# Unrestricted virtual braids and crystallographic braid groups

Paolo Bellingeri  Normandie Univ, UNICAEN, CNRS, LMNO, 14000 Caen, France
<EMAIL_ADDRESS>,  John Guaschi  Normandie Univ, UNICAEN, CNRS,
LMNO, 14000 Caen, France<EMAIL_ADDRESS>and  Stavroula Makri
Normandie Univ, UNICAEN, CNRS, LMNO, 14000 Caen, France
<EMAIL_ADDRESS>
(Date: 27th August 2024)

###### Abstract.

We show that the crystallographic braid group $B_{n}/[P_{n},P_{n}]$ embeds
naturally in the group of unrestricted virtual braids $UVB_{n}$, we give new
proofs of known results about the torsion elements of $B_{n}/[P_{n},P_{n}]$,
and we characterise the torsion elements of $UVB_{n}$.

###### Key words and phrases:

Braid groups, virtual and welded braid groups, unrestricted virtual braid
groups

###### 1991 Mathematics Subject Classification:

Primary 20F36; Secondary 20H15

##  1\. Introduction

Let $n\geq 1$. The group of unrestricted virtual braids, denoted throughout
this paper by $UVB_{n}$, was introduced by Kauffman and Lambropoulou in [22]
as the analogue of fused links in the setting of braids. The classification of
fused links is now well known. Such links are distinguished by their _virtual
linking number_ , see for instance [25], where they are considered as the
closure of unrestricted virtual braids, as well as [1] for their
classification in terms of Gauss diagrams. Unrestricted virtual braid groups
occur as natural quotients of virtual and welded braid groups. They appear for
instance in [20] in the study of local representations of welded braid groups,
where they are called _symmetric loop braid groups_ , and they may be
decomposed as a semi-direct product of a right-angled Artin group, which is in
fact the pure subgroup $UVP_{n}$ of $UVB_{n}$, by the symmetric group $S_{n}$
[3]. The main aim of this paper is to characterise the torsion elements of
$UVB_{n}$ using this decomposition, namely to show that any element of finite
order is a conjugate of an element of $S_{n}$ by an element of $UVP_{n}$.

The structure of this paper is as follows. In Section 2, we give presentations
of the virtual braid groups $VB_{n}$, of the welded braid groups $WB_{n}$, and
of $UVB_{n}$. In this way, $UVB_{n}$ may be viewed as a quotient of both
$VB_{n}$ and $WB_{n}$. We also recall two important results of [3] that
describe the structure of the pure unrestricted virtual braid group $UVP_{n}$
as a direct sum of copies of the free group $F_{2}$ on two generators, which
as mentioned above, allows us to decompose $UVB_{n}$ as a semi-direct product
of $UVP_{n}$ and $S_{n}$ in a natural way. A similar decomposition holds for
$VB_{n}$ and $WB_{n}$, but the canonical homomorphism $\eta\colon\thinspace
B_{n}\longrightarrow UVB_{n}$, where $B_{n}$ is the Artin braid group, is not
injective, which is in contrast with the nature of the corresponding
homomorphisms for $VB_{n}$ and $WB_{n}$. In Section 3, we study the image of
$\eta$, and in Proposition 3.1, we prove that it is isomorphic to the quotient
$B_{n}/[P_{n},P_{n}]$, where $P_{n}$ is the pure Artin braid group, and
$[P_{n},P_{n}]$ is its commutator subgroup. This quotient has been the subject
of recent study, one of the reasons being that it is a crystallographic group
[16, 17, 18]. The results of [16] have been generalised to quasi-Abelianised
quotients of complex reflection groups [4, 24], and to surface braid groups
[19]. This enables us to give an alternative proof in Proposition 3.5 and
Remark 3.6 of the fact that $B_{n}/[P_{n},P_{n}]$ embeds in the semi-direct
product $P_{n}/[P_{n},P_{n}]\rtimes S_{n}$. In the final section of the paper,
Section 4, we study the torsion elements of $UVB_{n}$. We apply the embedding
of Proposition 3.1 to give a new combinatorial proof in Theorem 4.1 of the
fact that there are no elements of even order in $B_{n}/[P_{n},P_{n}]$, and in
Theorem 4.3, we characterise the torsion elements of $UVB_{n}$ by showing that
every such element is conjugate to an element of $S_{n}$ by an element of
$UVP_{n}$. To our knowledge, it is not known whether a analogous result holds
for $VB_{n}$ and $WB_{n}$.

##  2\. Unrestricted virtual braids

In order to define unrestricted virtual braid groups, in this section we
recall first the notions of virtual and welded braid groups by exhibiting
their usual group presentations.

######  Definition 2.1.

The _virtual braid group_ $VB_{n}$ is the group defined by the group
presentation:

Generators:
$\\{\sigma_{1}^{\phantom{1}},\ldots,\sigma_{n-1}^{\phantom{1}},\rho_{1}^{\phantom{1}},\ldots,\rho_{n-1}^{\phantom{1}}\\}$

Relations:

$\displaystyle\sigma_{i}^{\phantom{1}}\sigma_{i+1}^{\phantom{1}}\sigma_{i}^{\phantom{1}}$
$\displaystyle=\sigma_{i+1}^{\phantom{1}}\sigma_{i}^{\phantom{1}}\sigma_{i+1}^{\phantom{1}}$
for $i=1,\dots,n-2$ (BR1)
$\displaystyle\sigma_{i}^{\phantom{1}}\sigma_{j}^{\phantom{1}}$
$\displaystyle=\sigma_{j}^{\phantom{1}}\sigma_{i}^{\phantom{1}}$ for $\lvert
i-j\rvert\geq 2$ (BR2)
$\displaystyle\rho_{i}^{\phantom{1}}\rho_{i+1}^{\phantom{1}}\rho_{i}^{\phantom{1}}$
$\displaystyle=\rho_{i+1}^{\phantom{1}}\rho_{i}^{\phantom{1}}\rho_{i+1}^{\phantom{1}}$
for $i=1,\dots,n-2$ (SR1)
$\displaystyle\rho_{i}^{\phantom{1}}\rho_{j}^{\phantom{1}}$
$\displaystyle=\rho_{j}^{\phantom{1}}\rho_{i}^{\phantom{1}}$ for $\lvert
i-j\rvert\geq 2$ (SR2) $\displaystyle\rho_{i}^{2}$ $\displaystyle=1$ for
$i=1,\dots,{n-1}$ (SR3)
$\displaystyle\sigma_{i}^{\phantom{1}}\rho_{j}^{\phantom{1}}$
$\displaystyle=\rho_{j}^{\phantom{1}}\sigma_{i}^{\phantom{1}}$ for $\lvert
i-j\rvert\geq 2$ (MR1)
$\displaystyle\rho_{i}^{\phantom{1}}\rho_{i+1}^{\phantom{1}}\sigma_{i}^{\phantom{1}}$
$\displaystyle=\sigma_{i+1}^{\phantom{1}}\rho_{i}^{\phantom{1}}\rho_{i+1}^{\phantom{1}}$
for $i=1,\dots,n-2$. (MR2)

A diagrammatic description of generators and relations of $VB_{n}$ may be
found for instance in [2, 5, 21]. For a topological interpretation, we refer
the reader to [10], and for an algebraic one (in terms of actions on root
systems) to [6]. Note that the relations (BR1)–(BR2) (resp. (SR1)–(SR3))
correspond to the usual relations of the Artin braid group $B_{n}$ (resp. the
symmetric group $S_{n}$) for the set
$\\{\sigma_{1}^{\phantom{1}},\ldots,\sigma_{n-1}^{\phantom{1}}\\}$ (resp. for
the set $\\{\rho_{1}^{\phantom{1}},\ldots,\rho_{n-1}^{\phantom{1}}\\}$), and
the remaining relations (MR1)–(MR2) are ‘mixed’ in the sense that they involve
generators of both of these sets.

Recall that the pure braid group $P_{n}$ is the kernel of the homomorphism
$\pi\colon\thinspace B_{n}\longrightarrow S_{n}$ defined on the generators
$\sigma_{1}^{\phantom{1}},\ldots,\sigma_{n-1}^{\phantom{1}}$ of $B_{n}$ by
$\pi(\sigma_{i})=s_{i}$ for all $i=1,\ldots n-1$, where $s_{i}$ is the
transposition $(i,i+1)$. Analogously, we define the _virtual pure braid group_
, denoted by $VP_{n}$, to be the kernel of the homomorphism
$\pi_{VP}\colon\thinspace VB_{n}\longrightarrow S_{n}$ that for all
$i=1,2,\dots,n-1$, maps the generators $\sigma_{i}^{\phantom{1}}$ and
$\rho_{i}^{\phantom{1}}$ to $s_{i}$. A group presentation for $VP_{n}$ is
given in [2]. Let $\iota\colon\thinspace S_{n}\longrightarrow VB_{n}$ be the
homomorphism defined by $\iota(s_{i})=\rho_{i}^{\phantom{1}}$ for
$i=1,\ldots,n-1$. Since $\iota$ is a section for $\pi_{VP}$, it follows that
$\iota$ is injective and that $VB_{n}$ is a semi-direct product of $VP_{n}$ by
$S_{n}$. The canonical homomorphism $\eta\colon\thinspace B_{n}\longrightarrow
VB_{n}$ defined by $\eta(\sigma_{i})=\sigma_{i}$ for $i=1,\ldots,n-1$ is
injective [5, 15, 21].

