The K(π,1)-conjecture for Artin groups

An Artin group \tiny GW is the fundamental group of the space \tiny YW which consists of regular orbits for the reflection action of a Coxeter group \tiny W on the complexified Tits cone. The best-known example is when \tiny W is the symmetric group, in which case \tiny GW is the classical braid group and \tiny YW is the configuration space of n different points in \tiny \mathbb{C}. A famous long-standing conjecture, attributed to Arnol'd, Brieskorn, Pham, and Thom, states that the orbit space \tiny YW is a \tiny K(Π,1)-space. The conjecture was proved for finite \tiny W by Deligne in a landmark paper ([Deligne, Invent. Math., '72]). Recently, in collaboration with Giovanni Paolini, we proved the conjecture for the next significant class of affine type Artin groups ([Paolini-Sal., Invent. Math, '21]). We discuss various aspects of our proof, which primarily utilizes combinatorial methods. In particular, we explore the relationship with the so-called dual approach to Artin groups, involving the development of Garside theory applied to non-crossing partition intervals. Some related interesting problems and conjectures can be posed for any Artin group. We have recently resolved these affirmatively for all rank three groups.