W*-rigidity for the von Neumann algebras of products of hyperbolic groups

We show that if ({Gamma = Gamma_1timesdotsbtimes Gamma_n}) is a product of ({{rm n} geq 2}) non-elementary ICC hyperbolic groups then any discrete group ({Lambda}) which is ({W^*})-equivalent to ({Gamma}) decomposes as a direct product of n ICC groups and does not decompose as a direct product of k ICC groups when ({{rm n} not= {rm k}}). 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 II₁ factors.