Rigidity results for wreath product II1 factors

We consider II1 factors of the form , where either (i) B is a non-hyperfinite II1 factor and G is an ICC amenable group or (ii) B is a weakly rigid II1 factor and G is ICC group and where G acts on by Bernoulli shifts. We prove that isomorphism of two such factors implies cocycle conjugacy of the corresponding Bernoulli shift actions. In particular, the groups acting must be isomorphic. As a consequence, we can distinguish between certain classes of group von Neumann algebras associated to wreath product groups.