W${}^{\ast }$-Rigidity for the von Neumann Algebras of Products of Hyperbolic Groups

Abstract

We show that if $\Gamma ={\Gamma }_1\times \cdots \times {\Gamma }_n$ is a product of $n\ge 2$ non-elementary ICC hyperbolic groups then any discrete group $\Lambda$ which is $W^{\ast }$-equivalent to $\Gamma$ decomposes as a $k$-fold direct sum exactly when $k=n$. This gives a group-level strengthening of Ozawa and Popa's unique prime decomposition theorem by removing all assumptions on the group $\Lambda$. This result in combination with Margulis' normal subgroup theorem allows us to give examples of lattices in the same Lie group which do not generate stably equivalent II${}_1$ factors.

References in corpus (1)

• W*-superrigidity for group von Neumann algebras of left-right wreath products

Cited by in corpus (7)

• Some Applications of Group Theoretic Rips Constructions to the Classification of von Neumann Algebras
• Tensor product decompositions of II${}_1$ factors arising from extensions of amalgamated free product groups
• Orbit equivalence rigidity for product actions
• Prime II${}_1$ factors arising from irreducible lattices in products of rank one simple Lie groups
• Product rigidity in von Neumann and C${}^{\ast }$-algebras via s-malleable deformations
• Classification of tensor decompositions for II${}_1$ factors
• Ozawa's class $\mathcal{S}$ for locally compact groups and unique prime factorization