# When the Lower Central Series Stops: A Comprehensive Study for Braid Groups and Their Relatives

Jacques Darné, Martin Palmer, Arthur Soulié

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Jacques Darné, Martin Palmer, Arthur Soulié. When the Lower Central Series Stops: A Comprehensive Study for Braid Groups and Their Relatives. Memoirs of the American Mathematical Society, 2025, 309 (1563), 10.1090/memo/1563. hal-05102004

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Distributed under a Creative Commons Attribution 4.0 International License
When the Lower Central Series Stops:
A Comprehensive Study for Braid Groups and
Their Relatives

Jacques Darné

Martin Palmer

Arthur Soulié

Author address:

INSTITUT DE RECHERCHE EN MATHÉMATIQUE ET EN PHYSIQUE, CHEMIN DU
CYCLOTRON 2, 1348 OTTIGNIES-LOUVAIN-LA-NEUVE, BELGIQUE

Email address: jacques.darne@normalesup.org

INSTITUTUL DE MATEMATICĂ SIMION STOILOW AL ACADEMIEI ROMÂNE, 21
CALEA GRIVIȚEI, 010702 BUCUREȘTI, ROMANIA

Email address: mpanghel@imar.ro

CENTER FOR GEOMETRY AND PHYSICS, INSTITUTE FOR BASIC SCIENCE
(IBS), 77 CHEONGAMRO, NAM-GU, POHANG-SI, GYEONGSANGBUK-DO, KOREA
790-784 & POSTECH, GYEONGSANGBUKDO, KOREA

Email address: artsou@hotmail.fr, arthur.soulie@ibs.re.kr
<figure><img src="image_2.png" /></figure>
# Contents

<table><tr><td>Introduction</td><td>vii</td></tr><tr><td>Table 1: The stable cases</td><td>xi</td></tr><tr><td>Table 2: The unstable cases for which the LCS stops</td><td>xii</td></tr><tr><td>Table 3: The unstable cases for which the LCS does not stop</td><td>xiii</td></tr><tr><td>Chapter 1. General recollections</td><td>1</td></tr><tr><td>1.1. Commutator calculus and lower central series</td><td>1</td></tr><tr><td>1.2. Lie rings of lower central series</td><td>3</td></tr><tr><td>1.3. Computing abelianisations from decompositions</td><td>4</td></tr><tr><td>Chapter 2. Strategy and first examples</td><td>5</td></tr><tr><td>2.1. Generation in degree one – first consequences</td><td>5</td></tr><tr><td>2.2. Artin groups</td><td>6</td></tr><tr><td>Chapter 3. Partitioned braids</td><td>9</td></tr><tr><td>3.1. Reminders: braids</td><td>9</td></tr><tr><td>3.2. Basic theory of partitioned braids</td><td>9</td></tr><tr><td>3.3. The lower central series</td><td>11</td></tr><tr><td>Chapter 4. Virtual and welded braids</td><td>17</td></tr><tr><td>4.1. Recollections</td><td>17</td></tr><tr><td>4.2. Notion of support of an element</td><td>28</td></tr><tr><td>4.3. Virtual braids</td><td>29</td></tr><tr><td>4.4. Welded braids</td><td>31</td></tr><tr><td>4.5. Extended welded braids</td><td>33</td></tr><tr><td>4.6. Partitioned virtual, welded and extended welded braids</td><td>34</td></tr><tr><td>Chapter 5. Variants on partitioned welded braids</td><td>41</td></tr><tr><td>5.1. Pure bipartite welded braids</td><td>41</td></tr><tr><td>5.2. Tripartite welded braids</td><td>43</td></tr><tr><td>5.3. Partitioned tripartite welded braids</td><td>45</td></tr><tr><td>5.4. The lower central series</td><td>46</td></tr><tr><td>Chapter 6. Braids on surfaces</td><td>55</td></tr><tr><td>6.1. Surfaces</td><td>55</td></tr><tr><td>6.2. Braids on surfaces: general theory</td><td>57</td></tr><tr><td>6.3. Presentations of surface braid groups</td><td>62</td></tr><tr><td>6.4. The lower central series of the whole group</td><td>69</td></tr><tr><td>6.5. Partitioned braids on surfaces</td><td>74</td></tr><tr><td>6.6. Partitions with small blocks</td><td>81</td></tr></table>
iv

CONTENTS

Appendix A. Some calculations of lower central series 101

A.1. Relative lower central series 101

A.2. Semi-direct products of abelian groups 102

A.3. Free products 105

A.4. Wreath products 107

Appendix B. Presentation of an extension 111

Bibliography 113
Abstract

Understanding the lower central series of a group is, in general, a difficult
task. It is, however, a rewarding one: computing the lower central series and
the associated Lie algebras of a group or of some of its subgroups can lead to a
deep understanding of the underlying structure of that group. Our goal here is to
showcase several techniques aimed at carrying out part of this task. In particular,
we seek to answer the following question: when does the lower central series stop?
We introduce a number of tools that we then apply to various groups related to
braid groups: the braid groups themselves, surface braid groups, groups of virtual
and welded braids, and *partitioned* versions of all of these groups. The path from
our general techniques to their application is far from being a straight one, and
some astuteness and tenacity is required to deal with all of the cases encountered
along the way. Nevertheless, we arrive at an answer to our question for each and
every one of these groups, save for one family of partitioned braid groups on the
projective plane. In several cases, we even compute completely the lower central
series. Some results about the lower central series of Artin groups are also included.

<footnote>Received by the editor February 16, 2022.</footnote>

<footnote n="2020">2020 Mathematics Subject Classification. Primary 20F14, 20F36, 20F38, 57M07.</footnote>

<footnote>*Key words and phrases.* lower central series, surface braid groups, welded and virtual braid groups, loop braid groups.</footnote>

<footnote>This paper was written while the first author was employed by the FNRS as a *Chargé de
recherches*. The first and third authors were partially supported by the ANR Projects ChroK
ANR-16-CE40-0003 and AlMaRe ANR-19-CE40-0001-01.</footnote>

<footnote>The second author was partially supported by a grant of the Romanian Ministry of Edu-
cation and Research, CNCS - UEFISCDI, project number PN-III-P4-ID-PCE-2020-2798, within
PNCDI III.</footnote>

<footnote>The third author was partially supported by a Rankin-Sneddon Research Fellowship of the
University of Glasgow and by the Institute for Basic Science IBS-R003-D1.</footnote>

<footnote>All three authors would like to thank the anonymous referee for very helpful comments and
suggestions on an earlier version of this work.</footnote>
<figure><img src="image_3.png" /></figure>
# Introduction

One of the most basic objects one needs to understand when studying the structure of a group $G$ is its lower central series (shortened to “LCS”) $G = \Gamma_1(G) \supseteq \Gamma_2(G) \supseteq \dots$. Its behaviour varies greatly from one group to another. For instance, if $G$ is perfect (i.e. all its elements can be written as products of commutators), its LCS is completely trivial; this holds for instance for mapping class groups of closed surfaces of genus $g \ge 3$ [Kor02, Thm 5.1]. On the contrary, if $G$ is nilpotent, or residually nilpotent, its LCS contains deep information about the structure of $G$; examples of residually nilpotent groups include free groups [MKS04, Chap. 5], pure braid groups [FR85, FR88], pure braid groups on surfaces [BB09b, BG16], pure welded braid groups [BP09, §5.5], and conjec- turally pure virtual braid groups [BMVW16]. The LCS is also deeply connected to the structure of the group ring of $G$. In particular, Quillen [Qui68] proved that if we consider the filtration of the group ring $\mathbb{Q}G$ by the powers of its augmentation ideal, then the associated graded algebra is isomorphic to the universal enveloping algebra of the Lie algebra $\mathcal{L}(G) \otimes \mathbb{Q}$, where $\mathcal{L}(G)$ is the graded Lie ring obtained from $\Gamma_*(G)$.

The amount of information one can hope to extract from the study of a LCS depends in the first place on whether or not it *stops* in the following sense:

**DEFINITION 0.1.** The LCS of a group $G$ is said to *stop* if there exists an integer $i \ge 1$ such that $\Gamma_i(G) = \Gamma_{i+1}(G)$. We say that it *stops at* $\Gamma_i$ if $i$ is the smallest integer for which this holds. Otherwise, we say the LCS *does not stop* or else that it *stops at* $\infty$.

It follows from the definition of the LCS (recalled in §1.1 below) that if $\Gamma_i(G) = \Gamma_{i+1}(G)$ for some $i \ge 1$, then $\Gamma_k(G) = \Gamma_{k+1}(G)$ for all $k \ge i$, whence our choice of terminology.

**Partitioned braid groups.** In this memoir, we study the LCS of the following families of groups, and their *partitioned* versions, in the sense described below:

* The Artin braid group $\mathbf{B}_n$,

* The virtual braid group $\mathbf{vB}_n$,

* The welded braid group $\mathbf{wB}_n$,

* The extended welded braid group $\tilde{\mathbf{wB}}_n$,

* The group $\mathbf{B}_n(S)$ of braids on any surface $S$.

Let $G_n$ denote one of the above groups. In each case, there is a notion of the underlying permutation of an element of $G_n$, corresponding to a canonical surjection $\pi: G_n \to \mathcal{S}_n$ to the symmetric group, from which we can define *partitioned* versions of $G_n$. Let us first fix our conventions concerning partitions of integers:
**DEFINITION 0.2.** Let $n \ge 1$ be an integer. A partition of $n$ is an $l$-tuple $\lambda = (n_1, \dots, n_l)$ of integers $n_i \ge 1$, for some $l \ge 1$ called the length of $\lambda$, such that $n$ is the sum of the $n_i$. Given such a $\lambda$, for $j \le l$, let us define $t_j := \sum_{i \le j} n_i$, including $t_0 = 0$. Then the set $b_j(\lambda) := \{t_{j-1} + 1, \dots, t_j\}$ is referred to as the $j$-th block of $\lambda$, and $n_i$ is called the size of the $i$-th block.

For $\lambda = (n_1, \dots, n_l)$ a partition of $n$, we consider the preimage

$$G_{\lambda} := \pi^{-1}(\mathfrak{S}_{\lambda}) = \pi^{-1}(\mathfrak{S}_{n_1} \times \dots \times \mathfrak{S}_{n_l}),$$

which is called the $\lambda$-partitioned version of $G_n$. There are two extremal situations: the trivial partition $\lambda = (n)$ simply gives the group $G_n$, whereas the discrete partition $\lambda = (1, 1, \dots, 1)$ corresponds to the subgroup of pure braids in $G_n$.

As we will see later on, for all the families of groups described above, the LCS of $G_n$ stops at $\Gamma_2$ or $\Gamma_3$ (when $n$ is at least 3 or 4, depending on the context), whereas the LCS of the subgroup of pure braids is a very complex object (in particular, it does not stop, when $n$ is at least 2 or 3). We can thus expect the partitioned braid groups $G_{\lambda}$ to display a range of intermediate behaviours when $\lambda$ varies, and this is indeed what we observe.

**Methods.** A fundamental tool in the study of LCS is the graded Lie ring structure on the associated graded $\mathcal{L}(G) := \bigoplus_{i \ge 1} \Gamma_i(G)/\Gamma_{i+1}(G)$. Namely, this is a graded abelian group endowed with a Lie bracket induced by commutators in $G$. It is always generated, as a Lie algebra over $\mathbb{Z}$, by its degree one piece, which is the abelianisation $G^{ab} = G/\Gamma_2(G)$. This often allows one to use disjoint support arguments to show that the LCS stops, when it does. Precisely, if one can show that pairs of generators of $G^{ab}$ have commuting representatives in $G$ (which is the case if they have representatives whose supports are disjoint, for a certain notion of support), then, by definition of the Lie bracket, they commute in $\mathcal{L}(G)$. In this case, $\mathcal{L}(G)$ is abelian, and it is generated by $G^{ab}$, which means that it is reduced to $G^{ab}$. In turn, that means that $\Gamma_i(G) = \Gamma_{i+1}(G)$ whenever $i \ge 2$. This kind of argument is used throughout the memoir. In particular, one can apply it readily to each of the $G_n$ above to show that its LCS stops at $\Gamma_2$ whenever $n$ is at least 3 or 4 (depending on the amount of space needed to have representatives with disjoint support for pairs of generators of $G^{ab}$).

Depending on the groups considered, having disjoint support will have a different meaning, but it will always imply that the elements considered commute. The most obvious definition is for mapping classes, which have disjoint support if they can be realised by homeomorphisms having disjoint support in the usual sense. This can be applied to usual braids, seen as mapping classes of the punctured disc. For braids on surfaces, one can generalise this by using a very similar idea: two such braids have disjoint support if they can be realised as geometric braids that do not move the same strands, and move them in disjoint regions of the surface. Equivalently, we view the surface braid group as a subgroup of the mapping class group of the punctured surface. The same definition can be used for welded braids, which can actually be seen as mapping classes (see [Dam17]). However, for the latter, it will be convenient to use another notion of support relying on their interpretation as automorphisms of free groups; with this point of view, the support is the list of generators involved in the definition of the automorphism (see Definition 4.29). Finally, for virtual braids, we will need a diagrammatic definition (Definition 4.32), listing the strands that really interact with the others in a diagram.
One other main line of argumentation for studying LCS is given by looking
for quotients whose LCS is well-understood. Namely, if we can find a quotient of
G whose LCS does not stop, then neither does the LCS of G; see Lemma 1.1.
Typically, we look for a quotient that is a semi-direct product of an abelian group
with $\mathbb{Z}$ or $\mathbb{Z}/2$, a free product of abelian groups or a wreath product of an abelian
group with some $\mathfrak{S}_\lambda$, whose LCS can be computed completely; see Appendix A.

Finally, a very important tool in our analysis is the study of the *quotient by the residue*. Precisely, if we denote by $\Gamma_\infty(G)$ (abbreviated $\Gamma_\infty$ when the context is clear) the intersection of the $\Gamma_i(G)$, the LCS of $G$ is “the same” as the LCS of $G/\Gamma_\infty$: each $\Gamma_i(G)$ is the preimage of $\Gamma_i(G/\Gamma_\infty)$ by the canonical projection, and this projection induces an isomorphism between $\mathcal{L}(G)$ and $\mathcal{L}(G/\Gamma_\infty)$. In particular, one of $\Gamma_*(G)$ and $\Gamma_*(G/\Gamma_\infty)$ stops if and only if the other does, which happens exactly when $G/\Gamma_\infty$ is nilpotent. Considering $G/\Gamma_\infty$ instead of $G$ can lead to very important simplifications. Let us illustrate this by an example, variations of which are used throughout the memoir. We know that $\Gamma_\infty = \Gamma_2$ for $\mathbf{B}_n$, and that $\Gamma_2$ contains the elements $\sigma_i\sigma_j^{-1}$ of $\mathbf{B}_n$. Thus whenever we have a morphism $\mathbf{B}_n \to G$, the subgroup $\Gamma_\infty(G)$ must contain the image of $\Gamma_\infty(\mathbf{B}_n)$, which contains the images of $\sigma_i\sigma_j^{-1}$, so all the $\sigma_i$ have the same image in $G/\Gamma_\infty$.

**Results.** Does the LCS stop? We give a complete answer to this question for all of the families of groups listed above, with the single exception of $\mathbf{B}_{2,m}(\mathbb{P}^2)$ with $m \ge 3$ (see Conjecture 6.94). We also obtain some amount of information on the associated Lie rings. In particular, we compute completely the Lie rings of partitioned braid groups on surfaces in the stable case. Besides their intrinsic value, these results have several applications, notably to the representation theory of braid groups and their relatives. See for instance [BGG17] and the work of the second and third authors [PS21], where the knowledge of the structure of such LCS is key in the construction and the study of representations of these groups using homological approaches. Furthermore, let us mention that one can see surface braid groups and their LCS as invariants of the surfaces themselves; as an application of this point of view, we recover the Riemann-Hurwitz formula for coverings of closed surfaces in Remark 6.46.

In addition to the families of groups introduced above, we also find out when
the LCS stops for what we call the *tripartite* welded braid groups $w\mathbf{B}(n_P, n_{S_+}, n_S)$,
which are the fundamental groups of the configuration spaces of $n_P$ points, $n_{S_+}$
oriented circles and $n_S$ unoriented circles in 3-space, where all the circles are un-
linked and unknotted. This is a generalisation of both $w\mathbf{B}_n = w\mathbf{B}(0, n, 0)$ and
$\tilde{w}\mathbf{B}_n = w\mathbf{B}(0, 0, n)$. We define partitioned versions $w\mathbf{B}(\lambda_P, \lambda_{S_+}, \lambda_S)$ of these
groups in the obvious way; see Definition 5.8. These groups are not only natu-
ral geometrical generalisations (obtained by mixing oriented circles, non-oriented
circles and moving points) of welded-type braid groups, they also have key applica-
tions for the homological constructions of representations of the groups $w\mathbf{B}_n$ and
$\tilde{w}\mathbf{B}_n$ introduced by [PS21]. In particular, the extension of the Burau representa-
tion to welded braid groups introduced by Vershinin [Ver01] (by assigning explicit
matrices to generators) may alternatively be defined by the methods of [PS21]
considering the abelianisation of $w\mathbf{B}(1, n, 0)$; see [PS22].

We summarise our results in three tables on pages xi–xiii. These are organised
as follows:
* In Table 1 are gathered the stable cases, which are those where the blocks of the partitions are large enough for the disjoint support argument described above to be applied readily.

* In Table 2 are gathered the cases where there are blocks in the partitions which are too small for the disjoint support argument to be applied readily, but not too many of them, so that the LCS still stops.

* In Table 3 are gathered the cases where the LCS does not stop.

Some of our results have already been obtained in the literature, with different methods. Namely, the question of whether or not the lower central series stops has already been studied:

* by Gorin and Lin [GL69] for $B_n$ and Kohno [Koh85] for the pure braid group $P_n = B_1, \dots, B_{n-1}$, which is moreover known to be residually nilpotent by Falk and Randell [FR85, FR88].

* by Bellingeri, Gervais and Guaschi [BGG08] for $B_n(S)$ where $S$ is a compact, connected, orientable surface with or without boundary.

* by Bellingeri and Gervais [BG16] for the pure surface braid group $P_n(S)$ where $S$ is a compact, connected, non-orientable surface with or without boundary and different from the projective plane $\mathbb{P}^2$.

* by Gonçalves and Guaschi [GG09a, GG09b] for $B_n(\mathbb{S}^2)$ and $B_n(\mathbb{S}^2 - \mathcal{P})$ where $\mathcal{P}$ is a finite set of points in $\mathbb{S}^2$.

* by Guaschi and de Miranda e Pereiro [GdMeP20] for $B_n(S)$ where $S$ is a compact, connected, non-orientable surface without boundary.

* by van Buskirk [vB66] and by Gonçalves and Guaschi [GG04b, GG11, GG07], both for the braid group on the projective plane $B_n(\mathbb{P}^2)$.

* by Bardakov and Bellingeri [BB09a] for the virtual braid group $vB_n$. The question of whether the pure virtual braid group $vP_n := vB_1, \dots, B_{n-1}$ is residually (torsion-free) nilpotent or not for $n \ge 4$ remains an open problem, while this property is proven for $vP_3$ by Bardakov, Mikhailov, Verшинin and Wu [BMVW16]. That the pure welded braid group $wP_n := wB_1, \dots, B_{n-1}$ is residually (torsion free) nilpotent is proven by Berceanu and Papadima [BP09, §5.5]. Results on the lower central series of the classical, virtual and welded pure braid groups and their relatives are collected in the survey paper of Suciu and Wang [SW17].

**Notation in the tables.** The letter $\lambda = (n_1, \dots, n_l)$ denotes a partition of $n$ of length $l \ge 1$. The letter $\mu$ denotes a partition that is either empty or whose blocks have size at least 3 for classical braids and surface braids, and at least 4 for virtual and welded braids. On the other hand, $\nu$ denotes any partition (possibly empty, unless stated otherwise).

In Table 2, the function $f$ is defined by $f(m) = \max\{v_2(m), 1\}$, where $v_2$ is the 2-adic valuation. The number $\epsilon$ is either 0 or 1 (the precise value may depend on the case, in particular on $m$, and is unknown, although we conjecture that it is always 1 for $m$ even and 0 for $m$ odd, so that $f(m) + \epsilon = v_2(m) + 1$ in all cases).

The letter $S$ denotes any connected surface (not necessarily compact nor orientable, and possibly with boundary). Six exceptional surfaces are mentioned in the tables, denoted by $\mathbb{D}$ (the disc), $\mathbb{D} - pt$ (the disc minus an interior point), $\mathbb{T}^2$ (the torus), $\mathbb{M}^2$ (the Möbius strip), $\mathbb{S}^2$ (the 2-sphere) and $\mathbb{P}^2$ (the projective plane). A surface $S$ is called *generic* if it is not one of these six exceptional surfaces.
The symbol (†) in front of a family of groups indicates that the result concerned is already partly known in the literature quoted above.

<table><caption>TABLE 1. The stable cases.</caption><thead><tr><th colspan="5">The stable cases</th></tr><tr><th>Family of groups</th><th>Partition</th><th>Stops at Γ<sub>k</sub></th><th>Ref.</th><th>Lie Alg.</th></tr></thead><tbody><tr><td>Classical braids B<sub>λ</sub></td><td>n<sub>i</sub> ≥ 3 (†)</td><td>k = 2</td><td>3.6 (3.7)</td><td>3.5</td></tr><tr><td rowspan="4">Surface braids B<sub>λ</sub>(S)</td><td>S ⊆ S<sup>2</sup></td><td>n<sub>i</sub> ≥ 3 (†)</td><td>k = 2</td><td>6.47</td></tr><tr><td>S ⊈ S<sup>2</sup>, orientable</td><td>n<sub>i</sub> ≥ 3 (†)</td><td>k = 3</td><td>§6.5.3</td></tr><tr><td rowspan="2">S non-orientable</td><td>l = 1, n<sub>1</sub> ≥ 3</td><td>k = 2</td><td>6.52</td></tr><tr><td>l ≥ 2, n<sub>i</sub> ≥ 3</td><td>k = 3</td><td>6.32</td></tr><tr><td>Virtual braids vB<sub>λ</sub></td><td></td><td></td><td></td><td></td></tr><tr><td>Welded braids wB<sub>λ</sub></td><td>n<sub>i</sub> ≥ 4</td><td>k = 2</td><td>4.47</td><td>4.46</td></tr><tr><td>Ext. welded braids w̃B<sub>λ</sub></td><td></td><td></td><td></td><td></td></tr><tr><td>Tripartite welded braids wB(λ<sub>P</sub>, λ<sub>S<sub>+</sub></sub>, λ<sub>S</sub>)</td><td>n<sub>i,P</sub> ≥ 3<br>n<sub>i,S<sub>+</sub></sub>, n<sub>i,S</sub> ≥ 4</td><td>k = 2</td><td>5.11 (5.12)</td><td>5.10</td></tr></tbody></table>
<table><caption>TABLE 2. The unstable cases for which the LCS stops.</caption><thead><tr><th colspan="5">The unstable cases for which the LCS stops</th></tr><tr><th>Family of groups</th><th>Partition</th><th>Stops at Γ<sub>k</sub></th><th>Ref.</th><th>Lie Alg.</th></tr></thead><tbody><tr><td rowspan="2">Classical braids B<sub>λ</sub></td><td>(2)</td><td rowspan="2">k = 2</td><td colspan="2">B<sub>2</sub> ≅ Z</td></tr><tr><td>(1, μ), (1, 1, μ)</td><td>3.6 (3.10, 3.11)</td><td>3.5</td></tr><tr><td rowspan="10">Surface braids<br>B<sub>λ</sub>(S)</td><td>S = D − pt</td><td>(1, μ)</td><td>k = 2</td><td>6.63</td><td>3.5</td></tr><tr><td rowspan="2">S = T<sup>2</sup></td><td>(1)</td><td>k = 2</td><td colspan="2">B<sub>1</sub>(T<sup>2</sup>) ≅ Z<sup>2</sup></td></tr><tr><td>(1, μ), μ ≠ ∅</td><td>k = 3</td><td>6.65</td><td>6.47 and 6.56</td></tr><tr><td>S = M<sup>2</sup></td><td>(1)</td><td>k = 2</td><td colspan="2">B<sub>1</sub>(M<sup>2</sup>) ≅ Z</td></tr><tr><td rowspan="3">S = S<sup>2</sup></td><td>(2) or (2, 1)</td><td rowspan="3">k = 2</td><td colspan="2">B<sub>2</sub>(S<sup>2</sup>) ≅ Z/2, B<sub>2,1</sub>(S<sup>2</sup>) ≅ Z/4</td></tr><tr><td>(1, μ), (1, 1, μ), (1, 1, 1, μ)</td><td>6.70</td><td>6.47</td></tr><tr><td>(2, m), m ≥ 3</td><td>k = f(m) + 1 + ϵ</td><td>6.73</td><td>–</td></tr><tr><td rowspan="3">S = P<sup>2</sup></td><td>(1, m), m ≥ 3</td><td>k = f(m) + 2 + ϵ</td><td>6.80</td><td>–</td></tr><tr><td>(1)</td><td>k = 2</td><td colspan="2">B<sub>1</sub>(P<sup>2</sup>) ≅ Z/2</td></tr><tr><td>(1, 1)</td><td>k = 3</td><td colspan="2">B<sub>1,1</sub>(P<sup>2</sup>) ≅ Q<sub>8</sub> (6.82)</td></tr><tr><td></td><td>(2)</td><td>k = 4</td><td colspan="2">B<sub>2</sub>(P<sup>2</sup>) ≅ Dic<sub>16</sub> (6.82)</td></tr><tr><td>Virtual braids vB<sub>λ</sub></td><td></td><td>(1, μ)</td><td>k = 2</td><td>4.47 (4.51)</td><td>4.46</td></tr><tr><td>Welded braids wB<sub>λ</sub></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="3">Tripartite welded braids<br>wB(λ<sub>P</sub>, λ<sub>S+</sub>, λ<sub>S</sub>)</td><td></td><td>((1, ..., 1, μ<sub>P</sub>), μ<sub>S+</sub>, μ<sub>S</sub>)</td><td rowspan="3">k = 2</td><td>5.11 (5.14)</td><td rowspan="3">5.10</td></tr><tr><td></td><td>((1, ..., 1, μ<sub>P</sub>), (1, μ<sub>S+</sub>), μ<sub>S</sub>)</td><td>5.11 (5.15, 5.16)</td></tr><tr><td></td><td>((2, ν<sub>P</sub>), ∅, μ<sub>S</sub>), μ<sub>S</sub> ≠ ∅</td><td>5.11 (5.21)</td><td>5.10 and 5.21</td></tr></tbody></table>
<table><caption>TABLE 3. The unstable cases for which the LCS does not stop.</caption><thead><tr><th colspan="3">The unstable cases for which the LCS does not stop</th></tr><tr><th>Family of groups</th><th>Partition</th><th>Ref.</th></tr></thead><tbody><tr><td rowspan="2">Classical braids B<sub>λ</sub></td><td>(1, 1, 1, ν) (†)</td><td>3.6 (3.8)</td></tr><tr><td>(2, ν), l ≥ 2</td><td>3.6 (3.12, 3.15, 3.18)</td></tr><tr><td rowspan="8">Surface braids B<sub>λ</sub>(S)</td><td>S generic</td><td>(1, ν), (2, ν) (†)</td><td>6.62</td></tr><tr><td rowspan="2">S = D − pt</td><td>(1, 1, ν)</td><td>6.63</td></tr><tr><td>(2, ν)</td><td>6.62</td></tr><tr><td rowspan="2">S = T<sup>2</sup></td><td>(1, 1, ν)</td><td>6.65</td></tr><tr><td>(2, ν) (†)</td><td>6.65 (6.62)</td></tr><tr><td rowspan="2">S = M<sup>2</sup></td><td>(1, ν), l ≥ 2</td><td>6.62, 6.67, 6.68</td></tr><tr><td>(2, ν)</td><td>6.62</td></tr><tr><td rowspan="2">S = S<sup>2</sup></td><td>(1, 1, 1, 1, ν)</td><td>6.70 (6.71)</td></tr><tr><td>(2, ν), l ≥ 3 or (2, 2)</td><td>6.70 (6.72, 6.75)</td></tr><tr><td rowspan="2">S = P<sup>2</sup></td><td>(1, ν), l ≥ 3</td><td>6.79</td></tr><tr><td>(2, ν), l ≥ 3 or (2, 2) or (2, 1)</td><td>6.83, 6.91, 6.88</td></tr><tr><td>Virtual braids vB<sub>λ</sub></td><td>(1, 1, ν), (2, ν), (3, ν)</td><td></td></tr><tr><td>Welded braids wB<sub>λ</sub></td><td></td><td>4.47</td></tr><tr><td>Ext. w. braids w̄B<sub>λ</sub></td><td>[(1, ν) with ν ≠ Ø], (2, ν), (3, ν)</td><td></td></tr><tr><td rowspan="4">Tripartite welded braids wB(λ<sub>P</sub>, λ<sub>S+</sub>, λ<sub>S</sub>)</td><td>(ν<sub>P</sub>, ν<sub>S+</sub>, (2, ν<sub>S</sub>)), (ν<sub>P</sub>, ν<sub>S+</sub>, (3, ν<sub>S</sub>))</td><td></td></tr><tr><td>(ν<sub>P</sub>, (2, ν<sub>S+</sub>), ν<sub>S</sub>), (ν<sub>P</sub>, (3, ν<sub>S+</sub>), ν<sub>S</sub>)</td><td>5.11</td></tr><tr><td>(ν<sub>P</sub>, (1, 1, ν<sub>S+</sub>), ν<sub>S</sub>)</td><td></td></tr><tr><td>(ν<sub>P</sub>, ν<sub>S+</sub>, (1, ν<sub>S</sub>)), l ≥ 2</td><td>5.11 (5.17)</td></tr><tr><td></td><td>((2, ν<sub>P</sub>), ν<sub>S+</sub>, ν<sub>S</sub>), ν<sub>S+</sub> ≠ Ø</td><td>5.11 (5.18)</td></tr></tbody></table>
<figure><img src="image_4.png" /></figure>
CHAPTER 1

General recollections

In this chapter, we recall some classical notions and tools to study the lower central series of groups. These will be used throughout the memoir.

