Results 1  10
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13
Subgroups of word hyperbolic groups in dimension 2
 Jour. London Math. Soc
, 1996
"... If G is a word hyperbolic group of cohomological dimension 2, then every subgroup of G of type FP2 is also word hyperbolic. Isoperimetric inequalities are denned for groups of type FP2 and it is shown that the linear isoperimetric inequality in this generalized context is equivalent to word hyperbol ..."
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Cited by 22 (10 self)
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If G is a word hyperbolic group of cohomological dimension 2, then every subgroup of G of type FP2 is also word hyperbolic. Isoperimetric inequalities are denned for groups of type FP2 and it is shown that the linear isoperimetric inequality in this generalized context is equivalent to word hyperbolicity. A sufficient condition for hyperbolicity of a general graph is given along with an application to 'relative hyperbolicity'. Finitely presented subgroups of Lyndon's small cancellation groups of hyperbolic type are word hyperbolic. Finitely presented subgroups of hyperbolic 1relator groups are hyperbolic. Finitely presented subgroups of free Burnside groups are finite in the stable range. 1.
Recognizing constant curvature discrete groups in dimension 3
 TRANS. AMER. MATH. SOC.
, 1998
"... We characterize those discrete groups G which can act properly discontinuously, isometrically, and cocompactly on hyperbolic 3space H 3 in terms of the combinatorics of the action of G on its space at infinity. The major ingredients in the proof are the properties of groups that are negatively cu ..."
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Cited by 20 (8 self)
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We characterize those discrete groups G which can act properly discontinuously, isometrically, and cocompactly on hyperbolic 3space H 3 in terms of the combinatorics of the action of G on its space at infinity. The major ingredients in the proof are the properties of groups that are negatively curved (in the large) (that is, Gromov hyperbolic), the combinatorial Riemann mapping theorem, and the SullivanTukia theorem on groups which act uniformly quasiconformally on the 2sphere.
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 11 (0 self)
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3. Why Call it Hyperbolic Geometry? 63 4. Understanding the OneDimensional Case 65
Noncommutative Poincaré Duality for Boundary Actions of Hyperbolic Groups
 J. Reine Angew. Math
, 2003
"... Abstract. For a large class of word hyperbolic groups Γ the cross product C ∗algebras C(∂Γ)⋊Γ, where ∂Γ denotes the Gromov boundary of Γ satisfy Poincaré duality in Ktheory. This class strictly contains fundamental groups of compact, negatively curved manifolds. We discuss the general notion of Po ..."
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Cited by 9 (6 self)
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Abstract. For a large class of word hyperbolic groups Γ the cross product C ∗algebras C(∂Γ)⋊Γ, where ∂Γ denotes the Gromov boundary of Γ satisfy Poincaré duality in Ktheory. This class strictly contains fundamental groups of compact, negatively curved manifolds. We discuss the general notion of Poincaré duality for C ∗algebras, construct the fundamental classes for the aforementioned algebras, and prove that KKproducts with these classes induce inverse isomorphisms. The BaumConnes Conjecture for amenable groupoids is used in a crucial way. 1.
BOUNDED COHOMOLOGY AND ISOMETRY GROUPS OF HYPERBOLIC SPACES
, 2005
"... Let X be an arbitrary hyperbolic geodesic metric space and let Γ be a countable subgroup of the isometry group Iso(X) of X. We show that if Γ is nonelementary and weakly acylindrical (this is a weak properness condition) then the second bounded cohomology groups H 2 b (Γ, R), H2 b (Γ, ℓp (Γ)) (1 ≤ ..."
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Cited by 8 (3 self)
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Let X be an arbitrary hyperbolic geodesic metric space and let Γ be a countable subgroup of the isometry group Iso(X) of X. We show that if Γ is nonelementary and weakly acylindrical (this is a weak properness condition) then the second bounded cohomology groups H 2 b (Γ, R), H2 b (Γ, ℓp (Γ)) (1 ≤ p < ∞) are infinite dimensional. Our result holds for example for any subgroup of the mapping class group of a nonexceptional surface of finite type which is not virtually abelian nor virtually splits as a direct product.
