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

Abstract

Abstract We provide two simple proofs of the fact that even Artin groups of FC-type are 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 let A Γ A_{\Gamma} be its associated Artin group; our new proof implies that if A T A_{T} is poly-free (resp. normally poly-free) for every clique 𝑇 in Γ, then A Γ A_{\Gamma} is poly-free (resp. normally poly-free). We prove similar results regarding the Farrell–Jones Conjecture for even Artin groups. In particular, we show that if A Γ A_{\Gamma} is an even Artin group such that each clique in Γ either has at most three 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 A_{\Gamma} satisfies the Farrell–Jones Conjecture. In addition, our methods enable us to obtain results for general Artin groups.