TY - CHAP

T1 - Uniformly recurrent subgroups

AU - Glasner, Eli

AU - Weiss, Benjamin

N1 - Publisher Copyright:
© 2015 American Mathematical Society.

PY - 2015

Y1 - 2015

N2 - 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.

AB - 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.

KW - Essentially free action

KW - Free group

KW - IRS

KW - Invariant minimal subgroups

KW - Stability group

KW - Stability system

KW - URS

UR - http://www.scopus.com/inward/record.url?scp=85106793025&partnerID=8YFLogxK

U2 - 10.1090/conm/631/12596

DO - 10.1090/conm/631/12596

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AN - SCOPUS:85106793025

T3 - Contemporary Mathematics

SP - 63

EP - 75

BT - Contemporary Mathematics

PB - American Mathematical Society

ER -