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

Abelian quandles and quandles with abelian structure group

“…Among C-groups one finds free and free abelian groups; Artin and Coxeter groups, and various generalisations thereof (virtual, welded, homotopy, etc.) [BPT23,Dar24]; cactus and twin groups, and various generalisations thereof [NNS24, Mos23, CN24]; Thompson's group F [Cho23,Szy24]; knot groups; structure groups and their finite quotients; and many more. Indeed, the commutation relation, for instance, can be interpreted as b −1 ab = a, the braid relation aba = bab as b −1 ab = c and a −1 ca = b (c being an extra generator), etc.…”

Representations of Flat Virtual Braids by Automorphisms of Free Group

“…Here, we use the term flat virtual kure braid since the term kure virtual braid group was used in [47] for kernel of the map π K : VB n → S n which is defined analogously to ν n : FVB n → S . The group FVK n = Ker(ν n ) also was denoted by FH n in [46] since it is the flat analog of the Rabenda's group H n from [48] (Prop.…”