On virtual cabling and a structure of 4-strand virtual pure braid group

This paper is dedicated to cabling on virtual braids. This construction gives a new generating set for the virtual pure braid group [Formula: see text]. We define simplicial group [Formula: see text] and its simplicial subgroup [Formula: see text] which is generated by [Formula: see text]. Consequently, we describe [Formula: see text] as HNN-extension and find presentation of [Formula: see text] and [Formula: see text]. As an application to classical braids, we find a new presentation of the Artin pure braid group [Formula: see text] in terms of the cabled generators.

On 3-strand singular pure braid group

In this paper, we study the singular pure braid group [Formula: see text] for [Formula: see text]. We find generators, defining relations and the algebraical structure of these groups. In particular, we prove that [Formula: see text] is a semi-direct product [Formula: see text], where [Formula: see text] is an HNN-extension with base group [Formula: see text] and cyclic associated subgroups. We prove that the center [Formula: see text] of [Formula: see text] is a direct factor in [Formula: see text].

Non-realizability of the pure braid group as area-preserving homeomorphisms

Let $\operatorname{Homeo}_{+}(D_{n}^{2})$ be the group of orientation-preserving homeomorphisms of $D^{2}$ fixing the boundary pointwise and $n$ marked points as a set. The Nielsen realization problem for the braid group asks whether the natural projection $p_{n}:\operatorname{Homeo}_{+}(D_{n}^{2})\rightarrow B_{n}:=\unicode[STIX]{x1D70B}_{0}(\operatorname{Homeo}_{+}(D_{n}^{2}))$ has a section over subgroups of $B_{n}$ . All of the previous methods use either torsion or Thurston stability, which do not apply to the pure braid group $PB_{n}$ , the subgroup of $B_{n}$ that fixes $n$ marked points pointwise. In this paper, we show that the pure braid group has no realization inside the area-preserving homeomorphisms using rotation numbers.

Planar pure braids on six strands

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.

On the Andreadakis Problem for Subgroups of $IA_n$

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.

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

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.

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

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 $1/n^2$.