1.1. Commutator calculus and lower central series

Let $G$ be a group. Recall that the lower central series (LCS) of $G$ is the descending sequence of normal subgroups $G = \Gamma_1(G) \supseteq \Gamma_2(G) \supseteq \cdots$ of $G$, also denoted by $\Gamma_*(G)$, defined by

$$
\Gamma_i(G) := \begin{cases} G & \text{if } i = 1, \\ [G, \Gamma_{i-1}(G)] & \text{if } i \ge 2, \end{cases}
$$

where [G, Γᵢ(G)] is the subgroup of G generated by all commutators [x, y] := xyx⁻¹y⁻¹ with x in G and y in Γᵢ₋₁(G). The subgroups Γᵢ(G) are fully invariant, and in particular normal in G. As a consequence, one can also think of the LCS as an ascending chain of quotients G/Γᵢ(G) of G. Recall that the abelianisation G<sup>ab</sup> of G is the first of these quotients, namely G/Γ₂(G). In general, G/Γ<sub>c+1</sub>(G) is the universal c-nilpotent quotient of G (recall that G is called c-nilpotent if Γ<sub>c+1</sub>(G) = {1}). The group G is called *residually nilpotent* if its residue Γ<sub>∞</sub>(G) := ⋂ Γᵢ(G) is equal to {1}. The quotient G/Γ<sub>∞</sub>(G) is the universal (and, in particular, the largest) residually nilpotent quotient of G.

The following lemma is folklore and will be used very often in the sequel. In particular, we will often use its contrapositive: if a group $G$ has a quotient whose LCS does not stop, then the LCS of $G$ does not stop either.

LEMMA 1.1. Let $H$ be a quotient of $G$. If $\Gamma_i(G) = \Gamma_{i+1}(G)$ for some $i$, then
$\Gamma_i(H) = \Gamma_{i+1}(H)$.

PROOF. For all $k \ge 1$, it follows from the definition of the LCS that $\Gamma_k H = \pi(\Gamma_k G)$. As a consequence, $\Gamma_{i+1} H = \Gamma_i H$ whenever $\Gamma_{i+1} G = \Gamma_i G$. $\square$

We now give two partial converses to Lemma 1.1. The first one (also the
most obvious one) is the case of a quotient by a subgroup having some finiteness
properties. Since most of the extensions that we consider in the sequel will not
satisfy the required hypothesis, we will use it only once, in the very simple case when
the kernel is cyclic of order two. Nevertheless, we state it in a general framework:

LEMMA 1.2. Let $K \hookrightarrow G \to H$ be a short exact sequence of groups. Suppose that there exists $l \ge 0$ such that every strictly decreasing central filtration of $K$ stops after at most $l$ steps, that is, if $K = K_1 \supseteq K_2 \supseteq \cdots \supseteq K_m$ is a nested sequence of subgroups satisfying $[K, K_i] \subset K_{i+1}$, then $m \le l$. Suppose moreover that for some $i \ge 1$, we have $\Gamma_{i+1} H = \Gamma_i H$. Then $\Gamma_{i+l+1} G = \Gamma_{i+l} G$.
PROOF. The filtration $K \cap \Gamma_*(G)$ is a central filtration of $K$, so it can strictly decrease only $l$ times. If $\Gamma_{i+1}H = \Gamma_iH$, then $\Gamma_i(H) = \Gamma_{i+k}(H)$ is the image of $\Gamma_{i+k}(G)$ in $H$, for all $k \ge 0$. Recall that if $L$ and $M$ are subgroups of $G$ such that $L \subseteq M$, then $L$ and $M$ are equal if and only if their image in $H$ and their intersection with $K$ are equal. As a consequence, for $\Gamma_{i+k}(G)$ to decrease when $k$ grows, its intersection with the kernel must decrease, which can happen at most $l$ times. So the LCS of $G$ must stop at most at $\Gamma_{i+l}$. $\square$

EXAMPLE 1.3. If $K$ is finite, such an $l$ clearly exists. In fact, for any central filtration $K_*$ on $K$, since the cardinal of $K/K_m$ is the product of the cardinals of the $K_i/K_{i+1}$, one can take $l$ to be the number of prime factors in the cardinal of $K$ (a bound that is optimal if $K$ is abelian). In particular, we will apply this with $K \cong \mathbb{Z}/2$ and $l = 1$ in the proof of Proposition 6.80.

REMARK 1.4. Finite groups are not the only ones for which the hypothesis holds. We could for instance apply the Lemma with $K$ simple, or more generally with $K$ perfect (with $l = 0$). Also, the class of groups $K$ satisfying this hypothesis is stable by extensions. In fact, an equivalent way of stating it is to ask that the maximal residually nilpotent quotient of $K$ (that is, $K/\Gamma_\infty(K)$) is finite.

The other partial converse to Lemma 1.1 that we will use concerns quotients by central subgroups. This case is a bit more subtle, and requires the following result, which can be useful when calculating quotients by residues.

PROPOSITION 1.5. Let $G$ be a group, and let $N$ be a normal subgroup of $G$. Suppose that for some $i \ge 2$, $N \cap \Gamma_i(G) = \{1\}$. Then the canonical morphism $\Gamma_\infty(G) \to \Gamma_\infty(G/N)$ is an isomorphism. In particular, $G$ is residually nilpotent if and only if $G/N$ is. Moreover, we have a short exact sequence:

$$N \hookrightarrow G/\Gamma_{\infty} \twoheadrightarrow (G/N)/\Gamma_{\infty}.$$

PROOF. Let $\pi: G \to G/N$ be the canonical projection. Since $N \cap \Gamma_\infty(G) = \{1\}$, the induced morphism $\Gamma_\infty(G) \to \Gamma_\infty(G/N)$ is injective. Let us show that it is surjective. Let $y \in \Gamma_\infty(G/N)$. Since $\Gamma_k(G/N) = \pi(\Gamma_k G)$ by definition of the LCS, there is, for each $k \ge 1$, some $x_k \in \Gamma_k(G)$ such that $\pi(x_k) = y$. Then $x_k x_{k+1}^{-1} \in N \cap \Gamma_k(G)$, which implies that $x_k x_{k+1}^{-1} = 1$ whenever $k \ge i$. Thus the sequence $(x_k)$ is stationary at $x := x_i$, which must be in $\Gamma_\infty(G)$ by definition of the $x_k$, and is sent to $y$ by $\pi$. This proves the first part of the Proposition. Now, let us consider the commutative diagram of groups:

<figure><img src="image_5.png" /></figure>

By the Nine Lemma, the bottom row must be a short exact sequence. $\square$

The following corollary provides the promised partial converse to Lemma 1.1:
COROLLARY 1.6. Let $G$ be a group and $A$ be a central subgroup of $G$. Suppose that for some $i \ge 2$, the canonical map $A \to G/\Gamma_i(G)$ is injective. If the LCS of $G/A$ stops at $\Gamma_k$, then the LCS of $G$ stops at $\Gamma_k$ or at $\Gamma_{k+1}$.

PROOF. The extension $A \hookrightarrow G/\Gamma_\infty \to (G/A)/\Gamma_\infty$ from Proposition 1.5 is a central one. As a consequence, if $(G/A)/\Gamma_\infty$ is $k$-nilpotent, then $G/\Gamma_\infty$ is nilpotent of class $k$ or $k+1$. $\square$

REMARK 1.7. In explicit examples, the hypothesis of Corollary 1.6 is most easily checked when $i = 2$, since we typically have a complete understanding of $G^{ab}$; see for example the proof of Proposition 6.73.

## 1.2. Lie rings of lower central series

We now recall the definition and basic properties of a key tool for studying the LCS of a group, namely its associated Lie ring. We refer the reader to [Laz54, Chap. 1] for further details.

Note that, for all $i \ge 1$, $[\Gamma_i(G), \Gamma_i(G)] \subseteq [G, \Gamma_i(G)] \subseteq \Gamma_{i+1}(G) \subseteq \Gamma_i(G)$. Thus, $\Gamma_i(G)$ is a normal subgroup of $G$, and the quotient $\mathcal{L}_i(G) := \Gamma_i(G)/\Gamma_{i+1}(G)$ is an abelian group. Moreover, one can show that $[\Gamma_i(G), \Gamma_j(G)] \subseteq \Gamma_{i+j}(G)$ for all $i, j \ge 1$, which is the crucial property allowing us to define the Lie ring associated with $\Gamma_*(G)$:

PROPOSITION 1.8 ([Laz54, Th. 2.1]). The graded abelian group defined by $\mathcal{L}(G) := \bigoplus_{i \ge 1} \mathcal{L}_i(G)$ is a Lie ring, with the Lie bracket induced by the commutator map of $\overline{G}$.

CONVENTION 1.9. Let $g$ be an element of $G$. If there is an integer $d$ such that $g \in \Gamma_d(G) - \Gamma_{d+1}(G)$, it is obviously unique. We then call $d$ the degree of $g$ with respect to $\Gamma_*(G)$. The notation $\overline{g}$ denotes the class of $g$ in some quotient $\mathcal{L}_i(G)$. If the integer $i$ is not specified, it is assumed that $i = d$, which means that $\overline{g}$ denotes the only non-trivial class induced by $g$ in $\mathcal{L}(G)$. If such a $d$ does not exist (that is, if $g \in \bigcap \Gamma_i(G)$), we say that $g$ has degree $\infty$ and we put $\overline{g} = 0$.

With this convention, the Lie bracket $[-, -]$ of $\mathcal{L}(G)$ is given by the collection of bilinear maps $\mathcal{L}_i(G) \times \mathcal{L}_j(G) \to \mathcal{L}_{i+j}(G)$ defined by:

$$ \forall x \in \mathcal{L}_i(G), \forall y \in \mathcal{L}_j(G), [\overline{x}, \overline{y}] := [x, y] \in \mathcal{L}_{i+j}(G). $$

The following lemma, which will be used several times in the sequel to identify $G/\Gamma_\infty$ for some group $G$, is one illustration of the use of Lie rings in studying the LCS:

LEMMA 1.10. Let $p: G \to Q$ be a surjective group morphism. If $Q$ is a residually nilpotent group, then the following conditions are equivalent:

* $\mathcal{L}(p): \mathcal{L}(G) \to \mathcal{L}(Q)$ is an isomorphism.

* $p$ induces an isomorphism $G/\Gamma_\infty \cong Q$.

PROOF. If $Q$ is residually nilpotent, $p$ induces a map $G/\Gamma_\infty \to Q$ between two residually nilpotent groups. Since $G \to G/\Gamma_\infty$ induces an isomorphism between the associated Lie rings, the statement for $G$ can be deduced from the statement for $G/\Gamma_\infty$. Thus, we can assume that $G$ is residually nilpotent and, under this hypothesis, we need to show that $p$ is an isomorphism if and only if $\mathcal{L}(p)$ is. Clearly, if $p$ is an isomorphism, then $\mathcal{L}(p)$ is too. Conversely, if $p$, which is surjective, is not
an isomorphism, then there is some non-trivial element $x$ in its kernel. Since $G$ is residually nilpotent, $x$ induces a non-trivial class $\bar{x}$ in $\mathcal{L}(G)$, which is sent to 0 by $\mathcal{L}(p)$. This implies that $\mathcal{L}(p)$ is not injective, which concludes our proof. $\square$

### 1.3. Computing abelianisations from decompositions

Let us recall some classical tools for computing the abelianisation from some decomposition of a given group. The abelianisation functor $G \mapsto G^{ab}$ is a left adjoint, hence right exact. In order to compute the abelianisation of an extension, one can say more. Given a short exact sequence of groups $H \hookrightarrow G \twoheadrightarrow K$, let us denote by $(H^{ab})_K$ the coinvariants of $H^{ab}$ with respect to the action of $K$ on $H^{ab}$ induced by conjugation in $G$.

**LEMMA 1.11.** *The short exact sequence $H \hookrightarrow G \twoheadrightarrow K$ induces the following exact sequence of abelian groups:*

$$(H^{ab})_K \to G^{ab} \to K^{ab} \to 0.$$

**PROOF.** The conjugation action of $G$ on $H^{ab}$ factors through $G/H = K$, hence $(H^{ab})_G = (H^{ab})_K$. Since we have an exact sequence $H^{ab} \to G^{ab} \to K^{ab} \to 0$, it suffices to show that the morphism $H^{ab} \to G^{ab}$ factors through $(H^{ab})_K$. It is equivariant with respect to the action of $G$ induced by conjugation (which is obviously trivial on $G^{ab}$), whence the result. $\square$

For split exact sequences, we can say even more:

**LEMMA 1.12.** *The abelianisation of a semidirect product $H \times K$ is isomorphic to the product $(H^{ab})_K \times K^{ab}$.*

**PROOF.** As a consequence of the usual formula $[x, yz] = [x, y](x[x, z]x^{-1})$, one sees that the commutator subgroup $[H \times K, H \times K]$ is normally generated by $[H, H]$, $[H, K]$ and $[K, K]$. We can take the quotient by these three sets of relations successively: $(H \times K)/[H, H]$ is isomorphic to $H^{ab} \times K$, then killing $[H, K]$ gives $(H^{ab})_K \times K$ and finally, $((H^{ab})_K \times K)/[K, K] \cong (H^{ab})_K \times K^{ab}$. $\square$
# Strategy and first examples

In this chapter, we present some general ideas used to decide whether the LCS stops or not. As a first example, we then apply these ideas to Artin groups.

## 2.1. Generation in degree one – first consequences

The Lie ring associated to the LCS has the following fundamental property:

**PROPOSITION 2.1.** The Lie ring $\mathcal{L}(G)$ is generated in degree one. That is, it is generated by the abelianisation $\mathcal{L}_1(G) = G^{ab}$ as a Lie algebra over $\mathbb{Z}$.

**PROOF.** It is a direct consequence of the definitions: the equality $\mathcal{L}_k(G) = [\mathcal{L}_1(G), \mathcal{L}_{k-1}(G)]$ is obtained directly from $\Gamma_k(G) = [\Gamma_1(G), \Gamma_{k-1}(G)]$, by passing to the appropriate quotients. $\square$

A first consequence of this is the following:

**COROLLARY 2.2.** Let $G$ be a group. If $G^{ab}$ is cyclic, then $\Gamma_2 G = \Gamma_3 G$.

**PROOF.** Proposition 2.1 implies that the Lie ring $\mathcal{L}(G)$ is a quotient of the free Lie ring on $G^{ab}$. Since $G^{ab}$ is cyclic, the latter is the abelian Lie ring consisting only of $G^{ab}$. As a consequence, $\Gamma_2 G / \Gamma_3 G = \mathcal{L}_2(G) = \{0\}$. $\square$

**EXAMPLE 2.3 (Braids).** Directly from their usual presentations, one computes the abelianisation of the braid groups: $\mathbf{B}_n^{ab} \cong \mathbb{Z}$ for $n \ge 2$. Thus $\Gamma_2(\mathbf{B}_n) = \Gamma_3(\mathbf{B}_n)$. This fact is originally due to Gorin and Lin [GL69], who proved it by different methods. This property of $\mathbf{B}_n$, which is also true for any quotient of $\mathbf{B}_n$ (such as the symmetric group $\mathfrak{S}_n$), may also be seen as a particular case of the computations for Artin groups below (Proposition 2.11).

**EXAMPLE 2.4 (Knot groups).** For any knot, the knot group has (infinite) cyclic abelianisation, thus its LCS stops at $\Gamma_2$. This generalises readily to the enveloping groups of any connected quandle; see for instance [BNS19, Prop. 3.3].

**EXAMPLE 2.5 (Automorphisms of free groups).** Consider the automorphism group Aut($\mathbf{F}_n$) of the free group on $n$ letters. The kernel $IA_n$ of the projection from Aut($\mathbf{F}_n$) onto Aut($\mathbf{F}_n^{ab}$) $\cong$ GL$_n(\mathbb{Z})$ is generated by the usual $K_{ij}$ and $K_{ijk}$ from [MKS04, §3.5], which are easily seen to be commutators of automorphisms. Thus Aut($\mathbf{F}_n$)$^{ab}$ $\cong$ GL$_n(\mathbb{Z})^{ab}$. Whenever $n \ge 3$, this group is cyclic of order two, so the LCS of Aut($\mathbf{F}_n$) stops at $\Gamma_2$, and so does the one of GL$_n(\mathbb{Z})$.

An easy generalisation of Corollary 2.2 is:

**COROLLARY 2.6.** Let $G$ be a group. Let $S$ be a generating set of $G^{ab}$. Suppose that, for each pair $(s, t) \in S^2$, we can find representatives $\tilde{s}, \tilde{t} \in G$ of $s$ and $t$ such that $\tilde{s}$ and $\tilde{t}$ commute. Then $\Gamma_2 G = \Gamma_3 G$.
PROOF. The Lie ring $\mathcal{L}(G)$ is generated by $S$. Moreover, the fact that $[\tilde{s}, \tilde{t}] = 1$ in $G$ readily implies that $[s, t] = 0$ in $\mathcal{L}(G)$. Since the brackets $[s, t]$ for $(s, t) \in S^2$ generate $\mathcal{L}_2(G) = [\mathcal{L}_1(G), \mathcal{L}_1(G)]$, we see that $\Gamma_2 G / \Gamma_3 G = \mathcal{L}_2(G) = \{0\}$. In fact, $\mathcal{L}(G)$ is an abelian Lie ring, reduced only to $\mathcal{L}_1(G) = G^{ab}$. $\square$

We have not made any effort to make the above corollary as general as possible. In particular, $\tilde{s}$ and $\tilde{t}$ may commute only up to an element of $\Gamma_3 G$, and the conclusion still holds. Also, one may think of similar statements showing that $\Gamma_3 G = \Gamma_4 G$, and so on. Weak as it may seem, our statement is already very useful. In particular, when applied to groups whose elements have a geometrical interpretation, it will often happen that $\tilde{s}$ and $\tilde{t}$ can be chosen with disjoint support (whatever this means, depending on the context – see for instance §4.2 for precise definitions in certain cases), which readily implies that they commute. We will sometimes need a more refined version of the above, but we will discuss it in each particular situation.

EXAMPLE 2.7 (Automorphisms of $\mathbf{F}_2$). As an example of a case where Corollary 2.6 does not work, but the same kind of technique does apply, let us consider Aut($\mathbf{F}_2$). We have mentioned that Aut($\mathbf{F}_n$)$^{ab} \cong GL_n(\mathbb{Z})^{ab}$; see Example 2.5. For $n = 2$, this is no longer cyclic, but isomorphic to $(\mathbb{Z}/2)^2$, generated by the (equivalences classes of the) automorphisms $\sigma$ and $\tau$ acting as follows (fixing free generators $x$ and $y$ of $\mathbf{F}_2$):

$$ \sigma(x) = y, \quad \sigma(y) = x, \quad \tau(x) = x^{-1}, \quad \tau(y) = y. $$

It follows that $\mathcal{L}_2(\mathrm{Aut}(\mathbf{F}_2))$ is generated by the (equivalence class of the) automorphism $\iota = [\sigma, \tau]$ acting by $\iota(x) = x^{-1}$ and $\iota(y) = y^{-1}$. It is easy to check that $\iota$ commutes with both $\sigma$ and $\tau$, so $\mathcal{L}_3(\mathrm{Aut}(\mathbf{F}_2)) = 0$. Thus, the LCS of Aut($\mathbf{F}_2$) stops at $\Gamma_3$, as does that of $GL_2(\mathbb{Z})$.

Let us spell out another useful consequence of Proposition 2.1:

COROLLARY 2.8. Let $G$ be a group and $d \ge 1$ be an integer. If $\mathcal{L}_k(G) = \Gamma_k G / \Gamma_{k+1} G$ is a $d$-torsion abelian group for some $k \ge 1$, then $\mathcal{L}_l(G) = \Gamma_l G / \Gamma_{l+1} G$ is too for all integers $l \ge k$.

PROOF. If an element $x \in \mathcal{L}(G)$ is of $d$-torsion, then for all $y \in \mathcal{L}(G)$, the bracket $[x, y]$ is too, because $d \cdot [x, y] = [d \cdot x, y] = 0$. Since $\mathcal{L}_{l+1}(G) = [\mathcal{L}_1(G), \mathcal{L}_l(G)]$, we get our result by induction on $l$. $\square$

EXAMPLE 2.9 (Virtual and welded braids). Let $G$ be a group such that $G^{ab} \cong \mathbb{Z} \times (\mathbb{Z}/d)$, where the factors are generated, respectively, by $u$ and $v$. Then $\mathcal{L}_2(G)$ is generated, as an abelian group, by $[u, v]$. Since $d \cdot [u, v] = [u, d \cdot v] = 0$, $\mathcal{L}_2(G)$ is of $d$-torsion, and then all the $\mathcal{L}_k(G)$, for $k \ge 2$, must be too. This applies, for instance, to the groups $\mathbf{vB}_2$ and $\mathbf{wB}_2$ (which are both isomorphic to $\mathbb{Z} * \mathbb{Z}/2$), with $d = 2$; see Chapter 4.

## 2.2. Artin groups

Let $S$ be a set, and $M = (m_{s,t})_{s \in S}$ be a Coxeter matrix, i.e. a symmetric matrix with coefficients in $\mathbb{N} \cup \{\infty\}$, with $m_{s,s} = 1$, and $m_{s,t} \ge 2$ if $s \ne t$. Let $A_M$ be the associated Artin group, defined by the presentation:

$$ A_M = \left\langle S \left| \begin{array}{c} \underbrace{\mathrm{stst}\cdots}_{m_{s,t}} = \underbrace{\mathrm{tstst}\cdots}_{m_{s,t}} \\ (s, t \in S, m_{s,t} \ne \infty) \end{array} \right. \right\rangle. $$
Let us consider the graph $\mathcal{G}$ whose incidence matrix is $M$ modulo 2. Namely, $\mathcal{G}$ is obtained by taking $S$ as its set of vertices and by drawing an edge between $s$ and $t$ whenever $m_{s,t}$ is an odd integer.

LEMMA 2.10. The group $A_M^{ab}$ is free abelian on the set $\pi_0(\mathcal{G})$ of connected components of $\mathcal{G}$.

PROOF. This is clear from the presentation: in $A_M^{ab}$, the relation between $s$ and $t$ becomes $\bar{s} = \bar{t}$ if $m_{s,t}$ is odd, and it becomes trivial if $m_{s,t}$ is even or $m_{s,t} = \infty$. $\square$

Let us now study the LCS of $A_M$. Suppose first that $\mathcal{G}$ is connected. Then $A_M^{ab}$ is cyclic, and Corollary 2.2 applies: the LCS of $A_M$ stops at $\Gamma_2$. This holds in particular for the classical braid group; see Example 2.3. Now, if $\mathcal{G}$ has several connected components, we need to study the interactions between the corresponding generators of the Lie ring of $A_M$. The simplest case happens when all the even $m_{s,t}$ are equal to 2. Then $A_M$ splits into the (restricted) direct product of the Artin groups corresponding to the connected components of our graph, thus $\mathcal{L}(A_M)$ is a direct sum of copies of $\mathbb{Z}$ (concentrated in degree one). Thus, we obtain a first result (which recovers [BGG08, Prop. 1] for spherical Artin groups):

PROPOSITION 2.11. If all the coefficients of $M$ are finite, and are either odd or equal to 2, then the LCS of $A_M$ stops at $\Gamma_2$.

In order to get a step further, we need to study more closely the interactions between the generators of the Lie ring of $A_M$ corresponding to the connected components of $\mathcal{G}$.

LEMMA 2.12. Let $s, t \in S$. If $m_{s,t} = 2k$ for some integer $k$, then $k \cdot [\bar{s}, \bar{t}] = 0$ in $\mathcal{L}(A_M)$.

PROOF. If $m_{s,t} = 2k$, then $(st)^k = (ts)^k$ in $A_M$. We can write this relation as $(st)^k(ts)^{-k} = 1$. We recall that modulo $\Gamma_3(A_M)$, commutators commute with any other element. Hence, modulo $\Gamma_3(A_M)$, we have:

$$1 = (st)^k(ts)^{-k} = (st)^{k-1}[s,t](ts)^{-(k-1)} = [s,t](st)^{k-1}(ts)^{-(k-1)} = \dots = [s,t]^k.$$

Thus, in $\mathcal{L}_2(A_M)$, we have $0 = k \cdot [\overline{s,t}] = k \cdot [\bar{s}, \bar{t}]$, as claimed. $\square$

From this, we deduce that for $x, y \in \pi_0(\mathcal{G})$, we have $d_{x,y} \cdot [x, y] = 0$ in $\mathcal{L}_2(A_M)$, where $d_{x,x} = 1$ and $d_{x,y} = \gcd\{\frac{m_{s,t}}{2} \mid s, t \in S, \bar{s} = x \text{ and } \bar{t} = y\}$ if $x \neq y$. In particular, this proves:

PROPOSITION 2.13. If all the coefficients of $M$ are finite, then $\Gamma_2(A_M)/\Gamma_3(A_M)$ is of d-torsion, where $d = \text{lcm}\{d_{x,y} \mid x, y \in \pi_0(\mathcal{G})\}$.

REMARK 2.14. Proposition 2.13 is more general than Proposition 2.11, which is recovered as a particular case where $d = 1$. In fact, for $d$ to be equal to 1 (which implies that the LCS stops at $\Gamma_2$), we do not need the even $m_{s,t}$ to be equal to 2, but only all the $d_{x,y}$ to be equal to 1.

REMARK 2.15. Proposition 2.13 does not say anything when at least one of the $d_{x,y}$ is infinite. For instance, when all the $m_{s,t}$ are infinite, $A_M$ is the free group on $S$, whose Lie ring is without torsion. More generally, Right-Angled Artin Groups (where all the $m_{s,t}$ are infinite or equal to 2) have torsion-free Lie rings [DK92]. Another example is the subgroup $KB_n$ of the virtual braid group $vB_n$ given by the
normal closure of $\mathbf{B}_n \subset \mathbf{vB}_n$ (denoted by $H_n$ in [BB09a]). This is an Artin group with all $m_{s,t}$ infinite or equal to 2 or 3 [BB09a, Prop. 17]; its abelianisation is free abelian and its LCS stops at $\Gamma_2$ for $n \ge 3$ [BB09a, Prop. 19].
# CHAPTER 3

## Partitioned braids

This chapter is devoted to the study of the LCS of the group of partitioned braids; see Definition 3.2 below. Our main results are summarised in Theorem 3.6. The group of partitioned braids is a subgroup of $\mathbf{B}_n$, which has already been studied notably by Manfredini [Man97] and by Bellingeri, Godelle and Guaschi [BGG17]. The former gave a presentation of this group, using the Reidemeister-Schreier method, which may be applied to this finite-index subgroup of Artin's braid group.

REMARK 3.1. One could use the aforementioned presentation of the partitioned braid group [Man97] to get generators, a calculation of the abelianisation, and (with a little bit of work) the stable case in the study of the LCS. However, we will avoid using this presentation altogether, for several reasons. Firstly, we want the present memoir to be as self-contained as possible. Secondly, even if a presentation is of some help in the study of the LCS of a group, there is only so much that one can deduce directly from it; in fact, not much more than a computation of the abelianisation, and that $\Gamma_2 = \Gamma_3$ when it holds. In particular, most of our non-stable results would not be simplified by using the presentation. Thirdly, and perhaps most importantly, to the best of our knowledge, nobody has written down a presentation of the other partitioned groups that we study later on. This could certainly be done using the Reidemeister-Schreier technique (at least when we have a presentation of the non-partitioned group), but this would require a fair amount of work, which we intend to avoid. We will do so precisely by generalising proofs that do not depend on the use of Manfredini's presentation.

### 3.1. Reminders: braids

We recall that the standard generators $\sigma_i$ and $A_{ij}$ of the braid group $\mathbf{B}_n$ and of the pure braid group $\mathbf{P}_n$ respectively, are the braids drawn in Figure 3.1. In Figure 3.2, they also appear in a “bird’s eye view” as paths of configurations.

### 3.2. Basic theory of partitioned braids

For a partition $\lambda = (n_1, \dots, n_l)$ of an integer $n$, we denote by $\mathfrak{S}_\lambda$ the subgroup $\mathfrak{S}_{n_1} \times \dots \times \mathfrak{S}_{n_l}$ of the symmetric group $\mathfrak{S}_n$. Let us now consider the braid group $\mathbf{B}_n$ on $n$ strands, and the usual projection $\pi: \mathbf{B}_n \to \mathfrak{S}_n$.

