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

A countable group Γ is called shift-minimal if every non-trivial measure preserving action of Γ weakly contained in the Bernoulli shift Γ y ([0, 1] , λ ) is free. We show that any group Γ whose reduced C∗-algebra Cr(Γ) admits a unique tracial state is shift-minimal, and that any group Γ admitting 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 Cr(Γ). Shift-minimal groups, fixed price 1, and the unique trace property Robin D. Tucker-Drob‡ Caltech