In this note we prove that every metric space $(X, d)$ of asymptotic dimmension at most $n$ is coarsely equivalent to a metric space $(Y, D)$ that satisfies the following property of Nagata:
For every $n+2$ points $y_1,..., y_{n+2}$ in $Y$ and for every $x$ in $Y$ there exist two different $i,j$ such that $D(y_i,y_j)\le D(x,y_i)$.
This solves problem 1400 of the book Open problems in Topology II.
对一类包含离散群的度量空间X，证明了X的渐近Assouad-Nagata维数AN-asdim X与X的Higson日冕关于X上的次线性粗结构的覆盖维数$\dim(ν_L X)$一致.然后我们应用这个事实来证明等式AN-asdim（X x R）= AN-asdim X + 1。我们注意到Gromov的渐近维数asdim的类似等式通常不成立[Dr 3]。此外，我们还构造了一个从无基点的渐近锥面到次线性Higson日冕的内射映射。
给定度量空间的函数$f\colon X\to Y$，它的{ \it渐近维数} $\asdim(f)$是$\asdim(A)$的上确界，使得$A\subset X$和$\asdim(f(A))=0$ .我们的主要结果是：\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}推广了Bell和Dranishnikov的一个结果，其中$f$是Lipschitz的，$X$是测地线的.我们为Assouad-Nagata维数$\dim_{AN}$和渐近Assouad-Nagata维数$\ANasdim$提供了\ref{ThmAInAbstract}的类似物。在线性控制的渐近尺寸$\Lasdim$的情况下，我们提供了反例Dranishnikov的问题列表中的三个问题。作为\ref{ThmAInAbstract}类似物的应用，我们证明了\begin{Thm} \label{ThmBInAbstract} If $1\to K\to G\to H\to 1$ is an exact sequence of groups and $G$ is fin…
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 show that every geodesic metric space admitting an injective continuous map into the plane as well as every planar graph has Nagata dimension at most two, hence asymptotic dimension at most two. This relies on and answers a question in a very recent work by Fujiwara and Papasoglu. We conclude that all three-dimensional Hadamard manifolds have Nagata dimension three. As a consequence, all such manifolds are absolute Lipschitz retracts.
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.
Affine Approximation in Finite Nagata Dimension and Applications to Lipschitz-free spaces
Hanyang University（汉阳大学）；Univ Gustave Eiffel, Univ Paris Est Creteil, CNRS, LAMA UMR8050（古斯塔夫·埃菲尔大学，巴黎东 Créteil 大学，法国国家科学研究中心，LAMA UMR8050）；Université de Franche-Comté, CNRS, LmB (UMR 6623)（弗朗什-孔泰大学，法国国家科学研究中心，LmB (UMR 6623)）；Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València（纯数学与应用数学大学研究所，巴伦西亚理工大学）
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…
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.
Asymptotic property C was introduced by Dranishnikov to study spaces with infinite asymptotic dimension. We show that asymptotic property C is preserved by infinite products. We also show that countable restricted direct products of countable groups with finite asymptotic dimension have asymptotic property C. Then we introduce hyperbolic property C, an infinite dimensional version of hyperbolic dimension.
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…
We prove extension theorems for several geometric properties such as asymptotic property C (APC), finite decomposition complexity (FDC), strict finite decomposition complexity (sFDC) which are weakenings of Gromov's finite asymptotic dimension (FAD). The context of all theorems is a finitely generated group $G$ with a word metric and a coarse quasi-action on a metric space $X$. We assume that the quasi-stabilizers have a property $P_1$, and $X$ has the same or sometimes a weaker property $P_2$. Then $G$ also has property $P_2$. We show some sample applications, discuss constraints to further g…
We introduce a geometric property complementary-finite asymptotic dimension (coas- dim). Similar with asymptotic dimension, we prove the corresponding coarse invariant theorem, union theorem and Hurewicz-type theorem.
We combine aspects of the notions of finite decomposition complexity and asymptotic property C into a notion that we call finite APC-decomposition complexity. Any space with finite decomposition complexity has finite APC-decomposition complexity and any space with asymptotic property C has finite APC-decomposition complexity. Moreover, finite APC-decomposition complexity implies property A for metric spaces. We also show that finite APC-decomposition complexity is preserved by direct products of groups and spaces, amalgamated products of groups, and group extensions, among other constructions.
The notion of the decomposition complexity was introduced in \cite{GTY} using a game theoretical approach. We introduce a notion of straight decomposition complexity and compare it with the original as well with the asymptotic property C. Then we define a game theoretical analog of Haver's property C in the classical dimension theory and compare it with the original.
We consider functions with an asymptotic mean value property, known to characterize harmonicity in Riemannian manifolds, in doubling metric measure spaces. We show that the strongly amv-harmonic functions are Hölder continuous for any exponent below one. More generally, we define the class of functions with finite amv-norm and show that functions in this class belong to a fractional Hajlasz-Sobolev space and their blow-ups satisfy the mean-value property. Furthermore, in the weighted Euclidean setting we find an elliptic PDE satisfied by amv-harmonic functions.
The entropy of a finite probability space $X$ measures the observable cardinality of large independent products $X^{\otimes n}$ of the probability space. If two probability spaces $X$ and $Y$ have the same entropy, there is an almost measure-preserving bijection between large parts of $X^{\otimes n}$ and $Y^{\otimes n}$. In this way, $X$ and $Y$ are asymptotically equivalent. It turns out to be challenging to generalize this notion of asymptotic equivalence to configurations of probability spaces, which are collections of probability spaces with measure-preserving maps between some of them. In…