The welded braid group $WB_{n}$ may be defined as a quotient of $VB_{n}$ by
adding the following family of relations to the presentation given in
Definition 2.1:

$\rho_{i}^{\phantom{1}}\sigma_{i+1}^{\phantom{1}}\sigma_{i}^{\phantom{1}}=\sigma_{i+1}^{\phantom{1}}\sigma_{i}^{\phantom{1}}\rho_{i+1}^{\phantom{1}},\
\text{for $i=1,\dots,n-2$,}$ (OC)

where OC stands for ‘Over Commuting’. Welded braid groups have several
different equivalent definitions [11]. In particular, they may be defined as
_basis conjugating automorphisms_ of free groups [14], from which it follows
that the homomorphism $\eta\colon\thinspace B_{n}\longrightarrow WB_{n}$
defined by $\eta(\sigma_{i})=\sigma_{i}$ for $i=1,\ldots,n-1$ is injective
[21]. As in the case of virtual braid groups, the _welded pure braid group_ ,
denoted by $WP_{n}$, is defined to be the kernel of the homomorphism
$\pi_{WP}\colon\thinspace WB_{n}\longrightarrow S_{n}$ given by sending the
generators $\sigma_{i}^{\phantom{1}}$ and $\rho_{i}^{\phantom{1}}$ of $WB_{n}$
to $s_{i}$ for all $i=1,2,\dots,n-1$. A group presentation for $WP_{n}$ may be
found in [8, 11, 26]. Abusing notation, we define $\iota\colon\thinspace
S_{n}\longrightarrow WB_{n}$ to be also the homomorphism defined by
$\iota(s_{i})=\rho_{i}^{\phantom{1}}$ for $i=1,\ldots,n-1$. As in the virtual
case, $\iota$ is a section for $\pi_{WP}$, and therefore $\iota$ is injective
and $WB_{n}$ is a semi-direct product of $WP_{n}$ by $S_{n}$. Note that the
symmetrical relations:

$\rho_{i+1}^{\phantom{1}}\sigma_{i}^{\phantom{1}}\sigma_{i+1}^{\phantom{1}}=\sigma_{i}^{\phantom{1}}\sigma_{i+1}^{\phantom{1}}\rho_{i}^{\phantom{1}},\
\text{for $i=1,\dots,n-2$},$ (UC)

where UC stands for ‘Under Commuting’, do not hold in $WB_{n}$ (see for
instance [7, 21]). Consequently, the following group, which was first defined
by Kauffman and Lambropoulou in [22], is a proper quotient of $WB_{n}$.

######  Definition 2.2.

The _unrestricted virtual braid group_ $UVB_{n}$ is the group defined by the
following group presentation:

Generators:
$\\{\sigma_{1}^{\phantom{1}},\ldots,\sigma_{n-1}^{\phantom{1}},\rho_{1}^{\phantom{1}},\ldots,\rho_{n-1}^{\phantom{1}}\\}$

Relations: the seven relations (BR1)–(BR2), (SR1)–(SR3) and (MR1)–(MR2) of
Definition 2.1, plus the relations of types (OC) and (UC).

As in the case of $VB_{n}$, let $\pi_{UVP}\colon\thinspace
UVB_{n}\longrightarrow S_{n}$ be the homomorphism defined by
$\pi_{UVP}(\sigma_{i})=\pi_{UVP}(\rho_{i})=s_{i}$ for $i=1,\ldots,n-1$. The
kernel of $\pi_{UVP}$ is the _unrestricted virtual pure braid group_ , denoted
by $UVP_{n}$. For $1\leq i,j\leq n$, $i\neq j$, we define the elements
$\lambda_{i,j}$ of $UVP_{n}$ as follows:

$\begin{cases}\lambda_{i,i+1}=\rho_{i}\sigma_{i}^{-1}&\text{for
$i=1,\ldots,n-1$}\\\ \lambda_{i+1,i}=\sigma_{i}^{-1}\rho_{i}&\text{for
$i=1,\ldots,n-1$}\\\
\lambda_{i,j}=\rho_{j-1}\rho_{j-2}\cdots\rho_{i+1}\lambda_{i,i+1}\rho_{i+1}\cdots\rho_{j-2}\rho_{j-1}&\text{for
$1\leq i<j-1\leq n-1$}\\\
\lambda_{j,i}=\rho_{j-1}\rho_{j-2}\cdots\rho_{i+1}\lambda_{i+1,i}\rho_{i+1}\cdots\rho_{j-2}\rho_{j-1}&\text{for
$1\leq i<j-1\leq n-1$.}\end{cases}$ (1)

######  Theorem 2.3 (Bardakov–Bellingeri–Damiani [3]).

The group $UVP_{n}$ admits the following presentation:
Generators: $\lambda_{i,j}$, where $1\leq i,j\leq n$, $i\neq j$.
Relations: The generators commute pairwise except for the pairs
$\lambda_{i,j}$ and $\lambda_{j,i}$ for all $1\leq i,j\leq n$, $i\neq j$.

It follows from Theorem 2.3 that the group $UVP_{n}$ is a right-angled Artin
group that is isomorphic to the direct product of the ${n(n-1)/2}$ free groups
$F_{i,j}$ of rank $2$, where $1\leq i<j\leq n$, and where
$\\{\lambda_{i,j},\lambda_{j,i}\\}$ is a basis of $F_{i,j}$. By convention, if
$1\leq i,j\leq n$ then we set $F_{j,i}=F_{i,j}$. As for $VB_{n}$, let
$\iota\colon\thinspace S_{n}\longrightarrow UVB_{n}$ be the homomorphism
defined by $\iota(s_{i})=\rho_{i}^{\phantom{1}}$ for $i=1,\ldots,n-1$. Then
$\iota$ is injective, and is a section for $\pi_{UVP}$. This leads to the
following natural decomposition of $UVB_{n}$.

######  Theorem 2.4 (Bardakov–Bellingeri–Damiani [3]).

The group $UVB_{n}$ is isomorphic to the semi-direct product $UVP_{n}\rtimes
S_{n}$, where $S_{n}$ acts on $UVP_{n}$ by permuting the indices of the
elements of the generating set of $UVP_{n}$ given in Theorem 2.3. More
precisely, for all $\lambda_{i,j}\in UVP_{n}$, where $1\leq i,j\leq n$, $i\neq
j$, and for all $s\in S_{n}$ we have:

$\iota(s)\,\lambda_{i,j}\,\iota(s)^{-1}=\lambda_{s(i),s(j)}.$ (2)

##  3\. Crystallographic groups

As we mentioned in Section 2, the Artin braid group $B_{n}$ embeds naturally
in $VB_{n}$ and $WB_{n}$ via the injective homomorphism $\eta$ defined in each
of these two cases. However this is no longer the case for the canonical
homomorphism $\eta\colon\thinspace B_{n}\longrightarrow UVB_{n}$ defined by
$\eta(\sigma_{i})=\sigma_{i}$ for $i=1,\ldots,n-1$. In this section, we show
that the image of $B_{n}$ under the homomorphism $\eta$ is isomorphic to the
quotient of $B_{n}$ by the commutator subgroup $[P_{n},P_{n}]$. This quotient
was introduced by Tits in [27] as _groupes de Coxeter étendus_ , and has been
studied more recently by various authors (see for example [4, 13, 16, 17, 18,
24]).

######  Proposition 3.1.

Let $\eta\colon\thinspace B_{n}\longrightarrow UVB_{n}$ be the canonical
homomorphism defined by $\eta(\sigma_{i})=\sigma_{i}$ for $1\leq i\leq n-1$.
Then $\eta(B_{n})$ is isomorphic to the group $B_{n}/[P_{n},P_{n}]$.

