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

TL;DR

This paper demonstrates that the Formanek-Procesi group, a non-linear semidirect product of two free groups, acts properly on a finite-dimensional CAT(0) cube complex, providing a new example with the Haagerup property.

Contribution

It provides the first example of a non-linear semidirect product of free groups with the Haagerup property, expanding understanding of group actions on CAT(0) spaces.

Findings

Formanek-Procesi group acts properly on a finite-dimensional CAT(0) cube complex

First example of a non-linear free-by-free group with the Haagerup property

Shows such groups can have complex geometric actions

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.