Amalgamated free product rigidity for group von Neumann algebras

We provide a fairly large family of amalgamated free product groups Gamma = Gamma(1) (*Sigma) Gamma(2) whose amalgam structure can be completely recognized from their von Neumann algebras. Specifically, assume that Gamma(i) is a product of two icc non-amenable bi-exact groups, and Sigma is icc amenable with trivial one-sided commensurator in Gamma(i), for every i = 1,2. Then Gamma satisfies the following rigidity property: any group Lambda such that L(Lambda) is isomorphic to L(Gamma) admits an amalgamated free product decomposition Lambda = Lambda(1 *Delta) Lambda(2) such that the inclusions L(Delta) subset of L(Lambda(i)) and L(Sigma) subset of L(Gamma(i)) 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. (C) 2018 Elsevier Inc. All rights reserved.