###### Proof.

Consider the homomorphism $\eta\colon\thinspace B_{n}\longrightarrow UVB_{n}$.
Since $P_{n}=\operatorname{\text{Ker}}(\pi)$ (resp.
$UVP_{n}=\operatorname{\text{Ker}}(\pi_{UVP})$), where $\pi\colon\thinspace
B_{n}\longrightarrow S_{n}$ (resp. $\pi_{UVP}\colon\thinspace
UVB_{n}\longrightarrow S_{n}$) is defined by $\pi(\sigma_{i})=s_{i}$ (resp.
$\pi_{UVP}(\sigma_{i})=\pi_{UVP}(\rho_{i})=s_{i}$) for all $1\leq i\leq n-1$,
it follows from the definition of $\eta\colon\thinspace B_{n}\longrightarrow
UVB_{n}$ that $\pi=\pi_{UVP}\circ\eta$, and thus
$\pi=\pi_{UVP}\left\lvert{}_{\eta(B_{n})}\right.\circ\eta$. Since $\pi$ is
surjective, the homomorphism
$\pi_{UVP}\left\lvert{}_{\eta(B_{n})}\right.\colon\thinspace\eta(B_{n})\longrightarrow
S_{n}$ is too. From the equality $\pi=\pi_{UVP}\circ\eta$, we obtain the
following commutative diagram of short exact sequences:

${1}$${P_{n}}$${B_{n}}$${S_{n}}$${1}$${1}$${UVP_{n}}$${UVB_{n}}$${S_{n}}$${1.}$$\scriptstyle{\eta\left\lvert{}_{P_{n}}\right.}$$\scriptstyle{\pi}$$\scriptstyle{\eta}$$\scriptstyle{\pi_{UVP}}$
(3)

We claim that
$\eta(P_{n})=\operatorname{\text{Ker}}(\pi_{UVP}\left\lvert{}_{\eta(B_{n})}\right.)$.
To prove the claim, the exactness of (3) implies that
$\eta(P_{n})\subset\operatorname{\text{Ker}}(\pi_{UVP}\left\lvert{}_{\eta(B_{n})}\right.)$.
Conversely, let
$x\in\operatorname{\text{Ker}}(\pi_{UVP}\left\lvert{}_{\eta(B_{n})}\right.)$.
Then $x\in\eta(B_{n})$, and there exists $y\in B_{n}$ such that $\eta(y)=x$.
The commutativity of (3) implies that $y\in P_{n}$, and so $x\in\eta(P_{n})$,
so
$\operatorname{\text{Ker}}(\pi_{UVP}\left\lvert{}_{\eta(B_{n})}\right.)\subset\eta(P_{n})$,
and the claim follows.

Considering $\eta$ and its restriction to $P_{n}$, we obtain the following
commutative diagram of short exact sequences:

${1}$${1}$${1}$${\operatorname{\text{Ker}}(\eta\left\lvert{}_{P_{n}}\right.)}$${P_{n}}$${\eta(P_{n})}$${1}$${1}$${\operatorname{\text{Ker}}(\eta)}$${B_{n}}$${\eta(B_{n})}$${1.}$${S_{n}}$${S_{n}}$${1}$${1}$$\scriptstyle{\eta\left\lvert{}_{P_{n}}\right.}$$\scriptstyle{\pi}$$\scriptstyle{\eta}$$\scriptstyle{\pi_{UVP}\left\lvert{}_{\eta(B_{n})}\right.}$
(4)

Note that the exactness of the rightmost column of (4) follows from the claim
of the previous paragraph. By a standard diagram-chasing argument, we conclude
from (4) that
$\operatorname{\text{Ker}}(\eta)=\operatorname{\text{Ker}}(\eta\left\lvert{}_{P_{n}}\right.)$.
Applying the isomorphism $UVB_{n}\cong UVP_{n}\rtimes S_{n}$ of Theorem 2.4
and using equation (1), we see that
$\eta(\sigma_{i})=\lambda_{i,i+1}^{-1}\rho_{i}$. Recall that the pure braid
group $P_{n}$ is generated by the set $\\{a_{ij}\mid 1\leq i<j\leq n\\}$,
where:

$\displaystyle a_{i,i+1}=\sigma_{i}^{2},$ $\displaystyle
a_{i,j}=\sigma_{j-1}^{\phantom{1}}\sigma_{j-2}^{\phantom{1}}\cdots\sigma_{i+1}^{\phantom{1}}\sigma_{i}^{2}\sigma_{i+1}^{-1}\cdots\sigma_{j-2}^{-1}\sigma_{j-1}^{-1}\mbox{,}\quad\mbox{for
}i+1<j\leq n.$

In [3, Corollary 2.8] it was shown that
$\eta(a_{i,j})=\lambda_{i,j}^{-1}\lambda_{j,i}^{-1}$ for $1\leq i+1<j\leq n$.
From the construction, we deduce that $[P_{n},P_{n}]$ is contained in
$\operatorname{\text{Ker}}(\eta\left\lvert{}_{P_{n}}\right.)$. Further,
$\eta(P_{n})$ is isomorphic to $P_{n}/[P_{n},P_{n}]=\mathbb{Z}^{n(n-1)/2}$
(see [3, Corollary 2.8]). It follows that $[P_{n},P_{n}]$ coincides with
$\operatorname{\text{Ker}}(\eta\left\lvert{}_{P_{n}}\right.)$. The statement
follows by combining this with the fact that
$\operatorname{\text{Ker}}(\eta)=\operatorname{\text{Ker}}(\eta\left\lvert{}_{P_{n}}\right.)$
and using the exactness of (4). ∎

######  Remark 3.2.

We can rephrase Proposition 3.1 by saying that the canonical homomorphism
$\eta\colon\thinspace B_{n}\longrightarrow UVB_{n}$ factors through an
embedding of $B_{n}/[P_{n},P_{n}]$ into $UVB_{n}$. On the other hand, by [9,
Theorem 2.5], the canonical homomorphism from $B_{n}$ to $VB_{n}$ induces an
embedding of $B_{n}/[P_{n},P_{n}]$ in $VB_{n}/[VP_{n},VP_{n}]$. Since
$VB_{n}/[VP_{n},VP_{n}]$ is isomorphic to $UVB_{n}/[UVP_{n},UVP_{n}]$ [9,
Theorem 4.1], the canonical homomorphism $B_{n}$ to $UVB_{n}$ gives rise to an
embedding of the quotient $B_{n}/[P_{n},P_{n}]$ in
$UVB_{n}/[UVP_{n},UVP_{n}]$. We therefore conclude that the composition of the
embedding of $B_{n}/[P_{n},P_{n}]$ in $UVB_{n}$ defined in the proof of
Proposition 3.1 and the canonical projection $q_{n}\colon\thinspace
UVB_{n}\longrightarrow UVB_{n}/[UVP_{n},UVP_{n}]$ is also injective.

We recall that an abstract group $\Gamma$ is said to be _crystallographic_ if
it can be realised as an extension of a free Abelian subgroup of $\Gamma$ of
maximal rank by a finite group [12, Theorem 2.1.4]. Using the following
result, in [16], Gonçalves–Guaschi–Ocampo proved that the group
$B_{n}/[P_{n},P_{n}]$ is a crystallographic group.

######  Proposition 3.3 (Gonçalves–Guaschi–Ocampo [16]).

Let $n\geq 2$. Then the following sequence:

${1}$${\mathbb{Z}^{n(n-1)/2}}$${B_{n}/[P_{n},P_{n}]}$${S_{n}}$${1,}$$\scriptstyle{\widehat{\pi}}$
(5)

is short exact, where $\widehat{\pi}$ is the homomorphism induced by
$\pi\colon\thinspace B_{n}\longrightarrow S_{n}$.

Note that the sequence (5) does not split [13, 16]. It follows from
Propositions 3.1 and 3.3 that the group $\eta(B_{n})$ is a crystallographic
subgroup of $UVB_{n}$. We remark also that $\operatorname{\text{Im}}(\eta)$ is
not normal in $UVB_{n}$, and that the normal closure of
$\operatorname{\text{Im}}(\eta)$ is of index $2$ in $UVB_{n}$. This follows
from the fact that in the quotient of $UVB_{n}$ by
$\operatorname{\text{Im}}(\eta)$, the generators $\rho_{i}$ are identified to
a single element due to relations (OC) and (UC) of Definition 2.2. The group
$UVB_{n}$ contains other crystallographic groups of rank $n(n-1)/2$ that are
not isomorphic to $\eta(B_{n})$. One such example is given in the following
proposition that is a straightforward consequence of Theorems 2.3 and 2.4

######  Proposition 3.4.

Let $C_{n}$ be the subgroup of $UVB_{n}$ generated by
$\\{\lambda_{i,j}\lambda_{j,i}^{-1}\\}$ for $1\leq i<j\leq n$ and by
$\rho_{i}$ for $i=1,\ldots,n$, and let $H_{n}$ be the subgroup of $C_{n}$
generated by $\\{\lambda_{i,j}\lambda_{j,i}^{-1}\\}$ for $1\leq i<j\leq n$.
The restriction of $\pi_{UVP}$ to $C_{n}$ defines the following split exact
sequence:

${1}$${H_{n}}$${C_{n}}$${S_{n}}$${1,}$$\scriptstyle{\pi_{UVP}{\big{|}}_{C_{n}}}$

In particular $C_{n}$ is crystallographic.

######  Proposition 3.5.

Let $H_{n}$ be the subgroup of $UVB_{n}$ defined in the statement of
Proposition 3.4, let $\langle\\!\langle H_{n}\rangle\\!\rangle$ its normal
closure in $UVB_{n}$, and let $\\{x_{i,j}\\}_{1\leq i<j\leq n}$ be a set of
generators of $\mathbb{Z}^{n(n-1)/2}$.

  1. (a)

The quotient $UVB_{n}/\langle\\!\langle H_{n}\rangle\\!\rangle$ is isomorphic
to the semi-direct product $\mathbb{Z}^{n(n-1)/2}\rtimes S_{n}$, where for any
$s\in S_{n}$, $s\cdot x_{i,j}=x_{s(i),s(j)}$, where we take $x_{i,j}=x_{j,i}$.

