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. © 2006 Elsevier Inc. All rights reserved.