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Unique Cartan decomposition for II_1 factors arising from arbitrary actions of free groups (1111.6951v3)

Published 29 Nov 2011 in math.OA, math.DS, and math.GR

Abstract: We prove that for any free ergodic probability measure preserving action \F_n \actson (X,\mu) of a free group on n generators \F_n, 2 \leq n \leq \infty, the associated group measure space II_1 factor $L\infty(X) \rtimes \F_n$ has L\infty(X) as its unique Cartan subalgebra, up to unitary conjugacy. We deduce that group measure space II_1 factors arising from actions of free groups with different number of generators are never isomorphic. We actually prove unique Cartan decomposition results for II_1 factors arising from arbitrary actions of a rather large family of groups, including all free products of amenable groups and their direct products.

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