We investigate isometric embeddings of finite metric trees into $(\mathbb{R}^n,d_{1})$ and $( \mathbb{R}^n, d_{\infty})$. We prove that a finite metric tree can be isometrically embedded into $(\mathbb{R}^n,d_{1})$ if and only if the number of its leaves is at most $2n$. We show that a finite star tree with at most $2^n$ leaves can be isometrically embedded into $(\mathbb{R}^{n}, d_{\infty})$ and a finite metric tree with more than $2^n$ leaves cannot be isometrically embedded into $(\mathbb{R}^{n}, d_{\infty})$. We conjecture that an arbitrary finite metric tree with at most $2^n$ leaves can be isometrically embedded into $(\mathbb{R}^{n}, d_{\infty})$.
In every dimension $n\ge 3$ we introduce a class of orthogonal graph-manifolds and prove that the fundamental group of any orthogonal graph-manifold quasi-isometrically embeds into a product of $n$ trees. As a consequence, we obtain that asymptotic and linearly-controlled asymptotic dimensions of such group are equal to $n$.
Main results: (a) If a metric space contains $2n$ elements, the transportation cost space on it contains a $1$-complemented isometric copy of $\ell_1^n$. (b) An example of a finite metric space whose transportation cost space contains an isometric copy of $\ell_\infty^4$. Transportation cost spaces are also known as Arens-Eells, Lipschitz-free, or Wasserstein $1$ spaces.
For every $n\in\mathbb N$ we construct a finite graph $G$ such that every orientation $\vec G$ of $G$ contains an isometric copy of any oriented tree on $n$ vertices, and evaluate the smallest possible cardinality of $G$. On the other hand, we prove that every graph $G$ admits an orientation containing no directed $ω$-paths of infinite diameter.
Commability is the finest equivalence relation between locally compact groups such that $G$ and $H$ are equivalent whenever there is a continuous proper homomorphism $G \to H$ with cocompact image. Answering a question of Cornulier, we show that all non-elementary locally compact groups acting geometrically on locally finite simplicial trees are commable, thereby strengthening previous forms of quasi-isometric rigidity for trees. We further show that 6 homomorphisms always suffice, and provide the first example of a pair of locally compact groups which are commable but without commation consis…
Using isometric embedding of metric trees into Banach spaces, this paper will investigate barycenters, type and cotype, and various measures of compactness of metric trees. A metric tree ($T$, $d$) is a metric space such that between any two of its points there is an unique arc that is isometric to an interval in $\mathbb{R}$. We begin our investigation by examining isometric embeddings of metric trees into Banach spaces. We then investigate the possible images $x_0=π((x_1+\ldots+x_n)/n)$, where $π$ is a contractive retraction from the ambient Banach space $X$ onto $T$ (such a $π$ always exist…
Quasi-isometric rigidity for random subsets in products of trees
School of Mathematical Sciences & Beijing International Center for Mathematical Research, Peking University（北京大学数学科学学院与北京国际数学研究中心）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Department of Mathematics, University of California, San Diego（加州大学圣地亚哥分校数学系）
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 show that for each n\ge 2 there is a quasi-isometric embedding of the hyperbolic space H^n in the product T^n=Tx...xT of n copies of a (simplicial) metric tree T. On the other hand, we prove that there is no quasi-isometric embedding H^2 --> TxR^m for any metric tree T and any m\ge 0.
We show that if a group G acts isometrically on a locally finite leafless R-tree inducing a two-transitive action on its ends, then this tree is determined by the action of G on the boundary. As a corollary we obtain that locally finite Euclidean buildings without tree or cone factors are determined by their complete building at infinity.
We prove that for every $p\in(1,\infty)$, $p\ne 2$, there exist a Banach space $X$ isomorphic to $\ell_p$ and a finite subset $U$ in $\ell_p$, such that $U$ is not isometric to a subset of $X$. This result shows that the finite isometric version of the Krivine theorem (which would be a strengthening of the Krivine theorem (1976)) does not hold.
