Commensurability of lattices in right-angled buildings
Sam Shepherd · Advances in Mathematics · 2024
Let Γ be a graph product of finite groups, with finite underlying graph, and let Δ be the associated right-angled building. We prove that a uniform lattice Λ in the cubical automorphism group Aut ( Δ ) is weakly commensurable to Γ if and only if all convex subgroups of Λ are separable. As a corollary, any two finite special cube complexes with universal cover Δ have a common finite cover. An important special case of our theorem is where Γ is a right-angled Coxeter group and Δ 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 Γ when Δ has the structure of a Fuchsian building.