Compact actions and uniqueness of the group measure space decomposition of II1 factors

We prove that any II1 factor L∞(X)⋊Γ arising from a compact, free, ergodic, probability measure preserving action of a countable group Γ with positive first ℓ2-Betti number, , has a unique group measure space Cartan subalgebra, up to unitary conjugacy.