Uniformly Recurrent Subgroups

We define the notion of uniformly recurrent subgroup, URS in short, which is a topological analog of the notion of invariant random subgroup (IRS), introduced in [2]. Our main results are as follows. (i) It was shown in [26] that for an arbitrary countable infinite group G, any free ergodic probability measure preserving G-system admits a minimal model. In contrast we show here, using URS’s, that for the lamplighter group there is an ergodic measure preserving action which does not admit a minimal model. (ii) For an arbitrary countable group G, every URS can be realized as the stability system of some topologically transitive G-system.