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 $L_\alpha(\Bbb Z^2) \rtimes \Gamma$,fortorsionfree,non-amenablesubgroups$\Gamma\subset SL(2, \Bbb Z)$and$\alpha = e^{2\pi i t}$,$t \not\in \Bbb Q$, implies isomorphism of the corresponding groups $\mathrm{\Gamma }$.