Related papers: Commutator subgroups and crystallographic quotient…

Let $n\ge 2$. Let $VB_n$ (resp. $VP_n$) be the virtual braid group (resp. the pure virtual braid group), and let $VT_n$ (resp. $PVT_n$) be the virtual twin group (resp. the pure virtual twin group). Let $\Pi$ be one of the following…

Let $VB_n$, resp. $WB_n$ denote the virtual, resp. welded, braid group on $n$ strands. We study their commutator subgroups $VB_n' = [VB_n, VB_n]$ and, $WB_n' = [WB_n, WB_n]$ respectively. We obtain a set of generators and defining relations…

Study of stable isotopy classes of a finite collection of immersed circles without triple or higher intersections on closed oriented surfaces is considered as a planar analogue of virtual knot theory, a far reaching generalisation of…

Twin groups and virtual twin groups are planar analogues of braid groups and virtual braid groups, respectively. These groups play the role of braid groups in the Alexander-Markov correspondence for the theory of stable isotopy classes of…

In the present paper we study structural aspects of certain quotients of braid groups and virtual braid groups. In particular, we construct and study linear representations $B_n\to {\rm GL}_{n(n-1)/2}\left(\mathbb{Z}[t^{\pm1}]\right)$,…

The virtual singular braid group arises as a natural common generalization of classical singular braid groups and virtual braid groups. In this paper, we study several algebraic properties of the virtual singular braid group $VSG_n$. We…

We introduce the universal virtual braid group $UV_n(c)$, which provides a unified algebraic framework for virtual braid--type structures with $c$ types of crossings and admits natural quotient maps onto the standard families in the…

Small Coxeter groups are precisely the ones for which the Tits representation is integral, which makes the study of their congruence subgroups relevant. The symmetric group $S_n$ has three natural extensions, namely, the braid group $B_n$,…

We show that the virtual singular braid monoid on $n$ strands embeds in a group $VSG_n$, which we call the virtual singular braid group on $n$ strands. The group $VSG_n$ contains a normal subgroup $VSPG_n$ of virtual singular pure braids.…

In this paper we study some subgroups and their decompositions in semi-direct product of the twisted virtual braid group $TVB_n$. In particular, the twisted virtual pure braid group $TVP_n$ is the kernel of an epimorphism of $TVB_n$ onto…

We show that the crystallographic braid group $B_n/[P_n,P_n]$ embeds naturally in the group of unrestricted virtual braids $UVB_n$, we give new proofs of known results about the torsion elements of $B_n/[P_n,P_n]$, and we characterise the…

We study the algebraic structures of the virtual singular braid monoid, $VSB_n$, and the virtual singular pure braid monoid, $VSP_n$. The monoid $VSB_n$ is the splittable extension of $VSP_n$ by the symmetric group $S_n$. We also construct…

Motivated by the notion of the multi-virtual braid group introduced by L. Kauffman and by the study of extensions of the well-known twin group T_n, n >= 2, we introduce a new group called the multi-virtual twin group M_kVT_n, where k >= 1…

Let $n\geq 2$. Let $VB_n$ (resp. $VP_n$) denote the virtual braid group (resp. virtual pure braid group), let $WB_n$ (resp. $WP_n$) denote the welded braid group (resp. welded pure braid group) and let $UVB_n$ (resp. $UVP_n$) denote the…

Let $UVB_n$ and $UVP_n$ be the unrestricted virtual braid group and the unrestricted virtual pure braid group on n strands respectively. We study the groups $UVB_n$ and $UVP_n$, and our main results are as follows: for $n\geq 5$, we give a…

In this paper we consider the cohomology of four groups related to the virtual braids of [Kauffman] and [Goussarov-Polyak-Viro], namely the pure and non-pure virtual braid groups (PvB_n and vB_n, respectively), and the pure and non-pure…

In this article, we introduce the algebraic definitions and presentations of the virtual singular twin monoid and virtual singular twin group, denoted by $VSTM_n$ and $VST_n$, respectively, for a positive integer $n$. These structures…

In this paper, we examine specific submonoids within the singular twisted virtual braid monoid $STVB_n$. Notably, we establish that the singular twisted virtual pure braid monoid $STVP_n$ serves as the kernel of an epimorphism from $STVB_n$…

By exploring simplicial structure of pure virtual braid groups, we give new connections between the homotopy groups of the 3-sphere and the virtual braid groups that are related to the theory of Brunnian virtual braids. The group structure…

We classify the (finite and infinite) virtually cyclic subgroups of the pure braid groups $P_{n}(RP^2)$ of the projective plane. The maximal finite subgroups of $P_{n}(RP^2)$ are isomorphic to the quaternion group of order 8 if $n=3$, and…