Results 1  10
of
25
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.
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
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.
ASYMPTOTIC CONES AND ASSOUADNAGATA DIMENSION
, 2008
"... Abstract. We prove the dimension of any asymptotic cone over a metric space (X, ρ) does not exceed the asymptotic AssouadNagata dimension asdimAN(X) of X. This improves a result of Dranishnikov and Smith [11] who showed dim(Y)≤asdimAN(X) for all separable subsets Y of special asymptotic cones Coneω ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
(Show Context)
Abstract. We prove the dimension of any asymptotic cone over a metric space (X, ρ) does not exceed the asymptotic AssouadNagata dimension asdimAN(X) of X. This improves a result of Dranishnikov and Smith [11] who showed dim(Y)≤asdimAN(X) for all separable subsets Y of special asymptotic cones Coneω(X), where ω is an exponential ultrafilter on natural numbers. We also show that AssouadNagata dimension of the discrete Heisenberg group equals its asymptotic dimension.
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.
DISCRETE GROUPS WITH FINITE DECOMPOSITION COMPLEXITY
"... Abstract. In [GTY] we introduced a geometric invariant, called finite decomposition complexity (FDC), to study topological rigidity of manifolds. In that article we proved the stable Borel conjecture for a closed aspherical manifold whose universal cover, or equivalently whose fundamental group, h ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
Abstract. In [GTY] we introduced a geometric invariant, called finite decomposition complexity (FDC), to study topological rigidity of manifolds. In that article we proved the stable Borel conjecture for a closed aspherical manifold whose universal cover, or equivalently whose fundamental group, has FDC. In this note we continue our study of FDC, focusing on permanence and the relation to other coarse geometric properties. In particular, we prove that the class of FDC groups is closed under taking subgroups, extensions, free amalgamated products, HNN extensions, and direct unions. As consequences we obtain further examples of FDC groups – all elementary amenable groups and all countable subgroups of almost connected Lie groups have FDC. 1.
SVARCMILNOR LEMMA: A PROOF BY DEFINITION
, 2006
"... Abstract. The famous ˇ SvarcMilnor Lemma says that a group G acting properly and cocompactly via isometries on a length space X is finitely generated and induces a quasiisometry equivalence g → g · x0 for any x0 ∈ X. We redefine the concept of coarseness so that the proof of the Lemma is automatic ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
Abstract. The famous ˇ SvarcMilnor Lemma says that a group G acting properly and cocompactly via isometries on a length space X is finitely generated and induces a quasiisometry equivalence g → g · x0 for any x0 ∈ X. We redefine the concept of coarseness so that the proof of the Lemma is automatic. Geometric group theorists traditionally restrict their attention to finitely generated groups equipped with a word metric. A typical proof of ˇ SvarcMilnor Lemma (see [5] or [1], p.140) involves such metrics. Recently, the study of large scale geometry of groups was expanded to all countable groups by usage of proper, leftinvariant metrics: in [6] such metrics were constructed and it was shown that they all induce the same coarse structure on a group (see also [2]). The point of this note is that a proper action of a group G on a space ought to be viewed as a geometric way of creating a coarse structure on G. That structure is not given by a proper metric but by something very similar; a pseudometric where only a finite set of points may be at mutual distance 0. From that point of view the proof of ˇ SvarcMilnor Lemma is automatic and the Lemma can be summarized as follows. There are two ways of creating coarse structures on countable groups: algebraic (via word or proper metrics) and geometric (via group actions), and both ways are equivalent. Definition 0.1. A pseudometric dX on a set X is called a largescale metric (or lsmetric) if for each x ∈ X the set {y ∈ X  dX(x, y) = 0} is finite. (X, dX) is called a largescale metric space (or an lsmetric space) if dX is an lsmetric.
Dimension theory: local and global
"... These survey lectures are devoted to a new subject of the large scale dimension theory which was initiated by Gromov as a part of asymptotic geometry. We are going to enter the large scale world and consider some new concepts, results and examples which are parallel in many cases to the correspondi ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
These survey lectures are devoted to a new subject of the large scale dimension theory which was initiated by Gromov as a part of asymptotic geometry. We are going to enter the large scale world and consider some new concepts, results and examples which are parallel in many cases to the corresponding elements of the standard (local) dimension theory. We start our presentation with the motivations. Lecture 1. MOTIVATONS and CONCEPTS 1.1. Big picture of the Novikov Conjecture. The Novikov Conjecture (NC) states that the higher signatures of a manifold are homotopy invariant. The higher signatures are the rational numbers of the type 〈L(M)∪ρ∗M(x), [M]〉, where [M] is the fundamental class of a manifold M, L is the Hirzebruch class, Γ = pi1(M), ρM: M → BΓ = K(Γ, 1) is a map classifying the universal cover of M and x ∈ H∗(BΓ;Q) is a rational cohomology class. The name ‘higher signature ’ is due to the Hirzebruch signature formula σ(M) = 〈L(M), [M]〉. It is known that the higher signatures are the only possible homotopy invariant characteristic numbers. It is convenient to formulate the NC for groups Γ instead of manifolds. We say that the Novikov Conjecture holds for a discrete group Γ if it holds for all manifolds M (closed, orientable) with the fundamental group pi1(M) = Γ. One of the reason for this is that the conjecture is verified for many large classes of groups. The other reason is that the Novikov Conjecture for the group can be reformulated in terms of the surgery exact sequence: The rational Wall assembly map lΓ ∗ : H∗(BΓ;Q) → L∗(pi)⊗Q is a monomorphism [Wa], [FRR],[KM].
CLASSIFYING HOMOGENEOUS ULTRAMETRIC SPACES UP TO COARSE EQUIVALENCE
, 801
"... Abstract. We prove that two homogeneous ultrametric spaces X, Y are coarsely equivalent if and only if Ent ♯ (X) = Ent ♯ (Y) where Ent ♯ (X) is the socalled sharp entropy of X. This classification implies that each homogeneous proper ultrametric space is coarsely equivalent to the antiCantor set ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We prove that two homogeneous ultrametric spaces X, Y are coarsely equivalent if and only if Ent ♯ (X) = Ent ♯ (Y) where Ent ♯ (X) is the socalled sharp entropy of X. This classification implies that each homogeneous proper ultrametric space is coarsely equivalent to the antiCantor set 2 <ω. For the proof of these results we develop a technique of towers which can have an independent interest.
LINEARLY CONTROLLED ASYMPTOTIC DIMENSION OF THE FUNDAMENTAL GROUP OF A GRAPHMANIFOLD
"... Abstract. We prove the estimate ℓasdimπ1(M) ≤ 7 for the linearly controlled asymptotic dimension of the fundamental group of any 3dimensional graphmanifold M. As applications, we show that the universal cover ĂM of M is an absolute Lipschitz retract and admits a quasisymmetric embedding into the ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. We prove the estimate ℓasdimπ1(M) ≤ 7 for the linearly controlled asymptotic dimension of the fundamental group of any 3dimensional graphmanifold M. As applications, we show that the universal cover ĂM of M is an absolute Lipschitz retract and admits a quasisymmetric embedding into the product of 8 metric trees. A motivation of the present work is a result of Bell and Dranishnikov [3, Theorem 1 ′] that the asymptotic dimension of a graphgroup whose vertex groups have finite asymptotic dimensions is also finite. The fundamental groups of graphmanifolds are graphgroups. However, it is unclear whether there exists an analog of the Bell–Dranishnikov