Classifying Virtually Special Tubular Groups

Abstract: A group is tubular if it acts on a tree with $\mathbb{Z}^2$ vertex stabilizers and $\mathbb{Z}$ edge stabilizers. We prove that a tubular group is virtually special if and only if it acts freely on a locally finite CAT(0) cube complex. Furthermore, we prove that if a tubular group acts freely on a finite dimensional CAT(0) cube complex, then it virtually acts freely on a three dimensional CAT(0) cube complex.