A classical result by Bieberbach says that uniform lattices acting on Euclidean spaces by isometries are virtually free abelian. On the other hand, uniform lattices acting on trees are virtually free. This motivates the study of commensurability classification of uniform lattices acting on $CAT(0)$ cube complexes associated with right-angled Artin groups (RAAG complexes). These complexes can be thought as “interpolations” between Euclidean spaces and trees. Uniform lattices acting on the same RAAG complex may not belong to the same commensurability class, as there are irreducible lattices acting on products of trees. However, we show that the tree times tree obstruction is the only obstruction for commmensurability of label-preserving lattices acting on RAAG complexes. Some connection of this problem with Haglund and Wise’s work on special cube complexes will be explained. It time permits, I will also discuss some applications to quasi-isometric rigidity of RAAGs.