Von Neumann equivalence and properly proximal groups

TL;DR

This paper introduces von Neumann equivalence as a new group relation, showing it preserves many analytic properties including proper proximality, and provides examples of groups with specific properties.

Contribution

It defines von Neumann equivalence, demonstrates its preservation of key properties, and explores its implications for group theory and operator algebras.

Findings

Von Neumann equivalence is coarser than measure and W*-equivalence.

Many properties like amenability, property (T), and the Haagerup property are preserved under this equivalence.

Proper proximality is also preserved, leading to new examples of groups with specific properties.

Abstract

We introduce a new equivalence relation on groups, which we call von Neumann equivalence, that is coarser than both measure equivalence and $W^*$-equivalence. We introduce a general procedure for inducing actions in this setting and use this to show that many analytic properties, such as amenability, property (T), and the Haagerup property, are preserved under von Neumann equivalence. We also show that proper proximality, which was defined recently by Boutonnet, Ioana, and the second author using dynamics, is also preserved under von Neumann equivalence. In particular, proper proximality is preserved under both measure equivalence and $W^*$-equivalence, and from this we obtain examples of non-inner amenable groups that are not properly proximal.