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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…

Let VB$_n$ be the virtual braid group on $n$ strands and let $\mathfrak{S}_n$ be the symmetric group on $n$ letters. Let $n,m \in \mathbb{N}$ such that $n \ge 5$, $m \ge 2$ and $n \ge m$. We determine all possible homomorphisms from VB$_n$…

Recently the first named author defined a 2-parametric family of groups $G_n^k$. Those groups may be regarded as analogues of braid groups. Study of the connection between the groups $G_n^k$ and dynamical systems led to the discovery of the…

We solve the isomorphism problem for braid groups on trees with $n = 4$ or 5 strands. We do so in three main steps, each of which is interesting in its own right. First, we establish some tools and terminology for dealing with computations…

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.…

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…

The singularities of the representation variety of $B_3$ are studied, where $B_3$ is the knot group on 3 strands. Specifically, we determine which semisimple representations are smooth points of this variety.

Let $M$ be a compact surface without boundary, and $n\geq 2$. We analyse the quotient group $B_n(M)/\Gamma_2(P_n(M))$ of the surface braid group $B_{n}(M)$ by the commutator subgroup $\Gamma_2(P_n(M))$ of the pure braid group $P_{n}(M)$. If…

The aim of the present note is to enhance groups $G_{n}^{3}$ and to construct new invariants of classical braids. In particular, we construct invariants valued in $G_{N}^{2}$ groups. In groups $G_{n}^{2}$, the identity problem is solved,…

We study the homology of $[P_n,P_n]$, the commutator subgroup of the pure braid group on $n$ strands, and show that $H_l([P_n,P_n])$ contains a free abelian group of infinite rank for all $1\leq l\leq n-2$. As a consequence we determine the…

We study the large-scale geometry of graph braid groups $\mathbb{B}_n(\mathsf{\Gamma})$, viewed as the fundamental groups of discrete configuration spaces $UD_n(\mathsf{\Gamma})$, which are special cube complexes in the sense of…

This article gives certain fibre bundles associated to the braid groups which are obtained from a translation as well as conjugation on the complex plane. The local coefficient systems on the level of homology for these bundles are given in…

This paper upbuilds the theoretical framework of orbit braids in $M\times I$ by making use of the orbit configuration space $F_G(M,n)$, which enriches the theory of ordinary braids, where $M$ is a connected topological manifold of dimension…

We ask if any finite type generalized braid group is a subgroup of some classical Artin braid group. We define a natural map from a given finite type generalized braid group to a classical braid group and ask if this map is an injective…

Homology of braid groups and Artin groups can be related to the study of spaces of curves. We completely calculate the integral homology of the family of smooth curves of genus $g$ with one boundary component, that are double coverings of…

We consider a certain modification of the group $G^3_n$ which describes dynamics of point configurations, in particular braids, and define a representation of the modified $G^3_n$. The braid representation induced is powerful enough to…

We prove that generic elements of braid groups are pseudo-Anosov, in the following sense: in the Cayley graph of the braid group with n $\ge$ 3 strands, with respect to Garside's generating set, we prove that the proportion of pseudo-Anosov…

Let B_n be the braid group on n > 3 strands. We prove that B_n modulo its center is co-Hopfian. We then show that any injective endomorphism of B_n is geometric in the sense that it is induced by a homeomorphism of a punctured disk. We…

We define a family of representations $\{\rho_n\}_{n\geq 0}$ of a pure braid group $P_{2k}$. These representations are obtained from an action of $P_{2k}$ on a certain type of $A_2$ web space with color $n$. The $A_2$ web space is a…

For a tree $G$, we study the changing behaviors in the homology groups $H_i(B_nG)$ as $n$ varies, where $B_nG := \pi_1($UConf$_n(G))$. We prove that the ranks of these homologies can be described by a single polynomial for all $n$, and…