DEFINITION 3.2. Let $n \ge 1$ be an integer, and let $\lambda = (n_1, \dots, n_l)$ be a partition of $n$. The corresponding partitioned braid group (also referred to as the group of $\lambda$-partitioned braids) is:

$$ \mathbf{B}_{\lambda} := \pi^{-1}(\mathfrak{S}_{\lambda}) = \pi^{-1}(\mathfrak{S}_{n_1} \times \dots \times \mathfrak{S}_{n_l}) \subseteq \mathbf{B}_n. $$
<figure><img src="image_6.png" /><figcaption>FIGURE 3.1. The standard generators $\sigma_i$ and $A_{ij}$ of, respectively, the braid group and the pure braid group.</figcaption></figure>

**LEMMA 3.3.** Let $\lambda = (n_1, \dots, n_l)$ be a partition of an integer $n \ge 1$. Then $\mathbf{B}_\lambda$ is the subgroup of $\mathbf{B}_n$ generated by:

* The $\sigma_\alpha$ for $1 \le \alpha \le n$ such that $\alpha$ and $\alpha+1$ are in the same block of $\lambda$.

* The $A_{\alpha\beta}$ for $1 \le \alpha < \beta \le n$ such that $\alpha$ and $\beta$ do not belong to the same block of $\lambda$.

**PROOF.** Consider the subgroup $G$ of $\mathbf{B}_n$ generated by these elements. Clearly, $G \subseteq \mathbf{B}_\lambda$, and we need to show that $G$ contains $\mathbf{B}_\lambda$. First, we see that $G$ contains $\mathbf{B}_{n_1} \times \dots \times \mathbf{B}_{n_l}$, which is generated by the chosen $\sigma_\alpha$. As a consequence, $\pi(G) = \mathfrak{S}_{n_1} \times \dots \times \mathfrak{S}_{n_l}$. Then, $G$ also contains the $A_{\alpha\beta}$, for all $1 \le \alpha < \beta \le n$. Indeed, the $A_{\alpha\beta}$ missing in the list of the statement are the ones with $\alpha$ and $\beta$ in the same block of $\lambda$, which are exactly the ones belonging to $\mathbf{B}_{n_1} \times \dots \times \mathbf{B}_{n_l}$. Thus, the pure braid group $\mathbf{P}_n$ is contained in $G$. Now, if $\beta$ is any $\lambda$-partitioned braid, then $\pi(\beta) \in \mathfrak{S}_{n_1} \times \dots \times \mathfrak{S}_{n_l}$, hence we can choose $g \in G$ such that $\pi(g) = \pi(\beta)$. Then $g^{-1}\beta$ is a pure braid, thus $g^{-1}\beta \in G$, which implies that $\beta \in G$. $\square$

**REMARK 3.4.** The generating set of $\mathbf{B}_\lambda$ described in Lemma 3.3 is clearly redundant, since $A_{\alpha\beta}$ is conjugate to $A_{\gamma\delta}$ if $\alpha$ (resp. $\beta$) is in the same block as $\gamma$ (resp. $\delta$). For instance, in $\mathbf{B}_{2,2}$, we have $\sigma_1 A_{13} \sigma_1^{-1} = A_{23}$. Since it is more convenient for our purpose, we prefer to keep it that way. Notice however that it is easy to extract a minimal set of generators: using the computation of $\mathbf{B}_\lambda^{ab}$ below, it is easy to show that keeping all the $\sigma_\alpha$ and choosing only one lift $A_{\alpha\beta}$ of each $a_{ij}$ (that is, one $A_{\alpha\beta}$ for each pair of blocks) does give a minimal set of generators of $\mathbf{B}_\lambda$. A similar remark (with a similar method for choosing minimal sets of generators) applies to the generating sets described later in the memoir for variants of partitioned braid groups.

We now compute $\mathbf{B}_\lambda^{ab}$, using the above generating set, together with the split projections corresponding to forgetting blocks of strands. These projections can be seen as particular cases of the projections from Proposition 6.15 below, applied to braids on the disc.

**PROPOSITION 3.5.** Let $\lambda = (n_1, \dots, n_l)$ be a partition of an integer $n \ge 1$. The abelianisation $\mathbf{B}_\lambda^{ab}$ is free abelian on the following basis:

* For each $i \in \{1, \dots, l\}$ such that $n_i \ge 2$, one generator $s_i$: this is the common class of the $\sigma_\alpha$ for $\alpha$ and $\alpha+1$ in the $i$-th block of $\lambda$.

* For each $i, j \in \{1, \dots, l\}$ such that $i < j$, one generator $a_{ij}$: this is the common class of the $A_{\alpha\beta}$ for $\alpha$ in the $i$-th block of $\lambda$ and $\beta$ in the $j$-th one.
PROOF. The abelianisation $\mathbf{B}_{\lambda}^{ab}$ is generated by the classes of the generators from Lemma 3.3. Moreover, we note that for any $i$, all the $\sigma_{\alpha}$ with $\alpha$ and $\alpha + 1$ in the $i$-th block of $\lambda$ (which exist only if $n_i \ge 2$) are conjugate to one another: for instance, if $n_1 \ge 3$, then $\sigma_2 = (\sigma_2\sigma_1)^{-1}\sigma_1(\sigma_2\sigma_1)$ is a conjugation relation inside $\mathbf{B}_{\lambda}$. Similarly, for each choice of $i < j$, all the $A_{\alpha\beta}$ with $\alpha$ in the $i$-th block and $\beta$ in the $j$-th one are conjugate to one another. Thus, the family described in the statement is well-defined and generates $\mathbf{B}_{\lambda}^{ab}$. Let us show that it is linearly independent, by using the projections obtained by forgetting strands.

Suppose that $\sum k_i s_i + \sum k_{ij} a_{ij} = 0$ for some integers $k_i$ ($i \le l$) and $k_{ij}$ ($i < j \le l$). Let us fix $i$ such that $n_i \ge 2$, and let us apply the canonical projection $\mathbf{B}_{\lambda}^{ab} \to \mathbf{B}_{n_i}^{ab} \cong \mathbb{Z}$ to the above relation. Since this projection kills all the generators save $s_i$, we get $k_i = 0$. This holds for all $i$, so we are left with the relation $\sum k_{ij} a_{ij} = 0$. To which, for any choice of $i < j$, we can apply the morphism $\mathbf{B}_{\lambda}^{ab} \to \mathbf{B}_{n_i, n_j}^{ab} \to \mathbf{B}_{n_i+n_j}^{ab} \cong \mathbb{Z}$. This kills all the $a_{kl}$, save $a_{ij}$ (which is sent to 2), hence $k_{ij} = 0$, whence the result. $\square$

### 3.3. The lower central series

This section is devoted to the proof of the following result, which states exactly when the LCS of the partitioned braid group stops. The proof requires different techniques for a number of different cases. We consider each case in turn and then combine the individual results into a proof of Theorem 3.6 at the very end of this section.

THEOREM 3.6. Let $n \ge 1$ be an integer, let $\lambda = (n_1, \dots, n_l)$ be a partition of $n$. The LCS of the partitioned braid group $\mathbf{B}_{\lambda}$:

* stops at $\Gamma_2$ if $n_i \ge 3$ for all $i$, save at most two indices for which $n_i = 1$.

* does not stop in all the other cases, except for $\mathbf{B}_2 \cong \mathbb{Z}$.

#### 3.3.1. The stable case: a disjoint support argument.

PROPOSITION 3.7. Let $n \ge 1$ be an integer, let $\lambda = (n_1, \dots, n_l)$ be a partition of $n$. Suppose that all the $n_i$ are at least 3. Then the LCS of $\mathbf{B}_{\lambda}$ stops at $\Gamma_2$.

PROOF. Consider the generating set for $\mathbf{B}_{\lambda}^{ab}$ described in Corollary 3.5. For any pair of such generators, it is possible to find lifts in $\mathbf{B}_{\lambda}$ having disjoint support (as mapping classes of the punctured disc), and thus commuting – see Figure 3.2 for the three cases that may arise. Then, we can apply Corollary 2.6 to prove our claim. $\square$

#### 3.3.2. Blocks of size 1.

LEMMA 3.8. If at least three blocks of the partition $\lambda$ are of size 1, then the LCS of $\mathbf{B}_{\lambda}$ does not stop.

PROOF. Under the hypothesis, there is a surjection $\mathbf{B}_{\lambda} \to \mathbf{B}_{1,1,1} \cong \mathbf{P}_3$ obtained by forgetting all the blocks save three blocks of size 1. The LCS of $\mathbf{P}_3 \cong \mathbf{F}_2 \times \mathbb{Z}$ does not stop, since it is an almost-direct product, and the LCS of $\mathbf{F}_2$ does not stop; see for instance [FR85]. As a consequence, the one of $\mathbf{B}_{\lambda}$ does not either by Lemma 1.1. $\square$

The cases with one or two blocks of size 1 (and still no blocks of size 2) are more difficult to handle. In both cases, we will have to use the following observation:
<figure><img src="image_7.png" /><figcaption>FIGURE 3.2. Choosing representatives with disjoint support for pairs of generators of the abelianisation of the partitioned braid group. Braces indicate points that lie in the same block of the partition.</figcaption></figure>

LEMMA 3.9. The quotient of $\mathbf{P}_3$ by the relation $[A_{13}, A_{23}] = 1$ is $\mathbf{P}_3^{ab} \cong \mathbb{Z}^3$.

PROOF. Let $N$ be the normal closure of $[A_{13}, A_{23}]$ in $\mathbf{P}_3$. We want to show that $N = \Gamma_2(\mathbf{P}_3)$. Clearly, $N \subseteq \Gamma_2(\mathbf{P}_3)$. To show the converse inclusion, we need to show that $\mathbf{P}_3/N$ is abelian. We check that the relations $A_{12}A_{13}A_{12}^{-1} = A_{23}^{-1}A_{13}A_{23}$ and $A_{12}^{-1}A_{23}A_{12} = A_{13}A_{23}A_{13}^{-1}$ hold in $\mathbf{P}_3$. As a consequence, modulo the relation $[A_{13}, A_{23}] = 1$ (that is, modulo $N$), we get $A_{12}A_{13}A_{12}^{-1} \equiv A_{13}$ and $A_{12}^{-1}A_{23}A_{12} \equiv A_{23}$. Thus $A_{12}$, $A_{13}$ and $A_{23}$ commute modulo $N$ and therefore $\mathbf{P}_3/N$ is abelian.
$\square$

PROPOSITION 3.10. Let $n \ge 1$ be an integer, let $\lambda = (1, n_2, \dots, n_l)$ be a partition of $n$, with $n_i \ge 3$ for $i \ge 2$. Then the LCS of $\mathbf{B}_\lambda$ stops at $\Gamma_2$.

PROOF. The case $l=2$ (i.e. $\lambda = (1, m)$ for some integer $m \ge 3$) works exactly like the stable case of Proposition 3.7. Namely, the two generators of the abelianisation do have lifts with disjoint supports, so that $\Gamma_2(\mathbf{B}_{1,m}) = \Gamma_3(\mathbf{B}_{1,m})$ by Corollary 2.6. This however does not work for $l > 2$.

Let us denote by $\mu$ the partition $(n_2, \dots, n_l)$ of $n-1$, so that $\lambda = (1, \mu)$. We are going to show that $\mathbf{B}_\lambda/\Gamma_\infty$ is abelian, which implies that $\Gamma_\infty = \Gamma_2$ for $\mathbf{B}_\lambda$.

It follows from Proposition 3.7 that $\Gamma_\infty = \Gamma_2$ for $\mathbf{B}_\mu$. As a consequence, the obvious morphism $\mathbf{B}_\mu \to \mathbf{B}_\lambda/\Gamma_\infty$ factors through $\mathbf{B}_\mu/\Gamma_2 \cong \mathbf{B}_\mu^{ab}$. Given the description of $\mathbf{B}_\mu^{ab}$ from Proposition 3.5, we have that modulo $\Gamma_\infty$, $\sigma_\alpha \equiv \sigma'_\alpha$ and $A_{\alpha\beta} \equiv A_{\alpha'\beta'}$ if $\alpha$ and $\alpha'$ (resp. $\beta$ and $\beta'$) are in the same block of $\mu$. Moreover, the corresponding classes $s_i$ and $a_{ij}$ commute with one another. In the same way, we can deduce from the case of $\mathbf{B}_{1,m}$ for $m \ge 3$ treated above that the class of a generator $A_{1\alpha}$ depends only on the block of $\alpha$. More precisely, we use the fact that the obvious morphism $\mathbf{B}_{1,n_i} \to \mathbf{B}_{1,\mu}/\Gamma_\infty$ factors through $\mathbf{B}_{1,n_i}^{ab}$, where all the $A_{1\alpha}$ are identified. Moreover, the corresponding $a_{1i}$ commutes with all the $s_p$ and $a_{pq}$ coming from $\mathbf{B}_\mu$, since they have lifts in $\mathbf{B}_\lambda$ with disjoint support.

We are left with showing that $a_{1i}$ commutes with $a_{1j}$ when $i < j$. This uses Lemma 3.9. Let us choose any $\alpha$ in the $i$-th block and any $\beta$ in the $j$-th one, and consider the morphism $\mathbf{P}_3 \to \mathbf{B}_\lambda$ induced by $1 \mapsto 1$, $2 \mapsto \alpha$ and $3 \mapsto \beta$. If we
compose it with the projection onto $\mathbf{B}_{\lambda}/\Gamma_{\infty}$, we get a morphism $f$ sending $A_{12}$ to $a_{1i}$, $A_{13}$ to $a_{1j}$, and $A_{23}$ to $a_{ij}$. Since $a_{ij}$ commutes with $a_{1j}$, $f$ sends $[A_{13}, A_{23}]$ to 1. Lemma 3.9 then implies that it factors through $\mathbf{P}_3^{ab}$, so that all the elements in its image commute, including $a_{1i}$ and $a_{1j}$. This finishes the proof that $\mathbf{B}_{\lambda}/\Gamma_{\infty}$ is abelian, whence equal to $\mathbf{B}_{\lambda}/\Gamma_2$. $\square$

**PROPOSITION 3.11.** Let $n \ge 1$ be an integer, let $\lambda = (1, 1, n_3, \dots, n_l)$ be a partition of $n$, with $n_i \ge 3$ for $i \ge 3$. Then the LCS of $\mathbf{B}_{\lambda}$ stops at $\Gamma_2$.

**PROOF.** Let $\mu$ denote the partition $(n_3, \dots, n_l)$ of $n-2$, so that $\lambda = (1, 1, \mu)$. It follows from Proposition 3.10 that $\Gamma_{\infty} = \Gamma_2$ for $\mathbf{B}_{1,\mu}$, hence both the obvious maps $\mathbf{B}_{1,\mu} \to \mathbf{B}_{\lambda}/\Gamma_{\infty}$ factor through $\mathbf{B}_{1,\mu}^{ab}$. From this, we deduce that $\mathbf{B}_{\lambda}/\Gamma_{\infty}$ is generated by elements $s_i$ ($3 \le i \le l$) and $a_{ij}$ ($1 \le i < j \le l$), and that the $a_{1j}$ (resp. the $a_{2j}$), the $s_p$ ($p \ge 1$) and the $a_{pq}$ ($q > p \ge 3$) commute with one another. Moreover, for all $i, j \ge 3$, $a_{1i}$ and $a_{2j}$ have lifts with disjoint support in $\mathbf{B}_{\lambda}$, so they commute (even when $i = j$).

We are left with showing that the class $a_{12}$ of $A_{12}$ commutes with all the other generators. A disjoint support argument shows that it commutes with the $s_k$ and $a_{kl}$ for $l > k \ge 3$. Now, for $i \ge 3$, let us choose $\alpha$ in the $i$-th block, and let us consider the morphism $\mathbf{P}_3 \to \mathbf{B}_{\lambda}$ induced by $1 \mapsto 1$, $2 \mapsto 2$ and $3 \mapsto \alpha$. If we compose it with the projection onto $\mathbf{B}_{\lambda}/\Gamma_{\infty}$, we get a morphism $f$ sending $A_{12}$ to $a_{12}$, $A_{13}$ to $a_{1i}$, and $A_{23}$ to $a_{2i}$. Since $a_{1i}$ commutes with $a_{2i}$, $f$ sends $[A_{13}, A_{23}]$ to 1. Lemma 3.9 then implies that it factors through $\mathbf{P}_3^{ab}$, so that all the elements in its image commute, showing that $a_{12}$ commutes with $a_{1i}$ and with $a_{2i}$. Thus, we have proved that $\mathbf{B}_{\lambda}/\Gamma_{\infty}$ is abelian, whence $\Gamma_{\infty}(\mathbf{B}_{\lambda}) = \Gamma_2(\mathbf{B}_{\lambda})$. $\square$

#### 3.3.3. Blocks of size 2.
When there is exactly one block of size 2 and no block of size 1, we get a complete description of the quotient of $\mathbf{B}_{\lambda}$ by its residue:

**PROPOSITION 3.12.** Let $n \ge 1$ be an integer, let $\lambda = (2, n_2, \dots, n_l)$ be a partition of $n$, with $n_i \ge 3$ if $i \ge 2$. Then $\mathbf{B}_{\lambda}/\Gamma_{\infty}$ decomposes as a direct product of $l(l-1)/2$ copies of $\mathbb{Z}$ with $\mathbb{Z}^{2(l-1)} \times \mathbb{Z}$, where $\mathbb{Z}$ acts via the involution exchanging the elements $e_{2i}$ and $e_{2i+1}$ of a basis of $\mathbb{Z}^{2(l-1)}$. In particular, if $l \ge 2$, then the LCS of $\mathbf{B}_{\lambda}$ does not stop.

**PROOF.** Let $\mu$ denote the partition $(n_2, \dots, n_l)$ of $n-2$, so that $\lambda = (2, \mu)$. Then the canonical projection $\mathbf{B}_{2,\mu} \to \mathfrak{S}_{2,\mu} \to \mathfrak{S}_2$ has $\mathbf{B}_{1,1,\mu}$ as its kernel. Moreover, $\Gamma_{\infty}(\mathbf{B}_{2,\mu})$ contains $\Gamma_{\infty}(\mathbf{B}_{1,1,\mu})$, which is equal to $\Gamma_2(\mathbf{B}_{1,1,\mu})$ by Proposition 3.11. We show that these are in fact equal. In order to do this, it is enough to show that $\mathbf{B}_{2,\mu}/\Gamma_2(\mathbf{B}_{1,1,\mu})$ is residually nilpotent. In fact, we are going to compute it completely.

First, let us remark that it makes sense to consider this quotient: $\Gamma_2(\mathbf{B}_{1,1,\mu})$ is a characteristic subgroup of $\mathbf{B}_{1,1,\mu}$, which is normal (of index 2) in $\mathbf{B}_{2,\mu}$, hence it is a normal subgroup of $\mathbf{B}_{2,\mu}$. Next, we can write $\mathbf{B}_{2,\mu}/\Gamma_2(\mathbf{B}_{1,1,\mu})$ as an extension:

$$ (3.1) \qquad \mathbf{B}_{1,1,\mu}^{ab} \stackrel{\sim}{\longrightarrow} \mathbf{B}_{2,\mu}/\Gamma_2(\mathbf{B}_{1,1,\mu}) \stackrel{\sim}{\longrightarrow} \mathfrak{S}_2. $$

We can use the method from Appendix B to get a presentation of the group $G := \mathbf{B}_{2,\mu}/\Gamma_2(\mathbf{B}_{1,1,\mu})$. Namely, we have a presentation of the kernel: $\mathbf{B}_{1,1,\mu}^{ab}$ is free abelian on the $s_i$ and the $a_{ij}$ indexed by the blocks of $(1, 1, \mu)$. We also know the action of $\mathfrak{S}_2$ on $\mathbf{B}_{1,1,\mu}^{ab}$ induced by conjugation by $\sigma_1$ in $\mathbf{B}_{2,\mu}$: it exchanges the $a_{1j}$ with the $a_{2j}$ with $j \ge 3$ and it acts trivially on all the other generators. Finally, we can lift
the only relation defining $\mathfrak{S}_2$ to $\sigma_1^2 = A_{12}$ in $\mathbf{B}_{2,\mu}$. As a consequence, we get the presentation:

$$ G = \left\{ \begin{array}{l} s, \\ s_i (3 \le i \le l+1), \\ a_{ij} (1 \le i < j \le l+1). \end{array} \middle| \begin{array}{l} \forall i, j, p, q, u, v, \\ \forall i \ge 1, \\ \forall j > i \ge 3, \\ \forall j \ge 3, \end{array} \begin{array}{l} [s_i, s_j] = [s_i, a_{pq}] = [a_{pq}, a_{uv}] = 1, \\ [s, s_i] = 1, \\ [s, a_{ij}] = 1, \\ sa_{1j}s^{-1} = a_{2j} \text{ and } sa_{2j}s^{-1} = a_{1j}, \\ s^2 = a_{12}. \end{array} \right\} $$

One can deduce from this presentation that $G$ decomposes as $\mathbb{Z}^N \times (\mathbb{Z}^{2(l-1)} \rtimes \mathbb{Z})$ where the first factor is free abelian on the $s_i$ ($i \ge 3$) and the $a_{ij}$ ($j > i \ge 3$) (hence $N = l(l-1)/2$), and the action of $\mathbb{Z}$ (free on $s$) on $\mathbb{Z}^{2(l-1)}$ (free abelian on the $a_{1j}$ and the $a_{2j}$) is given by $s$ exchanging the $a_{1j}$ and the $a_{2j}$. Checking that this holds is a matter of writing the appropriate split projections from the presentations of $G_l$ and its factors.

Finally, the decomposition of $G$ allows us to apply Proposition A.10 to compute its LCS. Namely, we apply it to $A = \langle a_{1j}, a_{2j} \rangle_{j \ge 3}$ (which is free abelian on these generators) endowed with the involution $\tau$ exchanging $a_{1j}$ with $a_{2j}$ for all $j$. Then $V = \mathrm{Im}(\tau - 1)$ is the free abelian subgroup generated by the $a_{1j} - a_{2j}$, and for $k \ge 2$, we have $\Gamma_k(G_l) = 2^{k-1}V$. In particular, this LCS does not stop. However its intersection is trivial: the group $G_l$ is residually nilpotent, which implies that $\Gamma_\infty(\mathbf{B}_{2,\mu}) = \Gamma_\infty(\mathbf{B}_{1,1,\mu})$, and finishes our proof. $\square$

**REMARK 3.13.** The Lie ring of $G$ (which identifies with the Lie ring of $\mathbf{B}_{2,\mu}$) can be completely computed, using Corollary A.11. Namely, it identifies with $\mathbb{Z}^N \times (L \times \mathbb{Z})$, where $L = \mathbb{Z}^{l-1} \oplus (\mathbb{Z}/2)^{l-1} \oplus (\mathbb{Z}/2)^{l-1} \oplus \dots$ and the action of the generator $t$ of $\mathbb{Z}$ on $L$ is via the degree-one map $\mathbb{Z}^{l-1} \to (\mathbb{Z}/2)^{l-1} \cong (\mathbb{Z}/2)^{l-1} \cong \dots$. In other words, as a Lie ring, $\mathcal{L}(\mathbf{B}_{2,\mu})$ admits the presentation via generators $t, X_1, \dots, X_{l-1}, Y_1, \dots, Y_N$ and relations:

$$ \left\{ \begin{array}{l} [Y_i, Y_j] = [Y_i, X_k] = [Y_i, t] = [X_k, X_l] = 0, \\ 2[t, X_i] = 0. \end{array} \right. $$

Let us now turn our attention to the case when there are two blocks of size 2 in the partition.

**PROPOSITION 3.14.** *The LCS of $\mathbf{B}_{2,2}$ does not stop.*

**PROOF.** Since $\mathbf{B}_{2,2}$ surjects onto $\mathbf{B}_{2,2}(\mathbb{S}^2)$, this is a direct consequence of Proposition 6.75 below, by an application of Lemma 1.1. Alternatively, one can adapt the proof of Proposition 6.75 to this case, getting that:

$$ \mathbf{B}_{2,2}/\langle A_{12}, A_{34}, \Gamma_2(\mathbf{P}_4) \rangle \cong (\mathbb{Z}^2)^{\otimes 2} \rtimes (\mathfrak{S}_2)^2, $$

where $\mathfrak{S}_2$ acts on $\mathbb{Z}^2$ by permutation of the factors. Then the methods of Appendix B can be used to compute completely the LCS of this group. $\square$

**COROLLARY 3.15.** *If at least two blocks of the partition $\lambda$ are of size 2, then the LCS of $\mathbf{B}_\lambda$ does not stop.*

**PROOF.** Under the hypothesis, there is a surjection $\mathbf{B}_\lambda \to \mathbf{B}_{2,2}$. Thus, this corollary is obtained from a direct application of Lemma 1.1. $\square$
**3.3.4. Blocks of size 1 and 2: study of $\mathbf{B}_{1,2}$.** We use that $\mathbf{B}_{1,2}$ is isomorphic to the Artin group of type $\mathbf{B}_2$, a classical fact of which we give a proof, for the sake of completeness.

**LEMMA 3.16.** *The group $\mathbf{B}_{1,2}$ is isomorphic to the Artin group of type $\mathbf{B}_2$, that is, to $G = \langle \sigma, x \mid (\sigma x)^2 = (x\sigma)^2 \rangle$. As a consequence, it is residually nilpotent, but not nilpotent. In particular, its LCS does not stop.*

**PROOF.** On the one hand, we can re-write the presentation of $G$ as:

$$G = \langle \sigma, x, y \mid (\sigma x)^2 = (x\sigma)^2, y = \sigma x \sigma^{-1} \rangle.$$

Then, modulo the second relation (which we conveniently rewrite $y\sigma = \sigma x$), the first one is equivalent to $(y\sigma)^2 = x(y\sigma)\sigma$, and in turn to $\sigma y\sigma^{-1} = y^{-1}xy$. Thus:

$$G = \langle \sigma, x, y \mid \sigma x \sigma^{-1} = y, \sigma y \sigma^{-1} = y^{-1} x y \rangle.$$

On the other hand, the projection $\mathbf{B}_{1,2} \to \mathbf{B}_2 \cong \mathbb{Z}$ splits, and its kernel identifies with the fundamental group of the disc minus two points, which is free on two generators $x$ and $y$. In fact, the corresponding action of $\mathbf{B}_2 \cong \langle \sigma \rangle$ on $\mathbf{F}_2$ is the usual Artin action. Which means exactly that the above relations are true in $\mathbf{B}_{1,2}$. Whence a surjection of $G$ onto $\mathbf{B}_{1,2}$, which induces a diagram with (split) short exact rows:

<figure><img src="image_8.png" /></figure>

Using the freeness of their target, one sees that the left and right vertical maps must be isomorphisms, hence so is the middle map, by the Five Lemma. The rest of the statement is then a reformulation of Proposition A.21. □

**REMARK 3.17.** A more precise result is given by Proposition A.23, which describes the Lie ring of the group $G$ (hence of $\mathbf{B}_{1,2}$). Notice, however, that this difficult calculation is not needed if one only wants to see that its LCS does not stop; see Remarks A.20 and A.22.

**COROLLARY 3.18.** *If the partition $\lambda$ has both a block of size 1 and a block of size 2, then the LCS of $\mathbf{B}_\lambda$ does not stop.*

**PROOF.** Apply Lemma 1.1 to a surjection $\mathbf{B}_\lambda \to \mathbf{B}_{1,2}$. □

<figure><img src="image_9.png" /></figure>

**PROOF OF THEOREM 3.6.** The first statement consists of Propositions 3.7, 3.10 and 3.11. The second one consists of the cases where $\lambda$ has at least three blocks of size 1 (Lemma 3.8), exactly one block of size 2 together with blocks of size at least 3 (Proposition 3.12), at least two blocks of size 2 (Corollary 3.15) or at least one block of size 1 and one block of size 2 (Corollary 3.18). □

<figure><img src="image_10.png" /></figure>
<figure><img src="image_11.png" /></figure>
CHAPTER 4

Virtual and welded braids

In this chapter, we study the LCS of the *virtual*, *welded* and *extended welded* braid groups, which are generalisations of the classical braid groups. We first review the definitions and properties of these three groups. Then we discuss the notion of support for their elements. Finally, we come to the study of the LCS of each of these groups, whose description is given, respectively, in Propositions 4.36, 4.38 and 4.42. We also study the LCS for the *partitioned* versions of each of these groups; the corresponding results are summed up as Theorem 4.47.

**4.1. Recollections**

Before diving into the study of their LCS, we review the definitions of the
groups involved and recall their basic properties.

NOTATION 4.1. For clarity, in group presentations, we write $a \rightleftharpoons b$ for the relation saying that $a$ and $b$ commute.

**4.1.1. Virtual braids.** Virtual knots were introduced by Kauffman at the end of the 1990s [Kau99]. Of course, the definition of virtual braids came with it, and the defining relations for the virtual braid group first appeared in [Kau00].

