  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).
