On balanced coronas of groups

Abstract

Let $G$ be an infinite group, $\kappa$ be an infinite cardinal, $\kappa \le \mid G\mid$ and let ${\mathcal{E}}_{\kappa }$ denotes a coarse structure on $G$ with the base $\{\{(x,y):y\in FxF\}:F\in [G{]}^{<\kappa }\}$. 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 $\kappa$.

coarse structure, slowly oscillating functions, balanced corona

References in corpus (1)

• A note on partitions of groups