This paper studies coarse compactifications and their boundary. We introduce two alternative descriptions to Roe's original definition of coarse compactification. One approach uses bounded functions on $X$ that can be extended to the boundary. They satisfy the Higson property exactly when the compactification is coarse. The other approach defines a relation on subsets of $X$ which tells when two subsets closure meet on the boundary. A set of axioms characterizes when this relation defines a coarse compactification. Such a relation is called large-scale proximity. Based on this foundational work we study examples for coarse compactifications Higson compactification, Freudenthal compactification and Gromov compactification. For each example we characterize the bounded functions which can be extended to the coarse compactification and the corresponding large-scale proximity relation. We provide an alternative proof for the property that the Higson compactification is universal among coarse compactifications. Furthermore the Freudenthal compactification is universal among coarse compactifications with totally disconnected boundary. If $X$ is hyperbolic geodesic proper then there is a closed embedding $ν(\mathbb R_+)\times \partial X\to ν(X)$. Its image is a retract of $ν(X)$ if $X$ is a tree.
We introduce the notion of controlled products on metric spaces as a generalization of Gromov products, and construct boundaries by using controlled products, which we call the Gromov boundaries. It is shown that the Gromov boundary with respect to a controlled product on a proper metric space complements the space as a coarse compactification. It is also shown that there is a bijective correspondence between the set of all coarse equivalence classes of controlled products and the set of all equivalence classes of coarse compactifications.
A coarse compactification of a proper metric space $X$ is any compactification of $X$ that is dominated by its Higson compactification. In this paper we describe the maximal coarse compactification of $X$ whose corona is of dimension $0$. In case of geodesic spaces $X$, it coincides with the Freundenthal compactification of $X$. As an application we provide an alternative way of extending the concept of the number of ends from finitely generated groups to arbitrary countable groups. We present a geometric proof of a generalization of Stallings' theorem by showing that any countable group of tw…
This paper is devoted to introducing coarse structures in a very simple way, namely as an equivalence relation on the set of simple ends. As an application we show that Gromov boundary of every hyperbolic space is an example of a Higson corona and each Freundenthal compactification is an example of a Higson compactification.
For any compact Hausdorff space $K$ we construct a canonical finitary coarse structure $\mathcal E_{X,K}$ on the set $X$ of isolated points of $K$. This construction has two properties: $\bullet$ If a finitary coarse space $(X,\mathcal E)$ is metrizable, then its coarse structure $\mathcal E$ coincides with the coarse structure $\mathcal E_{X,\bar X}$ generated by the Higson compactification $\bar X$ of $X$; $\bullet$ A compact Hausdorff space $K$ coincides with the Higson compactification of the coarse space $(X,\mathcal E_{X,K})$ if the set $X$ is dense in $K$ and the space $K$ is Frechet-Ur…
We present an idea of unifying small scale (topology, proximity spaces, uniform spaces) and large scale (coarse spaces, large scale spaces). It relies on an analog of multilinear forms from Linear Algebra. Each form has a large scale compactification and those include all well-known compactifications: Higson corona, Gromov boundary of hyperbolic spaces, the visual boundary of CAT(0)-spaces, \v Cech-Stone compactification, Samuel-Smirnov compactification, and Freudenthal compactification. As an application we get simple proofs of results generalizing well-known theorems from coarse topology. A …
We introduce the notion of (hybrid) large scale normal space and prove coarse geometric analogues of Urysohn's Lemma and the Tietze Extension Theorem for these spaces, where continuous maps are replaced by (continuous and) slowly oscillating maps. To do so, we first prove a general form of each of these results in the context of a set equipped with a neighbourhood operator satisfying certain axioms, from which we obtain both the classical topological results and the (hybrid) large scale results as corollaries. We prove that all metric spaces are hybrid large scale normal, and characterize thos…
A topology on a set $X$ is the same as a projection (i.e. an idempotent linear operator) $cl:2^X\to 2^X$ satisfying $A\subset cl(A)$ for all $A\subset X$. That's a good way to summarize Kuratowski's closure operator. Basic geometry on a set $X$ is a dot product $\cdot:2^X\times 2^X\to 2^Y$. Its equivalent form is an orthogonality relation on subsets of $X$. The optimal case is if the orthogonality relation satisfies a variant of parallel-perpendicular decomposition from linear algebra. We show that this concept unifies small scale (topology, proximity spaces, uniform spaces) and large scale (c…
John Roe \cite{Roe lectures}通过考虑$X\times X$的子集引入了任意集合$X$的粗糙结构。这个定义对分析家来说是自然的，但对拓扑学家和几何学家来说就有点难以理解了。本文通过一致有界族的概念引入了$X$上的大尺度结构，并证明了它们与$X$上的粗结构的等价性.这样，大尺度几何的所有基本概念（渐近维数、慢振荡函数、希格森紧化）都有了自然的定义，度量几何的基本结果也就转移到了粗几何中。
Let $X$ be a proper, non-compact CAT(-1) space, and $Γ$ a discrete cocompact subgroup of the isometries of $X$. We compactify the diagonal action of $Γ$ on $X \times X$ considering a domain of the horofunction boundary with respect to the maximum metric.
We formalize the concept of a family of metric spaces satisfying a coarse property uniformly and we generalize finite decomposition complexity of Erik Guentner, Romain Tessera, and Guoliang Yu. Of particular interest are results determining sufficient conditions for a metric space to satisfy Property A of Guoliang Yu.
We define two different weakenings of coarse amenability (also known as Yu's property A), namely fibred coarse amenability and coarse amenability at infinity. These two properties allow us to prove that a residually finite group is coarsely amenable if and only if some (or all) of its box spaces satisfy the weak properties. We then elaborate on a result of Willett by showing that graphs with large girth always satisfy fibred coarse amenability. Finally, we discuss some examples and counter-examples to these properties.
The aim of this paper is to investigate properties preserved and co-preserved by coarsely $n$-to-1 functions, in particular by the quotient maps $X\to X/\sim$ induced by a finite group $G$ acting by isometries on a metric space $X$. The coarse properties we are mainly interested in are related to asymptotic dimension and its generalizations: having finite asymptotic dimension, asymptotic Property C, straight finite decomposition complexity, countable asymptotic dimension, and metric sparsification property. We provide an alternative description of asymptotic Property C and we prove that the cl…
This paper discusses properties of the Higson corona by means of a quotient on coarse ultrafilters on a proper metric space. We use this description to show that the corona functor is faithful. This study provides a Künneth formula for twisted coarse cohomology. We obtain the Gromov boundary of a hyperbolic proper geodesic metric space as a quotient of its Higson corona.
In this paper we show that a countable, residually amenable group admits a proper isometric affine action on some uniformly convex Banach space if and only if one (or equivalently, all) of its box families admits a fibred cofinitely-coarse embedding into some uniformly convex Banach space.
