On balanced coronas of groups

Let $G$ be an infinite group, $\kappa$ be an infinite cardinal, $\kappa\leq \mid G\mid$andlet$$\mathcal{E}${\kappa}$denotesacoarsestructureon$G$withthebase${{ (x,y): y\in F x F}: F\in [G]^{<\kappa}}$. 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})$isasingleton\mathrm{.}If$\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 $\kappa$.