scholarly journals The concept of free group based on braid group

2018 ◽  
Vol 1116 ◽  
pp. 022034
Author(s):  
M K M Nasution ◽  
E Herawati ◽  
E Rosmaini
Keyword(s):  
Free Group ◽  
Braid Group

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.

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 Planar pure braids on six strands

2020 ◽  
Vol 29 (01) ◽  
pp. 1950097
Author(s):  
Jacob Mostovoy ◽  
Christopher Roque-Márquez
Keyword(s):  
Abelian Group ◽  
Fundamental Group ◽  
Free Product ◽  
Free Group ◽  
Configuration Space ◽  
Braid Group ◽  
Free Abelian Group ◽  
Braid Groups ◽  
Pure Braid Group

The group of planar (or flat) pure braids on [Formula: see text] strands, also known as the pure twin group, is the fundamental group of the configuration space [Formula: see text] of [Formula: see text] labeled points in [Formula: see text] no three of which coincide. The planar pure braid groups on 3, 4 and 5 strands are free. In this note, we describe the planar pure braid group on 6 strands: it is a free product of the free group on 71 generators and 20 copies of the free abelian group of rank two.

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 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.

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.

scholarly journals On the Andreadakis Problem for Subgroups of $IA_n$

2019 ◽  
Author(s):  
Jacques Darné
Keyword(s):  
Free Group ◽  
Braid Group ◽  
Lower Central Series ◽  
Central Series ◽  
Pure Braid Group ◽  
The Difference

Abstract Let $F_n$ be the free group on $n$ generators. Consider the group $IA_n$ of automorphisms of $F_n$ acting trivially on its abelianization. There are two canonical filtrations on $IA_n$: the 1st one is its lower central series $\Gamma _*$; the 2nd one is the Andreadakis filtration $\mathcal A_*$, defined from the action on $F_n$. The Andreadakis problem consists in understanding the difference between these filtrations. Here, we show that they coincide when restricted to the subgroup of triangular automorphisms and to the pure braid group.

SET-THEORETIC YANG–BAXTER SOLUTIONS VIA FOX CALCULUS

2006 ◽  
Vol 15 (08) ◽  
pp. 949-956 ◽  
Author(s):  
J. SCOTT CARTER ◽  
MASAHICO SAITO
Keyword(s):  
Free Group ◽  
Braid Group ◽  
Baxter Equation ◽  
Group Representations ◽  
Group Automorphisms ◽  
Free Group Automorphisms

We construct solutions to the set–theoretic Yang–Baxter equation using braid group representations in free group automorphisms and their Fox differentials. The method resembles the extensions of groups and quandles.

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).

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