The dual approach to the $K(\pi, 1)$ conjecture

Abstract

Dual presentations of Coxeter groups have recently led to breakthroughs in our understanding of affine Artin groups. In particular, they led to the proof of the $K(\pi, 1)$ conjecture and to the solution of the word problem. Will the "dual approach" extend to more general classes of Coxeter and Artin groups? In this paper, we describe the techniques used to prove the $K(\pi, 1)$ conjecture for affine Artin groups and we ask a series of questions that are mostly open beyond the spherical and affine cases.