Related papers: Braid Groups are Linear

We consider Thompson's groups from the perspective of mapping class groups of surfaces of infinite type. This point of view leads us to the braided Thompson groups, which are extensions of Thompson's groups by infinite (spherical) braid…

We introduce a family of groups $\Gamma_n^k$ for integer parameters $n>k$. These groups originate from discussion of braid groups on $2$-surfaces. On the other hand, they turn out to be related to 3-manifolds (in particular, they lead to…

We classify various types of graded extensions of a finite braided tensor category $\cal B$ in terms of its $2$-categorical Picard groups. In particular, we prove that braided extensions of $\cal B$ by a finite group $A$ correspond to…

The disjoint union of mapping class groups of surfaces forms a braided monoidal category $\mathcal M$, as the disjoint union of the braid groups $\mathcal B$ does. We give a concrete, and geometric meaning of the braiding $\beta_{r,s}$ in…

We prove that a finite braided tensor category A is invertible in the Morita 4-category BrTens of braided tensor categories if, and only if, it is non-degenerate. This includes the case of semisimple modular tensor categories, but also…

We investigate a group $B\_\bullet$ that includes Artin's braid group $B\_\infty$ and Thompson's group $F$. The elements of $B\_\bullet$ are represented by braids diagrams in which the distances between the strands are not uniform and,…

The space of unordered configurations of distinct points in the plane is aspherical, with Artin's braid group as its fundamental group. Remarkably enough, the space of ordered configurations of distinct points on the real projective line,…

We use the notion of Bridgeland stability condition and its associated metric to endow triangulated categories with extriangulated structures and study their extriangulated Grothendieck groups. This study is motivated by Khovanov-Seidel's…

In this paper the author finds explicitly all finite-dimensional irreducible representations of a series of finite permutation groups that are homomorphic images of Artin braid group.

We study the kernels of representations of mapping class groups of surfaces on twisted homologies of configuration spaces. We relate them with the kernel of a natural twisted intersection pairing: if the latter kernel is trivial then the…

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…

A function from configuration space to moduli space of surface may induce a homomorphism between their fundamental groups which are braid groups and mapping class groups of surface, respectively. This map $\phi: B_k \rightarrow…

We give a simple topological construction of the Burau representations of the loop braid groups. There are four versions: defined either on the non-extended or extended loop braid groups, and in each case there is an unreduced and a reduced…

We show that any homologically non-trivial Dehn twist of a compact surface F with boundary is the lifting of a half-twist in the braid group B_n, with respect to a suitable branched covering p : F -> B^2. In particular, we allow the surface…

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…

Let $n\geq 1$. The pro-unipotent completion of the pure braid group of $n$ points on a genus 1 surface has been shown to be isomorphic to an explicit pro-unipotent group with graded Lie algebra using two types of tools: (a) minimal models…

We study the Gassner representation of the pure braid group $P_n$ by considering its restriction to a free subgroup $F$. The kernel of the restriction is shown to lie in the subgroup $[\Gamma^3 F,\Gamma^2 F]$, sharpening a result of…

In the paper Blocked-braid Groups, submitted to Applied Categorical Structures, the present authors together with Davide Maglia introduced the blocked-braid groups BB_n on n strands, and proved that a blocked torsion has order either 2 or…

We give a method to produce faithful representations of the groups $G(n,m)=\langle X, Y \ \vert \ X^m = Y^n \rangle$ in $\mathrm{GL}_2(\mathbb{C}[t^{\pm 1}, q^{\pm 1}])$. These groups are Garside groups and the Garside normal forms of…

We introduce framed versions of the $L$-moves and prove a one move theorem for the extension of the Markov theorem for framed braids. We further introduce framed versions of the Hilden and Pure Hilden groups, we give presentations and we…