We investigate the geometry of the family $\cal M$ of isometry classes of compact metric spaces, endowed with the Gromov-Hausdorff metric. We show that sufficiently small neighborhoods of generic finite spaces in the subspace of all finite metric spaces with the same number of points are isometric to some neighborhoods in the space ${\mathbb R}^N_{\infty}$, i.e., in the space ${\mathbb R}^N$ with the norm $\|(x_1,\ldots,x_N)\|=\max_i|x_i|$. As a corollary, we get that each finite metric space can be isometrically embedded into $\cal M$ in such a way that its image belongs to a subspace consist…
The aim of this short paper is to offer a complete characterization of all (not necessarily surjective) isometric embeddings of the Wasserstein space $\mathcal{W}_p(\mathcal{X})$, where $\mathcal{X}$ is a countable discrete metric space and $0<p<\infty$ is any parameter value. Roughly speaking, we will prove that any isometric embedding can be described by a special kind of $\mathcal{X}\times(0,1]$-indexed family of nonnegative finite measures. Our result implies that a typical non-surjective isometric embedding of $\mathcal{W}_p(\mathcal{X})$ splits mass and does not preserve the shape of mea…
Let $(M,d)$ be a bounded countable metric space and $c>0$ a constant, such that $d(x,y)+d(y,z)-d(x,z) \ge c$, for any pairwise distinct points $x,y,z$ of $M$. For such metric spaces we prove that they can be isometrically embedded into any Banach space containing an isomorphic copy of $\ell_\infty$.
We prove that a nonempty closed and geodesically convex subset of the $l_{\infty}$ plane $\mathbb{R}^2_{\infty}$ is hyperconvex and we characterize the tight spans of arbitrary subsets of $\mathbb{R}^2_{\infty}$ via this property: Given any nonempty $X\subseteq\mathbb{R}^2_{\infty}$, a closed, geodesically convex and minimal subset $Y\subseteq\mathbb{R}^2_{\infty}$ containing $X$ is isometric to the tight span $T(X)$ of $X$.
We prove that for any given integer $c>0$ any metric space on $n$ points may be isometrically embedded into $l_{\infty}^{n-c}$ provided $n$ is large enough.
It is known that for every $α\geq 1$ there is a planar triangulation in which every ball of radius $r$ has size $Θ(r^α)$. We prove that for $α<2$ every such triangulation is quasi-isometric to a tree. The result extends to Riemannian 2-manifolds of finite genus, and to large-scale-simply-connected graphs. We also prove that every planar triangulation of asymptotic dimension 1 is quasi-isometric to a tree.
The paper is devoted to isometric Banach-space-theoretical structure of transportation cost (TC) spaces on finite metric spaces. The TC spaces are also known as Arens-Eells, Lipschitz-free, or Wasserstein spaces. A new notion of a roadmap pertinent to a transportation problem on a finite metric space has been introduced and used to simplify proofs for the results on representation of TC spaces as quotients of $\ell_1$ spaces on the edge set over the cycle space. A Tolstoi-type theorem for roadmaps is proved, and directed subgraphs of the canonical graphs, which are supports of maximal optimal …
We show that every inner metric space X is the metric quotient of a complete R-tree via a free isometric action, which we call the covering R-tree of X. The quotient mapping is a weak submetry (hence, open) and light. In the case of compact 1-dimensional geodesic space X, the free isometric action is via a subgroup of the fundamental group of X. In particular, the Sierpin'ski gasket and carpet, and the Menger sponge all have the same covering R-tree, which is complete and has at each point valency equal to the continuum. This latter R-tree is of particular interest because it is "universal" in…
We study extremal properties of finite ultrametric spaces $X$ and related properties of representing trees $T_X$. The notion of weak similarity for such spaces is introduced and related morphisms of labeled rooted trees are found. It is shown that the finite rooted trees are isomorphic to the rooted trees of nonsingular balls of special finite ultrametric spaces. We also found conditions under which the isomorphism of representing trees $T_X$ and $T_Y$ implies the isometricity of ultrametric spaces $X$ and $Y$.
