We use malleable deformations combined with spectral gap rigidity theory, in the framework of Popa's deformation/rigidity theory to prove unique tensor product decomposition results for II 1 factors arising as tensor product of wreath product factors. We also obtain a similar result regarding measure equivalence decomposition of direct products of such groups.Date: October 16, 2018.

We use deformation-rigidity theory in the von Neumann algebra framework to study probability measure preserving actions by wreath product groups. In particular, we single out large families of wreath product groups satisfying various types of orbit equivalence (OE) rigidity. For instance, we show that whenever H , K, Γ , Λ are icc, property (T) groups such that H Γ and K Λ admit stably orbit equivalent action σ and ρ such that σ | Γ , ρ| Λ , σ | H Γ , and ρ| K Λ are ergodic, then automatically σ Γ is stably orbit equivalent to ρ Λ and σ | H Γ is stably orbit equivalent to ρ| K Λ . Rigidity results for von Neumann algebras arising from certain actions of such groups (i.e. W * -rigidity results) are also obtained. Published by Elsevier Inc.