  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.
