Classifying virtually special tubular groups (Q1650828)

Summary: 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 \(\mathrm{CAT}(0)\) cube complex. Furthermore, we prove that if a tubular group acts freely on a finite dimensional \(\mathrm{CAT}(0)\) cube complex, then it virtually acts freely on a three dimensional \(\mathrm{CAT}(0)\) cube complex.