A unique decomposition result for HT factors with torsion free core

We prove that II1 factors M have a unique (up to unitary conjugacy) cross-product type decomposition around “core subfactors” N⊂M satisfying the property HT of [S. Popa, On a class of type II1 factors with Betti numbers invariants, Ann. of Math. (2) 163 (2006) 809–899] and a certain “torsion freeness” condition. In particular, this shows that isomorphism of factors of the form Lαi(Z2)⋊Fni, i=1,2, for Fni⊂SL(2,Z) free groups of rank ni and αj=e2πitj, tj∉Q, implies n1=n2.