Unique Cartan decomposition for II_1 factors arising from arbitrary   actions of hyperbolic groups
  We prove that for any free ergodic probability measure preserving action $\Gamma \actson (X,\mu)$ of a non-elementary hyperbolic group, or a lattice in a rank one simple Lie group, the associated group measure space II_1 factor $L^\infty(X) \rtimes \Gamma$ has L^\infty(X) as its unique Cartan subalgebra, up to unitary conjugacy. 
Sorin Popa and Stefaan Vaes
math.OA math.DS math.GR