Uniformly recurrent subgroups

By Eli Glasner, Barak Weiss. 2014

66 citations recorded by the source corpus.

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 a work of M. Abert, Y. Glasner and B. Virag. Our main results are as follows. (i) It was shown by B. Weiss 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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c7e811cf71d393c6e73f70c02b858b3d54886fda

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In Discrete Mathematics and Combinatorics

1. J. Peterson
2. M. Kennedy
3. U. Bader
4. P. Caprace
5. A. Erschler
6. D. Fisher

In Geometry and Topology

1. Cyril Houdayer
2. J. Peterson
3. R. Boutonnet
4. Tobias Hartnick
5. M. Kennedy
6. U. Bader
7. Adrian Ioana
8. S. Vaes

In Mathematical Physics

1. Cyril Houdayer
2. J. Peterson
3. M. Bjorklund
4. R. Boutonnet
5. Tobias Hartnick
6. M. Kennedy
7. U. Bader
8. Adrian Ioana

Citations in the corpus

• Charmenability of arithmetic groups of product type (2020, background): “Recall that a URS of is a minimal -invariant subset of Sub( ) (see [19]).”
• An intrinsic characterization of C*-simplicity (2015, background): “The notion of a uniformly recurrent subgroup of G was recently introduced by Glasner and Weiss [9] as a topological-dynamical analogue of the notion of an invariant random subgroup.”
• Approximate lattices (2016, background): “In the context of nonabelian groups, the study of invariant random subgroups and uniformly recurrent subgroups [1, 2, 25] has led to an intensive study of invariant measures on conjugation hulls, i.”