Results 1  10
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206
Rigidity of quasiisometries for symmetric spaces and Euclidean buildings
 Inst. Hautes Études Sci. Publ. Math
, 1997
"... 1.1 Background and statement of results An (L, C) quasiisometry is a map Φ: X − → X ′ between metric spaces such that for all x1, x2 ∈ X ..."
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Cited by 193 (28 self)
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1.1 Background and statement of results An (L, C) quasiisometry is a map Φ: X − → X ′ between metric spaces such that for all x1, x2 ∈ X
On knot Floer homology and lens space surgery
"... Abstract. In an earlier paper, we used the absolute grading on Heegaard Floer homology HF + to give restrictions on knots in S 3 which admit lens space surgeries. The aim of the present article is to exhibit stronger restrictions on such knots, arising from knot Floer homology. One consequence is th ..."
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Cited by 139 (13 self)
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Abstract. In an earlier paper, we used the absolute grading on Heegaard Floer homology HF + to give restrictions on knots in S 3 which admit lens space surgeries. The aim of the present article is to exhibit stronger restrictions on such knots, arising from knot Floer homology. One consequence is that all the nonzero coefficients of the Alexander polynomial of such a knot are ±1. This information in turn can be used to prove that certain lens spaces are not obtained as integral surgeries on knots. In fact, combining our results with constructions of Berge, we classify lens spaces L(p, q) which arise as integral surgeries on knots in S 3 with p  ≤ 1500. Other applications include bounds on the fourball genera of knots admitting lens space surgeries (which are sharp for Berge’s knots), and a constraint on threemanifolds obtained as integer surgeries on alternating knots, which is closely to related to a theorem of Delman and Roberts. 1.
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 110 (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
Monopoles and lens space surgeries
 ArXive:math.GT/0310164
, 2003
"... Abstract. Monopole Floer homology is used to prove that real projective threespace cannot be obtained from Dehn surgery on a nontrivial knot in the threesphere. To obtain this result, we use a surgery long exact sequence for monopole Floer homology, together with a nonvanishing theorem, which sh ..."
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Cited by 68 (9 self)
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Abstract. Monopole Floer homology is used to prove that real projective threespace cannot be obtained from Dehn surgery on a nontrivial knot in the threesphere. To obtain this result, we use a surgery long exact sequence for monopole Floer homology, together with a nonvanishing theorem, which shows that monopole Floer homology detects the unknot. In addition, we apply these techniques to give information about knots which admit lens space surgeries, and to exhibit families of threemanifolds which do not admit taut foliations. 1.
Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds
, 2008
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3manifolds with(out) metrics of nonpositive curvature
, 1992
"... Abstract. In the context of Thurstons geometrisation program we address the question which compact aspherical 3manifolds admit Riemannian metrics of nonpositive curvature. We prove that a Haken manifold with, possibly empty, boundary of zero Euler characteristic admits metrics of nonpositive curvat ..."
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Cited by 59 (10 self)
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Abstract. In the context of Thurstons geometrisation program we address the question which compact aspherical 3manifolds admit Riemannian metrics of nonpositive curvature. We prove that a Haken manifold with, possibly empty, boundary of zero Euler characteristic admits metrics of nonpositive curvature if the boundary is nonempty or if at least one atoroidal component occurs in its canonical topological decomposition. Our arguments are based on Thurstons Hyperbolisation Theorem. We give examples of closed graphmanifolds with linear gluing graph and arbitrarily many Seifert components which do not admit metrics of nonpositive curvature. 1
The Local Structure of Length Spaces With Curvature Bounded Above
, 1998
"... We show that a number of different notions of dimension coincide for length spaces with curvature bounded above. We then apply this result, showing that if X is a locally compact CAT (0) space with cocompact isometry group, then the dimension of the Tits boundary and the asymptotic cone(s) of X are ..."
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Cited by 51 (0 self)
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We show that a number of different notions of dimension coincide for length spaces with curvature bounded above. We then apply this result, showing that if X is a locally compact CAT (0) space with cocompact isometry group, then the dimension of the Tits boundary and the asymptotic cone(s) of X are determined by the maximal dimension of a flat in X. 1
Bounds on exceptional Dehn filling
 Geom. Topol
"... Abstract. We show that for a hyperbolic knot complement, there are at most 12 Dehn fillings which are not irreducible with infinite wordhyperbolic fundamental group. 1. ..."
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Cited by 49 (1 self)
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Abstract. We show that for a hyperbolic knot complement, there are at most 12 Dehn fillings which are not irreducible with infinite wordhyperbolic fundamental group. 1.
Algebraic limits of Kleinian groups which rearrange the pages of a book
 Zbl 0874.57012 MR 1411128
, 1996
"... Dedicated to Bernard Maskit on the occasion of his sixtieth birthday ..."
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Cited by 49 (15 self)
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Dedicated to Bernard Maskit on the occasion of his sixtieth birthday
The Virtual Haken Conjecture: experiments and examples
 GEOM. TOPOL
, 2002
"... A 3manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture says that every irreducible 3manifold with infinite fundamental group has a finite cover which is Haken. Here, we discuss two interrelated topics concerning this conjecture. First, we describe compu ..."
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Cited by 43 (2 self)
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A 3manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture says that every irreducible 3manifold with infinite fundamental group has a finite cover which is Haken. Here, we discuss two interrelated topics concerning this conjecture. First, we describe computer experiments which give strong evidence that the Virtual Haken Conjecture is true for hyperbolic 3manifolds. We took the complete HodgsonWeeks census of 10,986 smallvolume closed hyperbolic 3manifolds, and for each of them found finite covers which are Haken. There are interesting and unexplained patterns in the data which may lead to a better understanding of this problem. Second, we discuss a method for transferring the virtual Haken property under Dehn filling. In particular, we show that if a 3manifold with torus boundary has a Seifert fibered Dehn filling with hyperbolic base orbifold, then most of the Dehn filled manifolds are virtually Haken. We use this to show that every nontrivial Dehn surgery on the figure8 knot is virtually Haken.