Compact actions and uniqueness of the group measure space decomposition   of II$_1$ factors
  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. 
Adrian Ioana
math.OA math.FA math.GR