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

Abstract

We show that if Γ = Γ 1× ⋯ × Γ n is a product of n ≥ 2 non-elementary ICC hyperbolic groups then any discrete group Λ which is W∗-equivalent to Γ decomposes as a direct product of n ICC groups and does not decompose as a direct product of k ICC groups when n ≠ k. 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 II1 factors.