Commensurability of lattices in right-angled buildings

TL;DR

This paper establishes conditions under which lattices in right-angled buildings are commensurable to graph products of finite groups, linking geometric properties with algebraic separability and providing results on finite covers and quasi-isometric rigidity.

Contribution

It proves that weak commensurability of lattices corresponds to the separability of convex subgroups, extending to Davis complexes and right-angled Artin groups, and includes quasi-isometric rigidity results.

Findings

Lattices are weakly commensurable iff convex subgroups are separable.

Any two finite special cube complexes with the same universal cover have a common finite cover.

Quasi-isometric rigidity holds for certain right-angled buildings.

Abstract

Let $\Gamma$ 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 $\Gamma$ 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 $\Gamma$ 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 $\Gamma$ when $\Delta$ has the structure of a Fuchsian building.