4.1.1.1. *Presentations by generators and relations.* Virtual braids are best defined in the spirit of knot theory, using diagrams. However, the diagrammatic approach is nicely encoded by a group presentation, which makes for a convenient definition, which is both concise and well-suited to our purposes:

DEFINITION 4.2 ([Ver01, §2]). The virtual braid group $\mathbf{vB}_n$ is defined by generators $\sigma_1, \dots, \sigma_{n-1}, \tau_1, \dots, \tau_{n-1}$, and relations:

$$
\left\{
\begin{array}{l@{\qquad}l@{\qquad}l}
\text{relations of the classical braid group } \mathbf{B}_n \text{ for the } \sigma_i, & & \\
\text{relations of the symmetric group } \mathfrak{S}_n \text{ for the } \tau_i, & & \\
\text{mixed relations (R1) and (R2),} & &
\end{array}
\right.
$$

where the latter are:

$$
(4.1) \qquad \left\{
\begin{array}{ll}
(R1) & \sigma_i \rightleftharpoons \tau_k \quad \text{if } |i-k| \ge 2; \\
(R2) & \tau_i \tau_{i+1} \sigma_i = \sigma_{i+1} \tau_i \tau_{i+1} \quad \text{if } 1 \le i \le n-2.
\end{array}
\right.
$$

From the presentation, one sees immediately that there is a well-defined projec-
tion $\pi: \mathbf{vB}_n \to \mathfrak{S}_n$ sending both $\sigma_i$ and $\tau_i$ to the transposition $(i, i+1) \in \mathfrak{S}_n$. The
kernel of this projection is denoted by $\mathbf{vP}_n$ and called the *pure virtual braid group*.
Furthermore, this projection is split, via $(i, i+1) \mapsto \tau_i$. Thus, the $\tau_i$ generate a
copy of $\mathfrak{S}_n$ inside $\mathbf{vB}_n$, and we get a semi-direct product decomposition:

$$
\mathbf{vB}_n \cong \mathbf{vP}_n \times \mathfrak{S}_n.
$$
Using the Reidemeister-Schreier method, Bardakov gave the following presentation of the finite-index subgroup $\mathbf{vP}_n$ of $\mathbf{vB}_n$:

**PROPOSITION 4.3 ([Bar04]).** The pure virtual braid group $\mathbf{vP}_n$ admits the presentation given by generators $\chi_{ij}$, for $1 \le i \ne j \le n$, subject to the following relations, for $i, j, k, l$ pairwise distinct:

$$ (4.2) \qquad \left\{ \begin{array}{l} (\mathrm{vPR1}) \quad \chi_{ij} \iff \chi_{kl}; \\ (\mathrm{vPR2}) \quad \chi_{ij}\chi_{ik}\chi_{jk} = \chi_{jk}\chi_{ik}\chi_{ij}. \end{array} \right. $$

Here, the generator $\chi_{12}$ is $\tau_1\sigma_1$, and $\chi_{ij} = \sigma\chi_{12}\sigma^{-1}$, where $\sigma \in \mathfrak{S}_n \subset \mathbf{vB}_n$ is any permutation sending $i$ to 1 and $j$ to $n$. In particular, it is easy to deduce from the relations that $\chi_{i,i+1} = \tau_i\sigma_i$.

**COROLLARY 4.4.** $\mathbf{vP}_n^{ab}$ is free on the classes of the $\chi_{ij}$ for $1 \le i \ne j \le n$.

**COROLLARY 4.5.** The morphism $\iota: \mathbf{vB}_n \to \mathbf{vB}_{n+1}$ sending $\sigma_i$ to $\sigma_i$ and $\tau_i$ to $\tau_i$ is injective.

**PROOF.** The morphism $\iota$ is clearly well-defined. To show that it is injective, we notice that it restricts to the morphism from $\mathbf{vP}_n$ to $\mathbf{vP}_{n+1}$ sending $\chi_{ij}$ to $\chi_{ij}$. Proposition 4.3 ensures that this morphism has a well-defined retraction, which kills the $\chi_{i,n+1}$ and the $\chi_{n+1,i}$ and sends $\chi_{ij}$ to $\chi_{ij}$ if $i, j \le n$. From this, one infers easily that $\iota$ itself is an injection from $\mathbf{vB}_n = \mathbf{vP}_n \times_{\mathfrak{S}_n} \mathbf{vB}_{n+1} = \mathbf{vP}_{n+1} \times_{\mathfrak{S}_{n+1}} \mathbf{vB}_{n+1}$. $\square$

**REMARK 4.6.** With the diagrammatic interpretation introduced just below, $\iota$ adds a strand away from the other ones, whilst the retraction appearing in the proof corresponds to forgetting the last strand of pure (virtual) braid diagrams.

4.1.1.2. *A diagrammatic approach.* Let us now review how the above group presentation relates to the usual virtual braid diagrams.

A virtual braid diagram is made of $n$ curves in the plane, which are monotonous with respect to one direction (which we choose to be the vertical one), with fixed endpoints and only a finite number of singularities, which are transverse double points, plus some crossing information at the double points. Namely, the different types of crossings are depicted in Figure 4.1: at the classical crossings, one gives over/under information, and no crossing information is associated with virtual crossings.

**DEFINITION 4.7.** A diagrammatic virtual braid is a virtual braid diagram up to planar isotopies and:

* the classical and virtual Reidemeister-II and Reidemeister-III moves from Figures 4.3 and 4.4;

* the mixed Reidemeister-III move (V) from Figure 4.5.

The product of two diagrammatic virtual braids is induced by stacking diagrams on top of one another. This makes the set of diagrammatic virtual braids into a monoid. This monoid is easily seen to be generated by the elements $\sigma_i^{\pm 1}$ and $\tau_i$ depicted in Figure 4.2: every virtual braid diagram is isotopic to a stacking of diagrams with only one crossing. Then the Reidemeister-II moves (Figure 4.3) say that $\sigma_i^{-1}$ is indeed inverse to $\sigma_i$ and that $\tau_i^2 = 1$, so that the monoid is in fact a group. Finally, the relations of the presentation of $\mathbf{vB}_n$ correspond exactly to the Reidemeister-III moves (Figures 4.4 and 4.5) and to planar isotopy of diagrams. It
<figure><img src="image_12.png" /><figcaption>FIGURE 4.1. Crossings and decorations in braid diagrams.</figcaption></figure>

is in fact not very hard to turn this argument into a real proof that the group of diagrammatic virtual braids identifies with $\mathbf{vB}_n$.

CONVENTION 4.8. We fix the convention that reading a braid word from left to right corresponds to reading a braid diagram from top to bottom.

<figure><img src="image_13.png" /><figcaption>FIGURE 4.2. Diagrams for generators of $\mathbf{vB}_n$.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>FIGURE 4.3. Reidemeister-II moves.</figcaption></figure>

REMARK 4.9. There are variants of the above Reidemeister moves, which are consequences of the ones that we wrote. For example, the variant of (V) where the classical crossing is reversed follows from (V) and two instances of $(R_{II})$. Note that such deductions can also be seen as implications between relations in the group $\mathbf{vB}_n$. In fact, the set of moves that do not change the virtual braid is stable under some symmetries that correspond to transformations of the group. Namely, it is
<figure><img src="image_15.png" /><figcaption>FIGURE 4.4. Reidemeister-III moves.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>FIGURE 4.5. The mixed Reidemeister-III move.</figcaption></figure>

stable under top-bottom mirror image (which corresponds to passing to the inverses in $vB_n$), under left-right mirror image (which corresponds to the automorphism of $vB_n$ defined by $\sigma_i \mapsto \sigma_{n-i}$ and $\tau_i \mapsto \tau_{n-i}$) and under flipping classical crossings (which corresponds to the automorphism sending $\sigma_i$ to $\sigma_i^{-1}$ and fixing the $\tau_i$).

REMARK 4.10. The move (V) from Figure 4.5 and its variants alluded to in the previous remark can be encompassed in a general *detour move* that says that a piece of strand with only virtual crossings can be changed into any other such piece of strand with the same endpoints.

<figure><img src="image_17.png" /><figcaption>FIGURE 4.6. The standard generators $\chi_{ij}$ of pure virtual and welded braid groups. Notice that there are many other possible diagrams for them. For example, when $i < j$, any diagram with only one classical crossing where the $j$-th strand passes over the $i$-th strand from the left is a diagram of $\chi_{ij}$; see Remark 4.10.</figcaption></figure>
**4.1.2. Welded braids.** Welded braids were introduced at roughly the same time as virtual braids, by Fenn, Rimányi and Rourke in [FRR97], where it was shown that they could encode basis-conjugating automorphisms of the free group. However, they had appeared before under different guises. Their first appearance can be traced back to [Gol81], where they appeared as motions of unknotted circles in space and the link with basis-conjugating automorphisms of the free group was made. As basis-conjugating automorphisms of the free group, they were also studied by McCool in [McC86]. More recent important contributions to their study include [BWC07], [BH13] and the survey [Dam17] by Damiani, which gives a nice overview of the different interpretations of this group.

4.1.2.1. *Presentations by generators and relations.* We can give a definition of the welded braid group similar to Definition 4.2 above. In fact, we can see it as a quotient of the virtual braid group by one additional relation:

DEFINITION 4.11 ([FRR97]). The welded braid group $\mathbf{wB}_n$ is the quotient of the virtual braid group $\mathbf{vB}_n$ by the additional mixed relation:

$$ (4.3) \qquad (\text{R3}) \quad \sigma_i \sigma_{i+1} \tau_i = \tau_{i+1} \sigma_i \sigma_{i+1} \quad \text{if } 1 \le i \le n-2. $$

The projection $\pi: \mathbf{vB}_n \to \mathfrak{S}_n$ from §4.1.1.1 factorises through the quotient by this relation. In other words, the assignment $\sigma_i, \tau_i \mapsto (i, i+1) \in \mathfrak{S}_n$ also defines a projection from $\mathbf{wB}_n$ onto $\mathfrak{S}_n$. The kernel of this projection is denoted by $\mathbf{wP}_n$ and called the *pure welded braid group*. As in the case of virtual braids, $(i, i+1) \mapsto \tau_i$ defines a section of this projection, so that we get a semi-direct product decomposition:

$$ \mathbf{wB}_n \cong \mathbf{wP}_n \times_{\mathfrak{S}_n} \mathfrak{S}_n. $$

By a slight adaptation of the Reidemeister-Schreier method from [Bar04] (or an argument very similar to the proof of [BBD15, Th. 2.7]), one gets a presentation of $\mathbf{wP}_n$. In fact, the additional relation (R3) adds one relation to Bardakov's presentation of $\mathbf{vP}_n$ (Proposition 4.3), saying that $\chi_{ik}$ and $\chi_{jk}$ commute with each other. Modulo this relation, one can re-write the relation (vPR2) as a commutation relation (wPR3), to get:

PROPOSITION 4.12. The pure welded braid group $\mathbf{wP}_n$ admits the presentation given by generators $\chi_{ij}$, for $1 \le i \ne j \le n$, subject to the following relations, for i, j, k, l pairwise distinct:

$$ (4.4) \qquad \left\{ \begin{array}{l} (\text{wPR1}) \quad \chi_{ij} \iff \chi_{kl}; \\ (\text{wPR2}) \quad \chi_{ik} \iff \chi_{jk}; \\ (\text{wPR3}) \quad \chi_{ij} \iff \chi_{ik}\chi_{jk}. \end{array} \right. $$

This presentation first appeared in the work of McCool [McC86], which uses geometric group theory to show that it defines the subgroup of basis-conjugating automorphisms of the free group (see also §4.1.2.3 below).

We can deduce directly from this presentation that the abelianisation of $\mathbf{wP}_n$ is the same as the abelianisation of $\mathbf{vP}_n$ from Corollary 4.4:

COROLLARY 4.13. The canonical projection $\mathbf{vP}_n \to \mathbf{wP}_n$ induces an isomorphism $\mathbf{vP}_n^{ab} \cong \mathbf{wP}_n^{ab}$; both are free abelian on the classes $\bar{\chi}_{ij}$ for $1 \le i \ne j \le n$.

REMARK 4.14. Anticipating a bit the sequel, Corollary 4.13 can also be proved directly, without using the presentations from Theorems 4.3 and 4.12: showing
that the $\chi_{ij}$ generate $\mathbf{vP}_n$ (hence also $\mathbf{wP}_n$) is not difficult. Then, the linear independence of the classes of the $\chi_{ij}$ in $\mathbf{wP}_n^{ab}$ (hence also in $\mathbf{vP}_n^{ab}$) readily follows from the following fact: for any choice of $k \neq l$, there is a projection $\pi_{k,l}: \mathbf{wP}_n^{ab} \to \mathbf{wP}_2^{ab} \cong \mathbb{Z}^2$ (using $\mathbf{wP}_2 \cong \operatorname{Inn}(\mathbf{F}_2) \cong \mathbf{F}_2$ below) killing all the $\chi_{ij}$, save $\chi_{kl}$ and $\chi_{lk}$, which are sent to a basis of $\mathbb{Z}^2$. Such a projection is induced by the map $\mathbf{wP}_n \to \mathbf{wP}_2$ obtained by forgetting all the strands except the $j$-th and the $k$-th ones.

### REMARK 4.15.
Proposition 4.12 can be used to show that the morphism from $\mathbf{wB}_n$ to $\mathbf{wB}_{n+1}$ sending $\sigma_i$ to $\sigma_i$ and $\tau_i$ to $\tau_i$ is injective, by checking that its restriction to welded pure braids is a split injection, as in the proof of Corollary 4.5. This can also be deduced from the interpretation of welded braids as automorphisms of free groups (Corollary 4.20).

#### 4.1.2.2. A diagrammatic approach.
Welded braids can be represented as diagrams in exactly the same way as virtual braids above. The only thing that changes is that we add one Reidemeister move, called the *welded move* (W), saying that a strand may pass *above* a virtual crossing (Figure 4.7). But strands may not pass below virtual crossings, which should be thought of as *welded* to the plane below the diagram, whence the name of these braids.

### REMARK 4.16.
It is well-known [Nel01, Kan01] that adding the move (F) to welded knot theory makes the resulting knot theory trivial. However, the resulting link theory (the theory of *fused links* from [KL04]) is not quite trivial; this is connected to the fact that the quotient of $\mathbf{wB}_n$ by the relation (F) is not trivial either. It is however much less rich (and much easier to understand) than $\mathbf{wB}_n$. This quotient is studied in [BBD15], where its structure is completely determined.

### REMARK 4.17.
The variants of the move (W) from Figure 4.7 obtained by mirror images also hold, with the same arguments as in Remark 4.9. However, the variant obtained by changing the crossings does not hold, because its vertical mirror image would then hold too, and this would be the forbidden move (F), which does not hold (see Remark 4.22 below).

<figure><img src="image_18.png" /><figcaption>FIGURE 4.7. The welded move (W) and the forbidden move (F).</figcaption></figure>

#### 4.1.2.3. Welded braids as automorphisms of the free group.
One of the important features of welded braids, which we will use a lot in the sequel, is their interpretation as basis-conjugating automorphisms of the free group.
We use the following notations for automorphisms of the free group $\mathbf{F}_n$ on $n$ generators $x_1, \dots, x_n$:

$$ \tau_i: \begin{cases} x_i & \mapsto x_{i+1} \\ x_{i+1} & \mapsto x_i \end{cases} \quad \text{and} \quad \sigma_i: \begin{cases} x_i & \mapsto x_i x_{i+1} x_i^{-1} \\ x_{i+1} & \mapsto x_i \end{cases} $$

where it is implicit that the $x_k$ are fixed, for $k \neq i, i+1$.

**REMARK 4.18.** Our definition of $\sigma_i$ disagrees with the one in [Dam17]; instead, it corresponds to the inverse of the element called $\sigma_i$ by Damiani. This is because our Aut($\mathbf{F}_n$) is the opposite group of the group that she calls Aut($\mathbf{F}_n$), since we view composition of automorphisms of $\mathbf{F}_n$ from right to left, whereas she takes composition of automorphisms to go from left to right; see [Dam17, Rk. 6.3]. On the other hand, these considerations about the direction of composition do not affect the elements $\tau_i$ and $\rho_i$, since they are self-inverse. (We warn the reader, however, that our $\tau_i$ and $\rho_i$ correspond, respectively, to the elements denoted by $\rho_i$ and $\tau_i$ in [Dam17].)

It is easy to check that these elements satisfy the defining relations for $\mathbf{wB}_n$. The morphism from $\mathbf{wB}_n$ to Aut($\mathbf{F}_n$) that we obtain is in fact injective, as was shown by Fenn, Rimányi and Rourke towards the end of the century:

**THEOREM 4.19 ([FRR97]).** The morphism $\mathbf{wB}_n \to \text{Aut}(\mathbf{F}_n)$ sending $\tau_i$ and $\sigma_i$ to the corresponding automorphisms is a well-defined isomorphism onto its image, which consists of all automorphisms of the form:

$$ x_i \mapsto w_i x_{\sigma(i)} w_i^{-1}, \quad \text{with } w_1, \dots, w_n \in \mathbf{F}_n \text{ and } \sigma \in \mathfrak{S}_n. $$

**COROLLARY 4.20.** The morphism from $\mathbf{wB}_n$ to $\mathbf{wB}_{n+1}$ sending $\sigma_i$ to $\sigma_i$ and $\tau_i$ to $\tau_i$ is injective, and so is the morphism from $\mathbf{B}_n$ to $\mathbf{wB}_n$ (resp. to $\mathbf{vB}_n$) sending $\sigma_i$ to $\sigma_i$.

**PROOF.** All these morphisms are clearly well-defined. Theorem 4.19 identifies the first morphism with a restriction of the injection Aut($\mathbf{F}_n$) $\hookrightarrow$ Aut($\mathbf{F}_{n+1}$) that extends automorphisms by making them fix $x_{n+1}$, so this morphism is injective. Then the classical Artin representation identifies $\mathbf{B}_n$ with the stabiliser of $x_1 \cdots x_n \in \mathbf{F}_n$ in $\mathbf{wB}_n \subset \text{Aut}(\mathbf{F}_n)$ (see for instance [Bir74, Cor. 1.8.3]), identifying the second morphism with an inclusion of subgroups of Aut($\mathbf{F}_n$), so the second one is also injective. Finally, the morphism from $\mathbf{B}_n$ to $\mathbf{vB}_n$ can be composed with the quotient onto $\mathbf{wB}_n$ to give the previous morphism, so it must be injective too. $\square$

**REMARK 4.21.** Like in Remark 4.6, all the morphisms of Corollary 4.20 and of its proof admit obvious diagrammatic interpretations.

**REMARK 4.22.** The above morphism from $\mathbf{wB}_n$ to Aut($\mathbf{F}_n$) can also be used to show that the forbidden move (F) from Figure 4.7 (or, equivalently, the associated relation) does not hold in $\mathbf{wB}_n$: it suffices to compute the images of both sides in Aut($\mathbf{F}_n$) and to check that they are distinct. This implies that the welded move (W) (also from Figure 4.7) does not hold in $\mathbf{vB}_n$; in other words, $\mathbf{wB}_n$ is different from $\mathbf{vB}_n$. Indeed, one can see that relations (W) and (F) are exchanged by the last automorphism of $\mathbf{vB}_n$ defined in Remark 4.9 (flipping classical crossings), followed by the previous one (left-right mirror image). Thus (W) holds in $\mathbf{vB}_n$ if and only if (F) does. And the latter does not, since it does not hold in the quotient $\mathbf{wB}_n$.
One can check easily that the morphism from Theorem 4.19 identifies the generators $\chi_{ij}$ of the pure welded braid group $\mathbf{wP}_n$ with the automorphisms:

$$ \chi_{ij}: x_k \mapsto \begin{cases} x_j x_i x_j^{-1} & \text{if } k = i \\ x_k & \text{else.} \end{cases} $$

Given the presentation of $\mathbf{wP}_n$ from Proposition 4.12, one can see that Theorem 4.19 is in fact quite directly equivalent to McCool's theorem [McC86] saying that the subgroup of automorphisms of the form $x_i \mapsto w_i x_i w_i^{-1}$ admits the presentation from Proposition 4.12. The equivalence between the two statements boils down to the fact that $\mathbf{wB}_n/\mathbf{wP}_n \cong \mathfrak{S}_n$ is isomorphic to the quotient between the two subgroups of Aut($\mathbf{F}_n$), the latter being isomorphic to the group of permutation matrices via the canonical morphism Aut($\mathbf{F}_n$) $\to$ Aut($\mathbf{F}_n^{ab}$) $\cong$ GL$_n$(Z). The same kind of argument will be used later to deduce from the above statement a similar one for extended welded braids (see the proof of Theorem 4.27).

4.1.2.4. *A geometric interpretation.* Another very important feature of welded braids is their geometric interpretation as *loop braids*, which we describe now. These are motions of circles in 3-dimensional space, and can also be seen as tube braids in 4-dimensional space. For details, the reader in referred to [Dam17] and to the references therein (see in particular [Sat00, BH13]).

Let us consider the space $F_{n\mathfrak{S}^1}(\mathbb{D}^3)$ of configurations of $n$ unlinked, oriented circles in the unit 3-disc $\mathbb{D}^3$. By *oriented circles*, we mean that a configuration is defined up to *positive re-parametrisation* of the circles. The group $\mathfrak{S}_n$ acts on it by permuting the copies of $\mathfrak{S}^1$. Then the fundamental groups of $F_{n\mathfrak{S}^1}(\mathbb{D}^3)$ and $F_{n\mathfrak{S}^1}(\mathbb{D}^3)/\mathfrak{S}_n$ identify respectively with $\mathbf{wP}_n$ and $\mathbf{wB}_n$. Precisely, the correspondence between loops in $F_{n\mathfrak{S}^1}(\mathbb{D}^3)/\mathfrak{S}_n$ and the above generators of $\mathbf{wB}_n$ is given as follows:

* $\tau_i$ corresponds to swapping the circles $i$ and $i+1$ without either of them passing through the other;

* $\sigma_i$ corresponds to swapping the circles $i$ and $i+1$ while the $(i+1)$-th one goes through the $i$-th one in the direction determined by the orientation of the latter.

In fact, this identification is classically proven with automorphisms of free groups as an intermediate object. Namely, on the one hand, Theorem 4.19 identifies the combinatorial $\mathbf{wB}_n$ (with our definition via generators and relations, or via diagrams) to a subgroup of Aut($\mathbf{F}_n$). On the other hand, there is a morphism from the geometric $\mathbf{wB}_n$ to Aut($\mathbf{F}_n$), which is the Artin representation, adapted to this context. Precisely, it is a natural action of $\pi_1(F_{n\mathfrak{S}^1}(\mathbb{D}^3)/\mathfrak{S}_n)$ on the fundamental group of $\mathbb{R}^3$ with $n$ unlinked circles removed. This morphism is in fact an isomorphism onto its image [Gol81, Th. 5.3], which is the same subgroup of Aut($\mathbf{F}_n$), whence the result.

**REMARK 4.23.** We have been a little imprecise about what we mean by configurations of $n$ unlinked, oriented circles. One natural meaning is the space of all embeddings of $n$ disjoint copies of $S^1$ into the interior of $\mathbb{D}^3$ whose image is an unlink, equipped with the Whitney topology, quotiented by the action of orientation-preserving diffeomorphisms of the circles. Another natural meaning is the subspace of this space consisting of all such embedded unlinks where the embedded circles are *rigid*, in the sense that they are rotations, dilations and translations of the equator
$S^1 \subset S^2 = \partial\mathbb{D}^3$. The latter space has the advantage of being a finite-dimensional manifold, and it makes sense to speak of the unit vector normal to a circle of a configuration in this context. We will freely move between these two viewpoints in light of the theorem of Brendle and Hatcher [BH13, Th. 4.1], which says that these two models are homotopy equivalent.

**4.1.3. Extended welded braids.** The group $\tilde{w}\mathbf{B}_n$ of extended welded braids is slightly bigger than $w\mathbf{B}_n$, which it contains as a finite-index subgroup.

**4.1.3.1. Presentation by generators and relations.** Again, the definition of extended welded braids is motivated by both geometry and algebra, but we choose to define them using the group presentation. Precisely, it is a quotient of $w\mathbf{B}_n*(\mathbb{Z}/2)^n$ by some relations.

**DEFINITION 4.24.** The extended welded braid group $\tilde{w}\mathbf{B}_n$ admits a presentation with generators and relations of $w\mathbf{B}_n$ plus additional generators $\{\rho_1, \dots, \rho_n\}$ satisfying the relations of the abelian group $(\mathbb{Z}/2\mathbb{Z})^n$ and five extra mixed relations:

$$ (4.5) \qquad \left\{ \begin{array}{ll} (\text{R4}) & \sigma_i \stackrel{\leftarrow}{\leftrightarrow} \rho_k \quad \text{if } k \notin \{i, i+1\}; \\ (\text{R5}) & \tau_i \stackrel{\leftarrow}{\leftrightarrow} \rho_k \quad \text{if } k \notin \{i, i+1\}; \\ (\text{R6}) & \sigma_i \rho_i = \rho_{i+1} \sigma_i \quad \text{if } i \in \{1, \dots, n-1\}; \\ (\text{R7}) & \tau_i \rho_i = \rho_{i+1} \tau_i \quad \text{if } i \in \{1, \dots, n-1\}; \\ (\text{R8}) & \sigma_i \rho_{i+1} = \rho_i \tau_i \sigma_i^{-1} \tau_i \quad \text{if } i \in \{1, \dots, n-1\}. \end{array} \right. $$

The assignments $\sigma_i, \tau_i \mapsto (0, (i, i+1))$ and $\rho_i \mapsto (e_i, id)$ define a projection from $w\mathbf{B}_n$ onto the hyperoctahedral group $W_n := (\mathbb{Z}/2) \rtimes \mathfrak{S}_n = (\mathbb{Z}/2)^n \rtimes \mathfrak{S}_n$. One can adapt again the use of the Reidemeister-Schreier method from [Bar04] to get a presentation of the kernel. In fact, no additional generator or relation appears in the process, compared to the case of welded braids: one gets exactly the presentation of Theorem 4.4, so this kernel is exactly $w\mathbf{P}_n$, and we get a decomposition:

$$ \tilde{w}\mathbf{B}_n \cong w\mathbf{P}_n \times W_n. $$

**REMARK 4.25.** This is one way to show that the obvious morphism from $w\mathbf{B}_n$ to $\tilde{w}\mathbf{B}_n$ is injective using the group presentations (recall that $w\mathbf{B}_n \cong w\mathbf{P}_n \times \mathfrak{S}_n$). Alternatively, one could apply the Reidemeister-Schreier method directly to the image $G$ of $w\mathbf{B}_n$ in $\tilde{w}\mathbf{B}_n$: using that relations (R4) to (R8) allow one to “push the $\rho_i$ to the left” in words, and using the above projection to $W_n$ to distinguish them, one can see that the words $\rho_1^{\epsilon_1} \cdots \rho_n^{\epsilon_n}$ (with $\epsilon_i \in \{0, 1\}$) are a set of representatives for the quotient $\tilde{w}\mathbf{B}_n/G$ (which is not a group). Moreover, this set of representatives satisfies the Schreier condition, and by a very easy application of the Reidemeister-Schreier method, one gets that $G$ has the same presentation as $w\mathbf{B}_n$.

**4.1.3.2. A diagrammatic approach.** Extended welded braids can be represented by the same diagrams as $w\mathbf{B}_n$, with additional decorations (marked points) on the strands (see Figure 4.1). To the Reidemeister moves listed above for $w\mathbf{B}_n$, one adds a list of moves saying how the decorations interact with other decorations and with the different types of crossings: the four extended moves depicted in Figures 4.9 and 4.10. Exactly the same reasoning as above shows that the group of such diagrams up to planar isotopy and this list of moves is isomorphic to $\tilde{w}\mathbf{B}_n$, where the generator $\rho_i$ corresponds to the diagram from Figure 4.8.
<figure><img src="image_19.png" /><figcaption>FIGURE 4.8. The generator $\rho_i$</figcaption></figure>

REMARK 4.26. As above, the set of moves that do not change the underlying extended welded braids is stable by taking mirror images, for the same reasons as in Remark 4.17 (the automorphism encoding left-right mirror image being extended by $\rho_i \mapsto \rho_{n+1-i}$).