We study isometric actions of tree automorphism groups on the infinite-dimensional hyperbolic spaces. On the one hand, we exhibit a general one-parameter family of such representations and analyse the corresponding equivariant embeddings of the trees, showing that they are convex-cocompact and asymptotically isometric. On the other hand, focusing on the case of sufficiently transitive groups of automorphisms of locally finite trees, we classify completely all irreducible representations by isometries of hyperbolic spaces. It turns out that in this case our one-parameter family exhausts all non…
Warped cones are metric spaces introduced by John Roe from discrete group actions on compact metric spaces to produce interesting examples in coarse geometry. We show that a certain class of warped cones $\mathcal{O}_Γ(M)$ admit a fibred coarse embedding into a $L_p$-space ($1\leq p<\infty$) if and only if the discrete group $Γ$ admits a proper affine isometric action on a $L_p$-space. This actually holds for any class of Banach spaces stable under taking Lebesgue-Bochner $L_p$-spaces and ultraproducts, e.g., uniformly convex Banach spaces or Banach spaces with nontrivial type. It follows that…
We compute the Lipschitz-free spaces of subsets of the real line and characterize subsets of metric trees by the fact that their Lipschitz-free space is isometric to a subspace of $L_1$.
In this note, we prove optimal extension results for roughly isometric relations between metric (R-)trees and injective metric spaces. This yields sharp stability estimates, in terms of the Gromov-Hausdorff (GH) distance, for certain metric spanning constructions: The GH distance of two metric trees spanned by some subsets is smaller than or equal to the GH distance of these sets. The GH distance of the injective hulls, or tight spans, of two metric spaces is at most twice the GH distance between themselves.
Let $(X,d,\mathfrak{m})$ be a metric measure space. The study of the Wasserstein space $(\mathbb{P}_p(X),\mathbb{W}_p)$ associated to $X$ has proved useful in describing several geometrical properties of $X.$ In this paper we focus on the study of isometries of $\mathbb{P}_p(X)$ for $p \in (1,\infty)$ under the assumption that there is some characterization of optimal maps between measures, the so called Good transport behaviour $GTB_p$. Our first result states that the set of Dirac deltas is invariant under isometries of the Wasserstein space. Additionally we obtain that the isometry groups o…
It was shown by Nachbin in 1950 that an $n$-dimensional normed space $X$ is injective or equivalently is an absolute 1-Lipschitz retract if and only if $X$ is linearly isometric to $l_\infty^n$ (i.e., $\mathbb{R}^n$ endowed with the $l_{\infty}$-metric). We give an effective convex geometric characterization of injective convex polyhedra in $l_{\infty}^n$. As an application, we prove that if the set of solutions to a linear system of inequalities with at most two variables per inequality is non-empty, then it is injective when endowed with the $l_{\infty}$-metric.
We show that every Gromov hyperbolic group $\Ga$ admits a quasi-isometric embedding into the product of $(n+1)$ binary trees, where $n=\dim\di\Ga$ is the topological dimension of the boundary at infinity of $\Ga$.
We prove that every visual Gromov hyperbolic space X whose boundary at infinity has the finite capacity dimension n admits a quasi-isometric embedding into (n+1)-fold product of metric trees.
Let $G$ be a graph and $p \in [1, \infty]$. The parameter $f_p(G)$ is the least integer $k$ such that for all $m$ and all vectors $(r_v)_{v \in V(G)} \subseteq \mathbb{R}^m$, there exist vectors $(q_v)_{v \in V(G)} \subseteq \mathbb{R}^k$ satisfying $$\|r_v-r_w\|_p=\|q_v-q_w\|_p, \ \text{ for all }\ vw\in E(G).$$ It is easy to check that $f_p(G)$ is always finite and that it is minor monotone. By the graph minor theorem of Robertson and Seymour, there are a finite number of excluded minors for the property $f_p(G) \leq k$. In this paper, we determine the complete set of excluded minors for $f_…
We prove that the universal cover of any graph manifold quasi-isometrically embeds into a product of three trees. In particular we show that the Assouad-Nagata dimension of the universal cover of any closed graph manifold is 3, proving a conjecture of Smirnov.
