On the structural theory of II_1 factors of negatively curved groups, II: Actions by product groups

TL;DR

This paper develops structural results for von Neumann algebras from measure-preserving actions of product groups, demonstrating new examples of strongly solid factors and conditions for unique Cartan subalgebras, with implications for superrigidity.

Contribution

It extends methods to produce new strongly solid factors and analyzes Cartan subalgebras in von Neumann algebras from product group actions, including superrigidity results.

Findings

Weakly amenable groups in class S produce strongly solid factors.

Maximal abelian subalgebras have amenable normalizers in product group factors.

Certain lattice actions are virtually W*-superrigid.

Abstract

This paper includes a series of structural results for von Neumann algebras arising from measure preserving actions by product groups on probability spaces. Expanding upon the methods used earlier by the first two authors \cite{CS}, we obtain new examples of strongly solid factors as well as von Neumann algebras with unique or no Cartan subalgebra. For instance we show that every II$_1$ factor associated with a weakly amenable group in the class $\mathcal S$ of Ozawa is strongly solid, \cite{OzSolid}. There is also the following product version of this result: any maximal abelian $\star$-subalgebra of any II$_1$ factor associated with a finite product of weakly amenable groups in the class $\mathcal S$ of Ozawa has an amenable normalizing algebra. Finally, pairing some of these results with cocycle superrigidity results from \cite{IoaCSR}, it follows that compact actions by finite products of lattices in $Sp(n, 1)$, $n \geq2$, are virtually $W^*$-superrigid.