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Spectral theory, Hausdorff dimension and the topology of hyperbolic 3–manifolds
 J. GEOM. ANALYSIS
, 1998
"... Let M be a compact 3manifold whose interior admits a complete hyperbolic structure. We let Λ(M) be the supremum of λ0(N) where N varies over all hyperbolic 3manifolds homeomorphic to the interior of M. Similarly, we let D(M) be the infimum of the Hausdorff dimensions of limit sets of Kleinian gr ..."
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Cited by 13 (2 self)
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Let M be a compact 3manifold whose interior admits a complete hyperbolic structure. We let Λ(M) be the supremum of λ0(N) where N varies over all hyperbolic 3manifolds homeomorphic to the interior of M. Similarly, we let D(M) be the infimum of the Hausdorff dimensions of limit sets of Kleinian groups whose quotients are homeomorphic to the interior of M. We observe that Λ(M) = D(M)(2 − D(M)) if M is not handlebody or a thickened torus. We characterize exactly when Λ(M) = 1 and D(M) = 1 in terms of the characteristic submanifold of the incompressible core of M.
MAXIMALLY STRETCHED LAMINATIONS ON GEOMETRICALLY FINITE HYPERBOLIC MANIFOLDS
"... Abstract. Let Γ0 be a discrete group. For a pair (j, ρ) of representations of Γ0 into PO(n, 1) = Isom(Hn) with j injective and discrete and j(Γ0)\Hn geometrically finite, we study the set of (j, ρ)equivariant Lipschitz maps from the hyperbolic space Hn to itself that have minimal Lipschitz consta ..."
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Cited by 12 (7 self)
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Abstract. Let Γ0 be a discrete group. For a pair (j, ρ) of representations of Γ0 into PO(n, 1) = Isom(Hn) with j injective and discrete and j(Γ0)\Hn geometrically finite, we study the set of (j, ρ)equivariant Lipschitz maps from the hyperbolic space Hn to itself that have minimal Lipschitz constant. Our main result is the existence of a geodesic lamination that is “maximally stretched ” by all such maps when the minimal constant is at least 1. As an application, we generalize twodimensional results and constructions of Thurston and extend his asymmetric metric on Teichmüller space to a geometrically finite setting and to higher dimension. Another application is to actions of discrete subgroups Γ of
Remarks on Hausdorff Dimensions for Transient Limit Sets of Kleinian Groups
, 2003
"... In this paper we study discrepancy groups (dgroups), that are Kleinian groups whose exponent of convergence is strictly less than the Hausdorff dimension of their limit set. We show that the limit set of a dgroup always contains continuous families of fractal sets, each of which contains the set o ..."
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Cited by 9 (4 self)
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In this paper we study discrepancy groups (dgroups), that are Kleinian groups whose exponent of convergence is strictly less than the Hausdorff dimension of their limit set. We show that the limit set of a dgroup always contains continuous families of fractal sets, each of which contains the set of radial limit points and has Hausdorff dimension strictly less than the Hausdorff dimension of the whole limit set. Subsequently, we consider special dgroups which are normal subgroups of some geometrically finite Kleinian group. For these we obtain the result that their Poincaré exponent is always bounded from below by half of the Poincareacute; exponent of the associated geometrically finite group in which they are normal. Finally, we give a discussion of various examples of dgroups, which in particular also contains explicit constructions of these groups.
AdS/CFT Correspondence and Quotient Space Geometry
, 1999
"... We consider a version of the AdSd+1/CFTd correspondence, in which the bulk space is taken to be the quotient manifold AdSd+1/Γ with a fairly generic discrete group Γ acting isometrically on AdSd+1. We address some geometrical issues concerning the holographic principle and the UV/IR relations. It is ..."
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Cited by 6 (0 self)
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We consider a version of the AdSd+1/CFTd correspondence, in which the bulk space is taken to be the quotient manifold AdSd+1/Γ with a fairly generic discrete group Γ acting isometrically on AdSd+1. We address some geometrical issues concerning the holographic principle and the UV/IR relations. It is shown that certain singular structures on the quotient boundary S d /Γ can affect the underlying physical spectrum. In particular, the conformal dimension of the most relevant operators in the boundary CFT can increase as Γ becomes “large”. This phenomenon also has a natural explanation in terms of the bulk supergravity theory. The scalar twopoint function is computed using this quotient version of the AdS/CFT correspondence, which agrees with the expected result derived from conformal invariance of the boundary theory.
Discrete Lyapunov exponents and Hausdorff dimension
 Systems
, 1997
"... this paper we generalize these results to a discrete setting as follows. Let ! n ?= f1; :::; ng be an alphabet on n symbols. Denote by N;! n ? ..."
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Cited by 2 (1 self)
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this paper we generalize these results to a discrete setting as follows. Let ! n ?= f1; :::; ng be an alphabet on n symbols. Denote by N;! n ?
Geometry and dynamics in Gromov hyperbolic metric spaces I  with an emphasis . . .
, 2014
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Creative Commons License Resonances for
, 2008
"... Full metadata for this item is available in the St Andrews ..."
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