A countable group G is called shift-minimal if all non-trivial measure pre- serving actions of G weakly contained in the Bernoulli shift G ([0, 1]G,λG) are free. I will discuss the relation between shift-minimality and certain properties of the reduced C*-algebra of G, and present a proof that any group whose reduced C*-algebra admits a unique tracial state is shift-minimal. This implies shift-minimality for a wide variety of groups including all non-abelian free groups. I will outline how direct ergodic theoretic arguments also give more specific information about freeness properties of many shift-minimal groups. Several open questions will be discussed.