Ends and simple coarse structures

TL;DR

This paper introduces a simple approach to coarse structures using equivalence relations on simple ends and explores their connections to Gromov boundaries and Higson compactifications.

Contribution

It presents a novel, simplified method for defining coarse structures and relates them to important concepts in geometric group theory and topology.

Findings

Gromov boundary is an example of a Higson corona

Freundenthal compactification is a Higson compactification

Coarse structures can be characterized via equivalence relations on simple ends

Abstract

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.