A class of II1 factors with many non conjugate Cartan subalgebras

We construct a class of II1 factors M that admit unclassifiably many Cartan subalgebras in the sense that the equivalence relation of being conjugate by an automorphism of M is complete analytic, in particular non Borel. We also construct a II1 factor that admits uncountably many non isomorphic group measure space decompositions, all involving the same group G. So G is a group that admits uncountably many non stably orbit equivalent actions whose crossed product II1 factors are all isomorphic. (C) 2012 Elsevier Inc. All rights reserved.