scholarly journals GENERALIZED LONG-MOODY REPRESENTATIONS OF BRAID GROUPS

2008 ◽  
Vol 10 (supp01) ◽  
pp. 1093-1102
Author(s):  
STEPHEN BIGELOW ◽  
JIANJUN PAUL TIAN
Keyword(s):  
Braid Group ◽  
Hecke Algebra ◽  
Braid Groups ◽  
Pure Braid Group ◽  
Special Case

Long and Moody give a method of constructing representations of the braid groups Bn. We discuss some ways to generalize their construction. One of these gives representations of subgroups of Bn, including the Gassner representation of the pure braid group as a special case. Another gives representations of the Hecke algebra.

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 Minimal Asymptotic Translation Lengths of Torelli Groups and Pure Braid Groups on the Curve Graph

2018 ◽  
Vol 2020 (24) ◽  
pp. 9974-9987
Author(s):  
Hyungryul Baik ◽  
Hyunshik Shin
Keyword(s):  
Braid Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Braid Groups ◽  
Mapping Class ◽  
Torelli Group ◽  
Pure Braid Group ◽  
Curve Graph ◽  
Translation Length

Abstract In this paper, we show that the minimal asymptotic translation length of the Torelli group ${\mathcal{I}}_g$ of the surface $S_g$ of genus $g$ on the curve graph asymptotically behaves like $1/g$, contrary to the mapping class group ${\textrm{Mod}}(S_g)$, which behaves like $1/g^2$. We also show that the minimal asymptotic translation length of the pure braid group ${\textrm{PB}}_n$ on the curve graph asymptotically behaves like $1/n$, contrary to the braid group ${\textrm{B}}_n$, which behaves like $1/n^2$.

scholarly journals A BRAIDED SIMPLICIAL GROUP

2002 ◽  
Vol 84 (3) ◽  
pp. 645-662 ◽  
Author(s):  
JIE WU
Keyword(s):  
Group Action ◽  
Braid Group ◽  
Homotopy Group ◽  
Braid Groups ◽  
Subject Classification ◽  
Homotopy Groups ◽  
Simplicial Group ◽  
Pure Braid Group ◽  
Fixed Set ◽  
Braid Group Action

By studying the braid group action on Milnor's construction of the 1-sphere, we show that the general higher homotopy group of the 3-sphere is the fixed set of the pure braid group action on certain combinatorially described groups. This establishes a relation between the braid groups and the homotopy groups of the sphere.2000Mathematical Subject Classification: 20F36, 55P35, 55Q05, 55Q40, 55U10.

scholarly journals Classifying Spaces for the Family of Virtually Cyclic Subgroups of Braid Groups

2018 ◽  
Vol 2020 (5) ◽  
pp. 1575-1600
Author(s):  
Ramón Flores ◽  
Juan González-Meneses
Keyword(s):  
Braid Group ◽  
Braid Groups ◽  
Classifying Spaces ◽  
Classifying Space ◽  
Cyclic Subgroups ◽  
Cyclic Groups ◽  
Pure Braid Group ◽  
Minimal Dimension ◽  
The Family

Abstract We prove that, for n ≥ 3, the minimal dimension of a model of the classifying space of the braid group $B_{n}$, and of the pure braid group $P_{n}$, with respect to the family of virtually cyclic groups is n.

Presentation of pure braid groups

2001 ◽  
Vol 10 (04) ◽  
pp. 609-623 ◽  
Author(s):  
F. DIGNE ◽  
V. GOMI
Keyword(s):  
Coxeter Group ◽  
Braid Group ◽  
Type A ◽  
Braid Groups ◽  
Direct Products ◽  
Pure Braid Group ◽  
Combinatorial Methods

In this article we study by combinatorial methods presentations of the pure braid group associated with any Coxeter group. We generalize to all types the decomposition of the pure braid group into successive semi-direct products known in the case of type An.

scholarly journals THE CO-HOPFIAN PROPERTY OF SURFACE BRAID GROUPS

2013 ◽  
Vol 22 (10) ◽  
pp. 1350055 ◽  
Author(s):  
YOSHIKATA KIDA ◽  
SAEKO YAMAGATA
Keyword(s):  
Finite Index ◽  
Braid Group ◽  
Orientable Surface ◽  
Braid Groups ◽  
Index Subgroup ◽  
Closed Orientable Surface ◽  
Pure Braid Group ◽  
Injective Homomorphism ◽  
Finite Index Subgroup

Let g and n be integers at least 2, and let G be the pure braid group with n strands on a closed orientable surface of genus g. We describe any injective homomorphism from a finite index subgroup of G into G. As a consequence, we show that any finite index subgroup of G is co-Hopfian.

On braids and groups Gnk

2015 ◽  
Vol 24 (13) ◽  
pp. 1541009 ◽  
Author(s):  
Vassily Olegovich Manturov ◽  
Igor Mikhailovich Nikonov
Keyword(s):  
Dynamical Systems ◽  
Braid Group ◽  
Knot Theory ◽  
Free Groups ◽  
Braid Groups ◽  
The Other ◽  
Pure Braid Group ◽  
And Topology ◽  
Other Hand ◽  
Definition Of

In [Non-reidemeister knot theory and its applications in dynamical systems, geometry, and topology, preprint (2015), arXiv:1501.05208.] the first author gave the definition of [Formula: see text]-free braid groups [Formula: see text]. Here we establish connections between free braid groups, classical braid groups and free groups: we describe explicitly the homomorphism from (pure) braid group to [Formula: see text]-free braid groups for important cases [Formula: see text]. On the other hand, we construct a homomorphism from (a subgroup of) free braid groups to free groups. The relations established would allow one to construct new invariants of braids and to define new powerful and easily calculated complexities for classical braid groups.

scholarly journals A geometric interpretation of the normal closure of the braid group Bn in the braid group of the torus Bn(T)

2019 ◽  
Vol 28 (05) ◽  
pp. 1950038
Author(s):  
Liming Pang
Keyword(s):  
Braid Group ◽  
Commutator Subgroup ◽  
Geometric Interpretation ◽  
Braid Groups ◽  
Normal Closure ◽  
Geometric Description ◽  
Pure Braid Group

It had been proved by Birman and Goldberg that the normal closure of the pure braid group [Formula: see text] in the pure braid group of the torus [Formula: see text] is the commutator subgroup [Formula: see text]. In this paper, we are going to study the case of full braid groups: i.e. the normal closure of [Formula: see text] in [Formula: see text], which turns out to have an interesting geometric description.

Sign in / Sign up

Export Citation Format

Share Document