Haagerup property for wreath products constructed with Thompson's groups

TL;DR

This paper demonstrates that certain wreath products involving Thompson's group V and groups with the Haagerup property also possess the Haagerup property, expanding understanding of this property in complex group constructions.

Contribution

It proves that wreath products formed with Thompson's group V and any group with the Haagerup property also have the Haagerup property, using recent techniques by Jones.

Findings

Wreath products with Thompson's group V inherit the Haagerup property.

The approach applies to a large family of groups and groupoids.

The results extend the class of groups known to have the Haagerup property.

Abstract

Using recent techniques introduced by Jones we prove that a large family of discrete groups and groupoids have the Haagerup property. In particular, we show that if G is a discrete group with the Haagerup property, then the wreath product $\oplus_{Q_2}G\rtimes V$ obtained from the group G and the usual action of Thompson's group V on the dyadic rational $Q_2$ of the unit interval has the Haagerup property.