Amalgamated free product rigidity for group von Neumann algebras

We provide a fairly large family of amalgamated free product groups Γ=Γ1⁎ΣΓ2 whose amalgam structure can be completely recognized from their von Neumann algebras. Specifically, assume that Γi is a product of two icc non-amenable bi-exact groups, and Σ is icc amenable with trivial one-sided commensurator in Γi, for every i=1,2. Then Γ satisfies the following rigidity property: any group Λ such that L(Λ) is isomorphic to L(Γ) admits an amalgamated free product decomposition Λ=Λ1⁎ΔΛ2 such that the inclusions L(Δ)⊆L(Λi) and L(Σ)⊆L(Γi) are isomorphic, for every i=1,2. This result significantly strengthens some of the previous Bass-Serre rigidity results for von Neumann algebras. As a corollary, we obtain the first examples of amalgamated free product groups which are W⁎-superrigid.