We prove the dimension of any asymptotic cone over a metric space X does not exceed the asymptotic Assouad-Nagata dimension of X. This improves a result of Dranishnikov and Smith who showed that dim(Y) does not exceed asymptotic Assouad-Nagata dimension of X for all separable subsets Y of special asymptotic cones of X over an exponential ultrafilter.
We also show that Assouad-Nagata dimension of the discrete Heisenberg group equals its asymptotic dimension.
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 \cite{Buyalo1}) 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.
We study the Assouad dimension and the Nagata dimension of metric spaces. As a general result, we prove that the Nagata dimension of a metric space is always bounded from above by the Assouad dimension. Most of the paper is devoted to the study of when these metric dimensions of a metric space are locally given by the dimensions of its metric tangents. Having uniformly close tangents is not sufficient. What is needed in addition is either that the tangents have dimension with uniform constants independent from the point and the tangent, or that the tangents are unique. We will apply our result…
For a large class of metric space X including discrete groups we prove that the asymptotic Assouad-Nagata dimension AN-asdim X of X coincides with the covering dimension $\dim(ν_L X)$ of the Higson corona of X with respect to the sublinear coarse structure on X. Then we apply this fact to prove the equality AN-asdim(X x R) = AN-asdim 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 from the asymptotic cone without the basepoint to the sublinear Higson Corona.
In this work we study two problems about Assouad-Nagata dimension: 1) Is there a metric space of non zero Assouad-Nagata dimension such that all of its asymptotic cones are of Assouad-Nagata dimension zero? (Dydak and Higes) 2) Suppose $G$ is a locally finite group with a proper left invariant metric $d_G$. If $\dim_{AN}(G, d_G)>0$, is $\dim_{AN} (G, d_G)$ infinite? (Brodskiy, Dydak and Lang) The first question is answered positively not only for general metric spaces but also for discrete groups with proper left invariant metrics. The second question has a negative solution. We show that for …
Consider the wreath product $H\wr G$, where $H\ne 1$ is finite and $G$ is finitely generated. We show that the Assouad-Nagata dimension $\dim_{AN}(H\wr G)$ of $H\wr G$ depends on the growth of $G$ as follows: If the growth of $G$ is not bounded by a linear function, then $\dim_{AN}(H\wr G)=\infty$, otherwise $\dim_{AN}(H\wr G)=\dim_{AN}(G)\leq 1$.
Given a function $f\colon X\to Y$ of metric spaces, its {\it asymptotic dimension} $\asdim(f)$ is the supremum of $\asdim(A)$ such that $A\subset X$ and $\asdim(f(A))=0$. Our main result is \begin{Thm} \label{ThmAInAbstract} $\asdim(X)\leq \asdim(f)+\asdim(Y)$ for any large scale uniform function $f\colon X\to Y$. \end{Thm} \ref{ThmAInAbstract} generalizes a result of Bell and Dranishnikov in which $f$ is Lipschitz and $X$ is geodesic. We provide analogs of \ref{ThmAInAbstract} for Assouad-Nagata dimension $\dim_{AN}$ and asymptotic Assouad-Nagata dimension $\ANasdim$. In case of linearly cont…
We discuss a variation of Gromov's notion of asymptotic dimension that was introduced and named Nagata dimension by Assouad. The Nagata dimension turns out to be a quasisymmetry invariant of metric spaces. The class of metric spaces with finite Nagata dimension includes in particular all doubling spaces, metric trees, euclidean buildings, and homogeneous or pinched negatively curved Hadamard manifolds. Among others, we prove a quasisymmetric embedding theorem for spaces with finite Nagata dimension in the spirit of theorems of Assouad and Dranishnikov, and we characterize absolute Lipschitz re…
This paper is the first in a two-part series. In this paper, we prove that the Assouad-Nagata dimension of any finitely generated (but not necessarily finitely presented) $C'(1/6)$ group is at most 2. In the next paper, we use this result, along with techniques of classical small cancellation theory, to answer two open questions in the study of asymptotic and Assouad-Nagata dimension of finitely generated groups.
We show for a given metric space $(X,d)$ of asymptotic dimension $n$ that there exists a coarsely and topologically equivalent hyperbolic metric $d'$ of the form $d' = f \circ d$ such that $(X,d')$ is of asymptotic Assouad-Nagata dimension $n$. As a corollary we construct examples of spaces realising strict inequality in the logarithmic law for AN-asdim of a Cartesian product. One of them may be viewed as a counterexample to a specific kind of a Morita-type theorem for AN-asdim.
We prove that for all $k,m,n \in \mathbb N \cup \{\infty\}$ with $4 \leq k \leq m \leq n$, there exists a finitely generated group $G$ with a finitely generated subgroup $H$ such that the asymptotic dimension of $G$ is $k$, the Assouad-Nagata dimension of $G$ is $m$, and the Assouad-Nagata dimension of $H$ is $n$. This simultaneously answers two open questions in asymptotic dimension theory.