  2. (b)

Let $\theta\colon\thinspace B_{n}\longrightarrow UVB_{n}/\langle\\!\langle
H_{n}\rangle\\!\rangle$ be the composition of $\eta:B_{n}\longrightarrow
UVB_{n}$ and the projection $\pi_{H}\colon\thinspace UVB_{n}\longrightarrow
UVB_{n}/\langle\\!\langle H_{n}\rangle\\!\rangle$. The group $\theta(B_{n})$
is isomorphic to $B_{n}/[P_{n},P_{n}]$.

###### Proof.

Part (a) follows from Theorems 2.3 and 2.4 that imply that
$UVB_{n}/\langle\\!\langle H_{n}\rangle\\!\rangle=UVP_{n}/\langle\\!\langle
H_{n}\rangle\\!\rangle\rtimes S_{n}$. The identification
$\lambda_{i,j}=\lambda_{j,i}$ for $i\not=j$ implies that
$UVP_{n}/H_{n}=\mathbb{Z}^{n(n-1)/2}$, and it also defines the induced action
of $S_{n}$ on this quotient. The proof of Proposition 3.1 can be adapted to
prove part (b) (the image of $P_{n}$ by $\theta$ is a free Abelian group of
rank $n(n-1)/2$, and is thus isomorphic to $P_{n}/[P_{n},P_{n}]$). ∎

######  Remark 3.6.

The quotient $UVB_{n}/\langle\\!\langle H_{n}\rangle\\!\rangle$ coincides with
the semi-direct product considered in [27, Section 2.7] and in the Remark
following [23, Proposition 5.12]. Thus Proposition 3.5 yields a new proof of
the fact that the crystallographic braid group $B_{n}/[P_{n},P_{n}]$ embeds in
the semi-direct product $\mathbb{Z}^{n(n-1)/2}\rtimes S_{n}$.

##  4\. Torsion elements of $UVB_{n}$

Let $n\geq 3$. In [16], it was shown that $B_{n}/[P_{n},P_{n}]$ has torsion,
but that it possesses no elements of even order. Viewing $B_{n}/[P_{n},P_{n}]$
as the subgroup $\operatorname{\text{Im}}(\eta)$ of $UVB_{n}$ via Proposition
3.1, we may give an alternative proof of this latter fact. We do this by first
characterising the elements of $UVB_{n}$ of order $2$, and then showing that
these elements do not belong to $\operatorname{\text{Im}}(\eta)$.

######  Theorem 4.1.

Let $n\geq 3$.

  1. (a)

An element $v\in UVB_{n}$ is of order $2$ if and only if there exist
$\rho\in\operatorname{\text{Im}}(\iota)$ of order $2$ and $g\in UVP_{n}$ such
that $v=g\rho g^{-1}$.

  2. (b)

The elements of order $2$ of $UVB_{n}$ do not belong to
$\operatorname{\text{Im}}(\eta)$. In particular, $B_{n}/[P_{n},P_{n}]$ has no
elements of even order.

###### Proof.

Let $n\geq 3$.

  1. (a)

First observe that the given condition is clearly sufficient. Conversely,
suppose that $v\in UVB_{n}$ is of order $2$. Identifying $UVB_{n}$ with the
internal semi-direct product $UVP_{n}\rtimes\operatorname{\text{Im}}(\iota)$
using Theorem 2.4, there exist unique elements $w\in UVP_{n}$ and
$\rho\in\operatorname{\text{Im}}(\iota)$ such that $v=w\rho$, and the pair
$(w,\rho)$ is non trivial. Further, the fact that $UVP_{n}$ is torsion free
implies that $\rho\neq 1$. Since $v$ is of order $2$, we have
$1=v^{2}=(w\rho)^{2}=w\ldotp\rho w\rho^{-1}\ldotp\rho^{2}$, where $w\ldotp\rho
w\rho^{-1}\in UVP_{n}$ and $\rho^{2}\in\operatorname{\text{Im}}(\iota)$, from
which we deduce using Theorem 2.4 that $\rho$ is also of order $2$. Hence
$\pi_{UVP}(\rho)\in S_{n}$ is also of order $2$, and thus may be written as a
non-trivial product of disjoint transpositions. So there exists
$\tau\in\operatorname{\text{Im}}(\iota)$ such that
$\pi_{UVP}(\tau\rho\tau^{-1})=\sigma$, where $\sigma=(1,2)(3,4)\cdots(m,m+1)$,
and $1\leq m\leq n-1$ is odd. Setting $\widetilde{v}=\tau v\tau^{-1}$,
$\widetilde{w}=\tau w\tau^{-1}$ and $\widetilde{\rho}=\tau\rho\tau^{-1}$, it
follows that $\widetilde{v}=\widetilde{w}\widetilde{\rho}$, where
$\widetilde{v}\in UVB_{n}$ is of order $2$, $\widetilde{w}\in UVP_{n}$,
$\widetilde{\rho}\in\operatorname{\text{Im}}(\iota)$ is of order $2$,
$\pi_{UVP}(\widetilde{\rho})=\sigma$, and
$\widetilde{w}\ldotp\widetilde{\rho}\widetilde{w}\widetilde{\rho}^{-1}=1$. For
$1\leq i<j\leq n$, let $\pi_{i,j}\colon\thinspace UVP_{n}\longrightarrow
F_{i,j}$ denote the projection of $UVP_{n}$ onto $F_{i,j}$ (see Theorem 2.3
and the comments that follow it). Applying (2), we observe that conjugation by
an element of $UVB_{n}$ permutes the $\\{F_{i,j}\\}_{1\leq i<j\leq n}$. Let
$T$ denote the subset of $n(n-1)/2$ transpositions of $S_{n}$, and let
$T_{\sigma}=\\{(i,i+1)\in T\,\mid\,i\in\\{1,3,\ldots,m\\}\\}$. Then the
subgroup $\langle\sigma\rangle$ of $S_{n}$ of order $2$ acts on $T$ by
conjugation, and if $1\leq i<j\leq n$, the orbit $\mathcal{O}(i,j)$ of $(i,j)$
is equal to $\\{(i,j),(\sigma(i),\sigma(j))\\}$, which is equal to
$\\{(i,j)\\}$ if and only if either $(i,j)\in T_{\sigma}$ or if $m+2\leq
i<j\leq n$, and contains two elements otherwise. Further, for all $1\leq
i<j\leq n$, $\sigma(i)<\sigma(j)$ if and only if $(i,j)\notin T_{\sigma}$. Let
$\mathcal{T}$ be a transversal for this action of $\langle\sigma\rangle$ on
$T$. Using Theorem 2.3, we obtain:

$\widetilde{w}=\prod_{(i,j)\in\mathcal{T}}\prod_{(k,l)\in\mathcal{O}(i,j)}w_{k,l}(\lambda_{k,l},\lambda_{l,k}),$
(6)

where $w_{k,l}=w_{k,l}(\lambda_{k,l},\lambda_{l,k})\in F_{k,l}$, and this
decomposition is unique up to permutation of the factors. So:

$\displaystyle 1$
$\displaystyle=\widetilde{w}\ldotp\widetilde{\rho}\widetilde{w}\widetilde{\rho}\,^{-1}=\left(\prod_{(i,j)\in\mathcal{T}}\prod_{(k,l)\in\mathcal{O}(i,j)}w_{k,l}(\lambda_{k,l},\lambda_{l,k})\right)\widetilde{\rho}\left(\prod_{(i,j)\in\mathcal{T}}\prod_{(k,l)\in\mathcal{O}(i,j)}w_{k,l}(\lambda_{k,l},\lambda_{l,k})\right)\widetilde{\rho}\,^{-1}$
$\displaystyle=\left(\prod_{(i,j)\in\mathcal{T}}\prod_{(k,l)\in\mathcal{O}(i,j)}w_{k,l}(\lambda_{k,l},\lambda_{l,k})\right)\left(\prod_{(i,j)\in\mathcal{T}}\prod_{(k,l)\in\mathcal{O}(i,j)}w_{k,l}(\lambda_{\sigma(k),\sigma(l)},\lambda_{\sigma(l),\sigma(k)})\right)$
$\displaystyle=\prod_{(i,j)\in\mathcal{T}}\prod_{(k,l)\in\mathcal{O}(i,j)}w_{k,l}(\lambda_{k,l},\lambda_{l,k})\ldotp
w_{\sigma(k),\sigma(l)}(\lambda_{k,l},\lambda_{l,k}),$ (7)

where we have used the fact that $\sigma^{2}=1$. Note also that if $(i,j)\in
T_{\sigma}$ then $j=\sigma(i)>\sigma(j)=i$, and in this case, any term of the
form $w_{\sigma(i),\sigma(j)}(\lambda_{i,j},\lambda_{j,i})$ in (7) should be
interpreted as $w_{i,j}(\lambda_{j,i},\lambda_{i,j})$. The expression (7) is
written with respect to the direct product structure of $UVP_{n}$. It follows
from Theorem 2.3 that $w_{i,j}(\lambda_{i,j},\lambda_{j,i})\ldotp
w_{\sigma(i),\sigma(j)}(\lambda_{i,j},\lambda_{j,i})=1$ for all $1\leq i<j\leq
n$, and hence:

