Results 1  10
of
18
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 ..."
Abstract

Cited by 35 (10 self)
 Add to MetaCart
(Show Context)
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.
J.Smith, Asymptotic dimension of discrete groups
 Fund. Math
"... Abstract. We extend Gromov’s notion of asymptotic dimension of finitely generated groups to all discrete groups. In particular, we extend the Hurewicz type theorem proven in [BD2] to general groups. Then we use this extension to prove a formula for the asymptotic dimension of finitely generated sol ..."
Abstract

Cited by 26 (2 self)
 Add to MetaCart
(Show Context)
Abstract. We extend Gromov’s notion of asymptotic dimension of finitely generated groups to all discrete groups. In particular, we extend the Hurewicz type theorem proven in [BD2] to general groups. Then we use this extension to prove a formula for the asymptotic dimension of finitely generated solvable groups in terms of their Hirsch length. The main purpose of this paper is to extend the formula asdim Γ ≤ h(Γ) from [BD2], which gives an upper bound for the asymptotic dimension of nilpotent finitely generated groups in terms of the Hirsch length, to solvable groups. The main problem here is that even working with finitely generated groups one has to consider infinitely generated
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. ..."
Abstract

Cited by 23 (2 self)
 Add to MetaCart
(Show Context)
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.
Dimension of locally and asymptotically selfsimilar spaces
, 2005
"... We obtain two in a sense dual to each other results: First, that the capacity dimension of every compact, locally selfsimilar metric space coincides with the topological dimension, and second, that the asymptotic dimension of a metric space, which is asymptotically similar to its compact subspace c ..."
Abstract

Cited by 19 (0 self)
 Add to MetaCart
We obtain two in a sense dual to each other results: First, that the capacity dimension of every compact, locally selfsimilar metric space coincides with the topological dimension, and second, that the asymptotic dimension of a metric space, which is asymptotically similar to its compact subspace coincides with the topological dimension of the subspace. As an application of the first result, we prove the Gromov conjecture that the asymptotic dimension of every hyperbolic group G equals the topological dimension of its boundary at infinity plus 1, asdimG = dim∂∞G + 1. As an application of the second result, we construct Pontryagin surfaces for the asymptotic dimension, in particular, those are first examples of metric spaces X, Y with
Squeezing and higher algebraic Ktheory
 KTheory
"... Abstract. We prove that the Assembly map in algebraic Ktheory is split injective for groups of finite asymptotic dimension admitting a finite classifying space. 1. ..."
Abstract

Cited by 18 (6 self)
 Add to MetaCart
(Show Context)
Abstract. We prove that the Assembly map in algebraic Ktheory is split injective for groups of finite asymptotic dimension admitting a finite classifying space. 1.
A notion of geometric complexity and its applications to topological rigidity, Invent math DOI
, 2010
"... Abstract. We introduce a geometric invariant, called finite decomposition complexity (FDC), to study topological rigidity of manifolds. We prove for instance that if the fundamental group of a compact aspherical manifold M has FDC, and if N is homotopy equivalent to M, then M×Rn is homeomorphic to ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We introduce a geometric invariant, called finite decomposition complexity (FDC), to study topological rigidity of manifolds. We prove for instance that if the fundamental group of a compact aspherical manifold M has FDC, and if N is homotopy equivalent to M, then M×Rn is homeomorphic to N×Rn, for n large enough. This statement is known as the stable Borel conjecture. On the other hand, we show that the class of FDC groups includes all countable subgroups of GL(n,K), for any field K, all elementary amenable groups, and is closed under taking subgroups, extensions, free amalgamated products, HNN extensions, and direct unions. 1.
An étale approach to the Novikov Conjecture
 Comm. Pure Appl. Math
"... Abstract. We show that the rational Novikov conjecture for a group Γ of finite homological type follows from the mod 2 acyclicity of the Higson compactifcation of an EΓ. We then show that for groups of finite asymptotic dimension the Higson compactification is mod p acyclic for all p, and deduce the ..."
Abstract

Cited by 11 (7 self)
 Add to MetaCart
Abstract. We show that the rational Novikov conjecture for a group Γ of finite homological type follows from the mod 2 acyclicity of the Higson compactifcation of an EΓ. We then show that for groups of finite asymptotic dimension the Higson compactification is mod p acyclic for all p, and deduce the integral Novikov conjecture for these groups. Ten years ago, the most popular approach to the Novikov conjecture went via compactifications. If a compact aspherical manifold, say, has a universal cover which suitably equivariantly compactifies, already Farrell and Hsiang [FH] proved that the Novikov conjecture follows. Subsequent work by many authors weakened
DIMENSION OF ASYMPTOTIC CONES OF LIE GROUPS
, 2007
"... Abstract. We compute the covering dimension the asymptotic cone of a connected Lie group. For simply connected solvable Lie groups, this is the codimension of the exponential radical. As an application of the proof, we give a characterization of connected Lie groups that quasiisometrically embed in ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
Abstract. We compute the covering dimension the asymptotic cone of a connected Lie group. For simply connected solvable Lie groups, this is the codimension of the exponential radical. As an application of the proof, we give a characterization of connected Lie groups that quasiisometrically embed into a nonpositively curved metric space. 1.
Growth of the Asymptotic Dimension Function for Groups
, 2005
"... Abstract. It is relatively easy to construct a finitely generated group with infinite asymptotic dimension: the restricted wreath product of Z by Z provides an example. In light of this, it becomes interesting to consider the rate of growth of the asymptotic dimension function of a group. Loosely sp ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
(Show Context)
Abstract. It is relatively easy to construct a finitely generated group with infinite asymptotic dimension: the restricted wreath product of Z by Z provides an example. In light of this, it becomes interesting to consider the rate of growth of the asymptotic dimension function of a group. Loosely speaking, we measure the dimension on λscale and let λ increase to infinity to recover the asymptotic dimension. In this paper we consider how the asymptotic dimension function is affected by different constructions involving groups. 1.
ON ASYMPTOTIC ASSOUADNAGATA DIMENSION
, 2006
"... Abstract. For a large class of metric spaces X including discrete groups we prove that the asymptotic AssouadNagata dimension ANasdim X of X coincides with the covering dimension dim(νLX) of the Higson corona of X with respect to the sublinear coarse structure on X. Then we apply this fact to prov ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
(Show Context)
Abstract. For a large class of metric spaces X including discrete groups we prove that the asymptotic AssouadNagata dimension ANasdim X of X coincides with the covering dimension dim(νLX) of the Higson corona of X with respect to the sublinear coarse structure on X. Then we apply this fact to prove the equality ANasdim(X ×R) = ANasdim X +1. We note that the similar equality for Gromov’s asymptotic dimension asdim generally fails to hold [Dr3]. Additionally we construct an injective map ξ: coneω(X) \[x0] → νLX from the asymptotic cone without the base point to the sublinear Higson corona. The AssouadNagata dimension was introduced in the 80s by Assouad [As1],[As2] under the name Nagata dimension. Recently this notion was revived in the asymptotic geometry due to works of Lang and Schlichenmaier [LSch], and Buyalo and Lebedeva [Bu], [BL]. The concept takes into account the dimension of a metric space on all scales.