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).