Linear representations, crystallographic quotients, and twisted conjugacy of virtual Artin groups

TL;DR

This paper studies virtual Artin groups by constructing linear representations, analyzing crystallographic quotients, classifying torsion and conjugacy, and exploring twisted conjugacy, revealing structural properties and the $R_ infty$-property in certain cases.

Contribution

It introduces a linear representation for virtual Artin groups and characterizes their crystallographic quotients, torsion elements, and conjugacy relations, advancing understanding of their algebraic structure.

Findings

Constructed a linear representation of $VA[\Gamma]$.

Identified when $VA[\Gamma]/PVA[\Gamma]'$ is crystallographic.

Proved that right-angled virtual Artin groups have the $R_ infty$-property.

Abstract

Virtual Artin groups were recently introduced by Bellingeri, Paris, and Thiel as broad generalizations of the well-known virtual braid groups. For each Coxeter graph $\Gamma$, they defined the virtual Artin group $VA[\Gamma]$, which is generated by the corresponding Artin group $A[\Gamma]$ and the Coxeter group $W[\Gamma]$, subject to certain mixed relations inspired by the action of $W[\Gamma]$ on its root system $\Phi[\Gamma]$. There is a natural surjection $ \mathrm{VA}[\Gamma] \rightarrow W[\Gamma]$, with the kernel $PVA[\Gamma]$ representing the pure virtual Artin group. In this paper, we explore linear representations, crystallographic quotients, and twisted conjugacy of virtual Artin groups. Inspired from the work of Cohen, Wales, and Krammer, we construct a linear representation of the virtual Artin group $VA[\Gamma]$. As a consequence of this representation, we deduce that if $W[\Gamma]$ is a spherical Coxeter group, then $VA[\Gamma]/PVA[\Gamma]'$ is a crystallographic group of dimension $ |\Phi[\Gamma]|$ with the holonomy group $W[\Gamma]$. We also classify the torsion elements in $VA[\Gamma]/PVA[\Gamma]'$ and determine precisely when two elements are conjugate in this group. Further, we investigate twisted conjugacy, and prove that each right-angled virtual Artin group admit the $R_\infty$-property.