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197
Three dimensional manifolds, Kleinian groups and hyperbolic geometry
 BULL. AMER. MATH. SOC
, 1982
"... ..."
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 141 (4 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 78 (5 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 59 (14 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 42 (29 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.
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 42 (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 38 (22 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
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 35 (8 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.
Stabilization for mapping class groups of 3manifolds
"... Abstract. We prove that the homology of the mapping class group of any 3manifold stabilizes under connected sum and boundary connected sum with an arbitrary 3manifold when both manifolds are compact and orientable. The stabilization also holds for the quotient group by twists along spheres and dis ..."
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Cited by 33 (5 self)
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Abstract. We prove that the homology of the mapping class group of any 3manifold stabilizes under connected sum and boundary connected sum with an arbitrary 3manifold when both manifolds are compact and orientable. The stabilization also holds for the quotient group by twists along spheres and disks, and includes as particular cases homological stability for symmetric automorphisms of free groups, automorphisms of certain free products, and handlebody mapping class groups. Our methods also apply to manifolds of other dimensions in the case of stabilization by punctures. The main result of this paper is a homological stability theorem for mapping class groups of 3manifolds, where the stabilization is by connected sum with an arbitrary 3manifold. More precisely, we show that given any two compact, connected, oriented 3manifolds N and P with ∂N ̸ = ∅, the homology group Hi(π0Diff(N#P # · · · #P rel ∂N); Z) is independent of the number n of copies of P in the connected sum, as long as n ≥ 2i + 2, i.e. each homology group stabilizes with P. We also prove an analogous result for boundary connected sum, and a