The pure braid groups and their relatives (original) (raw)

The Chen groups of the pure braid group

Contemporary Mathematics, 1995

The Chen groups of a group are the lower central series quotients of its maximal metabelian quotient. We show that the Chen groups of the pure braid group P_n are free abelian, and we compute their ranks. The computation of these Chen groups reduces to the computation of the Hilbert series of a certain graded module over a polynomial ring, and the latter is carried out by means of a Groebner basis algorithm. This result shows that, for n >= 4, the group P_n is not a direct product of free groups.

On the Linear Representation of Braid Groups

Transactions of the American Mathematical Society, 1989

We give a new derivative of the Burau and Gassner representations of the braid and pure braid groups. Various applications are explored.

The BNS-invariant for the pure braid groups

Groups, Geometry, and Dynamics, 2015

In Bieri, Neumann and Strebel introduced a geometric invariant for discrete groups. In this article we compute and explicitly describe the BNS-invariant for the pure braid groups.

Wada's Representations of the Pure Braid Group of High Degree

2012

We consider the representation obtained by composing the embedding map of the pure braid group P n → P n+k and Wada's representation of degree n + k to get a linear representation P n → GL n+k (C[t ±1 1 ,. .. , t ±1 n+k ]), whose composition factors are to be determined. A similar work was done in a previous work in the case of the Gassner representation of P n .

On some groups related to the Braid Groups of type A

Annals of the University of Craiova - Mathematics and Computer Science Series, 2010

We prove that a family of groups R(n) forms the algebraic structure of an operad and that they admit a presentation similar to that of the Braid groups of type A. This result provides a new proof that the Braid Groups form an operad, a topic emphasized in ~\cite{16}~\cite{ulrike}. These groups proved to be useful in several problems which belong to different areas of Mathematics. Representations of R(n) came from a system of mixed Yang-Baxter type equations. We define the Hopf equation in braided monoidal categories and we prove that representations for our groups came from any braided Hopf algebra with invertible antipode. Using this result, we prove that there is a morphism from R(n) to the mapping class group Gamman,1\Gamma_{n,1}Gamman,1, using some results from 3-dimensional topology.

Virtual and universal braid groups, their quotients and representations

Journal of Group Theory, 2022

In the present paper, we study structural aspects of certain quotients of braid groups and virtual braid groups. In particular, we construct and study linear representations B n → GL n ⁢ ( n - 1 ) / 2 ⁢ ( Z ⁢ [ t ± 1 ] ) B_{n}\to\mathrm{GL}_{n(n-1)/2}(\mathbb{Z}[t^{\pm 1}]) , VB n → GL n ⁢ ( n - 1 ) / 2 ⁢ ( Z ⁢ [ t ± 1 , t 1 ± 1 , t 2 ± 1 , … , t n - 1 ± 1 ] ) \mathrm{VB}_{n}\to\mathrm{GL}_{n(n-1)/2}(\mathbb{Z}[t^{\pm 1},t_{1}^{\pm 1},t_{2}^{\pm 1},\ldots,t_{n-1}^{\pm 1}]) which are connected with the famous Lawrence–Bigelow–Krammer representation. It turns out that these representations induce faithful representations of the crystallographic groups B n / P n ′ B_{n}/P_{n}^{\prime} , VB n / VP n ′ \mathrm{VB}_{n}/\mathrm{VP}_{n}^{\prime} , respectively. Using these representations we study certain properties of the groups B n / P n ′ B_{n}/P_{n}^{\prime} , VB n / VP n ′ \mathrm{VB}_{n}/\mathrm{VP}_{n}^{\prime} . Moreover, we construct new representations and decompositions of the un...

Abstract commensurators of braid groups

Journal of Algebra, 2006

Let B n be the braid group on n ≥ 4 strands. We show that the abstract commensurator of B n is isomorphic to Mod(S) ⋉ (Q × ⋉ Q ∞ ), where Mod(S) is the extended mapping class group of the sphere with n + 1 punctures.