Coarse Embedding Research Articles

Finite determination for embeddings into Banach spaces: New proof, low distortion

The main goal of this paper is to improve the result of M. I. Ostrovskii [Embeddability of locally finite metric spaces into Banach spaces is finitely determined, Proc. Amer. Math. Soc. 140 (2012) 2721–2730.] on the finite determination of bilipschitz and coarse embeddability of locally finite metric spaces into Banach spaces. There are two directions of the improvement: (1) Substantial decrease of distortion (from about [Formula: see text] to 3 + [Formula: see text]) is achieved by replacing the barycentric gluing by the logarithmic spiral gluing. This decrease in the distortion is particularly important when the finite determination is applied to construction of embeddings. (2) Simplification of the proof: a collection of tricks employed in M. I. Ostrovskii [Embeddability of locally finite metric spaces into Banach spaces is finitely determined, Proc. Amer. Math. Soc. 140 (2012) 2721–2730.] is no longer needed; in addition to the logarithmic spiral gluing we use only a version of the Brunel–Sucheston argument used to prove the existence of spreading models.

On the expansiveness of coarse maps between Banach spaces and geometry preservation

We introduce a new notion of embeddability between Banach spaces. By studying the classical Mazur map, we show that it is strictly weaker than the notion of coarse embeddability. We use the techniques from metric cotype introduced by M. Mendel and A. Naor to prove results about cotype preservation and complete our study of embeddability between ℓp spaces. We confront our notion with nonlinear invariants introduced by N. Kalton, which are defined in terms of concentration properties for Lipschitz maps defined on countably branching Hamming or interlaced graphs. Finally, we address the problem of the embeddability into ℓ∞.

Coarse entropy of metric spaces

Coarse geometry studies metric spaces on the large scale. The recently introduced notion of coarse entropy is a tool to study dynamics from the coarse point of view. We prove that all isometries of a given metric space have the same coarse entropy and that this value is a coarse invariant. We call this value the coarse entropy of the space and investigate its connections with other properties of the space. We prove that it can only be either zero or infinity, and although for many spaces this dichotomy coincides with the subexponential–exponential growth dichotomy, there is no relation between coarse entropy and volume growth more generally. We completely characterise this dichotomy for spaces with bounded geometry and for quasi-geodesic spaces. As an application, we provide an example where coarse entropy yields an obstruction for a coarse embedding, where such an embedding is not precluded by considerations of volume growth.

Coarse Embeddability of Wasserstein Space and the Space of Persistence Diagrams

AbstractWe prove an equivalence between open questions about the embeddability of the space of persistence diagrams and the space of probability distributions (i.e. Wasserstein space). It is known that for many natural metrics, no coarse embedding of either of these two spaces into Hilbert space exists. Some cases remain open, however. In particular, whether coarse embeddings exist with respect to the p-Wasserstein distance for $$1\le p\le 2$$ 1 ≤ p ≤ 2 remains an open question for the space of persistence diagrams and for Wasserstein space on the plane. In this paper, we show that embeddability for persistence diagrams is equivalent to embeddability for Wasserstein space on $$\mathbb {R}^2$$ R 2 . When $$p > 1$$ p > 1 , Wasserstein space on $$\mathbb {R}^2$$ R 2 is snowflake universal (an obstruction to embeddability into any Banach space of non-trivial type) if and only if the space of persistence diagrams is snowflake universal.

Umbel convexity and the geometry of trees

For every p∈(0,∞), a new metric invariant called umbel p-convexity is introduced. The asymptotic notion of umbel convexity captures the geometry of countably branching trees, much in the same way as Markov convexity, the local invariant which inspired it, captures the geometry of bounded degree trees. Umbel convexity is used to provide a “Poincaré-type” metric characterization of the class of Banach spaces that admit an equivalent norm with Rolewicz's property (β). We explain how a relaxation of umbel p-convexity, called infrasup-umbel p-convexity, plays a role in obtaining compression rate bounds for coarse embeddings of countably branching trees. Local analogues of these invariants - fork p-convexity and infrasup-fork p-convexity - are introduced, and their relationship to Markov p-convexity and relaxations of the p-fork inequality is discussed. The metric invariants are estimated for a large class of Heisenberg groups, and in particular a parallelogram p-convexity inequality is proved for Heisenberg groups over p-uniformly convex Banach spaces. Finally, a new characterization of non-negative curvature is given.

Coarse directed limits of metric spaces admitting coarse embeddings into Hilbert spaces

For an inductive system of metric spaces that are coarsely embeddable into Hilbert spaces, its directed limit in the category of metric spaces may not admit a coarse embedding into a Hilbert space. In this article, we will show that the corresponding directed limit in the category of coarse spaces (with morphisms being bornologous maps) is always coarsely embeddable into a Hilbert space.

