Results 1  10
of
109
The WeilPetersson metric and volumes of 3dimensional hyperbolic convex cores
 J. Amer. Math. Soc
, 2001
"... Recent insights into the combinatorial geometry of Teichmüller space have shed new light on fundamental questions in hyperbolic geometry in 2 and 3 dimensions. Paradoxically, a coarse perspective on Teichmüller space appears to refine the analogy of Teichmüller geometry with the internal geometry of ..."
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Cited by 45 (6 self)
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Recent insights into the combinatorial geometry of Teichmüller space have shed new light on fundamental questions in hyperbolic geometry in 2 and 3 dimensions. Paradoxically, a coarse perspective on Teichmüller space appears to refine the analogy of Teichmüller geometry with the internal geometry of hyperbolic 3manifolds
Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds
, 2008
"... ..."
The Dirac operator on hyperbolic manifolds of finite volume
 J. Differential Geom
"... We study the spectrum of the Dirac operator on hyperbolic manifolds of finite volume. Depending on the spin structure it is either discrete or the whole real line. For link complements in S 3 we give a simple criterion in terms of linking numbers for when essential spectrum can occur. We compute the ..."
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Cited by 20 (1 self)
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We study the spectrum of the Dirac operator on hyperbolic manifolds of finite volume. Depending on the spin structure it is either discrete or the whole real line. For link complements in S 3 we give a simple criterion in terms of linking numbers for when essential spectrum can occur. We compute the accumulation rate of the eigenvalues of a sequence of closed hyperbolic 2 or 3manifolds degenerating into a noncompact hyperbolic manifold of finite volume. It turns out that in three dimensions there is no clustering at all.
Local connectivity, Kleinian groups and geodesics on the blowup of the torus
, 2001
"... Let N = H 3 =\Gamma be a hyperbolic 3manifold with free fundamental group 1 (N) = \Gamma = hA; Bi; such that [A; B] is parabolic. We show that the limit set of N is always locally connected. More precisely, ..."
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Cited by 17 (0 self)
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Let N = H 3 =\Gamma be a hyperbolic 3manifold with free fundamental group 1 (N) = \Gamma = hA; Bi; such that [A; B] is parabolic. We show that the limit set of N is always locally connected. More precisely,
Flat spacetimes with compact hyperbolic Cauchy surfaces
 J. Differential Geom
, 2005
"... Given a closed hyperbolic nmanifold M, we study the flat Lorentzian structures on M × R such that M ×{0} is a Cauchy surface. We show there exist only two maximal structures sharing a fixed holonomy (one future complete and the other one past complete). We study the geometry of those maximal spacet ..."
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Cited by 14 (5 self)
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Given a closed hyperbolic nmanifold M, we study the flat Lorentzian structures on M × R such that M ×{0} is a Cauchy surface. We show there exist only two maximal structures sharing a fixed holonomy (one future complete and the other one past complete). We study the geometry of those maximal spacetimes in terms of cosmological time. In particular, we study the asymptotic behaviour of the level surfaces of the cosmological time. As a byproduct, we get that no affine deformation of the hyperbolic holonomy ρ: π1(M) → SO(n, 1) of M acts freely and properly on the whole Minkowski space. The present work generalizes the case n = 2 treated by Mess, taking from a work of Benedetti and Guadagnini the emphasis on the fundamental rôle played by the cosmological time. In the last sections, we introduce measured geodesic stratifications on M, that in a sense furnish a good generalization of measured geodesic laminations in any dimension and we investigate relationships between measured stratifications on M and Lorentzian structures on M × R. 1.
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 14 (0 self)
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3. Why Call it Hyperbolic Geometry? 63 4. Understanding the OneDimensional Case 65
Hausdorff dimension and conformal dynamics I: Strong convergence of Kleinian groups
, 1997
"... This paper investigates the behavior of the Hausdorff dimensions of the limit sets Λn and Λ of a sequence of Kleinian groups Γn → Γ, where M = H 3 /Γ is geometrically finite. We show if Γn → Γ strongly, then: (a) Mn = H 3 /Γn is geometrically finite for all n ≫ 0, (b) Λn → Λ in the Hausdorff topolog ..."
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Cited by 11 (0 self)
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This paper investigates the behavior of the Hausdorff dimensions of the limit sets Λn and Λ of a sequence of Kleinian groups Γn → Γ, where M = H 3 /Γ is geometrically finite. We show if Γn → Γ strongly, then: (a) Mn = H 3 /Γn is geometrically finite for all n ≫ 0, (b) Λn → Λ in the Hausdorff topology, and (c) H. dim(Λn) → H. dim(Λ), if H. dim(Λ) ≥ 1. On the other hand, we give examples showing the dimension can vary discontinuously under strong limits when H. dim(Λ) < 1. Continuity can be recovered by requiring that accidental parabolics converge radially. Similar results hold for higherdimensional manifolds. Applications are
MEASURE HOMOLOGY AND SINGULAR HOMOLOGY ARE ISOMETRICALLY ISOMORPHIC
, 2006
"... Measure homology is a variation of singular homology designed by Thurston in his discussion of simplicial volume. Zastrow and Hansen showed independently that singular homology (with real coefficients) and measure homology coincide algebraically on the category of CWcomplexes. It is the aim of this ..."
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Cited by 10 (4 self)
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Measure homology is a variation of singular homology designed by Thurston in his discussion of simplicial volume. Zastrow and Hansen showed independently that singular homology (with real coefficients) and measure homology coincide algebraically on the category of CWcomplexes. It is the aim of this paper to prove that this isomorphism is isometric with respect to the ℓ 1seminorm on singular homology and the seminorm on measure homology induced by the total variation. This, in particular, implies that one can calculate the simplicial volume via measure homology – as already claimed by Thurston. For example, measure homology can be used to prove Gromov’s proportionality principle of simplicial volume.