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

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. Mart\'inez-P\'erez and L. Paris. More generally, let $\Gamma$ be a finite simplicial graph with all edges labelled by positive even integers and $A_\Gamma$ 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 $\Gamma$, then $A_\Gamma$ 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 $A_\Gamma$ is an even Artin group such that each clique in $\Gamma$ 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 $A_\Gamma$ satisfies the Farrell--Jones Conjecture. In addition, our methods enables us to obtain results for general Artin groups.