Haagerup property for wreath products constructed with Thompson's groups

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 $\Gamma$ is a discrete group with the Haagerup property, then the permutational restricted wreath product ${⨁}_{{\mathbb{Q}}_2}\Gamma ⋊V$ obtained from the group $\Gamma$ and the usual action of Richard Thompson's group $V$ on the dyadic rational ${\mathbb{Q}}_2$ of the unit interval has the Haagerup property.