ON THE COMPOSITION OF THE BURAU REPRESENTATION AND THE NATURAL MAP Bn → Bnk

2003 ◽  
Vol 02 (02) ◽  
pp. 169-175 ◽  
Author(s):  
MOHAMMAD N. ABDULRAHIM
Keyword(s):  
Free Group ◽  
Braid Group ◽  
Linear Representation ◽  
Burau Representation ◽  
Standard Representation ◽  
Linear Representations ◽  
Group B ◽  
Composition Factors

A lot of linear representations of the braid group, Bn, arise as a result of treating braids as automorphisms of a free group. In this paper, we consider the composition of F. R. Cohen's map Bn → Bnk and the embedding Bnk → Aut (Fnk). This gives us a linear representation of Bn whose composition factors are one copy of the Burau representation and k - 1 copies of the standard representation, a representation investigated by I. Sysoeva.

scholarly journals The Interplay between Linear Representations of the Braid Group

2007 ◽  
Vol 2007 ◽  
pp. 1-9
Author(s):  
Mohammad N. Abdulrahim ◽  
Nibal H. Kassem
Keyword(s):  
Free Group ◽  
Braid Group ◽  
Burau Representation ◽  
Standard Representation ◽  
Linear Representations ◽  
Standard Action ◽  
Composition Factors

We consider Wada's representation as a twisted version of the standard action of the braid group,Bn, on the free group withngenerators. Constructing a free group,Gnm, of ranknm, we compose Cohen's mapBn→Bnmand the embeddingBnm→Aut(Gnm)via Wada's map. We prove that the composition factors of the obtained representation are one copy of Burau representation andm−1copies of the standard representation after changing the parameterttotkin the definitions of the Burau and standard representations. This is a generalization of our previous result concerning the standard Artin representation of the braid group.

scholarly journals EXTENDING REPRESENTATIONS OF BRAID GROUPS TO THE AUTOMORPHISM GROUPS OF FREE GROUPS

2005 ◽  
Vol 14 (08) ◽  
pp. 1087-1098 ◽  
Author(s):  
VALERIJ G. BARDAKOV
Keyword(s):  
Free Group ◽  
Braid Group ◽  
Automorphism Groups ◽  
Free Groups ◽  
Linear Representation ◽  
Braid Groups ◽  
Burau Representation ◽  
Pure Braid Group

We construct a linear representation of the group IA (Fn) of IA-automorphisms of a free group Fn, an extension of the Gassner representation of the pure braid group Pn. Although the problem of faithfulness of the Gassner representation is still open for n > 3, we prove that the restriction of our representation to the group of basis conjugating automorphisms Cbn contains a non-trivial kernel even if n = 2. We construct also an extension of the Burau representation to the group of conjugating automorphisms Cn. This representation is not faithful for n ≥ 2.

scholarly journals A free subgroup in the image of the 4-strand Burau representation

2015 ◽  
Vol 24 (12) ◽  
pp. 1550065
Author(s):  
Stefan Witzel ◽  
Matthew C. B. Zaremsky
Keyword(s):  
Free Group ◽  
Braid Group ◽  
Free Subgroup ◽  
The Other ◽  
Metric Geometry ◽  
Burau Representation ◽  
Euclidean Building

It is known that the Burau representation of the 4-strand braid group is faithful if and only if certain matrices f and k generate a (non-abelian) free group. Regarding f and k as isometries of a Euclidean building, we show that f3 and k3 generate a free group. We give two proofs, one utilizing the metric geometry of the building, and the other using simplicial retractions.

On the Linearized Artin Braid Representation

1993 ◽  
Vol 02 (04) ◽  
pp. 399-412 ◽  
Author(s):  
F. Constantinescu ◽  
F. Toppan
Keyword(s):  
Free Group ◽  
Braid Group ◽  
Faithful Representation ◽  
Two Dimensional ◽  
Pure Braid Group ◽  
Linear Representations

We linearize the Artin representation of the braid group given by (right) automorphisms of a free group providing a linear faithful representation of the braid group. This result is generalized to obtain linear representations for the coloured braid groupoid and pure braid group too. Applications to some areas of two-dimensional physics are mentioned.