Fix $p\in[1,\infty)$, $K\in(0,\infty)$ and a probability measure $μ$. We prove that for every $n\in\mathbb{N}$, $\varepsilon\in(0,1)$ and $x_1,\ldots,x_n\in L_p(μ)$ with $\big\| \max_{i\in\{1,\ldots,n\}} |x_i| \big\|_{L_p(μ)} \leq K$, there exists $d\leq \frac{32e^2 (2K)^{2p}\log n}{\varepsilon^2}$ and vectors $y_1,\ldots, y_n \in \ell_p^d$ such that $$\forall \ i,j\in\{1,\ldots,n\}, \qquad \|x_i-x_j\|^p_{L_p(μ)}- \varepsilon \leq \|y_i-y_j\|_{\ell_p^d}^p \leq \|x_i-x_j\|^p_{L_p(μ)}+\varepsilon.$$ Moreover, the argument implies the existence of a greedy algorithm which outputs $\{y_i\}_{i=1}^n…
In this paper, we study the weighted Fermat-Frechet problem for a $\frac{N (N+1)}{2}-$tuple of positive real numbers determining $N$-simplexes in the $N$ dimensional $K$-Space ($N$-dimensional Euclidean space $\mathbb{R}^{N}$ if $K=0,$ the $N$-dimensional open hemisphere of radius $\frac{1}{\sqrt{K}}$ ($\mathbb{S}_{\frac{1}{\sqrt{K}}}^{N}$) if $K >0$ and the Lobachevsky space $\mathbb{H}_{K}^{N}$ of constant curvature $K$ if $K<0$). The (weighted) Fermat-Frechet problem is a new generalization of the (weighted) Fermat problem for $N$-simplexes. We control the number of solutions (weighted Ferm…
A metric graph is a pair $(G,d)$, where $G$ is a graph and $d:E(G) \to\mathbb{R}_{\geq0}$ is a distance function. Let $p \in [1,\infty]$ be fixed. An isometric embedding of the metric graph $(G,d)$ in $\ell_p^k = (\mathbb{R}^k, d_p)$ is a map $ϕ: V(G) \to \mathbb{R}^k$ such that $d_p(ϕ(v), ϕ(w)) = d(vw)$ for all edges $vw\in E(G)$. The $\ell_p$-dimension of $G$ is the least integer $k$ such that there exists an isometric embedding of $(G,d)$ in $\ell_p^k$ for all distance functions $d$ such that $(G,d)$ has an isometric embedding in $\ell_p^K$ for some $K$. It is easy to show that $\ell_p$-dim…
The paper is devoted to the geometry of transportation cost spaces and their generalizations introduced by Melleray, Petrov, and Vershik (2008). Transportation cost spaces are also known as Arens-Eells, Lipschitz-free, or Wasserstein $1$ spaces. In this work, the existence of metric spaces with the following properties is proved: (1) uniformly discrete infinite metric spaces transportation cost spaces on which do not contain isometric copies of $\ell_1$, this result answers a question raised by Cuth and Johanis (2017); (2) locally finite metric spaces which admit isometric embeddings only into…
Recently Kloeckner described the structure of the isometry group of the quadratic Wasserstein space $\mathcal{W}_2\left(\mathbb{R}^n\right)$. It turned out that the case of the real line is exceptional in the sense that there exists an exotic isometry flow. Following this line of investigation, we compute $\mathrm{Isom}\left(\mathcal{W}_p(\mathbb{R})\right)$, the isometry group of the Wasserstein space $\mathcal{W}_p(\mathbb{R})$ for all $p \in [1, \infty)\setminus\{2\}$. We show that $\mathcal{W}_2(\mathbb{R})$ is also exceptional regarding the parameter $p$: $\mathcal{W}_p(\mathbb{R})$ is is…
