The Haagerup property for wreath products

Abstract: A countable group has the Haagerup property, or is a-T-menable, if it admits a metrically proper isometric action on a Hilbert space. The class of a-T-menable groups has nice stability properties, but it is NOT closed under general semi-direct products. We prove that it is closed under wreath products (joint work with Y. de Cornulier and Y. Stalder). Combining this result with a recent result by Ozawa and Popa, we disprove (half of) a conjecture of M. Cowling on the coincidence between the class of a-T-menable groups and the class of groups with the completely bounded approximation property.