Weak Modularity and ${\tilde{A}}_n$ Buildings

Abstract

The ${\tilde{A}}_n$ Coxeter groups are known to not be systolic or cocompactly cubulated for $n\ge 3$. 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 emeddings of the 1-skeleta of ${\tilde{A}}_n$ Coxeter complexes into the Euclidean spaces ${\mathbb{R}}^{n+1}$. We also prove weak modularity for buildings of type ${\tilde{A}}_3$.

References in corpus (2)

• Weakly modular graphs and nonpositive curvature
• The (2,4,5) triangle Coxeter group is not systolic

Cited by in corpus (1)

• 2-dimensional Coxeter groups are biautomatic