On closure operations in the space of subgroups and applications

We establish some interactions between uniformly recurrent subgroups (URSs) of a group G and cosets topologies N on G associated to a family N of normal subgroups of G. We show that when N consists of finite index subgroups of G, there is a natural closure operation H clN (H) that associates to a URS H another URS clN (H), called the N-closure of H. We give a characterization of the URSs H that are N-closed in terms of stabilizer URSs. This has consequences on arbitrary URSs when G belongs to the class of groups for which every faithful minimal profinite action is topologically free. We also consider the largest amenable URS AG, and prove that for certain coset topologies on G, almost all subgroups H AG have the same closure. For groups in which amenability is detected by a set of laws (a property that is variant of the Tits alternative), we deduce a criterion for AG to be a singleton based on residual properties of G.