We prove that for any free ergodic probability measure-preserving action of a free group on n generators , the associated group measure space II1 factor has L∞(X) as its unique Cartan subalgebra, up to unitary conjugacy. We deduce that group measure space II1 factors arising from actions of free groups with different number of generators are never isomorphic. We actually prove unique Cartan decomposition results for II1 factors arising from arbitrary actions of a much larger family of groups, including all free products of amenable groups and their direct products.
S.P. was supported in part by NSF Grant DMS-1101718. S.V. was supported by ERC Starting Grant VNALG-200749, Research Programme G.0639.11 of the Research Foundation – Flanders (FWO) and KU Leuven BOF research grant OT/08/032.
"Unique Cartan decomposition for II1 factors arising from arbitrary actions of free groups." Acta Math. 212 (1) 141 - 198, 2014. https://doi.org/10.1007/s11511-014-0110-9