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27
Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces
, 2014
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Uniform embeddability of relatively hyperbolic groups
, 2005
"... Abstract. Let Γ be a finitely generated group which is hyperbolic relative to a finite family {H1,...,Hn} of subgroups. We prove that Γ is uniformly embeddable in a Hilbert space if and only if each subgroup Hi is uniformly embeddable in a Hilbert space. 1. ..."
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Cited by 23 (2 self)
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Abstract. Let Γ be a finitely generated group which is hyperbolic relative to a finite family {H1,...,Hn} of subgroups. We prove that Γ is uniformly embeddable in a Hilbert space if and only if each subgroup Hi is uniformly embeddable in a Hilbert space. 1.
Boundary amenability of relatively hyperbolic groups
, 2005
"... Abstract. Let K be a fine hyperbolic graph and Γ be a group acting on K with finite quotient. We prove that Γ is exact provided that all vertex stabilizers are exact. In particular, a relatively hyperbolic group is exact if all its peripheral groups are exact. We prove this by showing that the group ..."
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Cited by 21 (1 self)
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Abstract. Let K be a fine hyperbolic graph and Γ be a group acting on K with finite quotient. We prove that Γ is exact provided that all vertex stabilizers are exact. In particular, a relatively hyperbolic group is exact if all its peripheral groups are exact. We prove this by showing that the group Γ acts amenably on a compact topological space. We include some applications to the theories of group von Neumann algebras and of measurable orbit equivalence relations. 1.
An obstruction to the strong relative hyperbolicity of a group
 Journal of Group Theory
, 2006
"... We give a simple combinatorial criterion for a group that, when satisfied, implies the group cannot be strongly relatively hyperbolic. Our criterion applies to several classes of groups, such as surface mapping class groups, Torelli groups, and automorphism and outer automorphism groups of free grou ..."
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We give a simple combinatorial criterion for a group that, when satisfied, implies the group cannot be strongly relatively hyperbolic. Our criterion applies to several classes of groups, such as surface mapping class groups, Torelli groups, and automorphism and outer automorphism groups of free groups. MSC 20F67 (primary), 20F65 (secondary) 1
The class group of D
 M, J. Pure Appl. Algebra
, 1988
"... A simple criterion for nonrelative hyperbolicity and oneendedness ..."
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Cited by 11 (0 self)
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A simple criterion for nonrelative hyperbolicity and oneendedness
A simple criterion for nonrelative hyperbolicity and oneendedness of groups
"... We give a combinatorial criterion that implies both the nonstrong relative hyperbolicity and the oneendedness of a finitely generated group. We use this to show that many important classes of groups do not admit a strong relatively hyperbolic group structure and have one end. Applications include ..."
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We give a combinatorial criterion that implies both the nonstrong relative hyperbolicity and the oneendedness of a finitely generated group. We use this to show that many important classes of groups do not admit a strong relatively hyperbolic group structure and have one end. Applications include surface mapping class groups, the Torelli group, (special) automorphism and outer automorphism groups of most free groups, and the threedimensional Heisenberg group. Our final application is to Thompson’s group F. MSC 20F67 (primary), 20F65 (secondary) 1
Fillings, finite generation and direct limits of relatively hyperbolic groups
 GROUPS, GEOMETRY, AND DYNAMICS
, 2007
"... We examine the relationship between finitely and infinitely generated relatively hyperbolic groups. We observe that direct limits of relatively hyperbolic groups are in fact direct limits of finitely generated relatively hyperbolic groups. We use this (and known results) to prove the Strong Noviko ..."
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Cited by 8 (1 self)
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We examine the relationship between finitely and infinitely generated relatively hyperbolic groups. We observe that direct limits of relatively hyperbolic groups are in fact direct limits of finitely generated relatively hyperbolic groups. We use this (and known results) to prove the Strong Novikov Conjecture for the groups constructed by Osin in [17].
RELATIVE HYPERBOLICITY AND BOUNDED COHOMOLOGY
"... Abstract. Let Γ be a finitely generated group and Γ ′ = {Γi  i ∈ I} be a family of its subgroups. We utilize the notion of tuple (Γ, Γ ′ , X, V ′ ) that makes the statements and arguments for the pair (Γ, Γ ′ ) parallel to the nonrelative case, and define the snake metric dς on the set of edges o ..."
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Abstract. Let Γ be a finitely generated group and Γ ′ = {Γi  i ∈ I} be a family of its subgroups. We utilize the notion of tuple (Γ, Γ ′ , X, V ′ ) that makes the statements and arguments for the pair (Γ, Γ ′ ) parallel to the nonrelative case, and define the snake metric dς on the set of edges of a simplicial complex. The language of tuples and snake metrics seems to be convenient for dealing with relative hyperbolicity. For tuples, the properties of being finitely generated, finitely presented (cf. [28, 29]), of type Fn, of type F, and of having fine triangles are defined. Fine triangles are the ones that are “thin with respect to the snake metric”. Call a pair (Γ, Γ ′ ) hyperbolic if there is a finitely generated tuple (Γ, Γ ′ , X, V ′ ) with fine triangles and with X (1) fine. We give a definition of relative hyperbolicity of Γ with respect to Γ ′ which slightly generalizes the definition of Bowditch, and show that this notion coincides with hyperbolicity of the pair (Γ, Γ ′). We describe the snake resolution St ς (Γ, Γ ′), or the relative standard projective resolution. It is used to define both relative cohomology and relative bounded cohomology. We generalize the argument in [22, 23] to show that if (Γ, Γ ′ ) is hyperbolic then H 2 b (Γ, Γ ′ ; V) → H2 (Γ, Γ ′; V) is surjective for all bounded QΓmodules V. The same holds for bounded RΓmodules,