<figure><img src="image_20.png" /><figcaption>FIGURE 4.9. Three of the four extended welded moves</figcaption></figure>

4.1.3.3. Extended welded braids as automorphisms of the free group. The above morphism from $\mathbf{w}\mathbf{B}_n$ to Aut($\mathbf{F}_n$) can be extended to $\tilde{\mathbf{w}}\mathbf{B}_n$ by sending $\rho_i \in \tilde{\mathbf{w}}\mathbf{B}_n$ to the automorphism $\rho_i: x_i \mapsto x_i^{-1}$ of $\mathbf{F}_n$ (fixing the $x_k$ if $k \neq i$). The following result (see [Dam17, Th. 4.1]) is easily deduced from Theorem 4.19:

THEOREM 4.27. The morphism $\tilde{\mathbf{w}}\mathbf{B}_n \to \mathrm{Aut}(\mathbf{F}_n)$ sending $\tau_i$, $\sigma_i$ and $\rho_i$ to the corresponding automorphisms (as defined above) is a well-defined isomorphism onto its image, which consists of all automorphisms of the form:

$$x_i \mapsto w_i x_{\sigma(i)}^{\pm 1} w_i^{-1}, \quad \text{with } w_1, \dots, w_n \in \mathbf{F}_n \text{ and } \sigma \in \mathfrak{S}_n.$$

PROOF. Let $G$ denote the subgroup of Aut($\mathbf{F}_n$) consisting of automorphisms of the form described in the statement. It is easy to see that the morphism $f$ under scrutiny is well-defined and takes values in $G$. The canonical morphism $\mathrm{Aut}(\mathbf{F}_n) \to \mathrm{Aut}(\mathbf{F}_n^{\mathrm{ab}}) \cong \mathrm{GL}_n(\mathbb{Z})$ sends $G$ onto the group of signed permutation matrices, which is isomorphic to $W_n$, and this projection is split, $\langle \tau_i, \rho_i \rangle$ being the image of the obvious splitting. Moreover, the kernel $K$ of this projection is the subgroup of automorphisms of the form $x_i \mapsto w_i x_i w_i^{-1}$, and the morphism $f$

<figure><img src="image_21.png" /><figcaption>FIGURE 4.10. The fourth extended welded move</figcaption></figure>
restricts to the isomorphism between $\mathbf{wP}_n$ and $K$ from Theorem 4.19. Since $f$ also
induces an isomorphism $\tilde{\mathbf{wB}}_n/\mathbf{wP}_n \cong W_n \cong G/K$, it is an isomorphism between
$G=K \times W_n$ and $\tilde{\mathbf{wB}}_n = \mathbf{wP}_n \times W_n$. $\square$

4.1.3.4. A geometric interpretation. The geometric interpretation of welded braids from §4.1.2.4 holds for extended welded braids as well: they also correspond to motions of unknotted circles in space, if we allow the orientation of the circles to be flipped at the end of the motion. Precisely, the group $(\mathbb{Z}/2)^n$ acts on $F_{n\mathcal{S}^1}(\mathbb{D}^3)$, by changing the orientations of the different copies of $\mathcal{S}^1$. This action combines with the action of $\mathcal{S}_n$ from §4.1.2.4 into an action of the hyperoctahedral group $W_n = (\mathbb{Z}/2) \rtimes \mathcal{S}_n$. Then the fundamental group of $F_{n\mathcal{S}^1}(\mathbb{D}^3)/W_n$ identifies with $\tilde{\mathbf{wB}}_n$, where the correspondence between loops in $F_{n\mathcal{S}^1}(\mathbb{D}^3)/W_n$ and the above generators of $\tilde{\mathbf{wB}}_n$ is given as before for $\tau_i$ and $\sigma_i$, and $\rho_i$ corresponds to a loop that rotates the $i$-th circle by 180 degrees.

As in §4.1.2.4, this identification is classically proven with automorphisms of free group as an intermediate object. In fact, it can be seen as a slight enhancement of the identification of $\pi_1(F_{n\mathcal{S}^1}(\mathbb{D}^3)/\mathcal{S}_n)$ with the combinatorial $\mathbf{wB}_n$, in the same way as Theorem 4.27 above is an enhancement of Theorem 4.19. Precisely, the Artin representation extends naturally to an action of $\pi_1(F_{n\mathcal{S}^1}(\mathbb{D}^3)/W_n) =: G$ on $\mathbf{F}_n$. This gives a morphism $\alpha$ from $G$ to $\tilde{\mathbf{wB}}_n$, the latter being identified to a subgroup of Aut($\mathbf{F}_n$) via Theorem 4.27. The action of $W_n$ on $F_{n\mathcal{S}^1}(\mathbb{D}^3)$ is clearly free, so the quotient by $W_n$ is a covering, whence $\pi_1(F_{n\mathcal{S}^1}(\mathbb{D}^3)) =: H$ is a subgroup of $G$, and $G/H \cong W_n$. The restriction of $\alpha$ to $H$ is the isomorphism from [Gol81, Th. 5.3] (see §4.1.2.4) between $H$ and $\mathbf{wP}_n \subset \mathrm{Aut}(\mathbf{F}_n)$, and $\alpha$ is easily seen to induce an isomorphism between $G/H \cong W_n$ and $\tilde{\mathbf{wB}}_n/\mathbf{wP}_n \cong W_n$ (see the proof of Theorem 4.27 for the last isomorphism). Thus $\alpha$ is an isomorphism, as required.

Notice that the covering mentioned above is part of a commutative square of
coverings:

$$
\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
& F_{n\mathcal{S}^1}(\mathbb{D}^3) \arrow[r, "F_{n\mathcal{S}^1}(\mathbb{D}^3)/\mathcal{S}_n"] & \\
(4.6) \arrow[d, "F_{n\mathcal{S}^1}(\mathbb{D}^3)/(\mathbb{Z}/2)^n"] & (4.7) \arrow[d, "\tilde{\mathbf{w}}\mathbf{P}_n"]
\arrow[r, "F_{n\mathcal{S}^1}(\mathbb{D}^3)/W_n."] &
\end{tikzcd}
$$

The vertical maps in the diagram (4.6) can be thought of as “forgetting the orien-
tation of the circles”. If $\tilde{\mathbf{wP}}_n$ denotes the fundamental group of $F_{n\mathcal{S}^1}(\mathbb{D}^3)/(\mathbb{Z}/2)^n$, (the group of *pure* extended welded braids), by taking fundamental groups of (4.6), we get, directly from the geometric interpretation of these groups, a square of in-
jections:

$$
\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
& \mathbf{wP}_n \arrow[r] \arrow[d] & \mathbf{wB}_n \arrow[d] \\
(4.7) \arrow[r, "\mathbf{wP}_n"] & \mathbf{wB}_n \arrow[r, "\tilde{\mathbf{w}}P_n"] & \mathbf{wB}_n.
\end{tikzcd}
$$

REMARK 4.28. Recalling that $\mathbb{Z}/2 \cong \pi_1(\mathbb{P}^2)$ we can give a geometric interpre-
tation of the projection $\tilde{\mathbf{wB}}_n \to W_n = (\mathbb{Z}/2) \rtimes \mathcal{S}_n$ corresponding to the quotient
by $\mathbf{wP}_n$: a loop braid is sent to the collection of paths in $\mathbb{P}^2$ described by the unit
vector normal to each circle (see Remark 4.23), together with the permutation it
induces.
## 4.2. Notion of support of an element

In order to apply disjoint support arguments, we first need to discuss the notion of support for elements of these groups. We will use the following notion of support of an automorphism of $\mathbf{F}_n$.

**DEFINITION 4.29.** Let $\mathbf{F}_n$ be the free group on $n$ (fixed) generators $x_1, \dots, x_n$, and let $\varphi \in \mathrm{Aut}(\mathbf{F}_n)$. The support $\mathrm{Supp}(\varphi)$ of $\varphi$ is:

$\mathrm{Supp}(\varphi) := \{i \in \{1, \dots, n\} \mid \varphi(x_i) \neq x_i \text{ or } x_i \text{ appears in some } \varphi(x_j) \text{ for } j \neq i\}$,
where $x_i$ is said to appear in $w \in \mathbf{F}_n$ if $x_i^{\pm 1}$ is a letter in the reduced expression of $w$.

**EXAMPLE 4.30.** Using notations from §4.1.2.3 and §4.1.3.3, we have:

* $\mathrm{Supp}(\rho_\alpha) = \{\alpha\}$

* $\mathrm{Supp}(\sigma_\alpha) = \mathrm{Supp}(\tau_\alpha) = \{\alpha, \alpha + 1\}$

* $\mathrm{Supp}(\chi_{\alpha\beta}) = \{\alpha, \beta\}$

**FACT 4.31.** If $\varphi, \psi \in \mathrm{Aut}(\mathbf{F}_n)$ have disjoint support, then they commute.

**PROOF.** Observing that $\varphi(x_i) = x_i$ for $i \notin \mathrm{Supp}(\varphi)$, and that $\varphi(x_i)$ is a word in $\{x_j \mid j \in \mathrm{Supp}(\varphi)\}$ for $i \in \mathrm{Supp}(\varphi)$, it is easy to see that

$$ \psi(\varphi(x_i)) = \varphi(\psi(x_i)) = \begin{cases} \varphi(x_i) & \text{if } i \in \mathrm{Supp}(\varphi) \\ \psi(x_i) & \text{if } i \in \mathrm{Supp}(\psi) \\ x_i & \text{if } i \notin \mathrm{Supp}(\varphi) \cup \mathrm{Supp}(\psi), \end{cases} $$

for all $i \in \{1, \dots, n\}$. $\square$

<figure><img src="image_22.png" /></figure>

The groups $\mathbf{wB}_n$ and $\tilde{\mathbf{wB}}_n$ are naturally subgroups of $\mathrm{Aut}(\mathbf{F}_n)$ (see §4.1.2.3 and §4.1.3.3), so the above definition of support can be applied to (extended) welded braids. On the contrary, $\mathbf{vB}_n$ does not naturally embed into $\mathrm{Aut}(\mathbf{F}_n)$, so we need to use another definition of support for virtual braids. We do so using the diagrammatic description of virtual braids as equivalence classes of braid diagrams with both classical and virtual crossings (§4.1.1.2).

**DEFINITION 4.32.** Let $\varphi \in \mathbf{vB}_n$. The support $\mathrm{Supp}(\varphi) \subseteq \{1, \dots, n\}$ of $\varphi$ is the set of $i \in \{1, \dots, n\}$ such that either

* $i$ is not fixed by the permutation of $\{1, \dots, n\}$ induced by $\varphi$, or

* in every $n$-strand diagram representing $\varphi$, there is a classical crossing on the $i$-th strand.

Welded braids also admit a similar diagrammatic description (see §4.1.2.2), so one may define $\mathrm{Supp}(\varphi)$, for $\varphi \in \mathbf{wB}_n \subset \mathrm{Aut}(\mathbf{F}_n)$, either as in Definition 4.29 or as in Definition 4.32.

**LEMMA 4.33.** *These two definitions of $\mathrm{Supp}(\varphi)$ for $\varphi \in \mathbf{wB}_n \subset \mathrm{Aut}(\mathbf{F}_n)$ agree.*

**PROOF.** According to Definition 4.32, we have $i \notin \mathrm{Supp}(\varphi)$ if and only if $i$ is fixed by the permutation of $\{1, \dots, n\}$ induced by $\varphi$ and there is an $n$-strand diagram representing $\varphi$ with no classical crossings on the $i$-th strand. The first condition is equivalent to saying that $\varphi(x_i)$ is conjugate to $x_i$. The second condition splits into two statements: there is an $n$-strand diagram of $\varphi$ with no classical crossings involving the $i$-th strand as the upper strand and no classical crossings
involving the *i*-th strand as the lower strand. These two statements corresponds to saying that no $x_j$ (for $j \neq i$) appears in $\varphi(x_i)$ and that $x_i$ does not appear in any $\varphi(x_j)$ (for $j \neq i$). Combining the first of these statements with the condition that $\varphi(x_i)$ is conjugate to $x_i$, we obtain the condition that $\varphi(x_i) = x_i$. Hence $i \notin \text{Supp}(\varphi)$ if and only if $\varphi(x_i) = x_i$ and $x_i$ does not appear in any $\varphi(x_j)$ (for $j \neq i$). This is precisely Definition 4.29. $\square$

**REMARK 4.34.** Similarly, *extended welded braids* admit a diagrammatic description (§4.1.3.2), and so in this case one also has two possible definitions of $\text{Supp}(\varphi)$ using either Definition 4.32 or Definition 4.29. As in Lemma 4.33 for the non-extended case, these two definitions agree.

**FACT 4.35.** If two virtual braids have disjoint support, then they commute.

**PROOF.** Denote the given two virtual braids by $\varphi$ and $\psi$. It will suffice to show that they commute after conjugating them both by the same element of $\mathbf{vB}_n$. By conjugating with an appropriate element (involving only virtual crossings), we may therefore arrange that $\text{Supp}(\varphi) = \{1, \dots, r\}$ and $\text{Supp}(\psi) = \{r+1, \dots, r+s\}$ for some $r+s \le n$. By the virtual Reidemeister moves (specifically the *detour moves*), we may then arrange that we have diagrams $D_\varphi$, $D_\psi$ for $\varphi, \psi$ such that

$$D_{\varphi} = D'_{\varphi} \otimes \mathbb{I}_{n-r} \quad \text{and} \quad D_{\psi} = \mathbb{I}_{r} \otimes D'_{\psi} \otimes \mathbb{I}_{n-r-s},$$

where $\mathbb{I}_k$ denotes the trivial $k$-strand diagram and $\otimes$ denotes placing virtual braid diagrams side-by-side. By “vertically sliding the subdiagrams $D'_\varphi$ and $D'_\psi$”, we see that

$$ \begin{align*} D_{\varphi} \circ D_{\psi} &= (D'_{\varphi} \otimes \mathbb{I}_s \otimes \mathbb{I}_{n-r-s}) \circ (\mathbb{I}_r \otimes D'_{\psi} \otimes \mathbb{I}_{n-r-s}) \\ &= (\mathbb{I}_r \otimes D'_{\psi} \otimes \mathbb{I}_{n-r-s}) \circ (D'_{\varphi} \otimes \mathbb{I}_s \otimes \mathbb{I}_{n-r-s}) \\ &= D_{\psi} \circ D_{\varphi}. \tag*{\square} \end{align*} $$

There is another interpretation of the notion of *support* in the case of *welded* and *extended welded* braids, where we view elements as loops of configurations of unlinked, unknotted circles. This interpretation is closer to a usual geometric notion of support. In fact, it is exactly the same if one interprets these groups as mapping class groups, as in [Dam17]. Notice that we will often talk of the *support* of a mapping class, even if it is in fact not well-defined: only a *support* is, as the support of a homeomorphism in the mapping class. However, *having disjoint support* is well-defined for mapping classes (meaning that they have representatives with disjoint support), and we will often focus on the supports of particular representatives (the ones used to show that the mapping classes under scrutiny have disjoint support). For example, Figures 4.11 and 4.12 depict (representatives of) the same pair of elements, $\chi_{\beta,\alpha'}$ and $\sigma_\alpha$, representing generators $x_{ji}$ and $s_i$ of the abelianisation of the welded braid group. In each case the supports are shaded dark and light grey respectively: in the former case these are (disjoint) compact subsets of $\mathbb{R}^3$, in the latter case they are (disjoint) subcollections of strands.

## 4.3. Virtual braids

We now set out to study the lower central series of $\mathbf{vB}_n$. Let us first remark that we can compute $\mathbf{vB}_n^{ab}$ easily, using the presentation of Definition 4.2. Namely, for $n \ge 2$, $\mathbf{vB}_n^{ab} \cong \mathbb{Z} \oplus (\mathbb{Z}/2)$, where the first factor is generated by the common
<figure><img src="image_23.png" /><figcaption>FIGURE 4.11. The supports of $\sigma_\alpha$ and $\chi_{\beta,\alpha'}$ interpreted as loops of configurations of unlinks.</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>FIGURE 4.12. The supports of $\sigma_\alpha$ and $\chi_{\beta,\alpha'}$ interpreted diagrammatically.</figcaption></figure>

class $\sigma$ of the braid generators $\sigma_i$, and the second one is generated by the common class $\tau$ of the symmetric generators $\tau_i$.

PROPOSITION 4.36. The LCS of the virtual braid group $\mathbf{vB}_n$:

* stops at $\Gamma_2$ if $n \ge 4$.

* does not stop if $n = 2, 3$.

Moreover, $\mathbf{vB}_2$ is residually nilpotent, whereas $\mathbf{vB}_3$ is not.

PROOF. From the presentation of the virtual braid group $\mathbf{vB}_n$ (Definition 4.2), we deduce that in $\mathbf{vB}_n^{ab}$ all the $\sigma_i$ have the same class $\sigma$ and all the $\tau_i$ have the same class $\tau$.

Case 1: $n \ge 4$. Then $\sigma_1$ and $\tau_3$ represent respectively $\sigma$ and $\tau$, and they commute, since they have disjoint support; see Figure 4.13 for an illustration. Hence we can apply Corollary 2.6 to conclude that $\Gamma_2(\mathbf{vB}_n) = \Gamma_3(\mathbf{vB}_n)$.

Case 2: $n = 2$. The group $\mathbf{vB}_2 \cong \mathbb{Z} * \mathbb{Z}/2$ is residually nilpotent, but not nilpotent (Proposition A.19). In particular, its LCS does not stop (see also Remark A.20).

Case 3: $n = 3$. One can see directly from the presentation of $\mathbf{vB}_3$ that there is a well-defined surjection $\pi: \mathbf{vB}_3 \to \mathbf{vB}_2$, sending $\sigma_1$ and $\sigma_2$ to $\sigma_1$, and $\tau_1$ and $\tau_2$
to $\tau_1$. This induces a surjection $\mathcal{L}(\pi): \mathcal{L}(\mathbf{vB}_3) \to \mathcal{L}(\mathbf{vB}_2)$, hence the LCS of $\mathbf{vB}_3$ does not stop by Lemma 1.1.

Finally, the canonical map $\mathcal{G}_3 \to \mathbf{vB}_3$ sending $\tau_i$ to $\tau_i$ ($i \in \{1, 2\}$) splits, is thus injective, and $\mathcal{G}_3$ is not residually nilpotent. This implies that $\mathbf{vB}_3$ is not residually nilpotent. $\square$

We can describe precisely the behaviour of the LCS of $\mathbf{vB}_3$:

**PROPOSITION 4.37.** *The surjection $\pi: \mathbf{vB}_3 \to \mathbf{vB}_2$ described in the proof of Proposition 4.36 induces an isomorphism $\mathbf{vB}_3/\Gamma_\infty \cong \mathbf{vB}_2$. In particular, $\mathcal{L}(\pi)$ is an isomorphism between the associated Lie rings.*

**PROOF.** We first show that $\mathcal{L}(\pi)$ is an isomorphism. The surjection $\pi$ splits, a splitting $\iota$ being given by $\sigma_1 \mapsto \sigma_1$ and $\tau_1 \mapsto \tau_1$ and corresponds to adding a third strand. This splitting induces a surjection between abelianisations. Since the Lie rings associated to LCS are generated by their degree one, this implies that $\mathcal{L}(\iota)$ is surjective. Since $\mathcal{L}(\pi) \circ \mathcal{L}(\iota) = \mathcal{L}(\pi \circ \iota) = \text{id}$, we infer that $\mathcal{L}(\pi)$ and $\mathcal{L}(\iota)$ are isomorphisms $\mathcal{L}(\mathbf{vB}_3) \cong \mathcal{L}(\mathbf{vB}_2)$. Now, since $\mathbf{vB}_2 \cong \mathbb{Z} * \mathbb{Z}/2$ is residually nilpotent (Proposition A.19), the result follows directly from Lemma 1.10. $\square$

<figure><img src="image_25.png" /><figcaption>**FIGURE 4.13.** Disjoint-support representatives for the two generators of the abelianisation of the virtual braid group $\mathbf{vB}_n$ for $n \ge 4$. The supports are shaded in light grey.</figcaption></figure>

### 4.4. Welded braids

The definition of $\mathbf{wB}_n$ as a quotient of $\mathbf{vB}_n$ (Definition 4.11) allows us to deduce some results about the LCS of $\mathbf{wB}_n$ from their counterparts for $\mathbf{vB}_n$. This begins with noticing that $\mathbf{wB}_n^{ab} \cong \mathbf{vB}_n^{ab} \cong \mathbb{Z} \oplus (\mathbb{Z}/2)$.

**PROPOSITION 4.38.** *The LCS of the welded braid group $\mathbf{wB}_n$:*

* stops at $\Gamma_2$ if $n \ge 4$.

* does not stop if $n = 2, 3$.

Moreover, $\mathbf{wB}_2$ is residually nilpotent, whereas $\mathbf{wB}_3$ is not.

**PROOF.** *Case 1: $n \ge 4$.* The fact that $\Gamma_2(\mathbf{vB}_n) = \Gamma_3(\mathbf{vB}_n)$ directly implies that $\Gamma_2(\mathbf{wB}_n) = \Gamma_3(\mathbf{wB}_n)$. In fact, one can easily see that our disjoint support argument for $\mathbf{vB}_n$ can be used in $\mathbf{wB}_n$ too. See Figure 4.14 for an illustration from the point of view of “loop braids”; alternatively, see Figure 4.13 again, interpreted now as a diagram of welded braids instead of virtual braids.

*Case 2: $n = 2$. There is no additional relation: $\mathbf{wB}_2 \cong \mathbf{vB}_2 \cong \mathbb{Z} * \mathbb{Z}/2$.*

*Case 3: $n = 3$. We treat this tricky case as a separate result. Namely, Proposition 4.39 below gives a surjection from $\mathbf{wB}_3$ onto $\mathbb{Z}/2 * \mathbb{Z}/2$, whose LCS does not stop by Proposition A.16. The result thus follows from Lemma 1.1. $\square$*
<figure><img src="image_26.png" /><figcaption>FIGURE 4.14. Disjoint-support representatives for the two generators of the abelianisation of the welded braid group $wB_n$ for $n \ge 4$. The supports are shaded in light grey.</figcaption></figure>

The group $wB_3$ is a non-trivial quotient of $vB_3$, and our result on $vB_3$ cannot be transferred to $wB_3$, nor can we directly adapt the reasoning. Indeed, the surjection $vB_3 \to wB_2 \cong wB_2$ does not factor through the quotient by our additional relation, leaving us with no obvious surjection from $wB_3$ onto $wB_2$. However, we are still able to compute $wB_3/\Gamma_\infty$, and the situation is very similar to the case of virtual braids. In particular, the LCS does not stop, but $wB_3$ is not residually nilpotent.

**PROPOSITION 4.39.** The residue $\Gamma_\infty(wB_3)$ is normally generated by $\sigma_1\sigma_2^{-1}$ and $\tau_1\tau_2$. The quotient $G = wB_3/\Gamma_\infty$ has the following presentation:

$$G = \langle \sigma, \tau \mid \tau^2 = 1, \sigma^2\tau = \tau\sigma^2 \rangle.$$

Its centre $ZG$ is infinite cyclic, generated by $\sigma^2$, and:

$$G/ZG \cong \mathbb{Z}/2 * \mathbb{Z}/2.$$

**PROOF.** The canonical map $B_3 \to wB_3$ sends $\Gamma_\infty(B_3)$ to $\Gamma_\infty(wB_3)$. Since $\sigma_1\sigma_2^{-1} \in \Gamma_2(B_3) = \Gamma_\infty(B_3)$, we deduce that $\sigma_1\sigma_2^{-1} \in \Gamma_\infty(wB_3)$. The same holds with $\tau_1\tau_2$, which is the image of $\tau_1\tau_2 \in \Gamma_2(S_3) = \Gamma_\infty(S_3)$ by the canonical map $S_3 \to wB_3$. Thus, $\sigma_1\sigma_2^{-1}, \tau_1\tau_2 \in \Gamma_\infty(wB_3)$, implying that $wB_3/\Gamma_\infty$ is a quotient of the group $G$ obtained from $wB_3$ by adding the relations $\sigma_1 = \sigma_2$ and $\tau_1 = \tau_2$. We deduce from the usual presentation of $wB_3$ that $G$ is indeed described by the presentation of the statement. We are going to show that $G$ is residually nilpotent, which implies that $G$ is indeed equal to $wB_3/\Gamma_\infty$ (it is the biggest residually nilpotent quotient of $wB_3$). We first deduce from its presentation that the abelianisation of $G$ is given by:

$$G^{ab} \cong \mathbb{Z}\{\bar{\sigma}, \bar{\tau}\}/(2\bar{\tau}) \cong \mathbb{Z} \oplus \mathbb{Z}/2.$$

In particular, $\bar{\sigma}$ is not a torsion element of $G^{ab}$, which means that no non-trivial power of $\sigma$ belongs to $\Gamma_2(G)$.

Now, consider the element $\sigma^2 \in G$. It commutes with $\sigma$ and $\tau$, thus it is central. In particular, $\langle\sigma^2\rangle$ is a normal subgroup. Moreover, the quotient $G/\langle\sigma^2\rangle$ is described by:

$$G/\langle\sigma^2\rangle \cong \langle\sigma, \tau \mid \sigma^2 = \tau^2 = 1, \sigma^2\tau = \tau\sigma^2\rangle \cong \langle\sigma, \tau \mid \sigma^2 = \tau^2 = 1\rangle,$$

which is exactly $\mathbb{Z}/2 * \mathbb{Z}/2$. Proposition A.16 then implies that $G$ is residually nilpotent. Indeed, if $x \in \Gamma_\infty(G)$, then its image in $G/\langle\sigma^2\rangle \cong \mathbb{Z}/2 * \mathbb{Z}/2$ is in $\Gamma_\infty(\mathbb{Z}/2 * \mathbb{Z}/2) = \{1\}$. Thus $x \in \langle\sigma^2\rangle$. Since $\bar{\sigma}$ is not a torsion element in $G^{ab}$, we have $\langle\sigma^2\rangle \cap \Gamma_2 G = \{1\}$, whence $x = 1$.
Finally, we remark that $\mathbb{Z}/2*\mathbb{Z}/2$ is centreless, implying that $\mathbb{Z}G$ consists only of $\langle\sigma^2\rangle$. As $\bar{\sigma}$ is not a torsion element in $G^{ab}$, $\langle\sigma^2\rangle$ has to be infinite cyclic, and all our claims are proved. □

**REMARK 4.40.** Both $\mathbf{vB}_3$ and $\mathbf{wB}_3$ have non-stopping LCS, without being residually nilpotent. In this situation, one may continue the LCS to infinite ordinals by defining $\Gamma_{\alpha+1}(G) = [G, \Gamma_{\alpha}(G)]$ and $\Gamma_{\lambda}(G) = \bigcap_{\alpha<\lambda} \Gamma_{\alpha}(G)$ for limit ordinals $\lambda$. In particular, the first infinite term $\Gamma_{\omega}(G)$ is what we denote by $\Gamma_{\infty}(G)$. A result of Mal'cev [Mal49] says that groups may have LCS of any length: for any ordinal $\alpha$ there is a group $G_{\alpha}$ such that $\Gamma_{*}(G_{\alpha})$ stops precisely at $\alpha$. Such examples are in general “artificial”, but there are also naturally occurring groups $G$ whose LCS are long, meaning that $\Gamma_{\omega+1}(G) \neq \Gamma_{\omega}(G)$. For example, Levine [Lev91] gave an example of a finitely presented group whose LCS is long, and Cochran and Orr [CO98] gave many examples of 3-manifold groups whose LCS are long. In light of these results, one may wonder whether the LCS of $\mathbf{vB}_3$ and of $\mathbf{wB}_3$ are long.

In fact, in both cases, we have $[G, \Gamma_{\omega}(G)] = \Gamma_{\omega}(G)$, so their LCS are not long (they stop at $\omega$). Indeed, in both cases, $\Gamma_{\omega}(G)$ is normally generated by $\tau_1\tau_2^{-1}$ and $\sigma_1\sigma_2^{-1}$, belonging respectively to $\Gamma_2(\mathfrak{S}_3) = \Gamma_{\omega}(\mathfrak{S}_3)$ and $\Gamma_2(\mathbf{B}_3) = \Gamma_{\omega}(\mathbf{B}_3)$. Then:

$$ \tau_1\tau_2^{-1} \in \Gamma_3(\mathfrak{S}_3) = [\mathfrak{S}_3, \Gamma_2(\mathfrak{S}_3)] = [\mathfrak{S}_3, \Gamma_{\omega}(\mathfrak{S}_3)] \subseteq [G, \Gamma_{\omega}(G)]. $$

The same calculation implies that $[G, \Gamma_{\omega}(G)]$ contains $\sigma_1\sigma_2^{-1}$, so it must contain $\Gamma_{\omega}(G)$, as claimed. The same argument will lead to the same conclusion for $G = \tilde{\mathbf{w}}\mathbf{B}_3$; see Proposition 4.42 and Remark 4.43.

### 4.5. Extended welded braids

Let us now turn our attention to the LCS of $\tilde{\mathbf{w}}\mathbf{B}_n$. We first compute its abelianisation:

**LEMMA 4.41.** For $n \ge 2$, the abelianisation $\tilde{\mathbf{w}}\mathbf{B}_n^{ab}$ is isomorphic to $(\mathbb{Z}/2)^3$. A basis over $\mathbb{Z}/2$ is given by $\sigma, \tau$ and $\rho$, which are respectively the common class of the $\sigma_i$, the $\tau_i$ and the $\rho_i$.

**PROOF.** On can get this directly from the presentation of Definition 4.11. One can also see this as a particular case of Proposition 4.46 below. □

**PROPOSITION 4.42.** The LCS of the extended welded braid group $\tilde{\mathbf{w}}\mathbf{B}_n$:

* stops at $\Gamma_2$ if $n \ge 4$.

* does not stop if $n = 2, 3$.

**PROOF.** Case 1: $n \ge 4$. As in the proof of Proposition 4.36, we show that each pair of generators of $\tilde{\mathbf{w}}\mathbf{B}_n^{ab}$ may be represented by elements of $\tilde{\mathbf{w}}\mathbf{B}_n$ with disjoint support. For the first two generators in the list, this is possible since each of them may be supported inside a 3-ball in $\mathbb{D}^3$ containing two of the circles; see Figure 4.14. For one of the first two generators together with the third generator, this is possible even for $n \ge 3$, since the third generator may be supported in a 3-ball containing a single circle; see Figure 4.15.