$w_{\sigma(i),\sigma(j)}(\lambda_{\sigma(i),\sigma(j)},\lambda_{\sigma(j),\sigma(i)})=w_{i,j}^{-1}(\lambda_{\sigma(i),\sigma(j)},\lambda_{\sigma(j),\sigma(i)}).$
(8)

Suppose that $(i,j)\in T_{\sigma}$. Then
$w_{i,j}(\lambda_{i,j},\lambda_{j,i})\ldotp
w_{i,j}(\lambda_{j,i},\lambda_{i,j})=1$. Writing $a=\lambda_{i,j}$,
$b=\lambda_{j,i}$ and $w_{i,j}(a,b)=a^{k_{1}}b^{l_{1}}\cdots
a^{k_{m}}b^{l_{m}}$, where $m\geq 0$,
$k_{1},l_{1},\ldots,k_{m},l_{m}\in\mathbb{Z}$ and $l_{1},\ldots,k_{m}\neq 0$,
it follows from the relation $w_{i,j}(a,b)\ldotp w_{i,j}(b,a)=1$ that
$j_{q}=-k_{m+1-q}$ for all $q=1,\ldots,m$. Taking
$y_{i,j}(a,b)=a^{k_{1}}b^{-k_{m}}\cdots a^{k_{m/2}}b^{-k_{(m+2)/2}}$ (resp.
$y_{i,j}(a,b)=a^{k_{1}}b^{-k_{m}}\cdots
a^{k_{(m-1)/2}}b^{-k_{(m+3)/2}}a^{k_{(m+1)/2}}$) if $m$ is even (resp. odd),
we see that $w_{i,j}(a,b)=y_{i,j}(a,b)\ldotp y_{i,j}^{-1}(b,a)$. Thus
$w_{i,j}(\lambda_{i,j},\lambda_{j,i})=y_{i,j}(\lambda_{i,j},\lambda_{j,i})\ldotp
y_{i,j}^{-1}(\lambda_{j,i},\lambda_{i,j})$. Setting $z_{i,j}=w_{i,j}$ if
$(i,j)\notin T_{\sigma}$ and $z_{i,j}=y_{i,j}$ if $(i,j)\in T_{\sigma}$, it
follows from (6) and (8) that:

$\widetilde{v}=\widetilde{w}\widetilde{\rho}=\left(\prod_{(i,j)\in\mathcal{T}}z_{i,j}(\lambda_{i,j},\lambda_{j,i})z_{i,j}^{-1}(\lambda_{j,i},\lambda_{i,j})\right)\widetilde{\rho}.$
(9)

Now the terms $z_{i,j}(\lambda_{i,j},\lambda_{j,i})$ (resp.
$z_{i,j}^{-1}(\lambda_{j,i},\lambda_{i,j})$) appearing in (9) commute
pairwise, and setting
$\widetilde{g}=\prod_{(i,j)\in\mathcal{T}}z_{i,j}(\lambda_{i,j},\lambda_{j,i})\in
UVP_{n}$, we obtain:

$\widetilde{v}=\left(\prod_{(i,j)\in\mathcal{T}}z_{i,j}(\lambda_{i,j},\lambda_{j,i})\right)\left(\prod_{(i,j)\in\mathcal{T}}z_{i,j}^{-1}(\lambda_{j,i},\lambda_{i,j})\right)\widetilde{\rho}=\widetilde{g}\widetilde{\rho}\,\widetilde{g}^{-1}.$

Hence $v=\tau^{-1}\widetilde{v}\tau=g\rho g^{-1}$, where
$\rho\in\operatorname{\text{Im}}(\iota)$ is of order $2$ and
$g=\tau^{-1}\widetilde{g}\tau\in UVP_{n}$, which proves that the condition of
part (a) is also necessary.

  2. (b)

We start by characterising the elements of $\operatorname{\text{Im}}(\eta)$.
Let $v\in UVB_{n}$. Then $\pi_{UVP}(v)\in S_{n}$, and so
$\pi_{UVP}(v)=s_{i_{1}}\cdots s_{i_{r}}$, where $r\geq 0$, and for all
$j=1,\ldots,r$, $s_{i_{j}}\in\\{(k,k+1)\,\mid\,k=1,\ldots,n-1\\}$. Hence
$v=w\rho$, where $w\in UVP_{n}$ and
$\rho=\iota(\pi_{UVP}(v))=\rho_{i_{1}}\cdots\rho_{i_{r}}$. Let
$\beta=\sigma_{i_{1}}\cdots\sigma_{i_{r}}\in B_{n}$. Then by (1), we have
$\eta(\beta)=\sigma_{i_{1}}\cdots\sigma_{i_{r}}=\lambda_{i_{1},i_{1}+1}^{-1}\rho_{i_{1}}\cdots\lambda_{i_{r},i_{r}+1}^{-1}\rho_{i_{r}}=y\rho_{i_{1}}\cdots\rho_{i_{r}}=y\rho\in\operatorname{\text{Im}}(\eta)$,
where $y\in UVP_{n}$. It follows that:

$v\in\operatorname{\text{Im}}(\eta)\Longleftrightarrow(\eta(\beta))^{-1}v\in\operatorname{\text{Im}}(\eta)\Longleftrightarrow\rho^{-1}y^{-1}w\rho\in\operatorname{\text{Im}}(\eta).$

Now $y^{-1}w\in UVP_{n}$, so $\rho^{-1}y^{-1}w\rho\in UVP_{n}$, and it follows
that $v\in\operatorname{\text{Im}}(\eta)$ if and only if
$\rho^{-1}y^{-1}w\rho\in\eta(P_{n})$. From Section 3, $P_{n}$ is generated by
the set $\\{a_{ij}\,\mid\,1\leq i<j\leq n\\}$, and since
$\eta(a_{ij})=\lambda_{i,j}^{-1}\lambda_{j,i}^{-1}$ for all $1\leq i<j\leq n$,
we see that $v\in\operatorname{\text{Im}}(\eta)$ if and only if
$\rho^{-1}y^{-1}w\rho$ belongs to the free Abelian subgroup $\Gamma$ of
$UVP_{n}$ of rank $n(n-1)/2$ generated by the set
$\\{\lambda_{i,j}^{-1}\lambda_{j,i}^{-1}\,\mid\,1\leq i<j\leq n\\}$. In
particular, if $1\leq i<j\leq n$ and $\varepsilon_{i,j}\colon\thinspace
UVP_{n}\longrightarrow\mathbb{Z}$ denotes the evaluation homomorphism defined
by $\varepsilon_{i,j}(\lambda_{k,l})=1$ if $\\{k,l\\}=\\{i,j\\}$ and
$\varepsilon_{i,j}(\lambda_{k,l})=0$ otherwise, where $1\leq k,l\leq n$,
$k\neq l$, then $\varepsilon_{i,j}(\Gamma)=2\mathbb{Z}$ for all $1\leq i<j\leq
n$.

Now suppose on the contrary that $\operatorname{\text{Im}}(\eta)$ possesses an
element $v$ of order $2$. From part (a), there exist $g\in UVP_{n}$ and
$\rho\in\operatorname{\text{Im}}(\iota)$ of order $2$ such that $v=g\rho
g^{-1}$. Then $\pi_{UVP}(\rho)$ is also of order $2$, and so may be written as
a non-trivial product of transpositions whose supports are pairwise disjoint.
So there exist $1\leq r\leq n/2$ and distinct elements
$i_{1},j_{1},\ldots,i_{r},j_{r}$ of $\\{1,\ldots,n\\}$ such that $i_{k}<j_{k}$
for all $1\leq k\leq r$ and $\rho=\iota((i_{1},j_{1})\cdots(i_{r},j_{r}))$.
Let $\tau\in\operatorname{\text{Im}}(\iota)$ be such that
$\pi_{UVP}(\tau\rho\tau^{-1})=(1,2)\cdots(2r-1,2r)$, let
$\widetilde{\rho}=\tau\rho\tau^{-1}\in\operatorname{\text{Im}}(\iota)$, let
$\widehat{\tau}\in B_{n}$ be such that $\pi(\widehat{\tau})=\pi_{UVP}(\tau)$,
let $\beta=\sigma_{1}\sigma_{3}\cdots\sigma_{2r-1}$, and let
$\widehat{\beta}=\widehat{\tau}^{-1}\beta\widehat{\tau}$. Then
$\widetilde{\rho}=\rho_{1}\rho_{3}\cdots\rho_{2r-1}$ and
$\eta(\beta)=\sigma_{1}\sigma_{3}\cdots\sigma_{2r-1}=\lambda_{1,2}^{-1}\lambda_{3,4}^{-1}\cdots\lambda_{2r-1,2r}^{-1}\widetilde{\rho}$
by (1) and Theorem 2.3. Since $v\in\operatorname{\text{Im}}(\eta)$, it follows
that $(\eta(\widehat{\beta}))^{-1}\ldotp v\in\operatorname{\text{Im}}(\eta)$.
Hence:

