Asymptotic cones and Assouad-Nagata dimension (original) (raw)

Transfinite Asymptotic Dimension

2013

Asymptotic property C for metric spaces was introduced by Dranishnikos as generalization of finite asymptotic dimension - asdim. It turns out that this property can be viewed as transfinite extension of asymptotic dimension. The original definition was given by Radul. We introduce three equivalent definitions, show that asymptotic property C is closed under products (open problem stated "Open problems in topology II") and prove some other facts, i.e. by defining dimension of a family of metric spaces. Some examples of spaces enjoying countable trasfinite asymptotic dimension are given. We also formulate open problems and state "omega conjecture", which inspired most part of this paper.

A Hurewicz theorem for the Assouad-Nagata dimension

Journal of the London Mathematical Society, 2008

Given a function f : X → Y of metric spaces, its asymptotic dimension asdim(f ) is the supremum of asdim(A) such that A ⊂ X and asdim(f (A)) = 0. Our main result is

Addendum * Median structures on asymptotic cones and homomorphisms into mapping class groups

Proceedings of the London Mathematical Society, 2011

We prove that every asymptotic cone of a mapping class group has a bi-Lipschitz equivariant embedding into a product of real trees, with image a median subspace. We deduce several applications of this, one of which is that a group with Kazhdan's property (T) can only have finitely many pairwise non-conjugate homomorphisms into a mapping class group. Definition 2.3 (ultralimit of metric spaces). The ω-limit of the metric spaces (X n , dist n ) relative to the observation point e is the metric space obtained from Π e X n /ω by identifying all pairs of points x ω , y ω with D(x ω , y ω ) = 0; this space is denoted lim ω (X n , e). When there is no need to specify the ultrafilter ω then we also call lim ω (X n , e) ultralimit of (X n , dist n ) relative to e.

Local topological properties of asymptotic cones of groups

Algebraic & Geometric Topology, 2014

We define a local analogue to Gromov's loop division property which is use to give a sufficient condition for an asymptotic cone of a complete geodesic metric space to have uncountable fundamental group. As well, this property is used to understand the local topological structure of asymptotic cones of many groups currently in the literature.

Asymptotic dimension and small subsets in locally compact topological groups

Geometriae Dedicata, 2014

We prove that for a coarse space X the ideal S(X ) of small subsets of X coincides with the ideal D < (X ) = {A ⊂ X : asdim(A) < asdim(X )} provided that X is coarsely equivalent to a Euclidean space R n . Also we prove that for a locally compact Abelian group X , the equality S(X ) = D < (X ) holds if and only if the group X is compactly generated.

Assouad-Nagata dimension via Lipschitz extensions

Israel Journal of Mathematics, 2009

In the first part of the paper we show how to relate several dimension theories (asymptotic dimension with Higson property, asymptotic dimension of Gromov, and capacity dimension of Buyalo [7]) to Nagata-Assouad dimension. This is done by applying two functors on the Lipschitz category of metric spaces: microscopic and macroscopic. In the second part we identify (among spaces of finite Nagata-Assouad dimension) spaces of Nagata-Assouad dimension at most n as those for which the n-sphere S n is a Lipschitz extensor. Large scale and small scale analogs of that result are given.

Convergence Groups, Hausdorff Dimension, and a Theorem of Sullivan and Tukia

Geometriae Dedicata, 2004

We show that a discrete, quasiconformal group preserving H n has the property that its exponent of convergence and the Hausdorff dimension of its limit set detect the existence of a non-empty regular set on the sphere at infinity to H n. This generalizes a result due separately to Sullivan and Tukia, in which it is further assumed that the group act isometrically on H n , i.e. is a Kleinian group. From this generalization we are able to extract geometric information about infinite-index subgroups within certain of these groups.