On the structural theory of $\rm II_1$ factors of negatively curved groups

TL;DR

This paper proves that group factors of negatively curved groups are strongly solid and demonstrates the virtual W*-superrigidity of profinite actions of certain lattices, advancing the understanding of their structural properties.

Contribution

It introduces a new approach combining existing methods to establish strong solidity and superrigidity results for specific classes of groups and their actions.

Findings

Group factors of hyperbolic groups are strongly solid.

Profinite actions of lattices in Sp(n,1) are virtually W*-superrigid.

The paper develops a unified method for analyzing group von Neumann algebras.

Abstract

Ozawa showed that for any i.c.c., hyperbolic group, the associated group factor is solid. Developing a new approach that combines some methods of Peterson, Ozawa and Popa, and Ozawa, we strengthen this result by showing that these factors are strongly solid. Using our methods in cooperation with a cocycle superrigidity result of Ioana, we show that profinite actions of lattices in Sp(n,1), n>1, are virtually W*-superrigid.