Poly-freeness of Artin groups and the Farrell-Jones Conjecture

Abstract

We give two simple proofs of the fact that any even Artin groups of FC-type are normally poly-free which was recently established by R. Blasco-Garcia, C. Martínez-Pérez and L. Paris. More generally, let Γ be a finite simplicial graph with all edges labelled by positive even integers and A Γ be its associated Artin group, we show that if A T is poly-free (resp. normally poly-free) for every clique T in Γ, then A Γ is poly-free (resp. normally poly-free). We also prove similar results regarding the Farrell-Jones Conjecture for Artin groups. In particular, we show that if A Γ is an even Artin group such that each clique in Γ either has at most 3 vertices, has all of its labels at least 6, or is the join of these two types of cliques (the edges connecting the cliques are all labelled by 2), then A Γ satisfies the Farrell-Jones Conjecture. We also have some results for general Artin groups. of G. Our first theorem can be stated as follows.Theorem A. Let Γ be a finite simplicial graph with all edges labelled by positive even integers and A Γ be its associated Artin group. If A Γ is of FC-type, then it is normally polyfree. In general if A T is poly-free (resp. normally poly-free) for every clique T in Γ, then A Γ is poly-free (resp. normally poly-free).In [3, Question 2] and the discussions below it, Bestvina asks whether all Artin groups are virtually poly-free. In light of our main theorem, one might not need to pass to a finite index subgroup in the case of even Artin groups although our reduction does also work for