Dual structures on Coxeter and Artin groups of rank three

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.