$(\eta(\widehat{\beta}))^{-1}\ldotp
v=\eta(\widehat{\tau}^{-1})\widetilde{\rho}^{\,-1}\eta(\widehat{\tau})\ldotp\eta(\widehat{\tau}^{-1})\lambda_{2r-1,2r}\cdots\lambda_{3,4}\lambda_{1,2}\eta(\widehat{\tau})\ldotp
g\rho g^{-1}.$ (10)

Now using (3), we have
$\pi_{UVP}(\eta(\widehat{\tau}))=\pi(\widehat{\tau})=\pi_{UVP}(\tau)$, so:

$\eta(\widehat{\tau}^{-1})\lambda_{2r-1,2r}\cdots\lambda_{3,4}\lambda_{1,2}\eta(\widehat{\tau})=\lambda_{i_{r},j_{r}}\cdots\lambda_{i_{2},j_{2}}\lambda_{i_{1},j_{1}}.$
(11)

It follows from (10) and (11) that:

$\displaystyle(\eta(\widehat{\beta}))^{-1}\ldotp v$
$\displaystyle=\eta(\widehat{\tau}^{-1})\widetilde{\rho}^{\,-1}\eta(\widehat{\tau})\rho\ldotp\rho^{-1}\lambda_{i_{r},j_{r}}\cdots\lambda_{i_{2},j_{2}}\lambda_{i_{1},j_{1}}\rho\ldotp\rho^{-1}g\rho
g^{-1}$
$\displaystyle=\eta(\widehat{\tau}^{-1})\widetilde{\rho}^{\,-1}\eta(\widehat{\tau})\rho\lambda_{j_{r},i_{r}}\cdots\lambda_{j_{2},i_{2}}\lambda_{j_{1},i_{1}}\rho^{-1}g\rho\ldotp
g^{-1}$ (12)

Since $\pi_{UVP}(\eta(\widehat{\tau}))=\pi(\widehat{\tau})=\pi_{UVP}(\tau)$,
there exists $z\in UVP_{n}$ such that $\eta(\widehat{\tau})=\tau z$, and
hence:

$\displaystyle(\eta(\widehat{\beta}))^{-1}\ldotp v$
$\displaystyle=z^{-1}\tau^{-1}\widetilde{\rho}^{\,-1}\tau
z\rho\ldotp\lambda_{j_{r},i_{r}}\cdots\lambda_{j_{2},i_{2}}\lambda_{j_{1},i_{1}}\ldotp\rho^{-1}g\rho\ldotp
g^{-1}$
$\displaystyle=z^{-1}\ldotp\rho^{-1}z\rho\ldotp\lambda_{j_{r},i_{r}}\cdots\lambda_{j_{2},i_{2}}\lambda_{j_{1},i_{1}}\ldotp\rho^{-1}g\rho\ldotp
g^{-1}.$ (13)

The expression on the right-hand side of (13) is written as a product of
elements of $UVP_{n}$, which we will now project onto $F_{i_{1},j_{1}}$. Let
$\alpha(\lambda_{i_{1},j_{1}},\lambda_{j_{1},i_{1}})=\pi_{i_{1},j_{1}}(z)$ et
$\beta(\lambda_{i_{1},j_{1}},\lambda_{j_{1},i_{1}})=\pi_{i_{1},j_{1}}(g)$.
Using (2) and the fact that
$\pi_{UVP}(\rho)=(i_{1},j_{1})\cdots(i_{r},j_{r})$, by (13), we have:

$\pi_{i_{1},j_{1}}((\eta(\widehat{\beta}))^{-1}\ldotp
v)=(\alpha(\lambda_{i_{1},j_{1}},\lambda_{j_{1},i_{1}}))^{-1}\ldotp\alpha(\lambda_{j_{1},i_{1}},\lambda_{j_{1},i_{1}})\ldotp\lambda_{j_{1},i_{1}}\ldotp\beta(\lambda_{j_{1},i_{1}},\lambda_{i_{1},j_{1}})\ldotp(\beta(\lambda_{i_{1},j_{1}},\lambda_{j_{1},i_{1}}))^{-1},$

from which it follows that
$\varepsilon_{i_{1},j_{1}}(\eta(\widehat{\beta}))^{-1}\ldotp v)=1$. We
conclude from the previous paragraph that $\eta(\widehat{\beta}))^{-1}\ldotp
v\notin\Gamma$, and thus $v\notin\operatorname{\text{Im}}(\eta)$, which yields
a contradiction. Therefore $\operatorname{\text{Im}}(\eta)$ contains no
elements of order $2$, and we deduce from Proposition 3.1 that
$B_{n}/[P_{n},P_{n}]$ has no elements of even order.∎

######  Proposition 4.2.

If $n\geq 3$ then any torsion element of $UVB_{n}$ belongs to the normal
closure of $\operatorname{\text{Im}}(\iota)$ in $UVB_{n}$.

###### Proof.

From Definition 2.2, one may check that the quotient of $UVB_{n}$ by the
normal closure of $\operatorname{\text{Im}}(\iota)$ in $UVB_{n}$ is isomorphic
to $\mathbb{Z}$. If $g$ is a torsion element of $UVB_{n}$, its image in this
quotient is thus trivial, in other words $g$ belongs to the normal closure of
$\operatorname{\text{Im}}(\iota)$ in $UVB_{n}$. ∎

One may show in the same manner that the statement of Proposition 4.2 also
holds for $VB_{n}$ and $WB_{n}$. In the case of $UVB_{n}$, we may strengthen
the results of Theorem 4.1(a) and Proposition 4.2.

######  Theorem 4.3.

Let $n\geq 2$, and let $w$ be a torsion element of $UVB_{n}$ of order $r$.
Then there exists $s_{w}\in S_{n}$ of order $r$ such that $w$ is conjugate to
$\iota(s_{w})$ by an element of $UVP_{n}$.

Before proving Theorem 4.3, we define some notation, and we make study the
action of the symmetric group $S_{n}$ on the group $UVP_{n}$ that will used in
the proof. If $\tau\in S_{n}$, let $\operatorname{\text{Supp}}(\tau)$ denote
its support. Let $s\in S_{n}$, let $o(s)$ denote the order of $s$ and let
$G_{s}$ denote the cyclic subgroup of $S_{n}$ of order $o(s)$ generated by the
element $s$. By Theorem 2.4, the permutation $s$ acts on
$UVP_{n}=\langle\lambda_{i,j}\ |\ 1\leq i\neq j\leq n\rangle$ by permuting the
indices of the elements $\lambda_{i,j}$. To simplify the notation, if $g\in
UVP_{n}$ and $s\in S_{n}$, we set $s(g)=\iota(s)g\iota(s)^{-1}$, and we shall
identify $s$ with its image $\iota(s)$ in $UVB_{n}$. Let $s=s_{1}\cdots
s_{m(s)}$ be the cycle decomposition of $s$, where $1\leq m(s)\leq n$, and the
subsets
$\operatorname{\text{Supp}}(s_{1}),\ldots,\operatorname{\text{Supp}}(s_{m(s)})$
form a partition of the set $\\{1,\ldots,n\\}$ (in particular, some of the
$\operatorname{\text{Supp}}(s_{j})$, where $1\leq j\leq m(s)$, may be
singletons).

Let $I=\\{(i,j)\mid 1\leq i\neq j\leq n\\}$. The action of $G_{s}$ on $I$
gives rise to a partition of $I$ in $n(s)$ disjoint orbits, where
$n(s)\in\mathbb{N}$, and the cardinality of each such orbit is either the
order of one cycle or is the least common multiple of the orders of two cycles
of $s$. If $(i,j)\in I$, let $\mathcal{O}(i,j)$ denote its orbit, and let
$\lvert\mathcal{O}(i,j)\rvert$ denote the cardinality of $\mathcal{O}(i,j)$.
Clearly, $\lvert\mathcal{O}(j,i)\rvert=\lvert\mathcal{O}(i,j)\rvert$ for all
$(i,j)\in I$. Further, if $\mathcal{O}(j,i)=\mathcal{O}(i,j)$ then there exist
$1\leq q\leq m(s)$ and $p\in\mathbb{N}$ such that
$i,j\in\operatorname{\text{Supp}}(s_{q})$, $s_{q}^{p}(i)=j$ and
$s_{q}^{p}(j)=i$, from which it follows that
$\lvert\mathcal{O}(i,j)\rvert=o(s_{q})$ is even, and if $(k,l)\in I$ then
$(k,l)\in\mathcal{O}(i,j)$ if and only if $(l,k)\in\mathcal{O}(i,j)$. This
gives rise to a partition $\sqcup_{k=1}^{n(s)}\mathcal{O}_{k}$ of $I$, where
$n(s)\in\mathbb{N}$, that dominates the partition given by the action of
$G_{s}$ on $I$, and for $k=1,\ldots,n(s)$, $\mathcal{O}_{k}$ is defined by the
property that if $(i,j)\in I$, then $(i,j)\in\mathcal{O}_{k}$ if and only if
$\mathcal{O}_{k}=\mathcal{O}(i,j)\cup\mathcal{O}(j,i)$. Note that $n(s)$ is
the number of orbits of the action of $G_{s}$ on the set $\\{\\{i,j\\}\mid
1\leq i\neq j\leq n\\}$ of _unordered_ pairs of elements of
$\\{1,\ldots,n\\}$. Since
$\lvert\mathcal{O}(j,i)\rvert=\lvert\mathcal{O}(i,j)\rvert$, and
$\mathcal{O}(i,j)$ is even if $\mathcal{O}(j,i)=\mathcal{O}(i,j)$, it follows
that for all $1\leq k\leq n(s)$, the cardinality $\lvert\mathcal{O}_{k}\rvert$
of $\mathcal{O}_{k}$ is even, so $\lvert\mathcal{O}_{k}\rvert=2n_{k}$ for some
$n_{k}\in\mathbb{N}$. Let
$F_{\mathcal{O}_{k}}=\bigoplus_{(i,j)\in\mathcal{O}_{k}}F_{i,j}$, where as in
Section 2, we identify $F_{i,j}$ with $F_{j,i}$. Then:

$UVP_{n}=\bigoplus_{1\leq k\leq n(s)}F_{\mathcal{O}_{k}}.$ (14)

The following lemma is folklore.

