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 Abert, Glasner, and Virag (2014). Our main results are as follows. (i) It was shown in Weiss (2012) 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.