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

TL;DR

This paper proves that affine irreducible Coxeter groups of rank 3 satisfy the strong hull property, confirming a conjecture by Gaetz and Gao through geometric analysis of Coxeter complexes.

Contribution

It establishes the conjecture for a specific class of Coxeter groups by leveraging geometric methods, expanding understanding of convex hull properties in Coxeter group theory.

Findings

Confirmed the strong hull property for affine irreducible Coxeter groups of rank 3.

Reduced convex hull analysis to finitely many configurations using Coxeter complex geometry.

Validated the conjecture proposed by Gaetz and Gao for this class of groups.

Abstract

A conjecture proposed by Gaetz and Gao asserts that the Cayley graph of any Coxeter group satisfies 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 Coxeter complexes to reduce the analysis of convex hulls to finitely many manageable configurations.