######  Lemma 4.4.

Let $F_{2}=F_{2}(x,y)$ be the free group of rank $2$ freely generated by
$\\{x,y\\}$, and let $\alpha\colon\thinspace F_{2}\longrightarrow F_{2}$ be
the involutive automorphism defined by $\alpha(x)=y$ and $\alpha(y)=x$. Let
$w\in F_{2}$ be such that $w\alpha(w)=1$. Then there exists $u\in F_{2}$ such
that $w=u\alpha(u^{-1})$.

###### Proof.

Let $w\in F_{2}$ be such that $w\alpha(w)=1$. If $w=1$ then we may take $u=1$.
So suppose that $w\neq 1$. Since $w\alpha(w)=1$, $w$ cannot be written in
reduced form as a word starting and ending with a non-zero power of the same
generator. By replacing $w$ by $\alpha(w)$ if necessary, we may thus suppose
that $w=x^{\varepsilon_{1}}y^{\varepsilon_{2}}\cdots
y^{\varepsilon_{2t-1}}y^{\varepsilon_{2t}}\neq 1$, where
$\varepsilon_{1},\varepsilon_{2},\dots,\varepsilon_{2t}\in\mathbb{Z}\setminus\\{0\\}$,
and thus:

$1=w\alpha(w)=\underbrace{\big{(}x^{\varepsilon_{1}}y^{\varepsilon_{2}}\cdots
x^{\varepsilon_{2t-1}}y^{\varepsilon_{2t}}\big{)}}_{\neq
1}\cdot\underbrace{\big{(}y^{\varepsilon_{1}}x^{\varepsilon_{2}}\cdots
y^{\varepsilon_{2t-1}}x^{\varepsilon_{2t}}\big{)}}_{\neq 1}\in F_{2}.$

It follows that $\varepsilon_{2t-j}=-\varepsilon_{j+1}$ for $j=0,\ldots,2t-1$,
from which we conclude that $w=u\alpha(u^{-1})$, where
$u=x^{\varepsilon_{1}}y^{\varepsilon_{2}}\cdots y^{\varepsilon_{t}}$. ∎

In what follows, we shall identify $S_{n}$ with its image in $UVB_{n}$ by
$\iota$. In particular, if $s\in S_{n}$, then $\iota(s)$ shall be denoted
simply by $s$.

######  Lemma 4.5.

Let $n\geq 2$, let $s\in S_{n}$, let $\gamma_{1}\in F_{\mathcal{O}_{l}}$,
where $1\leq l\leq n(s)$, and let $w_{1}=\gamma_{1}s\in UVB_{n}$. If $w_{1}$
is of finite order then it is conjugate to $s$ by an element of
$F_{\mathcal{O}_{l}}$.

###### Proof.

Let $s\in S_{n}$, $\gamma_{1}\in F_{\mathcal{O}_{l}}$, where $1\leq l\leq
n(s)$, and $w_{1}\in UVB_{n}$ be as in the statement. By reordering the orbits
$\mathcal{O}_{1},\ldots,\mathcal{O}_{n(s)}$ if necessary, we may suppose that
$l=1$, so $\gamma_{1}\in F_{\mathcal{O}_{1}}$. The result is clear if
$o(w_{1})=1$. If $o(w_{1})=2$, then by Theorem 4.1(a), there exists $g\in
UVP_{n}$ such that $w_{1}=gsg^{-1}$. With respect to the decomposition (14),
let $g=\delta_{1}\cdots\delta_{n(s)}$, where for $k=1,\ldots,n(s)$,
$\delta_{k}\in F_{\mathcal{O}_{k}}$. Since $gsg^{-1}=\gamma_{1}s$, we have:

$\gamma_{1}=\delta_{1}\cdots\delta_{n(s)}\ldotp
s\,(\delta_{1}\cdots\delta_{n(s)})^{-1}s^{-1}=\delta_{1}\cdots\delta_{n(s)}\ldotp
s\delta_{n(s)}^{-1}s^{-1}\cdots s\delta_{1}^{-1}s^{-1}\in
F_{\mathcal{O}_{1}}.$ (15)

Now for all $k=1,\ldots,n(s)$, $F_{\mathcal{O}_{k}}$ is invariant under
conjugation by $s$, and identifying the components of (15) with respect to
(14), we see that $\delta_{k}\ldotp s\delta_{k}^{-1}s^{-1}=1$ for all $2\leq
k\leq n(s)$, or in other words, $\delta_{k}$ and $s$ commute. It follows from
(15) that $\gamma_{1}=\delta_{1}\ldotp s\delta_{1}^{-1}s^{-1}$, and thus
$w_{1}=\gamma_{1}s=\delta_{1}s\delta_{1}^{-1}$, where $\delta_{1}\in
F_{\mathcal{O}_{1}}$, as required.

So suppose that $o(w_{1})\geq 3$. For all $m\in\mathbb{N}$, we have:

$w_{1}^{m}=(\gamma_{1}s)^{m}=\gamma_{1}s\gamma_{1}s^{-1}s^{2}\gamma_{1}s^{-2}\cdots
s^{m-1}\gamma_{1}s^{-(m-1)}s^{m}=\gamma_{1}s(\gamma_{1})\cdots
s^{m-1}(\gamma_{1})s^{m}.$ (16)

Since $UVP_{n}$ is torsion free, it follows from (16) that $o(w_{1})=o(s)$. As
we mentioned above, $F_{\mathcal{O}_{1}}$ is invariant under conjugation by
$s$, so for all $j=0,1,\ldots,m-1$, the term
$s^{j}(\gamma_{1})=s^{j}\gamma_{1}s^{-j}$ of (16) belongs to
$F_{\mathcal{O}_{1}}$. Let $(i_{0},j_{0})\in\mathcal{O}_{1}$, where $1\leq
i_{0}<j_{0}\leq n$. If $\mathcal{O}(i_{0},j_{0})\neq\mathcal{O}(j_{0},i_{0})$
(resp. $\mathcal{O}(i_{0},j_{0})=\mathcal{O}(j_{0},i_{0})$), let
$\varepsilon=1$ (resp. $\varepsilon=2$). Then
$\lvert\mathcal{O}(i_{0},j_{0})\rvert=\varepsilon n_{1}$, and
$\lvert\mathcal{O}_{1}\rvert=2n_{1}$,
$F_{\mathcal{O}_{1}}=\bigoplus_{(i,j)\in\mathcal{O}_{1}}F_{i,j}=\bigoplus_{q=0}^{n_{1}-1}F_{s^{q}(i_{0}),s^{q}(j_{0})}$,
and for $j=0,\ldots,n_{1}-1$, there exists $u_{j}\in
F_{s^{j}(i_{0}),s^{j}(j_{0})}$ such that $\gamma_{1}=u_{0}\cdots u_{n_{1}-1}$.
Hence $u_{0},\ldots,u_{n_{1}-1}$ belong to distinct free factors of $UVP_{n}$,
so commute pairwise. Further:

$s^{k}(u_{j})\in F_{s^{j+k}(i_{0}),s^{j+k}(j_{0})}$ for all $k\in\mathbb{Z}$
and $j=0,\ldots,n_{1}-1$, (17)

where the superscript $j+k$ is taken modulo $n_{1}$. In particular, if $v\in
F_{s^{q}(i_{0}),s^{q}(j_{0})}$ for some $q=0,\ldots,n_{1}-1$, then
$s^{n_{1}}(v)=v$ (resp. $s^{n_{1}}(v)=\alpha(v)$, where
$\alpha\colon\thinspace F_{s^{q}(i_{0}),s^{q}(j_{0})}\longrightarrow
F_{s^{q}(i_{0}),s^{q}(j_{0})}$ is the automorphism that exchanges
$\lambda_{s^{q}(i_{0}),s^{q}(j_{0})}$ and
$\lambda_{s^{q}(j_{0}),s^{q}(i_{0})}$, as in Lemma 4.4), and thus
$s^{\varepsilon n_{1}}(v)=v$. It follows from this, the fact that
$o(w_{1})=o(s)$ and (16) that:

