Commensurability of lattices in right-angled buildings

Abstract

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.

Figures (8)

• Figure 1: An example of a 3-regular generalized 3-gon, known as the Heawood graph. This is a valid choice for $\mathcal{G}$ in parts \ref{['item:i']} and \ref{['item:ii']} of Theorem \ref{['thm:QI']}.
• Figure 2: An example of a simplicial complex $N$ and corresponding cubical cone $C(N)$.
• Figure 3: An example of the graph $\mathcal{G}$ and a section of the right-angled building $\Delta$. In this example the groups $G_i,G_j,G_k,G_l$ have orders $2,2,3,3$ respectively. One of the chambers is shown in bold, and its vertices are labeled by their types.
• Figure 4: A repeat of Figure \ref{['fig:building']}, but with the edges oriented to point upwards in the poset structure, so they form the Hasse diagram for $(\Delta^0,\leq)$. Four examples of level-equivalence classes are depicted, with colors red, orange, green and blue.
• Figure 5: A repeat of Figure \ref{['fig:leveladj']}, but with a parallelism class of (oriented) $i$-edges shown in red and a parallelism class of (oriented) $j$-edges shown in blue.

Theorems & Definitions (161)

• Theorem 1.1
• Remark 1.2
• Corollary 1.3
• Corollary 1.4
• Corollary 1.5
• Remark 1.6
• Corollary 1.7
• Corollary 1.9
• Corollary 1.10
• Theorem 1.11