The K(pi,1)-conjecture for Artin groups via combinatorial non-positive curvature

Abstract：

The K(pi,1)-conjecture for reflection arrangement complements, due to Arnold, Brieskorn, Pham, and Thom, predicts that certain complexified hyperplane complements associated to infinite reflection groups are Eilenberg MacLane spaces. We establish a close connection between a very simple property in metric graph theory about 4-cycles and the K(pi,1)-conjecture, via elements of non-positively curvature geometry. We also propose a new approach for studying the K(pi,1)-conjecture. As a consequence, we deduce a large number of new cases of Artin groups which satisfies the K(pi,1)-conjecture.