Amalgamated Free Product Rigidity for Group von Neumann Algebras

2017/05/20 by Chifan, Ionut, Ioana, Adrian · 2 citations
#20E06 #46L36 #FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1705.07350

Abstract

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 Γ_i is a product of two icc non-amenable bi-exact (e.g., hyperbolic) groups, and Σ is icc amenable and has trivial one-sided commensurator in Γ_i, for every i∈\1,2\. Then Γ satisfies the following rigidity property: any group \La such that L(\La) is isomorphic to L(\G) admits an amalgamated free product decomposition \La=\La₁∗_Δ\La₂ such that the inclusions L(Δ)⊆ L(\La_i) and L(Σ)⊆ L(\G_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.

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