# On balanced coronas of groups

## TL;DR

This paper investigates the structure of Higson's corona for infinite groups under certain coarse structures, showing it is a singleton in some cases and contains a complex space in others.

## Contribution

It characterizes the Higson's corona of infinite groups with specific coarse structures, revealing when it is trivial or contains a large ultrafilter space.

## Key findings

- Corona is singleton if <|G| or =|G| and  is singular.
- Corona contains a space of -uniform ultrafilters if =|G| and  is regular.
- Results depend on the cofinality of the cardinal .

## Abstract

Let $G$ be an infinite group, $\kappa$ be an infinite cardinal, $\kappa\leq \mid G\mid$ and let $\mathcal{E}_{\kappa}$ denotes a coarse structure on $G$ with the base $\{\{ (x,y): y\in F x F\}: F\in [G]^{<\kappa}\}$. We prove that if either $\kappa< \mid G\mid$ or $\kappa= \mid G\mid$ and $\kappa$ is singular then the Higson's corona $\nu _{\kappa} (G)$ of the coarse space $(G, \mathcal{E}_{\kappa})$ is a singleton. If $\kappa= \mid G\mid$ and $\kappa$ is regular then $\nu _{\kappa} (G)$ contains a copy of the space $U_{\kappa}$ of $\kappa$-uniform ultrafilters on $\kappa$.