The strong hull property for affine irreducible Coxeter groups of rank 3

TL;DR

The paper proves the strong hull property for the affine irreducible rank-3 Coxeter groups, namely $\widetilde{A}_2$, $\widetilde{C}_2$, and $\widetilde{G}_2$. It reinterprets convex hulls in Cayley graphs via the geometry of affine buildings, treating them as unions of minimal galleries to enable a finite-case analysis. For each type, the strong hull inequality $|\mathrm{Conv}(u,v)| \cdot |\mathrm{Conv}(v,w)| \geq |\mathrm{Conv}(u,v,w)|$ is verified through type-specific arguments: coordinate counting for $\widetilde{A}_2$, area-preserving translations for $\widetilde{C}_2$, and a reduction of $\widetilde{G}_2$ to the $\widetilde{A}_2$ case. The results suggest a scalable framework for extending the method to higher ranks in the future.

Abstract

A conjecture proposed by Gaetz and Gao asserts that the Cayley graph of any Coxeter group possesses the strong hull property. In this paper, we prove this conjecture for all affine irreducible Coxeter groups of rank 3. Our approach exploits the geometry of affine buildings to reduce the analysis of convex hulls to finitely many manageable configurations. These geometric reduction techniques offer a novel framework that may be applicable to higher-rank cases.

Figures (25)

• Figure 1: The triangulation of Euclidean space for rank-$3$ affine irreducible Coxeter groups
• Figure 2: The reflections generate $I_2(\infty)$
• Figure 3: The Cayley graph of $I_2(\infty)$
• Figure 4: The Cayley graph for $\widetilde{A}_2$
• Figure 5: Convex hull of $u$ and $v$ is shaded.

Theorems & Definitions (5)

• lemma 1
• theorem 1
• proof
• theorem 2: Strong hull property for type $\widetilde{A}_2$
• proof