A non-trivial example of a free-by-free group with the Haagerup property

Abstract

The aim of this note is to prove that the group of Formanek-Procesi acts properly isometrically on a finite dimensional CAT(0) cube complex. This gives a first example of a non-linear semidirect product between two non abelian free groups which satisfies the Haagerup property.Comment: 19 pages, 5 figures. With respect to the previous version, some proofs have been detailed and the dimension of the cube complex the group acts on has been correcte