Around subgroups of Artin groups: derived subgroups and acylindrical hyperbolicity in the even FC-case

TL;DR

This paper extends known results about right-angled Artin groups to certain Artin groups, exploring their subgroup structures, splittings, and hyperbolic properties, with implications for understanding their algebraic and geometric features.

Contribution

It generalizes key properties of RAAGs to a broader class of Artin groups, including subgroup freeness, splitting structures, and acylindrical hyperbolicity.

Findings

Derived subgroup of an Artin group is free iff the group is coherent.

Coherent Artin groups over non-complete graphs split as free amalgamated products.

Certain even Artin groups of FC-type are acylindrically hyperbolic.

Abstract

We generalize to (certain) Artin groups some results previously known for right-angled Artin groups (RAAGs). First, we generalize a result by Droms, B. Servatius, and H. Servatius, and prove that the derived subgroup of an Artin group is free if and only if the group is coherent. Second, coherent Artin groups over non complete graphs split as free amalgamated products along free abelian subgroups, and we extend to arbitrary Artin groups admitting such a splitting a recent result by Casals-Ruiz and the first author on finitely generated normal subgroups of RAAGs. Finally, we use splittings of even Artin groups of FC-type to generalize results of Minasyan and Osin on acylindrical hyperbolicity of their subgroups.