Coarse compactifications of proper metric spaces

TL;DR

This paper explores coarse compactifications of proper metric spaces, introduces new descriptions and axioms, and characterizes various examples including Higson, Freudenthal, and Gromov compactifications, highlighting their universal properties.

Contribution

It provides alternative definitions and axioms for coarse compactifications and characterizes their boundaries and functions, extending understanding of their universal properties.

Findings

Higson compactification is universal among coarse compactifications.

Freudenthal compactification is universal among those with totally disconnected boundary.

For hyperbolic geodesic spaces, a specific embedding relates boundary structures.

Abstract

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 $\nu(\mathbb R_+)\times \partial X\to \nu(X)$. Its image is a retract of $\nu(X)$ if $X$ is a tree.