Uniqueness of the group measure space decomposition for Popa's $\Cal H\Cal T$ factors

Abstract

We prove that every group measure space II$_1$ factor $L^{\infty}(X)\rtimes\Gamma$ coming from a free ergodic rigid (in the sense of [Po01]) probability measure preserving action of a group $\Gamma$ with positive first $\ell^2$--Betti number, has a unique group measure space Cartan subalgebra, up to unitary conjugacy. We deduce that many $\Cal H\Cal T$ factors, including the II$_1$ factors associated with the actions $\Gamma\curvearrowright \Bbb T^2$ and $\Gamma\curvearrowright$ SL$_2(\Bbb R)$/SL$_2(\Bbb Z)$, where $\Gamma$ is a non--amenable subgroup of SL$_2(\Bbb Z)$, have a unique group measure space Cartan subalgebra.