In this talk I will discuss how strong forms of non-amenability are reflected in the asymptotic behavior of a group's Bernoulli action. Central to the discussion will be the notion of shift-minimality: A countable group G is called shift-minimal if every non-trivial measure preserving action weakly contained in the Bernoulli shift of G is free. I will discuss the connection between shift-minimality and certain properties of the reduced C*-algebra of G, and indicate the proof that if G admits a free pmp action of cost >1 then there is a finite normal subgroup N of G such that G/N is shift-minimal.