A Bott periodicity theorem for ℓp-spaces and the coarse Novikov conjecture at infinity

We formulate and prove a Bott periodicity theorem for an ℓp-space (1≤p<∞). For a proper metric space X with bounded geometry, especially for a coarsely connected space, we introduce a version of K-homology at infinity and the Roe algebra at infinity and show that to prove the coarse Novikov conjecture, it suffices to prove the coarse assembly map at infinity is an injection. As a result, we show that the coarse Novikov conjecture holds for any metric space with bounded geometry which admits a fibred coarse embedding into an ℓp-space. These include all box spaces of a residually finite hyperbolic group, and a large class of warped cones of a compact metric space with an action by a hyperbolic group.

K-Theory of the maximal and reduced Roe algebras of metric spaces with A-by-CE coarse fibrations

Let [Formula: see text] be a discrete metric space with bounded geometry. In this paper, we show that if [Formula: see text] admits an “A-by-CE” coarse fibration, then the canonical quotient map [Formula: see text] from the maximal Roe algebra to the Roe algebra of [Formula: see text], and the canonical quotient map [Formula: see text] from the maximal uniform Roe algebra to the uniform Roe algebra of [Formula: see text], induce isomorphisms on [Formula: see text]-theory. A typical example of such a space arises from a sequence of group extensions [Formula: see text] such that the sequence [Formula: see text] has Yu’s property A, and the sequence [Formula: see text] admits a coarse embedding into Hilbert space. This extends an early result of Špakula and Willett [Maximal and reduced Roe algebras of coarsely embeddable spaces, J. Reine Angew. Math. 678 (2013) 35–68] to the case of metric spaces which may not admit a coarse embedding into Hilbert space. Moreover, it implies that the maximal coarse Baum–Connes conjecture holds for a large class of metric spaces which may not admit a fibered coarse embedding into Hilbert space.

The twisted coarse Baum-Connes conjecture with coefficients in coarsely proper algebras

We introduce a notion of (approximately) coarsely proper algebras for coarse embeddings of metric spaces to formulate and prove a version of twisted coarse Baum-Connes conjecture with coefficients in coarsely proper algebras. This may be regarded as a coarse geometric version of the generalized Green-Julg Theorem in the Baum-Connes conjecture for countable discrete groups. It also provides a conceptual framework for the Dirac-dual-Dirac method to the coarse Novikov conjecture for coarse embeddings into several different spaces, including Hilbert spaces, simply connected complete Riemannian manifolds with non-positive sectional curvature, Banach spaces with property (H) and Hilbert-Hadamard spaces. Moreover, for a group extension 1→N→G→Q→1, we show that if N is coarsely embeddable into Hilbert space and Q is coarsely embeddable into an admissible Hilbert-Hadamard space, then the coarse Novikov conjecture holds for G.

Strongly quasi-local algebras and their $K$-theories

In this paper, we introduce a notion of strongly quasi-local algebras. They are defined for each discrete metric space with bounded geometry, and sit between the Roe algebra and the quasilocal algebra. We show that strongly quasi-local algebras are coarse invariants, hence encoding coarse geometric information of the underlying spaces. We prove that for a discrete metric space with bounded geometry which admits a coarse embedding into a Hilbert space, the inclusion of the Roe algebra into the strongly quasi-local algebra induces an isomorphism in K -theory.

Probabilistic Coarsening for Knowledge Graph Embeddings

Knowledge graphs have risen in popularity in recent years, demonstrating their utility in applications across the spectrum of computer science. Finding their embedded representations is thus highly desirable as it makes them easily operated on and reasoned with by machines. With this in mind, we propose a simple meta-strategy for embedding knowledge graphs using probabilistic coarsening. In this approach, a knowledge graph is first coarsened before being embedded by an arbitrary embedding method. The resulting coarse embeddings are then extended down as those of the initial knowledge graph. Although straightforward, this allows for faster training by reducing knowledge graph complexity while revealing its higher-order structures. We demonstrate this empirically on four real-world datasets, which show that coarse embeddings are learned faster and are often of higher quality. We conclude that coarsening is a recommended prepossessing step regardless of the underlying embedding method used.

Non-surjective coarse embeddings of continuous functions spaces

We consider coarse embeddings of real continuous functions spaces. We show that such an embedding can be well approximated by an affine surjective isometry under a certain condition of almost surjectivity.

Poincaré profiles of Lie groups and a coarse geometric dichotomy

Poincaré profiles are analytically defined invariants, which provide obstructions to the existence of coarse embeddings between metric spaces. We calculate them for all connected unimodular Lie groups, Baumslag–Solitar groups and Thurston geometries, demonstrating two substantially different types of behaviour. For Lie groups, our dichotomy extends both the rank one versus higher rank dichotomy for semisimple Lie groups and the polynomial versus exponential growth dichotomy for solvable unimodular Lie groups. We provide equivalent algebraic, quasi-isometric and coarse geometric formulations of this dichotomy. As a consequence, we deduce that for groups of the form Ntimes S, where N is a connected nilpotent Lie group, and S is a rank one simple Lie group, both the growth exponent of N, and the conformal dimension of S are non-decreasing under coarse embeddings. These results are new even for quasi-isometric embeddings and give obstructions which in many cases improve those previously obtained by Buyalo–Schroeder.

