  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.

Operator Algebras;Functional Analysis;Group Theory