Virtual Artin groups

TL;DR

This paper introduces virtual Artin groups, generalizing virtual braid groups, and investigates their algebraic properties, subgroup structures, and topological aspects like the $K(\pi,1)$ conjecture, especially for spherical and affine types.

Contribution

It defines virtual Artin groups, computes presentations for key subgroups, and explores their algebraic and topological properties, including centers, word problem solutions, and cohomological dimensions.

Findings

The center of any virtual Artin group is trivial.

For spherical and affine types, certain subgroups are of the same type.

The word problem is solvable for these groups.

Abstract

Starting from the observation that the standard presentation of a virtual braid group mixes the standard presentation of the corresponding braid group with the standard presentation of the corresponding symmetric group and some mixed relations that mimic the action of the symmetric group on its root system, we define a virtual Artin group $VA[\Gamma]$ of a Coxeter graph $\Gamma$ mixing the standard presentation of the Artin group $A[\Gamma]$ with the standard presentation of the Coxeter group $W[\Gamma]$ and some mixed relations that mimic the action of $W[\Gamma]$ on its root system. By definition we have two epimorphisms $\pi_K:VA[\Gamma]\to W[\Gamma]$ and $\pi_P:VA[\Gamma]\to W[\Gamma]$ whose kernels are denoted by $KVA[\Gamma]$ and $PVA[\Gamma]$ respectively. We calculate presentations for these two subgroups. In particular $KVA[\Gamma]$ is an Artin group. We prove that the center of any virtual Artin group is trivial. In the case where $\Gamma$ is of spherical type or of affine type, we show that each free of infinity parabolic subgroup of $KVA[\Gamma]$ is also of spherical type or of affine type, and we show that $VA[\Gamma]$ has a solution to the word problem. In the case where $\Gamma$ is of spherical type we show that $KVA[\Gamma]$ satisfies the $K(\pi,1)$ conjecture and we infer the cohomological dimension of $KVA[\Gamma]$ and the virtual cohomological dimension of $VA[\Gamma]$. In the case where $\Gamma$ is of affine type we determine upper bounds for the cohomological dimension of $KVA[\Gamma]$ and for the virtual cohomological dimension of $VA[\Gamma]$.