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