arXiv subjects

Federica Gavazzi

Publications and source records attributed to Federica Gavazzi.

5 recordsLinked to original sources

arXiv2026

A Deligne complex for virtual Artin groups

Virtual Artin groups were recently introduced by Bellingeri-Paris-Thiel as a generalisation of virtual braid groups. In this article, we initiate a geometric study of these groups by constructing an analogue of the Deligne complex for virtual Artin groups, and we prove that it is CAT(0) for all locally reducible defining graphs (a class that contains in particular two-dimensional graphs and graphs without any label $3$, and which is generic in the sense of Goldsborough-Vaskou). As applications, we classify finite subgroups of locally reducible virtual Artin groups, showing that such groups are conjugated into an isomorphic copy of the corresponding Coxeter subgroup. We also prove an analogue of the $K(π, 1)$-conjecture for locally reducible virtual Artin groups: we show that these groups are virtually torsion-free and admit a cocompact model of classifying space for proper actions of minimal dimension, equal to the virtual cohomological dimension of the group.

math.GR↗

arXiv2026

Spaces Related to Virtual Artin Groups

This work explores the topological properties of virtual Artin groups, a recent extension of the ``virtual" concept - initially developed for braids - to all Artin groups, as introduced by Bellingeri, Paris, and Thiel. For any given Coxeter graph $Γ$, we define a CW-complex $Ω(Γ)$ whose fundamental group is isomorphic to the pure virtual Artin group $\mathrm{PVA}[Γ]$, which coincides with the pure virtual braid group when $Γ$ is $A_{n-1}$. This construction generalizes the previously studied BEER complex, originally defined for pure virtual braids, to all Coxeter graphs. We investigate the asphericity of $Ω(Γ)$ and demonstrate that it holds when $Γ$ is of spherical type or of affine type, thereby characterizing $Ω(Γ)$ as a classifying space for $\mathrm{PVA}[Γ]$. To achieve this, we establish a connection between $Ω(Γ)$ and the Salvetti complex associated with a specific Coxeter graph $\widehatΓ$ related to $Γ$, showing that they share a common covering space. This finding links the asphericity of $Ω(Γ)$ to the $K(π, 1)$-conjecture for Artin groups associated with $\widehatΓ$. Additionally, the paper introduces and studies almost parabolic (AP) reflection subgroups, which play a crucial role in constructing these complexes.

math.GR↗

arXiv2026

On Decomposability of Virtual Artin Groups

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.

math.GR↗

arXiv2026

Parabolic subgroups and word problem in virtual Artin groups

We begin by establishing two fundamental results on standard parabolic subgroups of virtual Artin groups. We first show that a standard parabolic subgroup is naturally isomorphic to a virtual Artin group. Second, we prove that the intersection of two standard parabolic subgroups is a standard parabolic subgroup. Our main result is that, if all free of infinity standard parabolic subgroups of a given virtual Artin group VA[Γ] have a solvable word problem, then VA[Γ] itself has a solvable word problem. It follows that virtual Artin groups of FC type and, more generally, of affine-FC type, have a solvable word problem. We also prove that, if a virtual Artin group VA[Γ] has a solvable word problem, then the strong membership problem for any standard parabolic subgroup in VA[Γ] is solvable.

math.GR↗

arXiv2024

Intersection of Parabolic Subgroups in Euclidean Braid Groups: a short proof

We give a short proof for the fact, already proven by Thomas Haettel, that the arbitrary intersection of parabolic subgroups in Euclidean Braid groups $A[\tilde{A}_n]$ is again a parabolic subgroup. To that end, we use that the spherical-type Artin group $A[B_{n+1}]$ is isomorphic to $A[\tilde{A}_n] \rtimes \mathbb{Z}$.

math.GR↗