**Case 2:** $n = 2$. For short, we denote $\sigma_1$ (resp. $\tau_1$) by $\sigma$ (resp. $\tau$). From the presentation of $\tilde{\mathbf{w}}\mathbf{B}_2$, we can see that the quotient by the relations $\rho_1 = \rho_2 = 1$ admits the presentation:

$$ \tilde{\mathbf{w}}\mathbf{B}_2/(\rho_1, \rho_2) = \langle\sigma, \tau \mid \tau^2 = 1, \sigma = \tau\sigma^{-1}\tau\rangle. $$
But this is a presentation of $\mathbb{Z} \times (\mathbb{Z}/2)$, whose LCS, computed in Corollary A.8, does not stop.

**Case 3:** $n = 3$. It is known that $\tilde{w}\mathbf{B}_3$ contains $\mathbf{B}_3$ as a subgroup (see for instance Remark 4.25 and Corollary 4.20), hence it cannot be residually nilpotent. In fact, $\Gamma_\infty(\tilde{w}\mathbf{B}_3)$ contains $\Gamma_\infty(\mathfrak{S}_3) = \Gamma_2(\mathfrak{S}_3)$ and $\Gamma_\infty(\mathbf{B}_3) = \Gamma_2(\mathbf{B}_3)$, which contain respectively $\tau_2\tau_1$ and $\sigma_2\sigma_1^{-1}$. Let us consider the quotient $G$ of $\tilde{w}\mathbf{B}_3$ by these two relations, and let us denote by $\tau$ (resp. $\sigma$) the common image of $\tau_1$ and $\tau_2$ (resp. $\sigma_1$ and $\sigma_2$) in $G$. We remark that in $G$, $\rho_2 = \tau_1\rho_1\tau_1$ and $\rho_3 = \tau_2\rho_2\tau_2$ are both identified with $\rho_1 = \tau_2\rho_1\tau_2 = \tau_1\rho_2\tau_1$, so we can also speak of their common class $\rho$. From the presentation of $\tilde{w}\mathbf{B}_3$, we get a presentation of $G$:

$$G = \langle \sigma, \tau, \rho \mid \rho^2 = \tau^2 = 1; \sigma, \tau \rightsquigarrow \rho; \sigma\rho = \rho\tau\sigma^{-1}\tau \rangle.$$

The second group of relations, saying $\rho$ is central, may be seen as *disjoint support* relations. Since $\rho$ is central, the last relation is equivalent to $\sigma = \tau\sigma^{-1}\tau$. As a consequence, this is a presentation of $\mathbb{Z}/2 \times (\mathbb{Z} \times (\mathbb{Z}/2))$, whose LCS does not stop, again thanks to Corollary A.8. $\square$

**REMARK 4.43.** In the third part of the proof, the fact that the quotient $G$ is residually nilpotent implies that it is the quotient by the whole of $\Gamma_\infty(\tilde{w}\mathbf{B}_3)$, which is thus normally generated by $\tau_2\tau_1$ and $\sigma_2\sigma_1^{-1}$. This also means that $\mathcal{L}(\tilde{w}\mathbf{B}_3) \cong \mathcal{L}(G)$, which is easy to compute completely; see Corollary A.8.

<figure><img src="image_27.png" /><figcaption>FIGURE 4.15. Disjoint-support representatives for two of the generators of the abelianisation of the extended welded braid group $\tilde{w}\mathbf{B}_n$ for $n \ge 3$. The supports are shaded in light grey.</figcaption></figure>

## 4.6. Partitioned virtual, welded and extended welded braids

We now turn our attention to partitioned virtual and welded braids, which are defined exactly in the same way as partitioned braids have been in Definition 3.2.

**4.6.1. Basic theory.** Let us consider the groups $\mathbf{vB}_n$, $\mathbf{wB}_n$ and $\tilde{w}\mathbf{B}_n$. Each one is endowed with a canonical projection $\pi$ onto $\mathfrak{S}_n$ (we use the same notation in the three cases). One can think of it as the replacement of every crossing in a diagram by virtual ones (and forgetting the mark points for extended welded braids); in terms of group presentations, it sends both $\sigma_i$ and $\tau_i$ to the transposition $(i, i+1)$, for each $i$ (and it kills the $\rho_i$).

**DEFINITION 4.44.** Let $n \ge 1$ be an integer and let $\lambda = (n_1, \dots, n_l)$ be a partition of $n$. The corresponding *partitioned virtual braid group* is:

$$\mathbf{vB}_{\lambda} := \pi^{-1}(\mathfrak{S}_{\lambda}) = \pi^{-1}(\mathfrak{S}_{n_1} \times \dots \times \mathfrak{S}_{n_l}) \subseteq \mathbf{vB}_{n}.$$
We define similarly the partitioned welded braid group $wB_λ := π^{-1}(\mathfrak{S}_λ)$ and the group of $λ$-partitioned extended welded braids $\tilde{w}B_λ = π^{-1}(\mathfrak{S}_λ)$.

Recalling that the elements $\chi_{\alpha\beta}$ introduced in §4.1.1.1 (see Figure 4.6) generate $vP_n$ (resp. $wP_n$), one can easily adapt the proof of Lemma 3.3 to the present context, giving a list of generators for $vB_λ$ (resp. $wB_λ$ and $\tilde{w}B_λ$):

LEMMA 4.45. Let $λ = (n_1, \dots, n_l)$ be a partition of an integer $n \ge 1$. Then $vB_λ$ (resp. $wB_λ$) is the subgroup of $vB_n$ (resp. $wB_n$) generated by:

* The $\sigma_α$ and the $\tau_α$ for $1 \le α \le n$ such that $α$ and $α+1$ are in the same block of $λ$.

* The $\chi_{\alpha\beta}$, for $1 \le α \ne β \le n$ such that $α$ and $β$ do not belong to the same block of $λ$.

Also, $\tilde{w}B_λ$ is the subgroup of $\tilde{w}B_n$ generated by the same set of generators, together with the $\rho_α$ for all $α \le n$.

In fact, the situation is even easier than the case of partitioned braids: for virtual (or welded) braids, the canonical projection onto $\mathfrak{S}_n$ splits, so that $vB_n \cong vP_n \times \mathfrak{S}_n$ (resp. $wB_n \cong wP_n \times \mathfrak{S}_n$); see §4.1.1.1 and §4.1.2.1. And this works for partitioned virtual/welded braids too, since this splitting restricts obviously to a splitting of $vB_λ \to \mathfrak{S}_λ$ (resp. of $wB_λ \to \mathfrak{S}_λ$), so that:

$$ (4.8) \qquad vB_λ \cong vP_n \times \mathfrak{S}_λ, $$

and likewise for welded braids. Extended welded braids behave much in the same way: the decomposition $\tilde{w}B_n \cong wP_n \times W_n$ from §4.1.3.1 restricts to a decomposition:

$$ (4.9) \qquad \tilde{w}B_λ \cong wP_n \times W_λ, $$

where $W_λ$ denotes the subgroup $(\mathbb{Z}/2) \rtimes \mathfrak{S}_λ$ of $W_n = (\mathbb{Z}/2) \rtimes \mathfrak{S}_n$.

We can use this, together with the classical calculation of $vP_n^{ab} \cong wP_n^{ab}$ from Corollary 4.13, to compute the abelianisations of $vB_λ$, $wB_λ$ and $\tilde{w}B_λ$:

PROPOSITION 4.46. Let $λ = (n_1, \dots, n_l)$ be a partition of an integer $n \ge 1$ and let $l'$ be the number of $i$ such that $n_i \ge 2$. Then:

$$ vB_λ^{ab} \cong wB_λ^{ab} \cong (\mathbb{Z}/2)^{l'} \times \mathbb{Z}^{l'} \times \mathbb{Z}^{l(l-1)} \quad \text{and} \quad \tilde{w}B_λ^{ab} \cong (\mathbb{Z}/2)^{l+2l'+l(l-1)}. $$

Precisely, the abelian group $vB_λ^{ab} \cong wB_λ^{ab}$ is generated by:

* For each $i \in \{1, \dots, l\}$ such that $n_i \ge 2$, one generator $s_i$ (resp. $t_i$), which is the common class of the $\sigma_α$ (resp. the $\tau_α$) for $α$ and $α+1$ in the $i$-th block of $λ$.

* For each $i, j \in \{1, \dots, l\}$ such that $i \ne j$, one generator $x_{ij}$, which is the common class of the $\chi_{\alpha\beta}$ for $α$ in the $i$-th block of $λ$ and $β$ in the $j$-th one.

The $t_i$ are a $\mathbb{Z}/2$-basis of the first factor, the $s_i$ are a $\mathbb{Z}$-basis of the second one, and the $x_{ij}$ are a $\mathbb{Z}$-basis of the third one.

A $(\mathbb{Z}/2)$-basis of $\tilde{w}B_λ^{ab} \cong (wB_λ^{ab} \otimes \mathbb{Z}/2) \times (\mathbb{Z}/2)^l$ is given by the same list of generators (which are of 2-torsion in $\tilde{w}B_λ^{ab}$), together with:

* For each $i \in \{1, \dots, l\}$, one generator $r_i$, which is the common class of the $\rho_α$ for $α$ in the $i$-th block of $λ$.
PROOF. We show our first statement for virtual braids; the exact same proof
works for $\mathbf{w}\mathbf{B}_\lambda$.

Applying Lemma 1.12 to the decomposition (4.8), we deduce that $\mathbf{v}\mathbf{B}_{\lambda}^{ab} \cong (\mathbf{v}\mathbf{P}_{n}^{ab})_{\mathfrak{S}_{\lambda}} \times \mathfrak{S}_{\lambda}^{ab}$. On the one hand, $\mathfrak{S}_{n_i}^{ab} \cong \mathbb{Z}/2$ (generated by the class of any transposition) for all $i$ such that $n_i \ge 2$, so that $\mathfrak{S}_{\lambda}^{ab} \cong (\mathbb{Z}/2)^l$ admits the $t_i$ as a $\mathbb{Z}/2$-basis. On the other hand, $\mathbf{v}\mathbf{P}_{n}^{ab}$ is freely generated by the set $X$ of the $\bar{\chi}_{\alpha\beta}$ with $1 \le \alpha \ne \beta \le n$, on which $\mathfrak{S}_{\lambda} \subseteq \mathfrak{S}_{n}$ acts by permuting the indices. Thus $(\mathbf{v}\mathbf{P}_{n}^{ab})_{\mathfrak{S}_{\lambda}}$ is free on the set $X/\mathfrak{S}_{\lambda}$. The latter is exactly the set of generators $x_{ij}$ described in the statement, plus one additional element for each $i$ such that $n_i \ge 2$, which is the common class $x_i$ of the $\chi_{\alpha\beta}$ for $\alpha$ and $\beta$ in the $i$-th block. Thus we get the decomposition of the statement, with a slightly different basis (the $x_i$ instead of the $s_i$). The relation $\chi_{\alpha,\alpha+1} = \tau_{\alpha}\sigma_{\alpha}$ implies that $s_i = x_i - t_i$, which is the change of basis we need to finish proving our first statement.

Let us now describe $\tilde{\mathbf{w}}\mathbf{B}_{\lambda}^{ab}$, using a similar computation. We deduce from the decomposition (4.9) that $\tilde{\mathbf{w}}\mathbf{B}_{\lambda}^{ab} \cong (\mathbf{w}\mathbf{P}_{n}^{ab})_{W_{\lambda}} \times W_{\lambda}^{ab}$. On the one hand, the group $W_{\lambda} = (\mathbb{Z}/2)\mathfrak{S}_{\lambda}$ is isomorphic to the product of the $(\mathbb{Z}/2)\mathfrak{S}_{n_i}$, whose abelianisation is $\mathbb{Z}/2 \times \mathbb{Z}/2$ (generated by $r_i$ and $s_i$) if $n_i \ge 2$, and $\mathbb{Z}/2$ (generated by $r_i$) if $n_i = 1$. Thus $W_{\lambda}^{ab} \cong (\mathbb{Z}/2)^{l+l'}$ gives a first factor in our formula. On the other hand the action of the generator $\rho_{\gamma}$ of $(\mathbb{Z}/2)^n \subset W_{\lambda}$ on $\mathbf{w}\mathbf{P}_{n}^{ab}$ is by fixing all the $\bar{\chi}_{\alpha\beta}$ for $\beta \ne \gamma$ and sending all $\bar{\chi}_{\alpha\gamma}$ to $-\bar{\chi}_{\alpha\gamma}$. One can check this from the presentation of Proposition 4.3: for instance, $\chi_{12} = \tau_1\sigma_1$, and $\rho_1\chi_{12}\rho_1^{-1} = \sigma_1^{-1}\tau_1 = \chi_{12}^{-1}$. As a consequence, $(\mathbf{v}\mathbf{P}_{n}^{ab})_{(\mathbb{Z}/2)^n} \cong \mathbf{v}\mathbf{P}_{n}^{ab} \otimes \mathbb{Z}/2$ is a free $\mathbb{Z}/2$-vector space on the $\bar{\chi}_{\alpha\beta}$, on which $\mathfrak{S}_n$ acts by permutation of the indices. Since $(\mathbf{w}\mathbf{P}_{n}^{ab})_{W_{\lambda}} = (\mathbf{w}\mathbf{P}_{n}^{ab})_{(\mathbb{Z}/2)\mathfrak{S}_{\lambda}} = ((\mathbf{v}\mathbf{P}_{n}^{ab})_{(\mathbb{Z}/2)^n})_{\mathfrak{S}_{\lambda}}$, the rest of the proof works exactly as for the previous case, except that we work with coefficients in $\mathbb{Z}/2$. $\square$

**4.6.2. The LCS.** The remainder of the section is devoted to the proof of the following theorem. Its proof splits into several cases, which we examine in the next few propositions; Theorem 4.47 is then deduced from these at the very end of this section.

THEOREM 4.47. Let $n \ge 1$ be an integer, and $\lambda = (n_1, \dots, n_l)$ be a partition of $n$. The LCS of the partitioned virtual (resp. welded) braid group $\mathbf{v}\mathbf{B}_\lambda$ (resp. $\mathbf{w}\mathbf{B}_\lambda$):

• stops at $\Gamma_2$ if $n_j = 1$ for at most one $j$, and $n_i \ge 4$ for all $i \ne j$.

• does not stop in all the other cases.

The LCS of the partitioned extended welded braid group $\tilde{\mathbf{w}}\mathbf{B}_{\lambda}$:

• stops at $\Gamma_2$ if $n_i \ge 4$ for all $i$.

• does not stop in all the other cases, except if $\lambda = (1)$, for $\tilde{\mathbf{w}}\mathbf{B}_1 \cong \mathbb{Z}/2$.

4.6.2.1. *The stable case.* The simplest case in the case when the size of every block is at least 4:

PROPOSITION 4.48. Let $\lambda$ be a partition of an integer $n$ into blocks of size at least 4. Then the LCS of $\mathbf{v}\mathbf{B}_\lambda$ (resp. of $\mathbf{w}\mathbf{B}_\lambda^{ab}$, resp. of $\tilde{\mathbf{w}}\mathbf{B}_\lambda^{ab}$) stops at $\Gamma_2$.

PROOF. As in the proofs of Propositions 4.36 and 4.38, we show that any pair of generators of $\mathbf{v}\mathbf{B}_{\lambda}^{ab}$ (resp. of $\mathbf{w}\mathbf{B}_{\lambda}^{ab}$, resp. of $\tilde{\mathbf{w}}\mathbf{B}_{\lambda}^{ab}$) may be represented by a pair of elements of the group having disjoint support (in the sense of Definition 4.32 for virtual braids, and of the equivalent sense of Definitions 4.29 and 4.32 for welded braids), which therefore commute by Facts 4.31 and 4.35. The generators we need
to consider are listed in Proposition 4.46: they are the $s_i$, the $t_i$, the $x_{ij}$ and (in the case of extended welded braids) the $r_i$. In each case, it is clear that the hypothesis $n_i \ge 4$ allows us to choose lifts with disjoint support among the generators of $\mathbf{vB}_\lambda$ (resp. of $\mathbf{wB}_\lambda$, resp. of $\tilde{\mathbf{wB}}_\lambda$) from Lemma 4.45. Indeed, the class of any such generator depends only on the blocks to which the indices involved belong, and the support of each generator contains at most two indices. For instance, if $\alpha$ denotes the minimum of the $i$-th block, $s_i$ and $t_i$ can be lifted respectively to $\sigma_\alpha$ (whose support is $\{\alpha, \alpha+1\}$) and to $\tau_{\alpha+2}$ (whose support is $\{\alpha+2, \alpha+3\}$). $\square$

REMARK 4.49 (Blocks of size 3). If $\lambda$ has blocks of size 3 (say, $n_1 = 3$), the LCS of $G_\lambda$ (with $G = \mathbf{vB}$, $\mathbf{wB}$ or $\tilde{\mathbf{wB}}$) does not stop: the disjoint support argument fails, since $t_1$ and $s_1$ do not have lifts with disjoint support. However, this failure is limited, and the argument can still be applied to understand the LCS of $G_{3,\mu}$ from the one of $G_\mu$ (where $\mu$ is any partition). Precisely, the generators $t_1$ and $s_1$ of the Lie ring do not commute with each other, but they do commute with every other generator. Using the canonical split injection of $G_3$ into $G_{3,\mu}$, we see that $t_1$ and $s_1$ (together with $r_1$ for extended welded braids) generate a copy of $\mathcal{L}(G_3)$, which is then a direct factor of $\mathcal{L}(G_{3,\mu})$. If all the blocks of $\mu$ have size at least 3, we can be even more precise: then the $2(l-1)$ elements $x_{1i}$ and $x_{i1}$ must be central. In this case, we get an isomorphism of Lie rings:

$$ \mathcal{L}(G_{3,\mu}) \cong \mathcal{L}(G_3) \times \mathbb{Z}^{2(l-1)} \times \mathcal{L}(G_\mu). $$

Moreover, $\mathcal{L}(G_3)$ is well-understood (see Propositions 4.37 and 4.39, and the proof of Proposition 4.42), so this allows us to compute $\mathcal{L}(G_\lambda)$ completely if all the blocks of $\lambda$ have size at least 3.

4.6.2.2. Adding one block of size 1. This is the tricky case, where the behaviour of $\tilde{\mathbf{wB}}_\lambda$ will differ from the other ones. For $\mathbf{wB}_\lambda$ and $\mathbf{vB}_\lambda$, we will show that the behaviour does not really differ from the stable case, using the following analogue of Lemma 3.9:

LEMMA 4.50. Let $G$ be the subgroup of $\mathbf{vP}_3$ generated by the elements $\chi_{12}, \chi_{13}$ and $\chi_{23}$. Then any two of the three relations

$$ [\chi_{12}, \chi_{23}] = 1 \quad [\chi_{13}, \chi_{23}] = 1 \quad [\chi_{12}, \chi_{13}] = 1 $$

imply the third. In other words, the quotient of $G$ by any two of these relations is abelian.

PROOF. The elements $\chi_{12}, \chi_{13}$ and $\chi_{23}$ satisfy the relation

$$ (4.10) \qquad \chi_{12}\chi_{13}\chi_{23}\chi_{12}^{-1}\chi_{13}^{-1}\chi_{23}^{-1} = 1 $$

in $\mathbf{vP}_3$; see [Bar04, Th. 1]. If we impose any two of the three relations above, then one of the three generators becomes central, and (4.10) becomes the relation stating that the other two generators commute. $\square$

PROPOSITION 4.51. Let $\mu$ be a partition of an integer $n$ into blocks of size at least 4. Then the LCS of $\mathbf{vB}_{1,\mu}$ stops at $\Gamma_2$, and so does the LCS of $\mathbf{wB}_{1,\mu}$.

PROOF. Notice first that the second statement follows directly from the first one, using Lemma 1.1, since $\mathbf{wB}_{1,\mu}$ is a quotient of $\mathbf{vB}_{1,\mu}$. Let us prove the first statement. The proof is similar to that of Proposition 3.10, with Lemma 4.50 replacing Lemma 3.9.
Let us denote by $\lambda = (1, n_2, \dots, n_l)$ the partition $(1, \mu)$ of $n+1$. The canonical morphism $\mathbf{vB}_\mu \hookrightarrow \mathbf{vB}_{1,\mu}$ induces a morphism:

$$ \mathbf{vB}_\mu^{ab} = \mathbf{vB}_\mu/\Gamma_\infty \to \mathbf{vB}_{1,\mu}/\Gamma_\infty, $$

where the equality on the left comes from the fact that all the blocks of $\mu$ have size at least 4 (Proposition 4.48). This implies that among the generators of $\mathbf{vB}_{1,\mu}$ from Proposition 4.45, the ones coming from $\mathbf{vB}_\mu$ pairwise commute modulo $\Gamma_\infty$. Moreover, this also implies that $\chi_{\alpha\beta}$ and $\chi_{\alpha'\beta'}$ have the same image $x_{ij}$ in $\mathbf{vB}_{1,\mu}/\Gamma_\infty$ if $\alpha$ and $\alpha'$ are in the $i$-th block of $\lambda$ and $\beta$ and $\beta'$ are in the $j$-th one, if $i, j \ge 2$. Similarly, $\sigma_\alpha$ and $\sigma_{\alpha'}$ (resp. $\tau_\alpha$ and $\tau_{\alpha'}$) have the same image $s_i$ (resp. $t_i$) in this quotient if $\alpha$ and $\alpha'$ are in the $i$-th block of $\lambda$, with $i \ge 2$. Finally, a generator of the form $\chi_{1\alpha}$ (resp. $\chi_{\alpha 1}$) is also sent to a class $x_{1i}$ (resp. $x_{i1}$) modulo $\Gamma_\infty$ depending only on the index $i \ge 2$ of the block containing $\alpha$. Indeed, let us suppose that $\alpha$ and $\alpha+1$ are in the $i$-th block of $\lambda$. This block has size at least 4, so $t_i$ has a representative $\tau_\beta$ in $\mathbf{vB}_{1,\mu}$ whose support does not contain $\alpha$, which then commutes with $\chi_{1\alpha}$. But $t_i$ is also the class of $\tau_\alpha$, and $\tau_\alpha\chi_{1\alpha}\tau_\alpha = \chi_{1,\alpha+1}$. Thus $\bar{\chi}_{1\alpha} = t_i\bar{\chi}_{1\alpha}t_i = \bar{\chi}_{1,\alpha+1}$. The same argument shows that $\bar{\chi}_{\alpha 1} = \bar{\chi}_{\alpha 1,1}$.

In order to prove our statement, we need to show that the $x_{1i}$ and the $x_{j1}$ commute with each other, and with all the other generators of $\mathbf{vB}_{1,\mu}/\Gamma_\infty$. The latter assertion is deduced from the fact that the pairs of generators involved have lifts to $\mathbf{vB}_{1,\mu}$ with disjoint support. The former one is a bit trickier, and requires the use of Lemma 4.50. Precisely, let $G$ denote the subgroup of the pure virtual braid group $\mathbf{vP}_3$ generated by the elements $\chi_{12}, \chi_{13}$ and $\chi_{23}$. In each case, we define a homomorphism $\theta: G \to \mathbf{vB}_{1,\mu}/\Gamma_\infty$ such that two of the elements $\chi_{12}, \chi_{13}$ and $\chi_{23}$ are sent to the elements of $\mathbf{vB}_{1,\mu}/\Gamma_\infty$ that we wish to show commute, and the third is sent to an element that commutes with both of them. It then follows from Lemma 4.50 that the first two elements commute. In detail, the four cases are as follows, for $i, j \ge 2$ with $i \ne j$:

* We wish to show that $x_{i1}$ and $x_{j1}$ commute. Let $\theta$ be induced by $1 \mapsto \alpha$, $2 \mapsto \beta$ and $3 \mapsto 1$, with $\alpha$ in the $i$-th block and $\beta$ in the $j$-th block of $\lambda$. The image of $\chi_{12}$ is $x_{ij}$, which commutes with $x_{i1} = \theta(\chi_{13})$ and $x_{j1} = \theta(\chi_{23})$, as we have deduced above from a disjoint support argument.

* The case of $x_{1j}$ and $x_{1i}$ is similar to the previous one, the roles of 1 and 3 being exchanged.

* Both $x_{i1}$ and $x_{1j}$ commute with $x_{ij}$, and again the same argument implies that they commute.

* Finally, $x_{i1}$ and $x_{1i}$ are the classes of $\chi_{\alpha 1}$ and $\chi_{1,\alpha+1}$ respectively, with $\alpha$ and $\alpha+1$ both in the $i$-th block of $\lambda$. Since $x_{i1}$ and $x_{1i}$ both commute with $\bar{\chi}_{\alpha,\alpha+1} = t_i s_i$, we can apply once again the same argument, $\theta$ being induced by $1 \mapsto 1$, $2 \mapsto \alpha$ and $3 \mapsto \alpha+1$.

We have therefore shown that all generators in a generating set for $\mathbf{vB}_{1,\mu}/\Gamma_\infty$ pairwise commute. Hence this quotient is abelian, in other words, $\Gamma_\infty = \Gamma_2$ for $\mathbf{vB}_{1,\mu}$. $\square$

On the contrary, for extended welded braids, adding one block of size 1 already prevents the LCS from stopping:

**PROPOSITION 4.52.** For any partition $\mu$ of an integer $n \ge 1$, the LCS of the partitioned extended welded braid group $\bar{\mathbf{vB}}_{1,\mu}$ does not stop.
PROOF. Forgetting every block of $\mu$ save one induces a surjection from $\tilde{w}\mathbf{B}_{1,\mu}$ onto $\tilde{w}\mathbf{B}_{1,m}$ for some $m \ge 1$, so by Lemma 1.1, it is enough to show that the LCS of $\tilde{w}\mathbf{B}_{1,m}$ does not stop. Let us consider $\tilde{w}\mathbf{B}_{1,m}$ as a subgroup of Aut($\mathbf{F}_{1+m}$). One can check on the generators of $\tilde{w}\mathbf{B}_{1,m}$ that the automorphisms in $\tilde{w}\mathbf{B}_{1,m}$ preserve the normal subgroup $N$ of $\mathbf{F}_{1+m}$ generated by all the elements $x_\alpha x_{\alpha+1}^{-1}$ for $2 \le \alpha \le m$, together with the $x_\alpha^2$ for $2 \le \alpha \le m+1$. For instance, $\tau_3(x_2x_3^{-1}) = x_2x_4^{-1} = (x_2x_3^{-1})(x_3x_4^{-1}) \in N$, $\rho_2(x_2x_3^{-1}) = x_2^{-1}x_3^{-1} = (x_2^2)^{-1}(x_2x_3^{-1}) \in N$, etc. As a consequence, the quotient of $\mathbf{F}_{1+m} \cong \mathbb{Z} * \mathbf{F}_m$ onto $\mathbb{Z} * \mathbb{Z}/2$ induces a well-defined morphism $\tilde{w}\mathbf{B}_{1,m} \to \mathrm{Aut}(\mathbb{Z} * \mathbb{Z}/2)$. Let us denote again by $x_1$ and $x_2$ the generators of $\mathbb{Z} * \mathbb{Z}/2$, which are the images of $x_1, x_2 \in \mathbf{F}_{1+m}$. The image $G$ of $\tilde{w}\mathbf{B}_{1,m}$ in $\mathrm{Aut}(\mathbb{Z} * \mathbb{Z}/2)$ is generated by $\rho_1, \chi_{12}$ and $\chi_{21}$, which are defined by the same formulas as the corresponding elements of $\mathrm{Aut}(\mathbf{F}_2)$, only with $x_2$ having square 1. We note that $\chi_{21}$ and $\chi_{12}$ are conjugation by $x_1$ and by $x_2$ on $\mathbb{Z} * \mathbb{Z}/2$ respectively, so they generate $\mathrm{Inn}(\mathbb{Z} * \mathbb{Z}/2) \cong \mathbb{Z} * \mathbb{Z}/2$, which is normal in $\mathrm{Aut}(\mathbb{Z} * \mathbb{Z}/2)$, whence also in $G$. Moreover, $\rho_1$ is not an inner automorphism and, since $\rho_1^2 = 1$, we have:

$$G = \langle \chi_{21}, \chi_{12} \rangle \rtimes \langle \rho_1 \rangle \cong (\mathbb{Z} * \mathbb{Z}/2) \rtimes (\mathbb{Z}/2).$$

Conjugation by $\rho_1$ fixes $\chi_{12}$ and sends $\chi_{21}$ to its inverse. Hence the quotient of $G$ by $\chi_{12}$ is isomorphic to $\mathbb{Z} \rtimes (\mathbb{Z}/2)$, whose LCS does not stop by Proposition A.16. $\square$

REMARK 4.53. If $m = 1$, then the same kind of calculation can be applied directly in Aut($\mathbf{F}_2$) to see that $\tilde{w}\mathbf{B}_{1,1} = \tilde{w}\mathbf{P}_2 \cong \mathbf{F}_2 \rtimes (\mathbb{Z}/2)^2$, where each generator of $(\mathbb{Z}/2)^2$ fixes one of the generators of $\mathbf{F}_2$, and sends the other one to its inverse. This group surjects onto $(\mathbb{Z} \rtimes (\mathbb{Z}/2)) \times (\mathbb{Z}/2)$ (by killing one of the generators of $\mathbf{F}_2$), so its LCS does not stop.