Lamplighter groups, median spaces and Hilbertian geometry

AbstractFrom any two median spaces $X$ and $Y$, we construct a new median space $X \circledast Y$, referred to as the diadem product of $X$ and $Y$, and we show that this construction is compatible with wreath products in the following sense: given two finitely generated groups $G,\,H$ and two (equivariant) coarse embeddings into median spaces $X,\,Y$, there exist a(n equivariant) coarse embedding $G\wr H \to X \circledast Y$. The construction offers a unified point of view on various questions related to the Hilbertian geometry of wreath products of groups.

Stochastic approximation of lamplighter metrics

We observe that embeddings into random metrics can be fruitfully used to study the L 1 $L_1$ -embeddability of lamplighter graphs or groups, and more generally lamplighter metric spaces. Once this connection has been established, several new upper bound estimates on the L 1 $L_1$ -distortion of lamplighter metrics follow from known related estimates about stochastic embeddings into dominating tree-metrics. For instance, every lamplighter metric on an n $n$ -point metric space embeds bi-Lipschitzly into L 1 $L_1$ with distortion O ( log n ) $O(\log n)$ . In particular, for every finite group G $G$ the lamplighter group H = Z 2 ≀ G $H=\mathbb {Z}_2\wr G$ bi-Lipschitzly embeds into L 1 $L_1$ with distortion O ( log log | H | ) $O(\log \log |H|)$ . In the case where the ground space in the lamplighter construction is a graph with some topological restrictions, better distortion estimates can be achieved. Finally, we discuss how a coarse embedding into L 1 $L_1$ of the lamplighter group over the d $d$ -dimensional infinite lattice Z d $\mathbb {Z}^d$ can be constructed from bi-Lipschitz embeddings of the lamplighter graphs over finite d $d$ -dimensional grids, and we include a remark on Lipschitz free spaces over finite metric spaces.

A coarse embedding theorem for homological filling functions

We demonstrate under appropriate finiteness conditions that a coarse embedding induces an inequality of homological Dehn functions. Applications of the main results include a characterization of what finitely presentable groups may admit a coarse embedding into a hyperbolic group of geometric dimension 2, characterizations of finitely presentable subgroups of groups with quadratic Dehn function with geometric dimension 2, and to coarse embeddings of nilpotent groups into other nilpotent groups of the same growth and into hyperbolic groups.

Coarse embeddings into products of trees

We give a short and elementary proof of the fact that every metric space of finite asymptotic dimension can be embedded into a finite product of trees.

Uniformly bounded fibred coarse embeddability and uniformly bounded a-T-menability

In this paper, we introduce the concept of uniformly bounded fibred coarse embeddability of metric spaces, generalizing the notion of fibred coarse embeddability defined by X. Chen, Q. Wang and G. Yu. Moreover, we show its relationship with uniformly bounded a-T-menability of groups. Finally, we give some examples to illustrate the differences between uniformly bounded fibred coarse embeddability and fibred coarse embeddability.

Slant products on the Higson–Roe exact sequence

We construct a slant product $/ \colon \mathrm{S}_p(X \times Y) \times \mathrm{K}_{1-q}(\mathfrak{c}^{\mathrm{red}}Y) \to \mathrm{S}_{p-q}(X)$ on the analytic structure group of Higson and Roe and the K-theory of the stable Higson corona of Emerson and Meyer. The latter is the domain of the co-assembly map $\mu^\ast \colon \mathrm{K}_{1-\ast}(\mathfrak{c}^{\mathrm{red}}Y) \to \mathrm{K}^\ast(Y)$. We obtain such products on the entire Higson--Roe sequence. They imply injectivity results for external product maps. Our results apply to products with aspherical manifolds whose fundamental groups admit coarse embeddings into Hilbert space. To conceptualize the class of manifolds where this method applies, we say that a complete $\mathrm{spin}^{\mathrm{c}}$-manifold is Higson-essential if its fundamental class is detected by the co-assembly map. We prove that coarsely hypereuclidean manifolds are Higson-essential. We draw conclusions for positive scalar curvature metrics on product spaces, particularly on non-compact manifolds. We also obtain equivariant versions of our constructions and discuss related problems of exactness and amenability of the stable Higson corona.

Property A and coarse embeddability for fuzzy metric spaces

Property A is a geometric property originally introduced for discrete metric spaces to provide a sufficient condition for coarse embeddability into Hilbert space, and it is defined via a Følner condition similar in spirit to the classical notion of amenability for groups. In this paper, we define property A for fuzzy metric spaces in the sense of George and Veeramani, show that it is an invariant in the coarse category of fuzzy metric spaces, and provide characterizations of it for uniformly locally finite fuzzy metric spaces. We also show that uniformly locally finite fuzzy metric spaces with property A are coarsely embeddable into Hilbert space.