A unique decomposition result for HT factors with torsion free core

Abstract

We prove that II 1 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 II 1 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 L α i ( Z 2 ) ⋊ F n i , i = 1 , 2 , for F n i ⊂ SL ( 2 , Z ) free groups of rank n i and α j = e 2 π i t j , t j ∉ Q , implies n 1 = n 2 .