Having dealt with all the tricky cases, we can finally prove the main result of this section:

PROOF OF THEOREM 4.47. The stable case, where we have $n_i \ge 4$ for all $i$, is dealt with in Proposition 4.48.

If $\lambda$ has one block of size 2 or 3, then there is a surjection (defined by forgetting all strands except for those corresponding to this block) from $\mathbf{v}\mathbf{B}_\lambda$ onto $\mathbf{v}\mathbf{B}_2$ or $\mathbf{v}\mathbf{B}_3$, thus we can use Lemma 1.1 to deduce from Proposition 4.36 that its LCS does not stop. The same argument applies to welded braids (resp. to extended welded braids), using Proposition 4.38 (resp. Proposition 4.42) instead of Proposition 4.36.

Similarly, if there are two blocks of size 1, then both $\mathbf{v}\mathbf{B}_\lambda$ and $\mathbf{w}\mathbf{B}_\lambda$ surject onto $\mathbf{v}\mathbf{B}_{1,1} = \mathbf{w}\mathbf{B}_{1,1} = \mathbf{v}\mathbf{P}_2 \cong \mathbf{F}_2$, whose LCS does not stop. Under this hypothesis, $\tilde{w}\mathbf{B}_\lambda$ surjects onto $\tilde{w}\mathbf{B}_{1,1} \cong \mathbf{F}_2 \rtimes (\mathbb{Z}/2)^2$, whose LCS does not stop either; see Remark 4.53.

Finally, the tricky cases where there is exactly one block of size 1 and no blocks of size 2 or 3 are dealt with in Propositions 4.51 and 4.52. $\square$
<figure><img src="image_28.png" /></figure>
# CHAPTER 5

## Variants on partitioned welded braids

This chapter is devoted to generalising the results of the previous chapter about (extended) welded braids. Precisely, we consider the configuration space of points, oriented circles and unoriented circles in 3-dimensional space; its fundamental group then consists of welded braids with three kinds of strands that interact with each other. We call them tripartite welded braids. Notice that if there is only one kind of strand, we recover the group of welded braids, or of extended welded braids (or the symmetric group if we keep only strands corresponding to motions of points, which is less interesting). We can consider partitioned versions of these groups, defined in the obvious way (see Definition 5.8 for details). Our goal is to understand when the LCS of a partitioned tripartite welded braid group stops, a goal that is fully reached in Theorem 5.11. This result contains the results of the previous chapter about the LCS of $wB_λ$ and $\tilde{w}B_λ$; we have kept these separate only for the sake of clarity. Indeed, considering this larger family of groups requires heavier notations (mostly to distinguish between the three different types of strands), and the cases to consider are more numerous. However, if one looks beyond the complexity of notations and the increased number of cases, one will observe that this generalisation does not fundamentally increase the complexity of the problem; in particular, all the methods that we use to solve it have already been used in the previous chapters.

Our first aim is to show that every tripartite welded braid group can in fact be identified with a subgroup of the group of extended welded braids. We first prove this identification for a finite-index subgroup consisting, in a certain sense, of pure braids (§5.1). We then extend it to the whole group (§5.2). It then applies to the subgroups of partitioned tripartite welded braids, which are introduced in §5.3. This identification allows us to treat tripartite welded braids as automorphisms of the free group, an interpretation that we put to good use in our study of the LCS of partitioned tripartite welded braid groups (§5.4).

### 5.1. Pure bipartite welded braids

We begin with introducing the pure version of the group of tripartite welded braids, obtained from configurations of points and oriented circles. Namely, let $k$ and $m$ be integers. The *pure bipartite* welded braid group $wP(k, m)$ is defined as the fundamental group of the configuration space $F_{k \sqcup m} S^1 (D^3)$ of $k$ points and $m$ oriented circles in the 3-ball $D^3$ (notations being as in §4.1.2.4). In order to study it, we are going to identify this group with a subgroup of $wP_{k+m} (\subset \text{Aut}(F_{k+m}))$. This is done by considering maps between configuration spaces. Let us first consider the map forgetting the $k$ points of each configuration.
our map must be an isomorphism onto its image. As a consequence the map $\mathbf{wP}(k, m) \to \mathbf{wP}_{k+m}$ under scrutiny is injective.

Now, let $G$ be the subgroup of $\mathbf{wP}_{k+m}$ generated by the $\chi_{\alpha\beta}$, for $1 \le \alpha \ne \beta \le n$ such that $\beta > k$. All these elements are seen to be images of elements of $\mathbf{wP}(k, m)$, so that $G \subseteq \mathbf{wP}(k, m)$. Moreover, $G$ contains the image of $\mathbf{P}_k(\mathbb{D}_m^3)$ (described explicitly in the above reasoning), and all of $\mathbf{wP}_m$, so that $G$ is equal to the whole of $\mathbf{wP}(k, m) = \mathbf{P}_k(\mathbb{D}_m^3) \times \mathbf{wP}_m$, and the proposition is proved. $\square$

**REMARK 5.3.** In the semi-direct product decomposition $\mathbf{wP}(k, m) = (\mathbf{F}_m)^k \times \mathbf{wP}_m$ from Lemma 5.1, the action of $\mathbf{wP}_m$ on $(\mathbf{F}_m)^k$ is diagonal, the action on each copy of $\mathbf{F}_m$ being the canonical one (corresponding to the injection $\mathbf{wP}_m \subset \mathrm{Aut}(F_m)$). One may see this by unravelling the geometric definitions of these objects; however, it is much easier to deduce this from Proposition 5.2, which allows us to see this action as a restriction of conjugation in $\mathbf{wP}_{k+m} \subset \mathrm{Aut}(\mathbf{F}_{k+m})$, and to compute it by writing explicit automorphisms of the free group.

**COROLLARY 5.4.** *The Artin representation $\mathbf{wP}_{k+m} \hookrightarrow \mathrm{Aut}(\mathbf{F}_{m+k})$ identifies $\mathbf{wP}(k, m)$ with the group of automorphisms of $\mathbf{F}_{k+m}$ of the form $x_i \mapsto w_i x_i w_i^{-1}$, with $w_i \in \langle x_{k+1}, \dots, x_{k+m} \rangle$ for all $i$.*

**PROOF.** Let $G$ be the subgroup of $\mathrm{Aut}(\mathbf{F}_{k+m})$ described by the condition of the statement. If we look only at its action of $x_{k+1}, \dots, x_{k+m}$, we see that every element of $G$ restricts to a basis-conjugating automorphism of $\langle x_{k+1}, \dots, x_{k+m} \rangle \cong \mathbf{F}_m$. This defines a morphism from $G$ to $\mathbf{wP}_m$, which admits a section, given by extending automorphisms by $x_i \mapsto x_i$ if $i \le k$. The kernel of this split projection is easily seen to be the copy of $(\mathbf{F}_m)^k$ generated by the $\chi_{ij}$ with $i \le k < j$ already described in the proof of Proposition 5.2. Moreover, since $\mathbf{wP}_m$ is (classically) generated by the $\chi_{ij}$ (for $1 \le i \ne j \le m$), the image of this section is generated by the $\chi_{ij}$, for $k < i \ne j \le k+m$. Thus $G$ is the semi-direct product of $\langle \chi_{ij} \rangle_{i \le k < j} \cong (\mathbf{F}_m)^k$ and $\langle \chi_{ij} \rangle_{k < i, j} \cong \mathbf{wP}_m$, so Proposition 5.2 implies that it is equal to $\mathbf{wP}(k, m)$. $\square$

**COROLLARY 5.5.** *The abelianisation $\mathbf{wP}(k, m)^{ab}$ is free abelian on the classes of the $\chi_{\alpha\beta}$, for $1 \le \alpha \ne \beta \le n$ such that $\beta > k$.*

**PROOF.** By Proposition 5.2, these classes generate $\mathbf{wP}(k, m)^{ab}$. Moreover, the canonical map $\mathbf{wP}(k, m)^{ab} \to \mathbf{wP}_{k+m}^{ab}$ sends these generators to linearly independent elements of $\mathbf{wP}_{k+m}^{ab}$ (see Proposition 4.12), so that they are a basis of $\mathbf{wP}(k, m)^{ab}$. $\square$

## 5.2. Tripartite welded braids

Given three integers $n_P, n_{S_+}$ and $n_S$, we define the group of tripartite welded braids by:

$$ \mathbf{wB}(n_P, n_{S_+}, n_S) := \pi_1 \left( F_{n_P \cup (n_{S_+} + n_S) \mathbb{S}^1} (\mathbb{D}^3) / (\mathfrak{S}_{n_P} \times \mathfrak{S}_{n_{S_+}} \times W_{n_S}) \right), $$

where the action by which we quotient is defined similarly to the ones in §4.1.2.4 and §4.1.3.4. In other words, we consider the fundamental group of the configuration space of $n_P$ points (unordered), $n_{S_+}$ oriented circles (unordered) and $n_S$ unoriented ones (unordered too), where all the circles are supposed unlinked and unknotted.

In order to study it, we identify this group with a subgroup of $\tilde{\mathbf{w}}\mathbf{B}_n$, where $n = n_P+n_{S_+}+n_S$. Since $\tilde{\mathbf{w}}\mathbf{B}_n$ identifies with a subgroup of $\mathrm{Aut}(\mathbf{F}_n)$ (see Theorem 4.27),
we will then be able to identify tripartite welded braids with automorphisms of $\mathbf{F}_n$, and this point of view will come in handy for doing explicit computations.

Let us consider the map from $F_{n_P \sqcup (n_{S_+} + n_S)\mathbb{S}^1}(\mathbb{D}^3)$ to $F_{n_S^1}(\mathbb{D}^3)$ defined, as above, by replacing each point in a configuration by a small oriented circle. By factorisation through quotients by the appropriate group actions, we get a map between the configuration spaces of which $\mathbf{wB}(n_P, n_{S_+}, n_S)$ and $\tilde{\mathbf{wB}}_n$ are the fundamental groups (this boils down to replacing points in configurations by small circles, and forgetting orientations of circles).

**PROPOSITION 5.6.** *The above map identifies $\mathbf{wB}(n_P, n_{S_+}, n_S)$ with the subgroup of $\tilde{\mathbf{wB}}_n$ with $n = n_P + n_{S_+} + n_S$ generated by:*

* the $\tau_\alpha$ for $\alpha \neq n_P, n_P + n_{S_+}$ and $1 \le \alpha < n$;

* the $\rho_\alpha$ for $n_P + n_{S_+} < \alpha \le n$;

* the $\chi_{\alpha\beta}$, for $1 \le \alpha \neq \beta \le n$ such that $\beta > n_P$.

**PROOF.** Consider the following commutative square of maps between configuration spaces:

$$ \begin{tikzcd} F_{n_P, (n_{S_+} + n_S)\mathbb{S}^1}(\mathbb{D}^3) \arrow[r] \arrow[d] & F_{n_P, (n_{S_+} + n_S)\mathbb{S}^1}(\mathbb{D}^3)/(\mathcal{G}_{n_P} \times \mathcal{G}_{n_{S_+}} \times W_{n_S}) \arrow[d] \\ F_{n_S^1}(\mathbb{D}^3) \arrow[r, "F_{n_S^1}(\mathbb{D}^3)/W_n"] & \end{tikzcd} $$

where the horizontal maps are regular coverings. These induce the left square of the following commutative diagram, whose rows are exact (thanks to the usual theory of coverings):

$$ \begin{tikzcd} \mathbf{wP}(n_P, n_{S_+} + n_S) \arrow[r] \arrow[d] & \mathbf{wB}(n_P, n_{S_+}, n_S) \arrow[r] \arrow[d] & \mathcal{G}_{n_P} \times \mathcal{G}_{n_{S_+}} \times W_{n_S} \arrow[d] \\ \mathbf{wP}_n \arrow[r, "W_n"] & \mathcal{G}_{n_{S_+}} \times \mathcal{G}_{n_S} \arrow[d] & \end{tikzcd} $$

The vertical map on the right is obviously injective. By Lemma 5.2, the one on the left is also injective. So, by the Five Lemma, the middle vertical map $\mathbf{wB}(n_P, n_{S_+}, n_S) \to \tilde{\mathbf{wB}}_n$ is too.

Now, let $G$ be the subgroup of $\tilde{\mathbf{wB}}_n$ generated by the elements listed in the statement. First, one easily sees that each element in the list is the image of some element of $\mathbf{wB}(n_P, n_{S_+}, n_S)$, so that $G \subseteq \mathbf{wB}(n_P, n_{S_+}, n_S)$. Moreover, $G$ contains $\mathbf{wP}(n_P, n_{S_+} + n_S)$ by Proposition 5.2, and it also contains $\mathcal{G}_{n_P} \times \mathcal{G}_{n_{S_+}} \times W_{n_S}$, which is generated by the $\tau_\alpha$ and the $\rho_\alpha$ contained in $G$. Thus, $G$ is all of their semi-direct product $\mathbf{wB}(n_P, n_{S_+}, n_S)$, and our statement is proved. $\square$

**COROLLARY 5.7.** *The Artin representation $\tilde{\mathbf{wB}}_n \hookrightarrow \mathrm{Aut}(\mathbf{F}_n)$ identifies the subgroup $\mathbf{wB}(n_P, n_{S_+}, n_S)$ with the group of automorphisms of $\mathbf{F}_n$ of the form*

$$ x_i \mapsto w_i x_{\sigma(i)}^{\epsilon_i} w_i^{-1}, $$
*with $\sigma \in \mathcal{G}_{n_P} \times \mathcal{G}_{n_{S_+}} \times \mathcal{G}_{n_S} \subseteq \mathcal{G}_n$, $w_i \in \langle x_{n_P+1}, \dots, x_n \rangle$ and $\epsilon_i \in \{\pm 1\}$ for all $i$, such that $\epsilon_i = 1$ if $i \le n_P + n_{S_+}$.

**PROOF.** Let $G$ be the subgroup of $\mathrm{Aut}(F_n)$ defined by the conditions of the statement. The canonical map from $\mathrm{Aut}(\mathbf{F}_n)$ to $\mathrm{Aut}(\mathbf{F}_n^{ab}) \cong \mathrm{GL}_n(\mathbb{Z})$ restricts to a projection of $G$ onto a group of signed permutation matrices isomorphic to
$\mathfrak{S}_{n_P} \times \mathfrak{S}_{n_{S_+}} \times W_{n_S}$. This projection is split, and the image of the obvious splitting is generated by the $\tau_\alpha$ and the $\rho_\alpha$ listed in the statement of Proposition 5.6. Moreover, Corollary 5.4 identifies the kernel of this projection as $\mathbf{wP}(n_P, n_{S_+}+n_S)$. Proposition 5.2 says that the latter is generated by the $\chi_{\alpha\beta}$ listed in Proposition 5.6. Thus, $G$ is generated by the elements listed in the statement of Proposition 5.6, so it is equal to $\mathbf{wB}(n_P, n_{S_+}, n_S)$. $\square$

## 5.3. Partitioned tripartite welded braids

Let us now introduce a partitioned version of this group. In order to do this, we use the canonical projection $\pi$ from $\mathbf{wB}(n_P, n_{S_+}, n_S)$ onto $\mathfrak{S}_{n_P} \times \mathfrak{S}_{n_{S_+}} \times \mathfrak{S}_{n_S}$, which identifies (using Proposition 5.6) with a restriction of $\pi$: $\tilde{\mathbf{wB}}_n \to \mathfrak{S}_n$.

**DEFINITION 5.8.** Let $\lambda_P, \lambda_{S_+}$ and $\lambda_S$ be partitions of integers $n_P, n_{S_+}$ and $n_S$ respectively. Let $\lambda = \lambda_P\lambda_{S_+}\lambda_S$ be their concatenation, which is a partition of $n = n_P + n_{S_+} + n_S$. The blocks of $\lambda$ will often be identified with blocks of $\lambda_P, \lambda_{S_+}$ and $\lambda_S$. The associated group of tripartite welded braids $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)$ is defined by:

$$ \mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S) = \pi^{-1}(\mathfrak{S}_\lambda) = \pi^{-1}(\mathfrak{S}_{\lambda_P} \times \mathfrak{S}_{\lambda_{S_+}} \times \mathfrak{S}_{\lambda_S}) \subseteq \mathbf{wB}(n_P, n_{S_+}, n_S) \subset \tilde{\mathbf{wB}}_n. $$

A direct adaptation of the proof of Lemma 3.3 to this context (using Proposition 5.2 to supply generators of the subgroup of pure braids) gives:

**LEMMA 5.9.** As a subgroup of $\tilde{\mathbf{wB}}_n$, $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)$ is generated by:

* the $\tau_\alpha$ for $1 \le \alpha < n$ such that $\alpha$ and $\alpha+1$ are in the same block of $\lambda$;

* the $\sigma_\alpha$ for $n_P < \alpha < n$ such that $\alpha$ and $\alpha+1$ are in the same block of $\lambda$;

* the $\rho_\alpha$ for $n_P + n_{S_+} < \alpha \le n$;

* the $\chi_{\alpha\beta}$, for $1 \le \alpha \ne \beta \le n$ with $\beta > n_P$, such that $\alpha$ and $\beta$ are not in the same block of $\lambda$.

We can compute the abelianisation of the group $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)$, like we did for partitioned groups before:

**PROPOSITION 5.10.** Let $\lambda_P, \lambda_{S_+}$ and $\lambda_S$ be partitions of integers $n_P, n_{S_+}$ and $n_S$ respectively, of respective length $l_P, l_{S_+}$ and $l_S$. Let us denote by $l'_P, l'_S$ and $l'_S$ the number of blocks of size at least two in each of these partitions. Let also $\lambda_P\lambda_{S_+}\lambda_S = \lambda = (n_1, \dots, n_l)$ be their concatenation, of length $l = l_P + l_{S_+} + l_S$, and with $l' = l'_P + l'_{S_+} + l'_S$ blocks of size at least two. Then:

$$ \mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)^{ab} \cong \mathbb{Z}^N \times (\mathbb{Z}/2)^M, $$

$$ \text{where } N = l'_{S_+} + l_{S_+}(l-1) \text{ and } M = l' + l'_S + l_{S_+l}. $$

Let us denote by $I_P$ the set $\{1, \dots, l_P\}$ of indices corresponding to blocks of $\lambda_P$, by $I_{S_+}$ the set $\{l_P+1, \dots, l_P+l_{S_+}\}$ and by $I_S$ the set $\{l_P+l_{S_+}+1, \dots, l\}$. With these notations, a basis of the first factor is given by:

* for each $i \in I_{S_+}$ such that $n_i \ge 2$, one generator $s_i$,

* for each $j \in I_{S_+}$ and each $i \in \{1, \dots, l\}$ such that $i \ne j$, one generator $x_{ij}$.

and a $\mathbb{Z}/2$-basis of the second factor by:

* for each $i \in I_S$ such that $n_i \ge 2$, one generator $s_i$,

* for each $i \in \{1, \dots, l\}$ such that $n_i \ge 2$, one generator $t_i$,

* for each $i \in I_S$, one generator $r_i$,
• for each $j \in I_S$ and each $i \in \{1, \dots, l\}$ such that $i \neq j$, one generator $x_{ij}$.
The generators are obtained as follows:

• $s_i$ (resp. $t_i$) is the common class of the $\sigma_\alpha$ (resp. $\tau_\alpha$) for $\alpha$ and $\alpha + 1$ in the $i$-th block of $\lambda$,

• $r_i$ is the common class of the $\rho_\alpha$ for $\alpha$ in the $i$-th block of $\lambda$,

• $x_{ij}$ is the common class of the $\chi_{\alpha\beta}$ for $\alpha$ in the $i$-th block of $\lambda$ and $\beta$ in the $j$-th one.

PROOF. The proof is very similar to that of Proposition 4.46, which is a particular case of the present general statement. Namely, we apply Lemma 1.12 to the decomposition:

$$ \mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S) \cong \mathbf{wP}(n_P, n_{S_+} + n_S) \rtimes (\mathcal{G}_{\lambda_P} \times \mathcal{G}_{\lambda_{S_+}} \times W_{\lambda_S}). $$

The abelianisation of the second factor identifies with $(\mathbb{Z}/2)^{l'+l_S}$, generated by the generators $t_i$ and $r_i$ of the statement. The abelianisation of $\mathbf{wP}(n_P, n_{S_+}+n_S)$ is free on the classes of the $\chi_{\alpha\beta}$ with $1 \le \alpha \ne \beta \le n$ such that $\beta > n_P$ by Corollary 5.5, and the action of $\mathcal{G}_{\lambda_P} \times \mathcal{G}_{\lambda_{S_+}} \times W_{\lambda_S}$ is by permutation of the indices. We note that this latter group identifies with $(\mathbb{Z}/2)^{n_S} \times \mathcal{G}_{\lambda}$, where $\mathcal{G}_{\lambda} = \mathcal{G}_{\lambda_P} \times \mathcal{G}_{\lambda_{S_+}} \times \mathcal{G}_{\lambda_S}$ acts on $(\mathbb{Z}/2)^{n_S}$ through the projection on its third factor (whose action permutes the factors). Thus, the coinvariants we need to compute are:

$$ (\mathbf{wP}(n_P, n_{S_+} + n_S)^{ab})_{(\mathbb{Z}/2)^{n_S} \rtimes \mathcal{G}_{\lambda}} \cong ((\mathbf{wP}(n_P, n_{S_+} + n_S)^{ab})_{(\mathbb{Z}/2)^{n_S}})_{\mathcal{G}_{\lambda}}. $$

The computation then continues exactly as in the proof of Proposition 4.46. In particular, $(\mathbf{wP}(n_P, n_{S_+} + n_S)^{ab})_{(\mathbb{Z}/2)^{n_S}}$ is a product of factors $\mathbb{Z}$ and $\mathbb{Z}/2$ generated by the $\bar{\chi}_{\alpha\beta}$, respectively with $n_P < \beta \le n_P + n_{S_+}$ and with $\beta > n_P + n_{S_+}$. The classes of these generators in the coinvariants give the generators $x_{ij}$ of the statement, together with the generators $s_i$, after the same change of basis as in the proof of Proposition 4.46. $\square$

From a geometrical point of view, the generators described in Proposition 5.10 correspond to the motions of points and circles in the 3-ball represented in Figure 5.1.

## 5.4. The lower central series

We now study the LCS of $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)$. In order to do so, we need to apply disjoint support arguments. These arguments are best understood by thinking of motions having disjoint support in the 3-ball, and the reader is advised to keep this point of view in mind. However, writing down precise arguments and explicit calculations is much easier when dealing with automorphisms of free groups, so we mainly identify elements of $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)$ with automorphisms of free groups in our proofs, using Proposition 5.6 and Theorem 4.27.

The remainder of this section is devoted to the proof of the following theorem. Sections §5.4.1–5.4.5 study each particular case of its statement; its proof is then deduced from these at the very end of §5.4.

**THEOREM 5.11.** Let $\lambda_P, \lambda_{S_+}$ and $\lambda_S$ denote partitions of integers $n_P, n_{S_+}$ and $n_S$ respectively, of respective lengths $l_P, l_{S_+}$ and $l_S$. Let also $\lambda_P \lambda_{S_+} \lambda_S = \lambda = (n_1, \dots, n_l)$ be their concatenation, of length $l = l_P + l_{S_+} + l_S$. We denote
<figure><img src="image_29.png" /><figcaption>FIGURE 5.1. The six types of generators for $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)^{ab}$ from Proposition 5.10. Types 2–4 each have two sub-types depending on whether the circles are (both) oriented or (both) un-oriented. The circle in Type 5 must be unoriented. Type 6 has four sub-types, depending on which of the circles are oriented. Type 2 generates a $\mathbb{Z}$ summand if both circles are oriented. Types 4 and 6 generate a $\mathbb{Z}$ summand if the non-moving circle is oriented. All other types generate a $\mathbb{Z}/2$ summand.</figcaption></figure>

by $I_P$ the set $\{1, \dots, l_P\}$ of indices corresponding to blocks of $\lambda_P$, by $I_{S_+}$ the set $\{l_P + 1, \dots, l_P + l_{S_+}\}$ and by $I_S$ the set $\{l_P + l_{S_+} + 1, \dots, l_P\}$.

Let us suppose that $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)$ is not trivial, i.e. $(\lambda_P, \lambda_{S_+}, \lambda_S)$ is not among $(\emptyset, \emptyset, \emptyset)$, $((1, 1, \dots, 1), \emptyset, \emptyset)$ and $(\emptyset, 1, \emptyset)$. Then the LCS of the group $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)$:

* stops at $\Gamma_2$ if $n_i \neq 2$ for all $i \in I_P$ and $n_i \ge 4$ for all $i \in I_{S_+} \cup I_S$, save for at most one $i \in I_{S_+}$ such that $n_i = 1$;

* stops at $\Gamma_3$ if $n_i = 2$ for at least one $i \in I_P$, $I_{S_+} = \emptyset$, $I_S \neq \emptyset$ and $n_i \ge 4$ for all $i \in I_S$;

* does not stop in all the other cases, except for $\mathbf{wB}(\emptyset, \emptyset, 1) = \tilde{\mathbf{wB}}_1 \cong \mathbb{Z}/2$.

**5.4.1. The stable case.** We begin with the cases where all the blocks of $\lambda$ are big enough to apply a disjoint support argument:

**PROPOSITION 5.12.** *The notations being as in Theorem 5.11, suppose that $n_i \ge 3$ for all $i \in I_P$ and $n_i \ge 4$ for all $i \in I_{S_+} \cup I_S$. Then the LCS of the group $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)$ stops at $\Gamma_2$.*

**PROOF.** The proof is an extension of that of Theorem 4.47. By Corollary 2.6, we just have to show that any pair of the generators of $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)^{ab}$ listed in Proposition 5.10 (and illustrated in Figure 5.1) has a pair of lifts in $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)$ with disjoint support. The assumption that $n_i \ge 3$ for all $i \in I_P$ and $n_i \ge 4$ for all $i \in I_{S_+} \cup I_S$ is exactly what we need for that, as one sees by direct inspection. For instance, a generator $x_{ij}$ has a lift whose support is $\{\alpha, \beta\}$ (namely, $\chi_{\alpha\beta}$), for any choice of $\alpha$ in the $i$-th block and $\beta$ in the $j$-th block of $\lambda$. Thus any $x_{kl}$ has a lift whose support is disjoint from some lift of $x_{ij}$, provided that the $i$-th and $j$-th blocks of $\lambda$ both have size at least 2. $\square$
**5.4.2. Blocks of points and oriented circles of size 1.** The result in the stable case can be extended to the cases where we add blocks of size 1 to $\lambda_P$, or one block of size 1 to $\lambda_{S_+}$, or both. These three cases correspond respectively to Propositions 5.14, 5.15 and 5.16 below. In each case, the strategy of proof is very similar to the proof of Proposition 4.51: most of the pairs of generators are already known to commute, using the previous cases and disjoint support arguments; the remaining ones are dealt with using Lemma 4.50 which, for welded braids, takes the simpler form:

**LEMMA 5.13.** Let $G$ be the subgroup of $\mathbf{wP}_3$ generated by the elements $\chi_{12}, \chi_{13}$ and $\chi_{23}$. Then $\Gamma_2(G)$ is normally generated by $[\chi_{12}, \chi_{23}]$ (resp. by $[\chi_{12}, \chi_{13}]$). In other words, the quotient of $H$ by the single relation $[\chi_{12}, \chi_{23}] = 1$ (resp. $[\chi_{12}, \chi_{13}] = 1$) is abelian.

**PROOF.** This is a direct corollary of Lemma 4.50, noticing that $G$ is a quotient of the subgroup $\langle \chi_{12}, \chi_{13}, \chi_{23} \rangle$ of $\mathbf{vP}_3$ where $[\chi_{13}, \chi_{23}]$ is already killed. $\square$

Let us first consider the case where blocks of size 1 are added to $\lambda_P$:

**PROPOSITION 5.14.** The notations being those of Theorem 5.11, if $n_i \ge 4$ for all $i \in I_{S_+} \cup I_S$ and $\lambda_P$ has no blocks of size 2, then the LCS of $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)$ stops at $\Gamma_2$.

**PROOF.** We argue like in the proof of Proposition 4.51.

Let us write $\lambda_P = (1, \dots, 1, \mu_P)$, where there are $k$ ones and each block of $\mu_P$ has size at least 3. There is an obvious homomorphism $\mathbf{wB}(\mu_P, \lambda_{S_+}, \lambda_S) \to \mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)$, which induces a homomorphism

$$ \mathbf{wB}(\mu_P, \lambda_{S_+}, \lambda_S)^{ab} = \mathbf{wB}(\mu_P, \lambda_{S_+}, \lambda_S)/\Gamma_\infty \to \mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)/\Gamma_\infty. $$

The equality on the left-hand side comes from Proposition 5.12. Similarly, for each block $i$ of the partition $\lambda_{S_+} \cup \lambda_S$, there is an obvious homomorphism from $\mathbf{wB}((1, \dots, 1), n_i, \emptyset)$ or $\mathbf{wB}((1, \dots, 1), \emptyset, n_i)$ to $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)$, inducing a homomorphism

$$ \mathbf{wB}((1, \dots, 1), n_i, \emptyset)^{ab} = \mathbf{wB}((1, \dots, 1), n_i, \emptyset)/\Gamma_\infty \to \mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)/\Gamma_\infty \\ \text{or } \mathbf{wB}((1, \dots, 1), \emptyset, n_i)^{ab} = \mathbf{wB}((1, \dots, 1), \emptyset, n_i)/\Gamma_\infty \to \mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)/\Gamma_\infty. $$

The equality on the left-hand side comes from the fact that the disjoint support argument used in the proof of Proposition 5.12 also works for showing that $\Gamma_2 = \Gamma_\infty$ in this particular case where there is only one block of circles.

