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. Martínez-Pérez 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.

13 pages; fixed a small gap in Section 2 found by the referee, typo corrections; to appear in J. Group Theory

References in corpus (7)

• Configuration Lie groupoids and orbifold braid groups
• Algebraic $K$-theory, assembly maps, controlled algebra, and trace methods
• Helly meets Garside and Artin
• Artin groups of infinite type: trivial centers and acylindical hyperbolicity
• Farrell-Jones Conjecture for free-by-cyclic groups
• Poly-freeness of large even Artin groups
• On the isomorphism problem for even Artin groups

Cited by in corpus (1)

• Configuration Lie groupoids and orbifold braid groups