Commensurability of lattices in right-angled buildings

Let I' be a graph product of finite groups, with finite underlying graph, and let Delta be the associated right-angled building. We prove that a uniform lattice Lambda in the cubical automorphism group Aut(Delta) is weakly commensurable to I' if and only if all convex subgroups of Lambda are separable. As a corollary, any two finite special cube complexes with universal cover Delta have a common finite cover. An important special case of our theorem is where I' is a right-angled Coxeter group and Delta is the associated Davis complex. We also obtain an analogous result for right-angled Artin groups. In addition, we deduce quasi-isometric rigidity for the group I' when Delta has the structure of a Fuchsian building. (c) 2024 Elsevier Inc. All rights reserved.