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

We show that if $\mathrm{\Gamma }={\mathrm{\Gamma }}_1\times \cdots \times {\mathrm{\Gamma }}_n$ is a product of $n\ge 2$ non-elementary ICC hyperbolic groups then any discrete group $\mathrm{\Lambda }$ which is $W^{\ast }$-equivalent to $\mathrm{\Gamma }$ 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 $\mathrm{\Lambda }$. 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.