Compact actions and uniqueness of the group measure space decomposition of II$_1$ factors

Abstract: We prove that any II$_1$ factor $L^{\infty}(X)\rtimes\Gamma$ arising from a compact, free, ergodic, probability measure preserving action of a countable group $\Gamma$ with positive first $\ell^2$-Betti number, has a unique group measure space Cartan subalgebra, up to unitary conjugacy.