Von Neumann equivalence and group approximation properties

TL;DR

This paper explores the invariance of various analytic properties of groups under von Neumann equivalence, extending known results and providing new proofs for properties like the approximation property and the Haagerup property.

Contribution

The paper proves that weak amenability, the weak Haagerup property, and the approximation property are invariant under von Neumann equivalence, expanding the class of properties known to be stable.

Findings

Weak amenability is invariant under vNE.

The weak Haagerup property is invariant under vNE.

The approximation property (AP) is stable under measure equivalence.

Abstract

The notion of von Neumann equivalence (vNE), which encapsulates both measure equivalence and $W^*$-equivalence, was introduced recently by Jesse Peterson, Lauren Ruth and the author. They showed that many analytic properties, such as amenability, property (T), the Haagerup property, and proper proximality are preserved under von Neumann equivalence. In this article, we expand on the list of properties that are stable under von Neumann equivalence, and prove that weak amenability, weak Haagerup property, and the approximation property (AP) are von Neumann equivalence invariants. In particular, we get that (AP) is stable under measure equivalence. Furthermore, our techniques give an alternate proof for vNE-invariance of the Haagerup property.