Shift-minimal groups, fixed price 1, and the unique trace property

Abstract

A countable group Γis called shift-minimal if every non-trivial measure preserving action of Γweakly contained in the Bernoulli shift of Γon ([0,1]^Γ,λ^Γ) is free. We show that any group Γwhose reduced C^*-algebra admits a unique tracial state is shift-minimal, and that any group Γadmitting a free measure preserving action of cost>1 contains a finite normal subgroup N such that Γ/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(Γ).