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

TL;DR

This paper reduces the $K( ext{pi},1)$-conjecture for Artin groups to cases with tree Coxeter diagrams, establishing new classes of groups satisfying the conjecture through actions on Bestvina complexes.

Contribution

It introduces a reduction of the $K( ext{pi},1)$-conjecture to tree-shaped Coxeter diagrams and constructs Artin group actions on Bestvina complexes for Garside groupoids.

Findings

Reduction of the conjecture to tree Coxeter diagrams

Identification of new Artin groups satisfying the conjecture

Construction of Artin group actions on Bestvina complexes

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.