Bestvina metric and tree reduction for $K(\pi,1)$-conjecture

Abstract

We reduce the $K(\pi,1)$-conjecture for all Artin groups to properties of Artin groups whose Coxeter diagrams are trees, from which we deduce new classes of Artin groups satisfying the $K(\pi,1)$-conjecture. This relies on constructing actions of Artin groups on Bestvina complexes of suitable Garside groupoids.