On the structure of relatively biexact group von Neumann algebras

TL;DR

This paper introduces a new technique at the von Neumann algebra level to classify subalgebras of group von Neumann algebras for biexact groups, extending previous results and combining with deformation-rigidity methods.

Contribution

Develops a novel approach to upgrade relative proper proximality to full proper proximality, enabling classification of subalgebras in biexact group von Neumann algebras.

Findings

Classifies subalgebras of $L\Gamma$ for biexact groups with almost malnormal subgroups

Establishes a structural absorption theorem for free products

Provides a generalized Kurosh type theorem for properly proximal von Neumann algebras

Abstract

Using computations in the bidual of $\mathbb{B}(L^2M)$ we develop a new technique at the von Neumann algebra level to upgrade relative proper proximality to full proper proximality. This is used to structurally classify subalgebras of $L\Gamma$ where $\Gamma$ is an infinite group that is biexact relative to a finite family of subgroups $\{\Lambda_i\}_{i\in I}$ such that each $\Lambda_i$ is almost malnormal in $\Gamma$. This generalizes the result of \cite{DKEP21} which classifies subalgebras of von Neumann algebras of biexact groups. By developing a combination with techniques from Popa's deformation-rigidity theory we obtain a new structural absorption theorem for free products and a generalized Kurosh type theorem in the setting of properly proximal von Neumann algebras.