A Unique Decomposition Result for HT Factors with Torsion Free Core

We prove that II$_1$ factors $M$ have a unique (up to unitary conjugacy) cross-product type decomposition around ``core subfactors'' $N \subset M$ satisfying the property HT and a certain ``torsion freeness'' condition. In particular, this shows that isomorphism of factors of the form $L_\alpha(\Bbb Z^2) \rtimes \Gamma$, for torsion free, non-amenable subgroups $\Gamma\subset SL(2, \Bbb Z)$ and $\alpha = e^{2\pi i t}$, $t \not\in \Bbb Q$, implies isomorphism of the corresponding groups $\Gamma$.