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

TL;DR

This paper demonstrates that $ ilde{A}_n$ Coxeter groups act on weakly modular graphs, a broader class than CAT(0) cube complexes, by embedding their complexes into Euclidean spaces, expanding understanding of their geometric actions.

Contribution

It introduces the concept of weak modularity for $ ilde{A}_n$ Coxeter groups and describes their embeddings into Euclidean spaces, providing new geometric insights.

Findings

$ ilde{A}_n$ Coxeter groups act on weakly modular graphs.

Canonical embeddings of Coxeter complexes into Euclidean spaces are constructed.

Weak modularity is established for buildings of type $ ilde{A}_3$.

Abstract

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