Virtual Artin groups
Paolo Bellingeri, Luis Paris, Anne-Laure Thiel

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 of a Coxeter graph mixing the standard presentation of the Artin group with the standard presentation of the Coxeter group and some mixed relations that mimic the action of on its root system. By definition we have two epimorphisms and whose kernels are denoted by and respectively. We calculate presentations for these two subgroups. In particular is an Artin group. We prove that the center…
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