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126
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.
Homotopy hyperbolic 3manifolds are hyperbolic
 Ann. of Math
, 2003
"... This paper introduces a rigorous computerassisted procedure for analyzing hyperbolic 3manifolds. This procedure is used to complete the proof of several longstanding rigidity conjectures in 3manifold theory as well as to ..."
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Cited by 58 (4 self)
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This paper introduces a rigorous computerassisted procedure for analyzing hyperbolic 3manifolds. This procedure is used to complete the proof of several longstanding rigidity conjectures in 3manifold theory as well as to
Comparing Heegaard splittings of nonHaken 3manifolds
 Topology
, 1996
"... A proof that geometrically compressible onesided splittings of nonHaken 3manifolds are stabilised is given. This is a generalisation of a recent proof for the case of RP 3 and uses a modification of these techniques. Combined with known results about geometrically incompressible surfaces, the mai ..."
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Cited by 43 (13 self)
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A proof that geometrically compressible onesided splittings of nonHaken 3manifolds are stabilised is given. This is a generalisation of a recent proof for the case of RP 3 and uses a modification of these techniques. Combined with known results about geometrically incompressible surfaces, the main result fully classifies onesided splittings of small Seifert fibred spaces and the (6,1) Dehn filling of Figure 8 knot space. Drawing on minimal surface theory, it can also be used to show that nonHaken hyperbolic 3manifolds have finitely many isotopy classes of onesided splittings of bounded genus. 1
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 33 (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.
On the geometric and topological rigidity of hyperbolic 3manifolds
 J. Amer. Math. Soc
, 1997
"... Abstract. A homotopy equivalence between a hyperbolic 3manifold and a closed irreducible 3manifold is homotopic to a homeomorphsim provided the hyperbolic manifold satisfies a purely geometric condition. There are no known examples of hyperbolic 3manifolds which do not satisfy this condition. One ..."
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Cited by 31 (2 self)
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Abstract. A homotopy equivalence between a hyperbolic 3manifold and a closed irreducible 3manifold is homotopic to a homeomorphsim provided the hyperbolic manifold satisfies a purely geometric condition. There are no known examples of hyperbolic 3manifolds which do not satisfy this condition. One of the central problems of 3manifold topology is to determine when a homotopy equivalence between two closed orientable irreducible 3manifolds is homotopic to a homeomorphism. If one of these manifolds is S 3, then this is Poincaré’s problem. The results of [Re], [Fr], [Ru], [Bo], and [HR] (see also [Ol]) completely solve this problem for maps between lens spaces. In particular there exist nonhomeomorphic but homotopy equivalent lens spaces (e.g. L(7,1) and L(7,2)), and there exist selfhomotopy equivalences not homotopic to homeomorphisms (e.g. the selfhomotopy equivalence of L(8,1) whose π1map is multiplication by 3). By Waldhausen [W] (resp. Scott [S]) a homotopy equivalence between a closed Haken 3manifold (resp. a Seifertfibred space with infinite π1) and an irreducible 3manifold can be homotoped to a homeomorphism. By Mostow [M] a homotopy equivalence between two closed hyperbolic 3manifolds can be homotoped to a homeomorphism and in fact an isometry. However, the general case of homotopy equivalence between a hyperbolic 3manifold and an irreducible 3manifold remains to be investigated. These problems and results should be contrasted with the conjecture [T] that a closed irreducible orientable 3manifold is either Haken, or Seifert fibred with infinite π1, or the quotient of S 3 by an orthogonal action, or the quotient of H 3 via a cocompact group of hyperbolic isometries. Theorem 1 [G2]. Let N be a closed, orientable, hyperbolic 3manifold containing an embedded hyperbolic tube of radius (log 3)/2 =.549306... about a closed geodesic. Then: (i) If f: M → N is a homotopy equivalence where M is an irreducible 3manifold, then f is homotopic to a homeomorphism. (ii) If f, g: M → N are homotopic homeomorphisms, then f is isotopic to g. (iii) The space of hyperbolic metrics on N is path connected.
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
Finiteness of classifying spaces of relative diffeomorphism groups of 3manifolds, Geometry and Topology 1
, 1997
"... The main theorem shows that if M is an irreducible compact connected orientable 3manifold with nonempty boundary, then the classifying space BDiff (M rel ∂M) of the space of diffeomorphisms of M which restrict to the identity map on ∂M has the homotopy type of a finite aspherical CWcomplex. This a ..."
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Cited by 13 (4 self)
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The main theorem shows that if M is an irreducible compact connected orientable 3manifold with nonempty boundary, then the classifying space BDiff (M rel ∂M) of the space of diffeomorphisms of M which restrict to the identity map on ∂M has the homotopy type of a finite aspherical CWcomplex. This answers, for this class of manifolds, a question posed by M. Kontsevich. The main theorem follows from a more precise result, which asserts that for these manifolds the mapping class group H(M rel ∂M) is built up as a sequence of extensions of free abelian groups and subgroups of finite index in relative mapping class groups of compact connected surfaces.
Canonical decompositions of 3manifolds
 Geom. Topol
, 1997
"... Abstract. We describe a new approach to the well known canonical decompositions of 3manifolds. 1. ..."
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Cited by 13 (0 self)
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Abstract. We describe a new approach to the well known canonical decompositions of 3manifolds. 1.