On the quotient of the braid group by commutators of transversal half-twists and its group actions

We consider braids on $m+n$ strands, such that the first $m$ strands are trivially fixed. We denote the set of all such braids by $B_{m,n}$. Via concatenation $B_{m,n}$ acquires a group structure. The objective of this paper is to find a presentation for $B_{m,n}$ using the structure of its corresponding pure braid subgroup, $P_{m,n}$, and the fact that it is a subgroup of the classical Artin group $B_{m+n}$. Then we give an irredundant presentation for $B_{m,n}$. The paper...

In this paper we compute the automorphism groups $\operatorname{Aut}(\mathbf{P}_n(\Sigma))$ and $\operatorname{Aut}(\mathbf{B}_n(\Sigma))$ of braid groups $\mathbf{P}_n(\Sigma)$ and $\mathbf{B}_n(\Sigma)$ on every orientable surface $\Sigma$, which are isomorphic to group extensions of the extended mapping class group $\mathcal{M}^*_n(\Sigma)$ by the transvection subgroup except for a few cases. We also prove that $\mathbf{P}_n(\Sigma)$ is always a characteristic subgroup o...

This work presents an approach towards the representation theory of the braid groups $B_n$. We focus on finite-dimensional representations over the field of Laurent series which can be obtained from representations of infinitesimal braids, with the help of Drinfeld associators. We set a dictionary between representation-theoretic properties of these two structures, and tools to describe the representations thus obtained. We give an explanation for the frequent apparition of u...

This article surveys many standard results about the braid group with emphasis on simplifying the usual algebraic proofs. We use van der Waerden's trick to illuminate the Artin-Magnus proof of the classic presentation of the algebraic mapping-class group of a punctured disc. We give a simple, new proof of the Dehornoy-Larue braid-group trichotomy, and, hence, recover the Dehornoy right-ordering of the braid group. We then turn to the Birman-Hilden theorem concerning bra...

In this paper we introduce distinct approaches to loop braid groups, a generalisation of braid groups, and unify all the definitions that have appeared so far in literature, with a complete proof of the equivalence of these definitions. These groups have in fact been an object of interest in different domains of mathematics and mathematical physics, and have been called, in addition to loop braid groups, with several names such as of motion groups, groups of permutation-conju...

We describe an algebraic proof of the well-known topological fact that $\pi_1(SO(n)) \cong Z/2Z$. The fundamental group of $SO(n)$ appears in our approach as the center of a certain finite group defined by generators and relations. The latter is a factor group of the braid group $B_n$, obtained by imposing one additional relation and turns out to be a nontrivial central extension by $Z/2Z$ of the corresponding group of rotational symmetries of the hyperoctahedron in dimension...

The word problem of a group is a very important question. The word problem in the braid group is of particular interest for topologists, algebraists and geometers. In previouse article we have looked at the braid group from a topological point of view, and thus using a new computerized representation of some elements of the fundamental group we gave a solution for its word problem. In this paper we will give an algorithm that will make it possible to transform the new present...

In this paper we define a new family of groups which generalize the {\it classical braid groups on} $\C $. We denote this family by $\{B_n^m\}_{n \ge m+1}$ where $n,m \in \N$. The family $\{ B_n^1 \}_{n \in \N}$ is the set of classical braid groups on $n$ strings. The group $B_n^m$ is the set of motions of $n$ unordered points in $\C^m$, so that at any time during the motion, each $m+1$ of the points span the whole of $\C^m$ as an affine space. There is a map from $B_n^m$ to ...

Let $n, k \geq 3$. In this paper, we analyse the quotient group $B\_n/\Gamma\_k(P\_n)$ of the Artin braid group $B\_n$ by the subgroup $\Gamma\_k(P\_n)$ belonging to the lower central series of the Artin pure braid group $P\_n$. We prove that it is an almost-crystallographic group. We then focus more specifically on the case $k=3$. If $n \geq 5$, and if $\tau \in N$ is such that $gcd(\tau, 6) = 1$, we show that $B\_n/\Gamma\_3 (P\_n)$ possesses torsion $\tau$ if and only if $...

Since the braid group was discovered by E. Artin, the question of its conjugacy problem has been solved by Garside and Birman, Ko and Lee. However, the solutions given thus far are difficult to compute with a computer, since the number of operations needed is extremely large. Meanwhile, random algorithms used to solve difficult problems such as primality of a number were developed, and the random practical methods have become an important tool. We give a random algorithm, alo...