Assouad-Nagata dimension of wreath products of groups

On Asymptotic Assouad-Nagata Dimension

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.

子式闭度量的Assouad-Nagata维数

Assouad-Nagata dimension of minor-closed metrics

该研究针对子式闭度量的Assouad-Nagata维数问题，通过分析子式闭度量的结构特征，确定了所有子式闭度量的Assouad-Nagata维数，其主定理统一推广了无权图与黎曼曲面相关维数的已有结论。

Assouad-Nagata dimension and gap for ordered metric spaces

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.

Hurewicz Theorem for Assouad-Nagata dimension

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…

Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions

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…

真子式封闭图类具有 Assouad-Nagata 维数 2

Proper Minor-Closed Classes of Graphs have Assouad-Nagata Dimension 2

本文将真子式封闭图类渐近维数为 2 的结论加强至更严格的 Assouad-Nagata 维数，并刻画细分封闭图类该维数有界的条件。

关于利普希茨单位分解与Assouad–Nagata维数

On Lipschitz partitions of unity and the Assouad--Nagata dimension

该研究推导了度量空间开覆盖下标准单位分解的利普希茨常数最优上界，分析了其ℓᵖ推广等的利普希茨性质，还给出了Assouad–Nagata维数的等价刻画。

Remarks on coarse triviality of asymptotic Assouad-Nagata dimension

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.

Assuad-Nagata dimension of nilpotent groups with arbitrary left invariant metrics

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.

子式闭族的渐近维数与曲面的Assouad-Nagata维数

Asymptotic Dimension of Minor-Closed Families and Assouad-Nagata Dimension of Surfaces

该研究证明真子式闭图族的渐近维数至多为2，解决了相关公开问题，还将结果推广至曲面的Assouad-Nagata维数及其他组合图类。

On Asymptotic Dimension With Linear Control

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.

A non-injective Assouad-type theorem with sharp dimension

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.

三维CAT(0)流形的渐近维数

Asymptotic dimension of 3-dimensional CAT(0) manifolds

该研究证明了带凸测地双射的真度量3-流形的Assouad-Nagata维数至多为3，进而将相关等周间隙定理从三维Hadamard流形推广到三维CAT(0)流形。

Dimension-Raising Maps in a Large Scale

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…

Universal consistency of the $k$-NN rule in metric spaces and Nagata dimension

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…

Embedding mapping class groups into finite products of trees

We prove the equivalence between a relative bottleneck property and being quasi-isometric to a tree-graded space. As a consequence, we deduce that the quasi-trees of spaces defined axiomatically by Bestvina-Bromberg-Fujiwara are quasi-isometric to tree-graded spaces. Using this we prove that mapping class groups quasi-isometrically embed into a finite product of simplicial trees. In particular, these groups have finite Assouad-Nagata dimension, direct embeddings exhibiting $\ell^p$ compression exponent $1$ for all $p\geq 1$ and they quasi-isometrically embed into $\ell^1(\mathbb N)$. We deduce…

Polynomial growth and asymptotic dimension

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.

Metric dimensions of minor excluded graphs and minor exclusion in groups

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) …

有限维顺从群

Finite dimensional amenable groups

本文证明有限 Assouad-Nagata 维数的顺从群满足 Shalom 的性质 $H_{FD}$，由此推出这类群既不是单群也不是挠群，并通过 Følner 偶直径的新估计加强了 Nowak 的相关结果。

双曲群视觉边界上的一个新共形度量

A new conformal gauge on the visual boundary of a hyperbolic group

本文通过对数重度量研究双曲群视觉边界，证明其与多项式Floyd边界双Lipschitz等价，保持Nagata维数和渐近维数，并检测单端性，但对数度量与视觉度量非拟对称等价。