Indecomposable continua as Higson coronae

TL;DR

This paper characterizes when the Higson corona of a space is an indecomposable continuum, linking it to coarse geometric properties and group end structures, with implications for finitely generated groups.

Contribution

It provides a characterization of spaces with indecomposable Higson coronae and relates this to coarse equivalence to natural numbers and group end structures.

Findings

Higson corona is indecomposable iff space is coarsely equivalent to natural numbers.

Characterizes groups with one or two ends via Higson corona decomposability.

Higson corona can be a topological sum of two indecomposable continua.

Abstract

In this paper, we consider spaces whose Higson coronae are indecomposable continua. We show that for a non-compact proper metric space $X$ which is coarsely geodesic and has coarse bounded geometry, the Higson corona of $X$ is an indecomposable continuum if and only if $X$ is coarsely equivalent to the space of natural numbers. Then we give characterizations of finitely generated groups that have one or two ends by decomposability/indecomposability of the components of their Higson coronae. we characterize it as a group whose Higson corona is a topological sum of two indecomposable continua.