Related papers: Property A and asymptotic dimension

We introduce a new quasi-isometry invariant of metric spaces called the hyperbolic dimension, hypdim, which is a version of the Gromov's asymptotic dimension, asdim. The hyperbolic dimension is at most the asymptotic dimension, however,…

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.

Let $X$ be a proper metric space and let $F$ be a finite group acting on $X$ by isometries. We show that the asymptotic dimension of $F\backslash X$ is the same as the asymptotic dimension of $X$.

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 a finite metric space with $|X|=n$. For a positive integer $k$ we define $A_k(X)$ to be the quotient set of all $k$-subsets of $X$ by isometry, and we denote $|A_k(X)|$ by $a_k$. The sequence $(a_1,a_2,\ldots,a_{n})$ is…

We show that if a subspace $A$ of a coarse $PD(n)$ metric space $X$ coarsely separates it, then it must have asymptotic dimension at least $n-1$.

Let $X$ be a geodesic metric space with $H_1(X)$ uniformly generated. If $X$ has asymptotic dimension one then $X$ is quasi-isometric to an unbounded tree. As a corollary, we show that the asymptotic dimension of the curve graph of a…

Let $(X, d)$ be a compact metric space and let $\mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I \colon \mathcal{M}(X) \to \R$ by \[I(\mu) = \int_X \int_X d(x,y) d\mu(x) d\mu(y),\] and set $M(X) = \sup…

We address pairs $(X, Y)$ of metric spaces with the following property: for every mapping $f: X \to Y$ the existence of points $x, y \in X$ with $d(f(x),f(y)) > d(x,y)$ implies the existence of $\widetilde{x}, \widetilde{y}\in X$ for which…

We prove a version of the countable union theorem for asymptotic dimension and we apply it to groups acting on asymptotically finite dimensional metric spaces. As a consequence we obtain the following finite dimensionality theorems. A) An…

We prove that for a coarse space $X$ the ideal $S(X)$ of small subsets of $X$ coincides with the ideal $D_<(X)$ of subsets $A\subset X$ of asymptotic dimension $asdim(A)<asdim(X)$ provided that $X$ is coarsely equivalent to an Euclidean…

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

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…

A definition of asymptotic flatness at spatial infinity in $d$ dimensions ($d\geq 4$) is given using the conformal completion approach. Then we discuss asymptotic symmetry and conserved quantities. As in four dimensions, in $d$ dimensions…

We develop a probabilistic framework for large-scale dimension bounds in metric geometry, based on padded decompositions, randomized ball carving on net graphs, and the Lov\'asz Local Lemma. For metric measure spaces with volume doubling…

We construct a universal space for the class of proper metric spaces of bounded geometry and of given asymptotic dimension. As a consequence of this result, we establish coincidence of the asymptotic dimension with the asymptotic inductive…

Let $(X,d)$ be an unbounded metric space and $\tilde r=(r_n)_{n\in\mathbb N}$ be a scaling sequence of positive real numbers tending to infinity. We define the pretangent space $\Omega_{\infty, \tilde r}^{X}$ to $(X, d)$ at infinity as a…

Let $(X, d)$ be a compact metric space and let $\mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I \colon \mathcal{M}(X) \to \R$ by $I(mu) = \int_X \int_X d(x,y) d\mu(x) d\mu(y)$, and set $M(X) = \sup…

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…

Let $\mathfrak{M}$ be a class of metric spaces. A metric space $Y$ is minimal $\mathfrak{M}$-universal if every $X\in\mathfrak{M}$ can be isometrically embedded in $Y$ but there are no proper subsets of $Y$ satisfying this property. We find…