamental group of the manifold to G. The invariant can be viewed as a state model on a Heegaard diagram or a triangulation of the manifold. The computation of the invariant involves tensor products and contractions of the structure tensors of the algebra. We show that every formal expression involving these tensors corresponds to a unique 3manifold modulo a wellunderstood equivalence. This raises the possibility of an algorithm which can determine whether two given 3manifolds are homeomorphic. 1
Algebraic limits of Kleinian groups which rearrange the pages of a book, Invent
 Math
, 1996
"... Dedicated to Bernard Maskit on the occasion of his sixtieth birthday ..."
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Cited by 35 (10 self)
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Dedicated to Bernard Maskit on the occasion of his sixtieth birthday
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.
Quasiisometries preserve the geometric decomposition of Haken manifolds
, 1995
"... . We prove quasiisometry invariance of the canonical decomposition for fundamental groups of Haken 3manifolds with zero Euler characteristic. We show that groups quasiisometric to Haken manifold groups with nontrivial canonical decomposition are finite extensions of Haken orbifold groups. As a by ..."
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Cited by 31 (7 self)
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. We prove quasiisometry invariance of the canonical decomposition for fundamental groups of Haken 3manifolds with zero Euler characteristic. We show that groups quasiisometric to Haken manifold groups with nontrivial canonical decomposition are finite extensions of Haken orbifold groups. As a byproduct we describe all 2dimensional quasiflats in the universal covers of nongeometric Haken manifolds. 1 Contents 1 Introduction 2 2 Preliminaries 4 2.1 Notations and conventions . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 3manifolds and their canonical decomposition . . . . . . . . . . . . . 5 2.3 Ultralimits and asymptotic cones . . . . . . . . . . . . . . . . . . . . 6 2.4 Busemann functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.5 Quasiisometric embeddings into piecewise Euclidean spaces . . . . . 7 3 Asymptotic cones of universal covers of Haken manifolds 8 3.1 Geometric components . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Separa...
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