On Decomposability of Virtual Artin Groups
Federica Gavazzi
TL;DR
The paper tackles the decomposability of virtual Artin groups VA[Γ] and their kernel KVA[Γ], proving indecomposability for all irreducible cases. It develops a framework based on ∞-connected Coxeter graphs, parabolic and colored Artin subgroups, and Bass–Serre theory to connect indecomposability of A[Γ] and CA[Γ] to that of VA[Γ] and KVA[Γ], including infinite-rank scenarios via direct limits. A key step is identifying KVA[Γ] with A[Ĝamma], where Ĝamma is ∞-connected, which yields KVA[Γ] indecomposable; the centralizer Z_{VA[Γ]}(KVA[Γ]) is shown to be trivial, enabling a robust indecomposability argument for VA[Γ] across center scenarios. Consequently, the Remak decomposition of VA[Γ] aligns with the decomposition of Γ into irreducible components, and automorphisms of VA[Γ] effectively reduce to automorphisms of its irreducible factors, up to finite permutations; this provides a practical route to understand Aut(VA[Γ]). In particular, the virtual braid groups VB_n are indecomposable, and the automorphism structure of VA[Γ] collapses to that of its irreducible constituents, guiding future analyses of VA[Γ] automorphisms.
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.