Classifying virtually special tubular groups

© 2018 European Mathematical Society. A group is tubular if it acts on a tree with Z2 vertex stabilizers and Z edge stabilizers. We prove that a tubular group being virtually special is equivalent to it acting freely on either a locally finite or finite dimensional 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.