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82
The Coarse BaumConnes Conjecture for Spaces Which Admit a Uniform Embedding into Hilbert Space
, 1998
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The Ktheoretic FarrellJones Conjecture for hyperbolic groups
 Invent. Math
"... Abstract. We prove the Ktheoretic FarrellJones Conjecture for hyperbolic groups with (twisted) coefficients in any associative ring with unit. ..."
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Cited by 41 (18 self)
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Abstract. We prove the Ktheoretic FarrellJones Conjecture for hyperbolic groups with (twisted) coefficients in any associative ring with unit.
Asymptotic topology
, 1999
"... Abstract. We establish some basic theorems in dimension theory and absolute extensor ..."
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Cited by 38 (13 self)
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Abstract. We establish some basic theorems in dimension theory and absolute extensor
A Hurewicztype theorem for asymptotic dimension and applications to geometric group theory
, 2004
"... Abstract. We prove an asymptotic analog of the classical Hurewicz theorem on mappings that lower dimension. This theorem allows us to find sharp upper bound estimates for the asymptotic dimension of groups acting on finitedimensional metric spaces and allows us to prove a useful extension theorem fo ..."
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Cited by 35 (10 self)
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Abstract. We prove an asymptotic analog of the classical Hurewicz theorem on mappings that lower dimension. This theorem allows us to find sharp upper bound estimates for the asymptotic dimension of groups acting on finitedimensional metric spaces and allows us to prove a useful extension theorem for asymptotic dimension. As applications we find upper bound estimates for the asymptotic dimension of nilpotent and polycyclic groups in terms of their Hirsch length. We are also able to improve the known upper bounds on the asymptotic dimension of fundamental groups of complexes of groups, amalgamated free products and the hyperbolization of metric spaces possessing the Higson property. 1.
Asymptotic dimension of relatively hyperbolic groups
, 2004
"... Abstract. Suppose that a finitely generated group G is hyperbolic relative to a collection of subgroups {H1,..., Hm}. We prove that if each of the subgroups H1,..., Hm has finite asymptotic dimension, then asymptotic dimension of G is also finite. 1. ..."
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Cited by 29 (2 self)
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Abstract. Suppose that a finitely generated group G is hyperbolic relative to a collection of subgroups {H1,..., Hm}. We prove that if each of the subgroups H1,..., Hm has finite asymptotic dimension, then asymptotic dimension of G is also finite. 1.
Universal spaces for asymptotic dimension
 Topology Appl
"... Abstract. We construct a universal space for the class of proper metric spaces of bounded geometry and of given asymptotic dimension. As a consequence of this result, we establish coincidence of the asymptotic dimension with the asymptotic inductive dimensions. 1. ..."
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Cited by 27 (4 self)
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Abstract. We construct a universal space for the class of proper metric spaces of bounded geometry and of given asymptotic dimension. As a consequence of this result, we establish coincidence of the asymptotic dimension with the asymptotic inductive dimensions. 1.
Niblo, Hilbert space compression and exactness of discrete groups
 J. Funct. Anal
"... We show that the Hilbert Space compression of any (unbounded) finite dimensional CAT(0) cube complex is 1 and deduce that any finitely generated group acting properly, cocompactly on a CAT(0) cube complex is exact, and hence has Yu’s Property A. The class of groups covered by this theorem include ..."
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Cited by 20 (2 self)
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We show that the Hilbert Space compression of any (unbounded) finite dimensional CAT(0) cube complex is 1 and deduce that any finitely generated group acting properly, cocompactly on a CAT(0) cube complex is exact, and hence has Yu’s Property A. The class of groups covered by this theorem includes free groups, finitely generated Coxeter groups, finitely generated right angled Artin groups, finitely presented groups satisfying the B(4)T(4) small cancellation condition and all those wordhyperbolic groups satisfying the B(6) condition. Another family of examples is provided by certain canonical surgeries defined by link diagrams.
The integral Ktheoretic Novikov conjecture for groups with finite asymptotic dimension
 Inventiones Math. 157 No
"... Abstract. The integral assembly map in algebraicKtheory is split injective for any geometrically finite discrete group with finite asymptotic dimension. The goal of this paper is to apply the techniques developed by the first author in [3] to verify the integral Novikov conjecture for groups with f ..."
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Cited by 20 (1 self)
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Abstract. The integral assembly map in algebraicKtheory is split injective for any geometrically finite discrete group with finite asymptotic dimension. The goal of this paper is to apply the techniques developed by the first author in [3] to verify the integral Novikov conjecture for groups with finite asymptotic dimension as defined by M. Gromov [9]. Recall that a finitely generated groupΓ can be viewed as a metric space with the word metric associated to a given presentation. Definition (Gromov). A family of subsets in a general metric spaceX is called ddisjoint if dist(V,V ′ ) = inf{dist(x,x ′)x ∈ V, x ′ ∈ V ′ }> d for all V, V ′. The asymptotic dimension ofX is defined as the smallest numbernsuch that for anyd>0there is a uniformly bounded coverUofX byn+1ddisjoint families of subsetsU=U 0 ∪...∪U n. It is known that asymptotic dimension is a quasiisometry invariant and so is an invariant of the finitely generated group, independent of the presentation. One saysΓ has finite asymptotic dimension if it does as the metric space with
On asymptotic dimension of groups
 2001), 57–71 (electronic). MR1808331 (2001m:20062), Zbl 1008.20039
"... Abstract. We prove a version of the countable union theorem for the asymptotic dimension and we apply it to groups acting on asymptotically finite dimensional metric spaces. As the consequence we obtain the following finite dimensionality theorems. A) An amalgamated product of asymptotically finite ..."
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Cited by 19 (6 self)
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Abstract. We prove a version of the countable union theorem for the asymptotic dimension and we apply it to groups acting on asymptotically finite dimensional metric spaces. As the consequence we obtain the following finite dimensionality theorems. A) An amalgamated product of asymptotically finite dimensional groups has a finite asymptotic dimension: asdim A ∗C B < ∞. B) Suppose that G ′ is an HNN extension of a group G with asdim G < ∞. Then asdim G ′ < ∞. C) Suppose that Γ is Davis ’ group constructed from a group π with asdim π < ∞. Then asdim Γ < ∞. The notion of the asymptotic dimension was introduced by Gromov [G1] as an asymptotic analog of Ostrand’s characterization of covering dimension. A metric space X has asymptotic dimension asdim X ≤ n if for an arbitrarily large number d one can find n + 1 uniformly bounded families U 0,..., U n of ddisjoint sets in X such that the union