Dual structures on Coxeter and Artin groups of rank three

TL;DR

This paper extends the theory of dual Coxeter and Artin groups to all rank-three Coxeter systems, establishing their topological and algebraic properties, including the $K( ext{pi},1)$ conjecture and solvability of the word problem.

Contribution

It introduces new geometric and combinatorial methods for rank-three Coxeter systems, proving key conjectures and properties for associated Artin groups.

Findings

Rank-three noncrossing partition posets are EL-shellable lattices

Associated Garside groups are isomorphic to standard Artin groups

Proved the $K( ext{pi},1)$ conjecture and solved the word problem for rank-three Artin groups

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 $K(\pi, 1)$ 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 $K(\pi, 1)$ conjecture and other open problems in the area.