W$^*$-Rigidity for the von Neumann Algebras of Products of Hyperbolic Groups

Abstract

We show that if $Γ= Γ_1\times\dotsb\times Γ_n$ is a product of $n\geq 2$ non-elementary ICC hyperbolic groups then any discrete group $Λ$ which is $W^*$-equivalent to $Γ$ 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 $Λ$. 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.