Weak Modularity and A˜n Coxeter Groups

Abstract

The ${\tilde{\mathit{A}}}_{\mathit{n}}$ Coxeter groups are known to not be systolic or cocompactly cubulated for $\mathit{n}\ge 3$. We prove that these groups act geometrically on weakly modular graphs, a weak notion of nonpositive curvature generalizing the 1-skeleta of $\mathrm{CAT}(0)$ cube complexes and systolic complexes. To prove weak modularity, we describe the canonical embeddings of the 1-skeleta of ${\tilde{\mathit{A}}}_{\mathit{n}}$ Coxeter complexes into the Euclidean spaces ${\mathbb{R}}^{\mathit{n}+1}$. We also prove that the 1-skeleta of buildings of type ${\tilde{\mathit{A}}}_3$ are weakly modular.