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 G is a discrete group with the Haagerup property, then the wreath product ${\oplus }_{Q_2}G⋊V$ 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.