2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/69834We 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 $L_α(\Bbb Z^2) \rtimes Γ$, for torsion free, non-amenable subgroups $Γ\subset SL(2, \Bbb Z)$ and $α= e^{2πi t}$, $t \not\in \Bbb Q$, implies isomorphism of the corresponding groups $Γ$.8 pages; minor corrections 02/17/04, 03/17/04Operator Algebras46L10, 46L35, 46L40text