Shift-minimal groups, fixed price 1, and the unique trace property
Abstract
A countable group \Gamma is called shift-minimal if every non-trivial 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 shift-minimal, 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 shift-minimal. Any shift-minimal group in turn is shown to have trivial amenable radical. Recurrence arguments are used in studying invariant random subgroups of a wide variety of shift-minimal 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, shift-minimality, C^*-simplicity, and uniqueness of tracial state on C^*_r(\Gamma).
- Publication:
-
arXiv e-prints
- 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
- E-Print:
- 55 pages, 1 figure