Shiftminimal groups, fixed price 1, and the unique trace property
Abstract
A countable group \Gamma is called shiftminimal if every nontrivial measure preserving action of \Gamma weakly contained in the Bernoulli shift of \Gamma on ([0,1]^\Gamma ,\lambda ^\Gamma) is free. We show that any group \Gamma whose reduced C^*algebra admits a unique tracial state is shiftminimal, and that any group \Gamma admitting a free measure preserving action of cost>1 contains a finite normal subgroup N such that \Gamma /N is shiftminimal. Any shiftminimal group in turn is shown to have trivial amenable radical. Recurrence arguments are used in studying invariant random subgroups of a wide variety of shiftminimal groups. We also examine continuity properties of cost in the context of infinitely generated groups and equivalence relations. A number of open questions are discussed which concern cost, shiftminimality, C^*simplicity, and uniqueness of tracial state on C^*_r(\Gamma).
 Publication:

arXiv eprints
 Pub Date:
 November 2012
 DOI:
 10.48550/arXiv.1211.6395
 arXiv:
 arXiv:1211.6395
 Bibcode:
 2012arXiv1211.6395T
 Keywords:

 Mathematics  Group Theory;
 Mathematics  Dynamical Systems;
 Mathematics  Operator Algebras;
 Mathematics  Probability;
 37A15;
 37A20;
 37A25;
 37A50;
 37A55;
 43A07;
 60B99
 EPrint:
 55 pages, 1 figure