Dehn functions and products of groups
 Trans. Amer. Math. Soc
, 1993
"... Abstract. If G is a finitely presented group then its Dehn function—or its isoperimetric inequality—is of interest. For example, G satisfies a linear isoperimetric inequality iff G is negatively curved (or hyperbolic in the sense of Gromov). Also, if G possesses an automatic structure then G satisfi ..."
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Cited by 6 (0 self)
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Abstract. If G is a finitely presented group then its Dehn function—or its isoperimetric inequality—is of interest. For example, G satisfies a linear isoperimetric inequality iff G is negatively curved (or hyperbolic in the sense of Gromov). Also, if G possesses an automatic structure then G satisfies a quadratic isoperimetric inequality. We investigate the effect of certain natural operations on the Dehn function. We consider direct products, taking subgroups of finite index, free products, amalgamations, and HNN extensions. The study of isoperimetric inequalities for finitely presented groups can be approached in two different ways. There is the geometric approach (see [Gr]). Given a finitely presented group G, choose a compact Riemannian manifold M with fundamental group being G. Then consider embedded circles which
The BaumConnes conjecture, noncommutative Poincaré duality, and the boundary of the free group
 Int. J. Math. Math. Sci
"... Abstract. For every hyperbolic group Γ with Gromov boundary ∂Γ, one can form the cross product C ∗algebra C(∂Γ)⋊Γ. For each such algebra we construct a canonical Khomology class, which induces a Poincaré duality map K∗(C(∂Γ) ⋊ Γ) → K ∗+1 (C(∂Γ) ⋊ Γ). We show that this map is an isomorphism in t ..."
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Cited by 4 (3 self)
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Abstract. For every hyperbolic group Γ with Gromov boundary ∂Γ, one can form the cross product C ∗algebra C(∂Γ)⋊Γ. For each such algebra we construct a canonical Khomology class, which induces a Poincaré duality map K∗(C(∂Γ) ⋊ Γ) → K ∗+1 (C(∂Γ) ⋊ Γ). We show that this map is an isomorphism in the case of Γ = F2 the free group on two generators. We point out a direct connection between our constructions and the BaumConnes Conjecture and eventually use the latter to deduce our result. 2000 Mathematics Subject Classification 46L80 1.
THE RATIO SET OF THE HARMONIC MEASURE OF A RANDOM WALK ON A HYPERBOLIC GROUP
, 2006
"... Abstract. We consider the harmonic measure on the Gromov boundary of a nonamenable hyperbolic group defined by a finite range random walk on the group, and study the corresponding orbit equivalence relation on the boundary. It is known to be always amenable and of type III. We determine its ratio se ..."
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Cited by 3 (1 self)
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Abstract. We consider the harmonic measure on the Gromov boundary of a nonamenable hyperbolic group defined by a finite range random walk on the group, and study the corresponding orbit equivalence relation on the boundary. It is known to be always amenable and of type III. We determine its ratio set by showing that it is generated by certain values of the Martin kernel. In particular, we show that the equivalence relation is never of type III0.
Dynamical stability in Lagrangian systems
 Proceedings of the NATO Advanced Study Institute on Hamiltonian
, 1995
"... The notion of stability in Dynamical Systems refers to dynamical behavior that persists ..."
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Cited by 1 (1 self)
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The notion of stability in Dynamical Systems refers to dynamical behavior that persists
A real variable characterization of Gromov hyperbolicity of flute surfaces
, 2007
"... Abstract. In this paper we give a characterization of the Gromov hyperbolicity of trains (a large class of Denjoy domains which contains the flute surfaces) in terms of the behavior of a real function. This function describes somehow the distances between some remarkable geodesics in the train. This ..."
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Cited by 1 (1 self)
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Abstract. In this paper we give a characterization of the Gromov hyperbolicity of trains (a large class of Denjoy domains which contains the flute surfaces) in terms of the behavior of a real function. This function describes somehow the distances between some remarkable geodesics in the train. This theorem has several consequences; in particular, it allows to deduce a result about stability of hyperbolicity, even though the original surface and the modified one are not quasiisometric.