Related papers: Unrestricted virtual braids and crystallographic b…

Path algebras are a convenient way of describing decompositions of tensor powers of an object in a tensor category. If the category is braided, one obtains representations of the braid groups $B_n$ for all $n\in \N$. We say that such…

The semigroup of convex bodies in ${\mathbb R}^n$ with Minkowski addition has a canonical embedding into an abelian group; its elements have been called virtual convex bodies. Geometric interpretations of such virtual convex bodies have…

We inspect the normal subgroup structure of the braided Thompson groups Vbr and Fbr. We prove that every proper normal subgroup of Vbr lies in the kernel of the natural quotient Vbr \onto V, and we exhibit some families of interesting such…

Given a knot K in an integral homology sphere with exterior N_K, there is a natural action of the cyclic group Z/n on the space of SL(n,C) representations of the knot group \pi_1(N_K), and this induces an action on the SL(n,C) character…

In this article we present an unpublished proof of W. Thurston that pure braid groups have the congruence subgroup property.

Following previous work on congruence subgroups and crystallographic braid groups, we study the lower central series of congruence braid groups related to the braid group $B_3$, showing in particular that corresponding quotients are almost…

The Burau representation enables to define many other representations of the braid group $B_n$ by the topological operation of ``cabling braids''. We show here that these representations split into copies of the Burau representation itself…

In this paper we study the kernel of the homomorphism $B_{g,n} \to B_n$ of the braid group $B_{g,n}$ in the handlebody $\mathcal{H}_g$ to the braid group $B_n$. We prove that this kernel is a semi-direct product of free groups. Also, we…

Some curious structural similarities between a recent braid- and Hurwitz algebraic description of the unbroken internal symmetries for a single generations of Standard Model fermions were recently identified. The non-trivial braid groups…

We give a method to produce representations of the braid group $B_n$ of $n-1$ generators ($n\leq \infty$). Moreover, we give sufficient conditions over a non unitary representation for being of this type. This method produces examples of…

We show that each of the Artin groups of type $B_n$ and $D_n$ can be presented as a semidirect product $F \rtimes {\cal B}_n$, where $F$ is a free group and ${\cal B}_n$ is the $n$-string braid group. We explain how these semidirect product…

The braid group $B_{n}$, endowed with Artin's presentation, admits two distinguished involutions. One is the anti-automorphism ${\rm{rev}}: B_{n} \to B_{n}$, $v \mapsto \bar{v}$, defined by reading braids in the reverse order (from right to…

We give a representation-theoretic bijection between rooted labeled forests with $n$ vertices and complete exceptional sequences for the quiver of type $A_n$ with straight orientation. The ascending and descending vertices in the forest…

Based on a normal form for braid group elements suggested by Dehornoy, we prove several representations of braid groups by automorphisms of a free group to be faithful. This includes a simple proof of the standard Artin's representation…

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…

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 $i$,…

In 1996 E. Formanek classified all the irreducible complex representations of $B_n$ of dimension at most $n-1,$ where $B_n$ is the Artin braid group on $n$ strings. In this paper we extend this classification to the representations of…

It is known that the pure braid groups are residually torsion-free nilpotent. This property is however widely open for the most obvious generalizations of these groups, like pure Artin groups and like fundamental groups of hyperplane…

We show that branched coverings of surfaces of large enough genus arise as characteristic maps of braided surfaces that is, lift to embeddings in the product of the surface with $\mathbb R^2$. This result is nontrivial already for…

We propose two definitions of configuration Lie groupoids and in both the cases we prove a Fadell-Neuwirth type fibration theorem for a class of Lie groupoids. We show that this is the best possible extension, in the sense that, for the…