All Questions
Tagged with geometric-group-theory braid-groups
12 questions
7
votes
0
answers
120
views
Normal subgroups of pure braid groups stable under strand bifurcation
$\DeclareMathOperator\PB{PB}\DeclareMathOperator\B{B}$Let $\PB_n$ be the $n$-strand pure braid group. For each $1\le k\le n$, let $\kappa_k^n \colon \PB_n \to \PB_{n+1}$ be the monomorphism that takes ...
- 3,740
21
votes
0
answers
473
views
Are braid groups known to not be linear over $\mathbb{Z}$?
$\DeclareMathOperator\GL{GL}$It is known that every braid group $B_n$ embeds as a subgroup of $\GL_m(\mathbb{Z}[q^{\pm 1},t^{\pm 1}])$, where $m=n(n-1)/2$ (see Krammer - Braid groups are linear). This ...
- 3,740
2
votes
0
answers
139
views
Image of the pure braid group under the Artin presentation into the automorphism group of the nilpotent quotient of a free group?
As I know, it is unknown that the image of the mapping class group of the surface and its Johnson filtration under the higher Johnson homomorphisms.
There are a relationship between the mapping class ...
- 347
8
votes
1
answer
264
views
Geometric intuition behind Garside's paper?
I apologize in advance for a somewhat wishy-washy question. I just read the paper "The Braid Group and Other Groups" by F. A. Garside in which he solves the conjugacy problem for the braid ...
- 5,349
8
votes
2
answers
695
views
Braid groups and Kazhdan's property (T)
In Nica's dissertation Group actions on median spaces, we can read the following assertion:
Braid groups do not contain infinite subgroups satisfying Kazhdan's property (T).
This is used in order to ...
- 8,401
12
votes
1
answer
285
views
Artin groups of type $D_n$ as mapping class groups?
According to Allcock (Braid Pictures for Artin groups, https://arxiv.org/abs/math/9907194), the Artin group $A(D_n$) of type $D_n$ may be realized as an index 2 subgroup of the orbifold fundamental ...
- 516
10
votes
0
answers
351
views
Finite quotients of surface braid groups
Let $\Sigma_b$ be a closed orientable surface of genus $b \geq 2$, and denote by $\mathsf{P}_2(\Sigma_b)$ the pure braid group with two strands on $\Sigma_b$.
There is a braid $A_{12} \in \Sigma_b$ ...
- 66.3k
5
votes
1
answer
787
views
The action of the mapping class group of a punctured disk on the boundary at infinity of the universal cover
Let $\mathbb{D}\subset\mathbb{C}$ be the unit disk, and remove $n\geq 2$ of its points $P$. The resulting object will be called the punctured disk $\mathbb{D}_n$ in the following. I am interested in ...
- 153
7
votes
1
answer
526
views
Generators of pure braid groups of arbitrary Coxeter groups
Let $W$ be an arbitrary Coxeter group, and let $A$ be the associated Artin-Tits braid group, with standard Coxeter generators $\sigma_i\in A$. Let $P$ be the "pure braid group", the kernel of the ...
- 213
1
vote
0
answers
278
views
Homology of spherical braid groups
By the spherical braid group, I mean the fundamental group of the configuration space of distinct unordered points in $S^2$. I am wondering what is known about the group homology of the spherical ...
- 116
2
votes
0
answers
211
views
A question of braid words
Let $(W,S)$ be a Coxeter group, let $B(W,S)$ be the corresponding braid or Artin-Tits group. Set $S=\{s_1,\dots, s_n\}$ and denote by $\bf{S}=\{\sigma_1,\dots, \sigma_n\}$ the corresponding generators ...
- 534
17
votes
1
answer
832
views
Loop spaces and infinite braids
The Artin braid groups $B_n$ and the symmetric groups $S_n$ are closely related by the maps $1 \to P_n \to B_n \to S_n \to 1$. The infinite symmetric group has interesting interactions with homotopy ...
- 2,734