Related papers: Commutator Subgroups of Welded Braid Groups

The group described in this paper appeared while studying fundamental groups of complements of branch curves. It turned out that a certain quotient of the braid group acts on those fundamental groups and studying this action is essential…

E. Artin described all irreducible representations of the braid group B_k to the symmetric group S(k). We strengthen some of his results and, moreover, exhibit a complete picture of homomorphisms of B_k to S(n) for n<2k+1. We show that the…

We exhibit a new presentation of the (equilateral) Von Dyck groups $D(2,3,n), \ n\ge 3$, in terms of two generators of order $n$ satisfying three relations, one of which is Artin's braid relation. By dropping the relation which fixes the…

Each element of the commutator subgroup of a group can be represented as a product of commutators. The minimal number of factors in such a product is called the commutator length of the element. The commutator length of a group is defined…

Let $G$ be a profinite group. We prove that the commutator subgroup $G'$ is finite-by-procyclic if and only if the set of all commutators of $G$ is contained in a union of countably many procyclic subgroups.

Let n\geq 3. We classify the finite groups which are realised as subgroups of the sphere braid group B_n(S^2). Such groups must be of cohomological period 2 or 4. Depending on the value of n, we show that the following are the maximal…

We give a complete classification of homomorphisms from the braid group on $n$ strands to the braid group on $2n$ strands when $n$ is at least 5. We also classify endomorphisms of the braid group on 4 strands, as well as homomorphisms from…

Given a group G and positive integers k,n, we let B_n=B_n(G) denote the set of all elements x in G such that |x^G|\leq n, and we say that G satisfies the (k,n)-covering condition for commutators if there is a subset S in G such that |S|\leq…

Hopf braces have been introduced as a Hopf-theoretic generalization of skew braces. Under the assumption of cocommutativity, these algebraic structures are equivalent to matched pairs of actions on Hopf algebras, that can be used to produce…

Let $SB_n$ be the singular braid group generated by braid generators $\sigma_i$ and singular braid generators $\tau_i$, $1 \leq i \leq n-1$. Let $ST_n$ denote the group that is the kernel of the homomorphism that maps, for each $i$,…

For every positive integer $n$, we construct, using algebraic groups, an infinite family of irreducible algebraic varieties $X$,whose automorphism group ${\rm Aut}(X)$ contains the automorphism group ${\rm Aut}(F_n)$ of a free group $F_n$…

We construct a braid group action on quantum covering groups. We further use this action to construct a PBW basis for the positive half in finite type which is pairwise-orthogonal under the inner product. This braid group action is induced…

The virtual braid group $VB_n$, the virtual twin group $VT_n$ and the virtual triplet group $VL_n$ are extensions of the symmetric group $S_n$, which are motivated by the Alexander-Markov correspondence for virtual knot theories. The…

Let $w$ be a multilinear commutator word. In the present paper we describe recent results that show that if $G$ is a profinite group in which all $w$-values are contained in a union of finitely (or in some cases countably) many subgroups…

In this paper we give a description of the generators of the prime level congruence subgroups of braid groups. Also, we give a new presentation of the symplectic group over a finite field, and we calculate symmetric quotients of the prime…

Let $w$ be a multilinear commutator word, that is, a commutator of weight $n$ in $n$ different group variables. It is proved that if $G$ is a profinite group in which all pronilpotent subgroups generated by $w$-values are periodic, then the…

Let $G$ be a finite $p$-group whose derived subgroup $G'$ can be generated by $2$ elements. If $G'$ is abelian, Guralnick proved that every element of $G'$ is a commutator. In this paper, we prove that the condition that $G'$ should be…

We introduce a notion of "freely braided element" for simply laced Coxeter groups. We show that an arbitrary group element $w$ has at most $2^{N(w)}$ commutation classes of reduced expressions, where $N(w)$ is a certain statistic defined in…

In the present paper the representation of the virtual braid group $VB_n$ into the automorphism group of free product of the free group and free abelian group is constructed. This representation generalizes the previously constructed ones.…

Let $m,n$ be positive integers and $w$ a multilinear commutator word. Assume that $G$ is a finite group having subgroups $G_1,\ldots,G_m$ whose union contains all $w$-values in $G$. Assume further that all elements of the subgroups…