Uniformly recurrent subgroups and simple C⁎-algebras

We study uniformly recurrent subgroups (URS) introduced by Glasner and Weiss [18]. Answering their query we show that any URS Z of a finitely generated group is the stability system of a minimal Z-proper action. We also show that for any sofic URS Z there is a Z-proper action admitting an invariant measure. We prove that for a URS Z all Z-proper actions admits an invariant measure if and only if Z is coamenable. In the second part of the paper we study the separable C⁎-algebras associated to URS's. We prove that if a URS is generic then its C⁎-algebra is simple. We give various examples of generic URS's with exact and nuclear C⁎-algebras and an example of a URS Z for which the associated simple C⁎-algebra is not exact and not even locally reflexive, in particular, it admits both a uniformly amenable trace and a nonuniformly amenable trace.