Unique Cartan decomposition for II₁ 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.