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

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. Mart'inez-P'erez and L. Paris. More generally, let $\mathrm{\Gamma }$ be a finite simplicial graph with all edges labelled by positive even integers and $A_{\mathrm{\Gamma }}$ be its associated Artin group; our new proof implies that if $A_T$ is poly-free (resp. normally poly-free) for every clique $T$ in $\mathrm{\Gamma }$, then $A_{\mathrm{\Gamma }}$ 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 $A_{\mathrm{\Gamma }}$ is an even Artin group such that each clique in $\mathrm{\Gamma }$ 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 $A_{\mathrm{\Gamma }}$ satisfies the Farrell–Jones Conjecture. In addition, our methods enables us to obtain results for general Artin groups.