Uniformly recurrent subgroups. (English) Zbl 1332.37014

Glasner, Eli; Weiss, Benjamin

Bhattacharya, Siddhartha (ed.) et al., Recent trends in ergodic theory and dynamical systems. International conference in honor of S. G. Dani’s 65th birthday, Vadodara, India, December 26–29, 2012. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-0931-9/pbk; 978-1-4704-2219-6/ebook). Contemporary Mathematics 631, 63-75 (2015).

Summary: 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 [M. Abért et al., Duke Math. J. 163, No. 3, 465–488 (2014; Zbl 1344.20061)]. Our main results are as follows. (i) It was shown in [the second author, Contemp. Math. 567, 249–264 (2012; Zbl 1279.37010)] 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.
For the entire collection see [Zbl 1309.37003].

MSC:

37B20	Notions of recurrence and recurrent behavior in topological dynamical systems
37A15	General groups of measure-preserving transformations and dynamical systems
20E05	Free nonabelian groups
20E15	Chains and lattices of subgroups, subnormal subgroups
57S20	Noncompact Lie groups of transformations

Keywords:

invariant minimal subgroups; URS; IRS; stability group; stability system; essentially free action; free group

Citations:

Zbl 1344.20061; Zbl 1279.37010