Let us consider the generators $\tau_\alpha, \sigma_\alpha, \rho_\alpha, \chi_{\alpha\beta}$ of $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)$ introduced in Lemma 5.9. Using Proposition 5.10, we deduce from the existence of the morphisms obtained above that their classes modulo $\Gamma_\infty$ depend only on the blocks to which $\alpha$ and $\beta$ belong. In other words, if $\alpha$ and $\alpha'$ belong to the same block of the partition, then $\sigma_\alpha = \sigma_{\alpha'}$ in $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)/\Gamma_\infty$, etc. Once we know this, it is easy to use a disjoint support argument to see that modulo $\Gamma_\infty$, all of these generators commute pairwise, with the possible exception of $\chi_{\alpha\beta}$ and $\chi_{\alpha\beta'}$, where $1 \le \alpha \le k$ (so it corresponds to a block of points of size 1) and where $\beta$ and $\beta'$ lie in distinct blocks of $\lambda_{S_+} \cup \lambda_S$. We deal with these as follows. Let us suppose that $\beta$ and $\beta'$ both lie in $\lambda_{S_+}$ (the argument in the other cases is similar). Consider the “adding strands” homomorphism

$$ (5.1) \quad \mathbf{wB}(1, (1, 1), \emptyset) \to \mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S) \to \mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)/\Gamma_\infty $$
induced by $1 \mapsto \alpha$, $2 \mapsto \beta$ and $3 \mapsto \beta'$. We know that the images of the elements $\chi_{12}$ and $\chi_{23}$ of the domain commute in $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)/\Gamma_\infty$. Thus the homomorphism (5.1) factors through the quotient of the left-hand side by $[\chi_{12}, \chi_{23}]$. Then Lemma 5.13 implies that the image $\chi_{\alpha\beta}$ of $\chi_{12}$ commutes with the image $\chi_{\alpha\beta'}$ of $\chi_{13}$ in $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)/\Gamma_\infty$. We have now shown that all the generators of $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)/\Gamma_\infty$ commute pairwise, so it is abelian. Thus $\Gamma_2 = \Gamma_\infty$ for $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)$. $\square$

The result in the stable case (Proposition 5.12) can also be extended to the case where $\lambda_{S_+}$ has one block of size 1, thus generalising the case of $\mathbf{wB}_{1,\mu}$ from Proposition 4.51:

**PROPOSITION 5.15.** *The notations being as in Theorem 5.11, suppose that $n_i \ge 3$ for all $i \in I_P$ and $n_i \ge 4$ for all $i \in I_{S_+} \cup I_S$, save for one $i \in I_{S_+}$ such that $n_i = 1$. Then the LCS of the group $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)$ stops at $\Gamma_2$.*

**PROOF.** Up to some permutation of the blocks, we can suppose that $\lambda_{S_+} = (1, \mu)$ for some partition $\mu$ whose blocks have size at least 4. Then, as in the proof of Proposition 4.51, we have a morphism:

$$ \mathbf{wB}(\lambda_P, \mu, \lambda_S)^{ab} = \mathbf{wB}(\lambda_P, \mu, \lambda_S)/\Gamma_\infty \to \mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)/\Gamma_\infty, $$

where the first equality follows from Proposition 5.12. Starting from this, the generalisation of the proof of Proposition 4.51 to this context is straightforward. $\square$

In fact, the same result holds when there are both some blocks of points of size 1 and one block of oriented circles of size 1, the size of the other blocks being in the stable range:

**PROPOSITION 5.16.** *The notations being as in Theorem 5.11, suppose that $n_i \ne 2$ for all $i \in I_P$, and $n_i \ge 4$ for all $i \in I_{S_+} \cup I_S$, save for one $i \in I_{S_+}$ such that $n_i = 1$. Then the LCS of the group $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)$ stops at $\Gamma_2$.*

**PROOF.** Let us denote by $\mu_P$ the partition obtained from $\lambda_P$ by removing blocks of size 1, and by $\mu_{S_+}$ the one obtained from $\lambda_{S_+}$ by removing the block of size 1. Then we have two morphisms:

$$ \mathbf{wB}(\mu_P, \lambda_{S_+}, \lambda_S)^{ab} = \mathbf{wB}(\mu_P, \lambda_{S_+}, \lambda_S)/\Gamma_\infty \to \mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)/\Gamma_\infty, $$

$$ \mathbf{wB}(\lambda_P, \mu_{S_+}, \lambda_S)^{ab} = \mathbf{wB}(\lambda_P, \mu_{S_+}, \lambda_S)/\Gamma_\infty \to \mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)/\Gamma_\infty, $$

where the equalities on the left come from Propositions 5.14 and 5.15. As a consequence, the classes of the $\chi_{\alpha\beta}$, the $\tau_\alpha$, the $\sigma_\alpha$ and the $\rho_\alpha$ generating $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)$ in the quotient by $\Gamma_\infty$ depend only on the blocks containing $\alpha$ and $\beta$. Indeed, this is obvious when $\alpha$ and $\beta$ both belong to blocks of size one, which is the only case where it cannot directly be deduced from the above description of the abelianisations (Proposition 5.10). As usual, when $\alpha$ is in the $i$-th block and $\beta$ in the $j$-th one, we denote these classes by $x_{ij}$, $t_i$, $s_i$ and $r_i$ respectively.

Now, let us consider two generators among these. Most of the time, both of them belong to the image of one of the morphisms above, whence they commute. When this does not hold, a disjoint support argument shows that they still commute, except when one of them is $x_{ij}$ for $i \in I_P$ and $j \in I_{S_+}$ such that $n_i = n_j = 1$, and the other one is either $x_{ik}$, $x_{jk}$ or $x_{kj}$, for some $k \in I_{S_+} \cup I_S$ different from $j$. We now show that these commute too, reasoning as in the proof of Proposition 4.51. Let us denote by $\alpha$ (resp. $\beta$) the only element of the $i$-th block (resp. the $j$-th one),
and by $\gamma, \gamma'$ two distinct elements of the k-th block (which must be of size at least 4).

* $x_{ij}$ and $x_{kj}$ always commute, since $\chi_{\alpha\beta}$ commutes with $\chi_{\gamma\beta}$.

* $x_{ij}$ and $x_{ik}$ are the images of $\chi_{13}$ and $\chi_{12}$ by a morphism from $\langle\chi_{12}, \chi_{23}, \chi_{13}\rangle$ sending $\chi_{23}$ to $x_{kj}$. Since $x_{ik}$ and $x_{kj}$ are the classes of $\chi_{\alpha\gamma}$ and $\chi_{\gamma'\beta}$, whose support is disjoint, they commute. Hence we can apply Lemma 5.13 to conclude that $x_{ij}$ and $x_{ik}$ commute.

* $x_{ij}$ and $x_{jk}$ are also in such a homomorphic image, and we know from the previous case that $x_{ij}$ and $x_{ik}$ commute, so we can apply Lemma 5.13 again to deduce that $x_{ij}$ and $x_{jk}$ commute too.

Finally, we have shown that all the generators of $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)/\Gamma_\infty$ commute pairwise. Thus this group is abelian, which means exactly that $\Gamma_\infty = \Gamma_2$ for $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)$. $\square$

**5.4.3. Small blocks of unoriented circles.** We already know that if $\lambda_S$ has at least one block of size 2 or 3, then $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)$ surjects onto $\tilde{\mathbf{wB}}_2$ or $\tilde{\mathbf{wB}}_3$, whose LCS does not stop. The next proposition, which generalises Proposition 4.52, deals with the case where $\lambda_S$ has at least one block of size 1:

**PROPOSITION 5.17.** Suppose that $\lambda_S$ has at least one block of size 1, and there is at least another block in $\lambda = \lambda_P\lambda_{S_+}\lambda_S$. Then the LCS of $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)$ does not stop.

**PROOF.** The group $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)$ surjects either onto $\mathbf{wB}(m, 0, 1)$, onto $\mathbf{wB}(0, m, 1)$ or onto $\mathbf{wB}(0, 0, (m, 1))$, for some $m \ge 1$. We need to show that the LCS of each of these three groups does not stop. The case of $\mathbf{wB}(0, 0, (m, 1)) = \tilde{\mathbf{wB}}_{1,m}$ has already been dealt with; see Proposition 4.52. The method used there can be adapted to the other two cases. Indeed, both $\mathbf{wB}(m, 0, 1)$ and $\mathbf{wB}(0, m, 1)$ identify with a subgroup of $\mathrm{Aut}(\mathbf{F}_{m+1})$ consisting of automorphisms preserving the normal closure $N$ of the $x_\alpha x_{\alpha+1}^{-1}$ for $1 \le \alpha \le m-1$, so they both project onto a subgroup $G$ of $\mathrm{Aut}(\mathbf{F}_{m+1}/N) = \mathrm{Aut}(\mathbf{F}_2)$. In the first case, $G$ is generated by $\rho_2$ and $\chi_{12}$, and is isomorphic to $\mathbb{Z} \times (\mathbb{Z}/2)$, whose LCS does not stop by Proposition A.16. In the second case, $G$ is generated by $\rho_2$, $\chi_{12}$ and $\chi_{21}$. Then $\langle\chi_{12}, \chi_{21}\rangle = \mathrm{Inn}(\mathbf{F}_2) \cong \mathbf{F}_2$ is a normal subgroup of $G$, and:

$$G = \langle\chi_{12}, \chi_{21}\rangle \rtimes \langle\rho_2\rangle \cong \mathbf{F}_2 \rtimes (\mathbb{Z}/2),$$

where the action of $\rho_2$ fixes $\chi_{21}$ and sends $\chi_{12}$ to its inverse. Then the quotient of $G$ by $\chi_{21}$ is isomorphic to $\mathbb{Z} \times (\mathbb{Z}/2)$, whose LCS does not stop by Proposition A.16. $\square$

**5.4.4. Blocks of points of size 2 with oriented circles.** We now turn to the cases with blocks of points of size 2 which have not already been covered. The following result takes care of a large part of them:

**PROPOSITION 5.18.** Suppose that $\lambda_P$ has at least one block of size 2, and that $\lambda_{S_+} \neq \emptyset$. Then the LCS of $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)$ does not stop.

**PROOF.** Since $\lambda_{S_+}$ is non-empty, it contains some block of size $m \ge 1$. Then $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)$ surjects onto $\mathbf{wB}(2, m, 0)$, so it is enough to show that the LCS of $\mathbf{wB}(2, m, 0)$ does not stop. The latter is the subgroup of $\mathrm{Aut}(\mathbf{F}_{2+m})$ generated by $\tau_\alpha$ and $\sigma_\alpha$ for $3 \le \alpha \le m+1$, $\chi_{1\alpha}$ and $\chi_{2\alpha}$ for $3 \le \alpha \le m+2$, and $\tau_1$. All these automorphisms preserve the subgroup $N$ normally generated by the $x_\alpha x_{\alpha+1}^{-1}$ for
$3 \le \alpha \le m+1$, as one can check by direct inspection: for instance, $\sigma_{\alpha}(x_{\alpha-1}x_{\alpha}^{-1}) = x_{\alpha-1}(x_{\alpha}x_{\alpha+1}x_{\alpha}^{-1})^{-1} = (x_{\alpha-1}x_{\alpha}^{-1}) \cdot x_{\alpha}(x_{\alpha}x_{\alpha+1}^{-1})x_{\alpha}^{-1}$. As a consequence, they induce automorphisms of the quotient $\mathbf{F}_{2+m}/N \cong \mathbf{F}_3$. This defines a morphism from $\mathbf{wB}(2, m, 0)$ to Aut($\mathbf{F}_3$) sending $\tau_{\alpha}$ and $\sigma_{\alpha}$ to the identity, all the $\chi_{1\alpha}$ to $\chi_{13}$, all the $\chi_{2\alpha}$ to $\chi_{23}$, and $\tau_1$ to $\tau_1$. The image of this projection is thus $\langle \tau_1, \chi_{13}, \chi_{23} \rangle = \mathbf{wB}(2, 1, 0)$. Moreover, since $\chi_{23} = \tau_2\sigma_2$ and thus $\tau_1\chi_{23}\tau_1 = \chi_{13}$ (see §4.1.2.3), one easily sees that

$$ \mathbf{wB}(2, 1, 0) = \langle \chi_{13}, \chi_{23} \rangle \times \langle \tau_1 \rangle \cong \mathbb{Z}^2 \times (\mathbb{Z}/2), $$

where the action of $\mathfrak{S}_2$, which is conjugation by $\tau_1$, exchanges the two generators $\chi_{13}$ and $\chi_{23}$ of $\mathbb{Z}^2$. Thus, this semi-direct product identifies with $\mathbb{Z} \rtimes \mathfrak{S}_2$, whose LCS does not stop (by Corollary A.29), so neither does the LCS of $\mathbf{wB}(2, m, 0)$, as claimed. $\square$

**REMARK 5.19.** One can give a more conceptual argument for the preservation of $N$ by $\mathbf{wB}(2, m, 0)$. Namely, these automorphisms act on $\mathbf{F}_{2+m} \cong \mathbf{F}_2 * \mathbf{F}_m$ in a block triangular fashion (they stabilise $\mathbf{F}_m$), and the automorphisms they induce on $\mathbf{F}_m$ preserve the characteristic subgroup $\Gamma_2(\mathbf{F}_m)$. Then their action on the quotient $\mathbf{F}_m^{\text{ab}} \cong \mathbb{Z}^m$ factors through the canonical action of $\mathfrak{S}_m$, so the projection onto $\mathbb{Z}$ identifying all the generators of $\mathbf{F}_m$ is equivariant. Whence a well-defined induced action of $\mathbf{wB}(2, m, 0)$ on $\mathbf{F}_2 * \mathbb{Z} \cong \mathbf{F}_3$.

**REMARK 5.20.** Concretely, the non-stopping of the LCS in this case is witnessed by iterated commutators of elements of Type 4 and Type 1 from Figure 5.1. Here, we are viewing the pictures in Figure 5.1 as elements of the group $G = \mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)$ itself, rather than as elements of its abelianisation $G^{\text{ab}} = \mathcal{L}_1$.

**5.4.5. Blocks of points of size 2 without oriented circles.** We now consider the LCS of $\mathbf{wB}(\lambda_P, \emptyset, \lambda_S)$. If there are no blocks in $\lambda_S$, then we are considering $\mathbf{wB}(\lambda_P, \emptyset, \emptyset) \cong \mathfrak{S}_{\lambda_P}$, whose LCS stops at $\Gamma_2$ (except if $\lambda_P = (1, \dots, 1)$, for which $\mathbf{wB}(\lambda_P, \emptyset, \emptyset) = \{1\}$). On the contrary, if there are no blocks in $\lambda_P$, then we are considering $\mathbf{wB}(\emptyset, \emptyset, \lambda_S) \cong \tilde{\mathbf{wB}}_{\lambda_P}$, whose LCS has been studied above (in Proposition 4.47).

Suppose now that there is at least one block in both $\lambda_P$ and $\lambda_S$. If $\lambda_S$ has a block of size 1, 2 or 3, we already know that the LCS of $\mathbf{wB}(\lambda_P, \emptyset, \lambda_S)$ does not stop (see §5.4.3); we can thus assume that the blocks of $\lambda_S$ have size at least 4. Under this hypothesis, Proposition 5.14 takes care of the case where $\lambda_P$ has no block of size 2 (then the LCS stops). The remaining case is covered by the following.

**PROPOSITION 5.21.** Let $\lambda_P$ be any partition having at least one block of size 2 and let $\lambda_S$ be a non-empty partition whose blocks have size at least 4. Then the LCS of $\mathbf{wB}(\lambda_P, \emptyset, \lambda_S)$ stops at $\Gamma_3$.

Moreover, $\mathcal{L}_2(\mathbf{wB}(\lambda_P, \emptyset, \lambda_S))$ has a $(\mathbb{Z}/2)$-basis consisting of the $[t_i, x_{ij}]$, for $i \in I_P$ with $n_i = 2$, and $j \in I_S$, and all the other brackets of two of the generators of $\mathbf{wB}(\lambda_P, \emptyset, \lambda_S)^{\text{ab}}$ described in Proposition 5.10 are trivial in the Lie ring.

**PROOF.** Generators of $\mathcal{L}_1 = \mathbf{wB}(\lambda_P, \emptyset, \lambda_S)^{\text{ab}}$ are described in Proposition 5.10 and Figure 5.1. Most of the pairs of such generators can be lifted to elements of $\mathbf{wB}(\lambda_P, \emptyset, \lambda_S)$ with disjoint support. The only ones for which this is not possible are:

* $t_i$ and $x_{ij}$ for $j \in I_S$ and $i \in I_P$ such that $n_i = 2$.
• $x_{ij}$ and $x_{ik}$ for $j, k \in I_S$ and $i \in I_P$ such that $n_i = 1$.

We first use Proposition 5.14 to show that $[x_{ij}, x_{ik}] = 0$ in the Lie ring. In order to do this, let us denote by $\mu_P$ the partition obtained from $\lambda_P$ by removing the blocks of size 2. Proposition 5.14 implies that $\mathcal{L}(\mathbf{w}\mathbf{B}(\mu_P, \emptyset, \lambda_S))$ is abelian. Since $x_{ij}$ and $x_{ik}$ are in the image of this Lie ring inside $\mathcal{L}(\mathbf{w}\mathbf{B}(\lambda_P, \emptyset, \lambda_S))$, their bracket must be trivial.

We now know that $\mathcal{L}_2$ is generated by the commutator $z_{ij} := [t_i, x_{ij}]$, for $j \in I_S$ and $i \in I_P$ such that $n_i = 2$ (note that this also works when $\lambda_P$ does not have blocks of size 2; then there are no $z_{ij}$, hence $\mathcal{L}_2 = 0$, and we recover the case $I_{S+} = \emptyset$ of Proposition 5.14). Before going further in our analysis of the Lie ring, let us examine the element $z_{ij}$ (with $i, j$ as above). If the $i$-th block is $\{\alpha, \alpha+1\}$ and $\beta$ is in the $j$-th block, $z_{ij}$ is the class of the commutator $\zeta_{\alpha\beta} := [\tau_\alpha, \chi_{\alpha\beta}]$, which is the automorphism:

$$ \zeta_{\alpha\beta} : \left\{ \begin{array}{rcl} x_{\alpha} & \mapsto & x_{\beta}^{-1} x_{\alpha} x_{\beta} \\ x_{\alpha+1} & \mapsto & x_{\alpha+1} \end{array} \right. \mapsto \left\{ \begin{array}{rcl} x_{\alpha}^{-1} x_{\alpha+1} x_{\beta} & \mapsto & x_{\beta}^{-1} x_{\alpha+1} x_{\beta} \\ x_{\alpha} & \mapsto & x_{\beta}^{-1} x_{\alpha} x_{\beta} \end{array} \right. \mapsto \left\{ \begin{array}{rcl} x_{\beta}^{-1} x_{\alpha} x_{\beta}, \\ x_{\beta} x_{\alpha+1} x_{\beta}^{-1} \end{array} \right. $$

where all the $x_\gamma$ with $\gamma \notin \{\alpha, \alpha+1\}$ are fixed. The corresponding motion is drawn in Figure 5.2. We note that $\zeta_{\alpha\beta} = \chi_{\alpha+1,\beta}\chi_{\alpha\beta}^{-1}$ (which can be seen directly, or deduced from $\tau_\alpha\chi_{\alpha\beta}\tau_\alpha^{-1} = \chi_{\alpha+1,\beta}$).

Let us turn our attention to $\mathcal{L}_3$. It is generated by the commutators of the generators $z_{ij} = \zeta_{\alpha\beta}$ of $\mathcal{L}_2$ with the generators of $\mathcal{L}_1$. Since Supp($\zeta_{\alpha\beta}$) = $\{\alpha, \alpha+1, \beta\}$ (where $\beta$ can be chosen anywhere in the $j$-th block for $\zeta_{\alpha\beta}$ to represent $z_{ij}$), one can choose a lift having disjoint support with $\zeta_{\alpha\beta}$ for each generator of $\mathcal{L}_1$, save $x_{ik}$ (for $k \in I_S$) and $t_i$. If $k = j$, we remark that $x_{ij}$ does have a lift commuting with $\zeta_{\alpha\beta}$, albeit not for reasons of support. Indeed, it is the class of $\chi_{\alpha\beta}$, which clearly commutes with $\zeta_{\alpha\beta}$, since both are automorphisms conjugating $x_\alpha$ and $x_{\alpha+1}$ by powers of the same element $x_\beta$. Thus the only generators of $\mathcal{L}_3$ which are possibly non-trivial are the $[t_i, z_{ij}]$ and the $[x_{ik}, z_{ij}]$ for $k \neq j$.

We first show that $[x_{ik}, z_{ij}]$ vanishes, when $k \neq j$. Let $\gamma$ be in the $k$-th block of $\lambda$. Then $[x_{ik}, z_{ij}]$ is the class of the commutator $[\chi_{\alpha\gamma}, \zeta_{\alpha\beta}]$. Since $\zeta_{\alpha\beta} = \chi_{\alpha+1,\beta}\chi_{\alpha\beta}^{-1}$ and $\chi_{\alpha\gamma}$ commutes with $\chi_{\alpha+1,\beta}$, this commutator is equal to $[\chi_{\alpha\gamma}, \chi_{\alpha\beta}^{-1}]$. We have seen above that $\chi_{\alpha\gamma}$ commutes with $\chi_{\alpha\beta}$ modulo $\Gamma_\infty$. Thus, the class of $[\chi_{\alpha\gamma}, \chi_{\alpha\beta}^{-1}]$ is trivial in the Lie ring, which means that $[x_{ik}, z_{ij}] = 0$.

We now show that $[t_i, z_{ij}] = 0$, using explicit computations. We have:

$$ \tau_{\alpha} \zeta_{\alpha\beta} \tau_{\alpha}^{-1} : \left\{ \begin{array}{rcl} x_{\alpha} & \mapsto & x_{\alpha+1} \\ x_{\alpha+1} & \mapsto & x_{\alpha} \end{array} \right. \mapsto \left\{ \begin{array}{rcl} x_{\beta} x_{\alpha+1} x_{\beta}^{-1} & \mapsto & x_{\beta} x_{\alpha} x_{\beta}^{-1}, \\ x_{\beta}^{-1} x_{\alpha} x_{\beta} & \mapsto & x_{\beta}^{-1} x_{\alpha+1} x_{\beta} \end{array} \right. $$

which means that $\tau_\alpha \zeta_{\alpha\beta} \tau_\alpha^{-1} = \zeta_{\alpha\beta}^{-1}$. We also have:

$$ \rho_{\beta} \zeta_{\alpha\beta} \rho_{\beta}^{-1} : \left\{ \begin{array}{rcl} x_{\alpha} & \mapsto & x_{\alpha} \\ x_{\alpha+1} & \mapsto & x_{\alpha+1} \\ x_{\beta} & \mapsto & x_{\beta}^{-1} \end{array} \right. \mapsto \left\{ \begin{array}{rcl} x_{\beta}^{-1} x_{\alpha} x_{\beta} & \mapsto & x_{\beta}^{-1} x_{\alpha+1} x_{\beta}^{-1}, \\ x_{\beta}^{-1} x_{\alpha+1} x_{\beta} & \mapsto & x_{\beta}^{-1} x_{\alpha+1} x_{\beta} \\ x_{\beta} & \mapsto & x_{\beta}^{-1} \end{array} \right. $$

whence $\rho_\beta \zeta_{\alpha\beta} \rho_\beta^{-1} = \zeta_{\alpha\beta}^{-1} = \tau_\alpha \zeta_{\alpha\beta} \tau_\alpha^{-1}$. Finally:

$$ [t_i, z_{ij}] = [\tau_\alpha, \zeta_{\alpha\beta}] = [\rho_\beta, \zeta_{\alpha\beta}] = [r_j, z_{ij}] = 0, $$

where the last equality comes from the above disjoint support argument. This finishes the proof that $\mathcal{L}_3 = 0$. Notice that the above calculation also implies
$\Gamma_3 \ni [\tau_\alpha, \zeta_{\alpha\beta}] = \zeta_{\alpha\beta}^{-2}$, so that $z_{ij} = \bar{\zeta}_{\alpha\beta}$ is of order 2 in $\mathcal{L}_2$. Thus, in order to prove our last claim, we need to show that the $z_{ij}$ are linearly independant over $\mathbb{Z}/2$. For each fixed choice of $j \in I_S$ and $i \in I_P$ such that $n_i = 2$, we have a projection from $\mathbf{wB}(\lambda_P, \emptyset, \lambda_S)$ onto $\mathbf{wB}(2, \emptyset, n_j)$ which induces a morphism between the $\mathcal{L}_2$ killing all the $z_{kl}$ save $z_{ij}$, which is sent to $z_{12}$. Thus, we only need to show that for $n \ge 4$, the element $z_{12}$ is non-trivial in $\mathcal{L}_2(\mathbf{wB}(2, \emptyset, n))$, which is then isomorphic to $\mathbb{Z}/2$. We do this by constructing one further projection. Elements of $\mathbf{wB}(2, \emptyset, n)$ are automorphisms of $\mathbf{F}_{2+n}$ preserving the normal subgroup generated by the $x_\alpha x_{\alpha+1}$ together with the $x_\alpha^2$ ($\alpha \ge 3$). Whence a well-defined morphism $\mathbf{wB}(2, \emptyset, n) \to \mathrm{Aut}(\mathbf{F}_2 * (\mathbb{Z}/2))$. The image of this morphism is easily seen to be isomorphic to $W_2 = (\mathbb{Z}/2) \rtimes \mathfrak{S}_2$ (where the images of $\chi_{13}$ and $\chi_{23}$ generate $(\mathbb{Z}/2)^2$ and the image of $\tau_1$ generates $\mathfrak{S}_2$), and the element $\zeta_{13} = \chi_{23}\chi_{13}^{-1}$ is sent to the generator $(1, 1, \mathrm{id})$ of $\Gamma_2(W_2)$ (see Proposition A.10 for the calculation of $\Gamma_*(W_2)$). Finally, the morphism induced between the Lie rings sends $z_{12}$ to the generator of $\mathcal{L}_2(W_2) \cong \mathbb{Z}/2$, whence the result. $\square$

<figure><img src="image_30.png" /><figcaption>FIGURE 5.2. Generators of $\mathcal{L}_2 = \Gamma_2/\Gamma_3$ for $G = \mathbf{wB}(\lambda_P, \emptyset, m)$, where each block of $\lambda_P$ has size $\ge 2$ and $m \ge 4$. There is one generator corresponding to the commutator $\zeta_{\alpha,\beta}$, represented by a loop of the form pictured, for each block of $\lambda_P$ of size exactly 2. This generalises to $G = \mathbf{wB}(\lambda_P, \emptyset, \lambda_S)$ where each block of $\lambda_S$ has size $\ge 4$, and the generators are indexed by $(b, x)$ where $b$ is a block of points of size exactly 2 and $x$ is a block of circles.</figcaption></figure>

PROOF OF THEOREM 5.11. The first statement is Proposition 5.16, which is a generalisation of Propositions 5.15 and 5.14, which each in turn generalise Proposition 5.12.

Among the remaining cases, here are the ones for which we can deduce our conclusion directly from the calculations above (Propositions 4.38 and 4.42), by exhibiting a quotient of $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)$ whose LCS does not stop:

* If $\lambda_S$ has one block of size 2 or 3, then it surjects onto $\tilde{\mathbf{wB}}_2$ or $\tilde{\mathbf{wB}}_3$.

* If $\lambda_S$ has at least two blocks of size 1, then it surjects onto $\tilde{\mathbf{wB}}_{1,1} = \tilde{\mathbf{wP}}_2 \cong \mathbf{F}_2 \times (\mathbb{Z}/2)^2$.

* If $\lambda_{S_+}$ has one block of size 2 or 3, then it surjects onto $\mathbf{wB}_2$ or $\mathbf{wB}_3$.

* If $\lambda_{S_+}$ has at least two blocks of size 1, then it surjects onto $\mathbf{wB}_{1,1} = \mathbf{wP}_2$.

In all of these cases, Lemma 1.1 allows us to conclude that the LCS of the partitioned tripartite welded braid group $\mathbf{wB}(\lambda_P, \lambda_{S_+}, \lambda_S)$ does not stop.
Moreover, if $\lambda_S$ has one block of size 1 which is not the only block of $\lambda$, then Proposition 5.17 ensures that the LCS of $wB(\lambda_P, \lambda_{S_+}, \lambda_S)$ does not stop. Then the only remaining case is the one where $\lambda_P$ has at least one block of size 2 and the blocks of $\lambda_S$ are of size at least 4. This last case is covered by Proposition 5.18 when $I_{S_+}$ is not empty and by Proposition 5.21 when $I_{S_+}$ is empty. $\square$