Related papers: On asymptotic dimension and a property of Nagata

The purpose of this note is to characterize the asymptotic dimension $asdim(X)$ of metric spaces $X$ in terms similar to Property A of Yu: If $(X,d)$ is a metric space and $n\ge 0$, then the following conditions are equivalent: [a.]…

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…

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 $G$…

We obtain two in a sense dual to each other results: First, that the capacity dimension of every compact, locally self-similar metric space coincides with the topological dimension, and second, that the asymptotic dimension of a metric…

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…

A metric space $(X,d)$ has the de Groot property $GP_n$ if for any points $x_0,x_1,...,x_{n+2}\in X$ there are positive indices $i,j,k\le n+2$ such that $i\ne j$ and $d(x_i,x_j)\le d(x_0,x_k)$. If, in addition, $k\in\{i,j\}$ then $X$ is…

Given a metric space $X$ of finite asymptotic dimension, we consider a quasi-isometric invariant of the space called dimension function. The space is said to have asymptotic Assouad-Nagata dimension less or equal $n$ if there is a linear…

Assouad-Nagata dimension addresses both large and small scale behaviors of metric spaces and is a refinement of Gromov's asymptotic dimension. A metric space $M$ is a minor-closed metric if there exists an (edge-)weighted graph $G$…

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…

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…

We construct a class of metric spaces whose transfinite asymptotic dimension and complementary-finite asymptotic dimension are both $\omega+k$ for any $k\in\mathbb{N}$, where $\omega$ is the smallest infinite ordinal number and a metric…

A nonnegative number d_infinity, called asymptotic dimension, is associated with any metric space. Such number detects the asymptotic properties of the space (being zero on bounded metric spaces), fulfills the properties of a dimension, and…

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…

We construct a metric space whose transfinite asymptotic dimension and complementary-finite asymptotic dimension $2\omega+1$.

We prove that a metric space with subexponential asymptotic dimension growth has Yu's property A.

For every countable ordinal number $\xi$, we construct a metric space $X_{\xi}$ whose transfinite asymptotic dimension and complementary-finite asymptotic dimension are both $\xi$.

Let $(X, d)$ be an unbounded metric space. To investigate the asymptotic behavior of $(X, d)$ at infinity, one can consider a sequence of rescaling metric spaces $(X, \frac{1}{r_n} d)$ generated by given sequence $(r_n)_{n \in \mathbb N}$…

In this paper, we investigate a proper CAT(0) space $(X,d)$ which is homeomorphic to $R^2$ and we show that the asymptotic dimension $asdim (X,d)$ is equal to 2.

We introduce matrix algebra of subsets in metric spaces and we apply it to improve results of Yamauchi and Davila regarding Asymptotic Property C. Here is a representative result: Suppose $X$ is an $\infty$-pseudo-metric space and $n\ge 0$…

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…