On Relative Property (T) and Haagerup's Property

TL;DR

This paper investigates the relationships between three properties related to countable discrete groups and their von Neumann algebras, demonstrating that certain implications between these properties do not hold in general.

Contribution

The paper shows that the converses of known implications between properties (1), (2), and (3) for groups and their von Neumann algebras are false, clarifying the independence of these properties.

Findings

Implication (2) to (1) is false.

Implication (3) to (2) is false.

Counterexamples demonstrate the independence of properties.

Abstract

We consider the following three properties for countable discrete groups $\Gamma$: (1) $\Gamma$ has an infinite subgroup with relative property (T), (2) the group von Neumann algebra $L\Gamma$ has a diffuse von Neumann subalgebra with relative property (T) and (3) $\Gamma$ does not have Haagerup's property. It is clear that (1) $\Longrightarrow$ (2) $\Longrightarrow$ (3). We prove that both of the converses are false.