PURE BRAIDS AS AUTOMORPHISMS OF FREE GROUPS

2005 ◽  
Vol 04 (04) ◽  
pp. 435-440
Author(s):  
MOHAMMAD N. ABDULRAHIM
Keyword(s):  
Braid Group ◽  
Free Groups ◽  
Linear Representation ◽  
One Dimensional ◽  
Pure Braid Group ◽  
Direct Sum ◽  
Automorphisms Of Free Groups ◽  
Diagonal Representation ◽  
Composition Factors

We study the composition of F. R. Cohen's map Pn → Pnk with the Gassner representation, where Pn is the pure braid group. This gives us a linear representation of Pn whose composition factors are one copy of the Gassner representation of Pn and k - 1 copies of a diagonal representation, hence a direct sum of one-dimensional representations.

scholarly journals ON AN ACTION OF THE BRAID GROUP B2g+2 ON THE FREE GROUP F2g

2013 ◽  
Vol 23 (04) ◽  
pp. 819-831 ◽  
Author(s):  
CHRISTIAN KASSEL
Keyword(s):  
Free Group ◽  
Braid Group ◽  
Modular Group ◽  
Group B ◽  
Special Case

We construct an action of the braid group B2g+2 on the free group F2g extending an action of B4 on F2 introduced earlier by Reutenauer and the author. Our action induces a homomorphism from B2g+2 into the symplectic modular group Sp 2g(ℤ). In the special case g = 2 we show that the latter homomorphism is surjective and determine its kernel, thus obtaining a braid-like presentation of Sp4(ℤ).

Some free groups of matrices and the Burau representation of B4

1991 ◽  
Vol 110 (2) ◽  
pp. 225-228
Author(s):  
Siegfried Moran
Keyword(s):  
Free Group ◽  
Braid Group ◽  
Free Groups ◽  
Burau Representation

Using a criterion due to J. Tits [4] we shall show that it is easy to give a pair of conjugate matrices in GLn(ℂ) which freely generate a free group of rank two. The difficulty lies in producing two such matrices whose freeness does not follow directly from the known freeness of two 2 × 2 matrices. Finally we show that the well known problem as to whether the Burau representation of Artin's braid group B4 is faithful turns out to be arbitrarily close to being true in a large number of ways. Thanks are due to the referee for his helpful comments.

Multi-variable Burau Matrices and Labeled Line Configurations

1995 ◽  
Vol 04 (02) ◽  
pp. 235-262 ◽  
Author(s):  
Rudi Penne
Keyword(s):  
Braid Group ◽  
Pairwise Disjoint ◽  
Linear Representation ◽  
Alexander Polynomial ◽  
Path Model ◽  
Burau Representation ◽  
Line Configuration

We study labeled configurations of n pairwise disjoint lines in projective 3-space, up to “rigid isotopy”. To this end, we introduce the “Labeled Braid Group”, and give a linear representation for it, which can be regarded as a labeled version of the Burau representation. We give a topological path model for these multi-variable matrices, and use them to compute the Gassner matrix of a pure braid and the Alexander polynomial of the link associated with a labeled line configuration.

AUTOMORPHISMS OF BRAID GROUPS ON S2, T2, P2 AND THE KLEIN BOTTLE K

2008 ◽  
Vol 17 (01) ◽  
pp. 47-53 ◽  
Author(s):  
PING ZHANG
Keyword(s):  
Braid Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Klein Bottle ◽  
Closed Surface ◽  
Group Extension ◽  
Braid Groups ◽  
Mapping Class ◽  
Group B ◽  
Central Automorphisms

It is shown that for the braid group Bn(M) on a closed surface M of nonnegative Euler characteristic, Out (Bn(M)) is isomorphic to a group extension of the group of central automorphisms of Bn(M) by the extended mapping class group of M, with an explicit and complete description of Aut (Bn(S2)), Aut (Bn(P2)), Out (Bn(S2)) and Out (Bn(P2)).

scholarly journals The normal closure of a power of a half-twist has infinite index in the mapping class group of a punctured sphere

2019 ◽  
Vol 11 (02) ◽  
pp. 273-292
Author(s):  
Charalampos Stylianakis
Keyword(s):  
Free Group ◽  
Braid Group ◽  
Mapping Class Group ◽  
Abelian Subgroup ◽  
Infinite Order ◽  
Class Group ◽  
Normal Closure ◽  
Mapping Class ◽  
Infinite Index ◽  
Half Twist

In this paper we show that the normal closure of the [Formula: see text]th power of a half-twist has infinite index in the mapping class group of a punctured sphere if [Formula: see text] is at least five. Furthermore, in some cases we prove that the quotient of the mapping class group of the punctured sphere by the normal closure of a power of a half-twist contains a free abelian subgroup. As a corollary we prove that the quotient of the hyperelliptic mapping class group of a surface of genus at least two by the normal closure of the [Formula: see text]th power of a Dehn twist has infinite order, and for some integers [Formula: see text] the quotient contains a free group. As a second corollary we recover a result of Coxeter: the normal closure of the [Formula: see text]th power of a half-twist in the braid group of at least four strands has infinite index. Our method is to reformulate the Jones representation of the mapping class group of a punctured sphere, using the action of Hecke algebras on [Formula: see text]-graphs, as introduced by Kazhdan–Lusztig.

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