On balanced coronas of groups

Abstract

Let G be an infinite group, κ be an infinite cardinal, κ≤|G| and let Eκ denote a coarse structure on G with the base {{(x,y):y∈FxF}:F∈[G]<κ}. We prove that if either κ<|G| or κ=|G| and κ is singular then the Higson's corona νκ(G) of the coarse space (G,Eκ) is a singleton. If κ=|G| and κ is regular then νκ(G) contains a copy of the space Uκ of κ-uniform ultrafilters on κ.