We prove that all spaces of finite Assouad-Nagata dimension admit a good order for Travelling Salesman Problem, and provide sufficient conditions under which the converse is true. We formulate a conjectural characterisation of spaces of finite $AN$-dimension, which would yield a gap statement for the efficiency of orders on metric spaces. Under assumption of doubling, we prove a stronger gap phenomenon about all orders on a given metric space.
Suppose $G$ is a countable, not necessarily finitely generated, group. We show $G$ admits a proper, left-invariant metric $d_G$ such that the Assouad-Nagata dimension of $(G,d_G)$ is infinite, provided the center of $G$ is not locally finite. As a corollary we solve two problems of A.Dranishnikov.
We construct a countable p-local group with a proper invariant metric whose Assouad-Nagata dimension is strictly greater than the asymptotic dimension with linear control. This solves Problem 8.6 from the list [Dr]. We study asymptotic dimension with linear control $\ell$-$asdim_ω$ that depends on a fixed ultrafilter on the naturals. It turns out that the Assouad-Nagata dimension is the supremum of $\ell$-$asdim_ω$ on all $ω$ and the asymptotic dimension with linear control is the minimum.
Lipschitz light maps, defined by Cheeger and Kleiner, are a class of non-injective "foldings" between metric spaces that preserve some geometric information. We prove that if a metric space $(X,d)$ has Nagata dimension $n$, then its "snowflakes" $(X,d^ε)$ admit Lipschitz light maps to $\mathbb{R}^n$ for all $0<ε<1$. This can be seen as an analog of a well-known theorem of Assouad. We also provide an application to a new variant of conformal dimension.
We introduce the notion of pseudo-cones of metric spaces as a generalization of both of the tangent cones and the asymptotic cones. We prove that the Assouad dimension of a metric space is bounded from below by that of any pseudo-cone of it. We exhibit a example containing all compact metric spaces as pseudo-cones, and examples containing all proper length spaces as tangent cones or asymptotic cones.
Hurewicz's dimension-raising theorem states that for every n-to-1 map f : X \to Y, dim Y =< dim X + n holds. In this paper we introduce a new notion of finite-to-one like map in a large scale setting. Using this notion we formulate a dimension-raising type theorem for the asymptotic dimension and the asymptotic Assouad-Nagata dimension. It is also well-known as Hurewicz's finite-to-one mapping theorem that dim X =< n if and only if there exists an (n + 1)-to-1 map from a 0-dimensional space onto X. We formulate a finite-to-one mapping type theorem for the asymptotic dimension and the asymptoti…
The $k$ nearest neighbour learning rule (under the uniform distance tie breaking) is universally consistent in every metric space $X$ that is sigma-finite dimensional in the sense of Nagata. This was pointed out by Cérou and Guyader (2006) as a consequence of the main result by those authors, combined with a theorem in real analysis sketched by D. Preiss (1971) (and elaborated in detail by Assouad and Quentin de Gromard (2006)). We show that it is possible to give a direct proof along the same lines as the original theorem of Charles J. Stone (1977) about the universal consistency of the $k$-N…
Bonamy et al \cite{BBEGLPS} showed that graphs of polynomial growth have finite asymptotic dimension. We refine their result showing that a graph of polynomial growth strictly less than $n^{k+1}$ has asymptotic dimension at most $k$. As a corollary Riemannian manifolds of bounded geometry and polynomial growth strictly less than $n^{k+1}$ have asymptotic dimension at most $k$. We show also that there are graphs of growth $<n^{1+ε}$ for any $ε>0$ and infinite asymptotic Assouad-Nagata dimension.
We introduce the idea of semigroup-controlled asymptotic dimension. This notion generalizes the asymptotic dimension and the asymptotic Assouad-Nagata dimension in the large scale. There are also semigroup controlled dimensions for the small scale and the global scale. Many basic properties of the asymptotic dimension theory are satisfied by a semigroup-controlled asymptotic dimension. We study how these new dimensions could help in the understanding of coarse embeddings and uniform embeddings. In particular we have introduced uncountable many invariants under quasi-isometries and uncountable …
An infinite graph G is minor excluded if there is a finite graph that is not a minor of G. We prove that minor excluded graphs have finite Assouad-Nagata dimension and study minor exclusion for Cayley graphs of finitely generated groups. Our main results and observations are: (1) minor exclusion is not a group property: it depends on the choice of generating set; (2) a group with one end has a generating set for which the Cayley graph is not minor excluded; (3) there are groups that are not minor excluded for any set of generators; (4) minor exclusion is preserved under free products; and (5) …
