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16
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 69 (6 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
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 37 (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.
Heegaard splittings, the virtually Haken Conjecture, and Property (τ)
, 2002
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Hyperbolic geometry
 In Flavors of geometry
, 1997
"... 3. Why Call it Hyperbolic Geometry? 63 4. Understanding the OneDimensional Case 65 ..."
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Cited by 15 (0 self)
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3. Why Call it Hyperbolic Geometry? 63 4. Understanding the OneDimensional Case 65
Expansion complexes for finite subdivision rules
 I, Conform. Geom. Dyn
"... Abstract. This paper gives applications of earlier work of the authors on the use of expansion complexes for studying conformality of finite subdivision rules. The first application is that a onetile rotationally invariant finite subdivision rule (with bounded valence and mesh approaching 0) has an ..."
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Cited by 8 (5 self)
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Abstract. This paper gives applications of earlier work of the authors on the use of expansion complexes for studying conformality of finite subdivision rules. The first application is that a onetile rotationally invariant finite subdivision rule (with bounded valence and mesh approaching 0) has an invariant partial conformal structure, and hence is conformal. The paper next considers onetile single valence finite subdivision rules. It is shown that an expansion map for such a finite subdivision rule can be conjugated to a linear map, and that the finite subdivision rule is conformal exactly when this linear map is either a dilation or has eigenvalues that are not real. Finally, an example is given of an irreducible finite subdivision rule that has a parabolic expansion complex and a hyperbolic expansion complex. 1.
Compactifying sufficiently regular covering spaces of compact 3manifolds
 Proc. Amer. Math. Soc
"... Abstract. In this paper it is proven that if the group of covering translations of the covering space of a compact, connected, P 2irreducible 3manifold corresponding to a nontrivial, finitelygenerated subgroup of its fundamental group is infinite, then either the covering space is almost compact ..."
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Abstract. In this paper it is proven that if the group of covering translations of the covering space of a compact, connected, P 2irreducible 3manifold corresponding to a nontrivial, finitelygenerated subgroup of its fundamental group is infinite, then either the covering space is almost compact or the subgroup is infinite cyclic and has normalizer a nonfinitelygenerated subgroup of the rational numbers. In the first case additional information is obtained which is then used to relate Thurston’s hyperbolization and virtual bundle conjectures to some algebraic conjectures about certain 3manifold groups. 1.
Topological Geodesics and Virtual Rigidity
"... We introduce the notion of a topological geodesic in a 3manifold. Under suitable hypotheses on the fundamental group, for instance wordhyperbolicity, topological geodesics are shown to have the useful properties of, and play the same role in several applications as, geodesics in negatively curved ..."
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We introduce the notion of a topological geodesic in a 3manifold. Under suitable hypotheses on the fundamental group, for instance wordhyperbolicity, topological geodesics are shown to have the useful properties of, and play the same role in several applications as, geodesics in negatively curved spaces. This permits us to obtain virtual rigidity results for 3manifolds.
Virtual Haken 3manifolds and Dehn filling
, 2000
"... The study of 3manifolds splits nicely into the cases of finite fundamental groups and infinite fundamental groups. Concerning 3manifolds with infinite fundamental groups, the following important conjecture due to Waldhausen [34] is well known. ..."
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Cited by 1 (0 self)
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The study of 3manifolds splits nicely into the cases of finite fundamental groups and infinite fundamental groups. Concerning 3manifolds with infinite fundamental groups, the following important conjecture due to Waldhausen [34] is well known.
Drilling long geodesics in hyperbolic 3manifolds
 In preparation
, 2006
"... Given a complete hyperbolic 3manifold one often wants to compare the original metric to a complete hyperbolic metric on the complement of some simple closed geodesic in the manifold. In some cases this can be done by interpolating between the two metrics using hyperbolic conemanifolds. We refer to ..."
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Given a complete hyperbolic 3manifold one often wants to compare the original metric to a complete hyperbolic metric on the complement of some simple closed geodesic in the manifold. In some cases this can be done by interpolating between the two metrics using hyperbolic conemanifolds. We refer to such a deformation as drilling and results which