On Decomposability of Virtual Artin Groups

Abstract

A group is called decomposable if it can be expressed as a direct product of two proper subgroups, and indecomposable otherwise. This paper explores the decomposability of virtual Artin groups, which were introduced by Bellingeri, Paris, and Thiel as a generalization of classical Artin groups within the framework of virtual braid theory. We establish that for any connected Coxeter graph Γ, the associated virtual Artin group VA[Γ] is indecomposable. Specifically, virtual braid groups are indecomposable. As a consequence of the indecomposability result, we deduce that studying the automorphism group of a virtual Artin group reduces to analyzing the automorphism groups of its irreducible components.

33 pages, 4 figures