Dual structures on Coxeter and Artin groups of rank three

TL;DR

This work extends the dual Coxeter/Artin framework to rank-three (hyperbolic) Coxeter groups, proving that the noncrossing partition poset $[1,w]$ is a lattice and EL-shellable, and that the dual Artin group is isomorphic to the standard Artin group. It develops the interval complex $K_W$ and a hierarchy of subcomplexes, then uses discrete Morse theory to deform $K_W$ onto $X_W'$, a finite complex homotopy equivalent to the orbit configuration space $Y_W$, thereby establishing the $K(\pi,1)$ conjecture for rank-three Artin groups and showing they are Garside with a solvable word problem. The authors also prove the triviality of the center in the non-spherical case and discuss how these constructions illuminate open problems in general Artin groups. The results significantly extend the reach of the dual approach beyond affine types and suggest avenues for complete resolutions of key conjectures in the broader Artin group landscape.

Abstract

We extend the theory of dual Coxeter and Artin groups to all rank-three Coxeter systems, beyond the previously studied spherical and affine cases. Using geometric, combinatorial, and topological techniques, we show that rank-three noncrossing partition posets are EL-shellable lattices and give rise to Garside groups isomorphic to the associated standard Artin groups. Within this framework, we prove the $K(π, 1)$ conjecture, the triviality of the center, and the solubility of the word problem for rank-three Artin groups. Some of our constructions apply to general Artin groups; we hope they will help develop complete solutions to the $K(π, 1)$ conjecture and other open problems in the area.

Figures (13)

• Figure 1: Examples of arrangements of rank-three hyperbolic Coxeter groups in the Poincaré model. Each picture is captioned with the labels $(m_1, m_2, m_3)$ of the corresponding Coxeter diagram. The hyperbolic plane is tiled by triangles with angles $\frac{\pi}{m_1}$, $\frac{\pi}{m_2}$, and $\frac{\pi}{m_3}$.
• Figure 2: Arrangements of irreducible spherical and affine Coxeter groups of rank three. Each picture is captioned with the standard name (following the classification of spherical and affine Coxeter groups) and the labels $(m_1, m_2, m_3)$ of the corresponding Coxeter diagram. In each case, the sphere or plane is tiled by triangles with angles $\frac{\pi}{m_1}, \frac{\pi}{m_2}, \frac{\pi}{m_3}$. The Coxeter groups of the spherical cases (top three) are the symmetry groups of the five platonic solids (tetrahedron, cube/octahedron, and dodecahedron/icosahedron, respectively).
• Figure 3: Proof of \ref{['lemma:translations']} (in the half-plane model).
• Figure 4: Reflection arrangements of \ref{['fig:hyperbolic-arrangements']} with the axis of a Coxeter element $w = abc$ (orange dashed line). The axial vertices are colored based on their orbit under the action of the infinite cyclic group generated by $w$ (see \ref{['lemma:axial-vertices-orbits']}).
• Figure 5: Axes of the Coxeter elements $abc$ (in orange), $bca$ (in blue), and $cab$ (in purple). The orthic triangle of the chamber delimited by $\mathop{\mathrm{\textsc{Fix}}}\nolimits(a), \mathop{\mathrm{\textsc{Fix}}}\nolimits(b)$, and $\mathop{\mathrm{\textsc{Fix}}}\nolimits(c)$ is highlighted in gray.

Theorems & Definitions (98)

• Lemma 2.1
• proof
• Remark 2.2
• Theorem 2.3: forman1998morsechari2000discretebatzies2002discrete
• Theorem 2.4: Patchwork theorem kozlov2007combinatorial
• Theorem 2.5: bestvina1999dehornoy1999gaussiancharney2004bestvinamccammond2005introductiondehornoy2015foundations
• Conjecture 2.6: $K(\pi, 1)$ conjecture
• Definition 3.1
• Lemma 3.2
• proof