$\displaystyle 1$ $\displaystyle=w_{1}^{\varepsilon
n_{1}o(s)}=(w_{1}^{\varepsilon n_{1}})^{o(s)}=(\gamma_{1}s(\gamma_{1})\cdots
s^{\varepsilon n_{1}-1}(\gamma_{1})s^{\varepsilon
n_{1}})^{o(s)}=(\gamma_{1}s(\gamma_{1})\cdots s^{\varepsilon
n_{1}-1}(\gamma_{1}))^{o(s)}s^{\varepsilon n_{1}o(s)}$
$\displaystyle=(\gamma_{1}s(\gamma_{1})\cdots s^{\varepsilon
n_{1}-1}(\gamma_{1}))^{o(s)}.$

Since $UVP_{n}$ is torsion free, we conclude that
$\gamma_{1}s(\gamma_{1})\cdots s^{\varepsilon n_{1}-1}(\gamma_{1})=1$. Hence:

$u_{0}\cdots u_{n_{1}-1}s\big{(}u_{0}\cdots u_{n_{1}-1}\big{)}\cdots
s^{\varepsilon n_{1}-1}\big{(}u_{0}\cdots u_{n_{1}-1}\big{)}=1.$ (18)

Applying (17), and projecting (18) into $F_{1,2}$, we see that:

$1=\prod_{i=0}^{\varepsilon-1}s^{in_{1}}(u_{0})s^{in_{1}+1}\big{(}u_{n_{1}-1}\big{)}\cdots
s^{(i+1)n_{1}-1}\big{(}u_{1}\big{)}=\prod_{i=0}^{\varepsilon-1}s^{in_{1}}\big{(}u_{0}s\big{(}u_{n_{1}-1}\big{)}\cdots
s^{n_{1}-1}\big{(}u_{1}\big{)}\big{)}.$ (19)

We deduce from (19) (resp. from (19) and Lemma 4.4) that
$u_{0}=\prod_{j=1}^{n_{1}-1}s^{n_{1}-j}\big{(}u_{j}^{-1}\big{)}$ (resp. that
there exists $\widetilde{u}\in F_{1,2}$ such that
$u_{0}=\widetilde{u}s^{n_{1}}(\widetilde{u}^{-1})\big{(}\prod_{j=1}^{n_{1}-1}s^{n_{1}-j}\big{(}u_{j}^{-1}\big{)}\big{)}$).
Using the fact that $u_{1},\ldots,u_{n_{1}-1}$ commute pairwise, it follows
that:

$\gamma_{1}=u_{0}u_{1}\cdots u_{n_{1}-1}=\beta
s^{n_{1}-1}\big{(}u_{1}^{-1}\big{)}\cdots s\big{(}u_{n_{1}-1}^{-1}\big{)}\cdot
u_{n_{1}-1}\cdots u_{1},$ (20)

where $\beta=1$ (resp. $\beta=\widetilde{u}s^{n_{1}}(\widetilde{u}^{-1})$). In
what follows, if $a,b\in UVB_{n}$, we shall write $a\sim b$ if $a$ and $b$ are
conjugate by an element of $UVP_{n}$. Let us show by induction that:

$\gamma_{1}\cdot s\sim\beta s^{n_{1}-1}\big{(}u_{1}^{-1}\big{)}\cdots
s^{t+1}\big{(}u_{n_{1}-(t+1)}^{-1}\big{)}s^{t}\big{(}u_{n_{1}-t}^{-1}\big{)}\cdot
u_{n_{1}-t}\cdots u_{1}\cdot s$ (21)

for all $t=1,\ldots,n_{1}$. If $t=1$ then the result follows directly from
(20). So suppose that (21) holds for some $1\leq t\leq n_{1}-1$. Let
$M_{0}=\prod_{c=1}^{n_{1}-1}s^{c}(\widetilde{u})$, and for $1\leq t\leq
n_{1}-1$, let $M_{t}=\prod_{c=1}^{t-1}s^{t-c}\big{(}u_{n_{1}-t}\big{)}$. Note
that $M_{1}=1$, and that $M_{0}=1$ if
$\mathcal{O}(i_{0},j_{0})\neq\mathcal{O}(j_{0},i_{0})$. By (17), for
$c=1,\ldots,t-1$, $s^{t-c}\big{(}u_{n_{1}-t}\big{)}\in
F_{s^{n_{1}-c}(i_{0}),s^{n_{1}-c}(j_{0})}$, so for $1\leq t\leq n_{1}-1$, the
factors of $M_{t}$ commute pairwise, and
$M_{t}\in\bigoplus_{c=n_{1}-t+1}^{n_{1}-1}F_{s^{c}(i_{0}),s^{c}(j_{0})}$.
Thus:

$M_{t}^{-1}s^{t}\big{(}u_{n_{1}-t}^{-1}\big{)}=\left(\prod_{c=1}^{t-1}s^{t-c}\big{(}u_{n_{1}-t}\big{)}\right)^{-1}s^{t}\big{(}u_{n_{1}-t}^{-1}\big{)}=\left(\prod_{c=1}^{t-1}s^{c}\big{(}u_{n_{1}-t}^{-1}\big{)}\right)s^{t}\big{(}u_{n_{1}-t}^{-1}\big{)}=\prod_{c=1}^{t}s^{c}\big{(}u_{n_{1}-t}^{-1}\big{)}.$
(22)

and since $s^{t}\big{(}u_{n_{1}-t}^{-1}\big{)}\in F_{i_{0},j_{0}}$ and
$u_{n_{1}-t}\cdots
u_{1}\in\bigoplus_{i=1}^{n_{1}-t}F_{s^{i}(i_{0}),s^{i}(j_{0})}$, we see that
$M_{t}^{-1}s^{t}\big{(}u_{n_{1}-t}^{-1}\big{)}$ commutes with
$u_{n_{1}-t}\cdots u_{1}$. So by (21) and (22), we have:

$\displaystyle\gamma_{1}\cdot s$ $\displaystyle\sim\beta
s^{n_{1}-1}\big{(}u_{1}^{-1}\big{)}\cdots
s^{t+1}\big{(}u_{n_{1}-(t+1)}^{-1}\big{)}M_{t}M_{t}^{-1}s^{t}\big{(}u_{n_{1}-t}^{-1}\big{)}\cdot
u_{n_{1}-t}\cdots u_{1}\cdot s$ $\displaystyle=\beta
s^{n_{1}-1}\big{(}u_{1}^{-1}\big{)}\cdots
s^{t+1}\big{(}u_{n_{1}-(t+1)}^{-1}\big{)}M_{t}\cdot u_{n_{1}-t}\cdots
u_{1}\left(\prod_{c=1}^{t}s^{c}\big{(}u_{n_{1}-t}^{-1}\big{)}\right)\cdot s$
$\displaystyle=\beta s^{n_{1}-1}\big{(}u_{1}^{-1}\big{)}\cdots
s^{t+1}\big{(}u_{n_{1}-(t+1)}^{-1}\big{)}\left(\prod_{c=1}^{t}s^{t-c}\big{(}u_{n_{1}-t}\big{)}\right)u_{n_{1}-(t+1)}\cdots
u_{1}\cdot s\cdot\prod_{c=0}^{t-1}s^{c}\big{(}u_{n_{1}-t}^{-1}\big{)},$ (23)

where we have used the fact that $M_{t}\cdot
u_{n_{1}-t}=\prod_{c=1}^{t}s^{t-c}\big{(}u_{n_{1}-t}\big{)}$. Now by (17),
$\prod_{c=1}^{t}s^{t-c}\big{(}u_{n_{1}-t}\big{)}\in\bigoplus_{c=n_{1}-t}^{n_{1}-1}F_{s^{c}(i_{0}),s^{c}(j_{0})}$,
and since $\beta s^{n_{1}-1}\big{(}u_{1}^{-1}\big{)}\cdots
s^{t+1}\big{(}u_{n_{1}-(t+1)}^{-1}\big{)}\in F_{i_{0},j_{0}}$, we see that
these two terms commute, and it follows from (23) that:

$\gamma_{1}\cdot
s\sim\left(\prod_{c=1}^{t}s^{t-c}\big{(}u_{n_{1}-t}\big{)}\right)\beta
s^{n_{1}-1}\big{(}u_{1}^{-1}\big{)}\cdots
s^{t+1}\big{(}u_{n_{1}-(t+1)}^{-1}\big{)}u_{n_{1}-(t+1)}\cdots u_{1}\cdot
s\cdot\prod_{c=0}^{t-1}s^{c}\big{(}u_{n_{1}-t}^{-1}\big{)}.$ (24)

One may check that $\prod_{c=1}^{t}s^{t-c}\big{(}u_{n_{1}-t}\big{)}$ is the
inverse of $\prod_{c=0}^{t-1}s^{c}\big{(}u_{n_{1}-t}^{-1}\big{)}$, and using
(24), we conclude that (21) holds for $t+1$. Taking $t=n_{1}$ in (21), we
obtain $\gamma_{1}\cdot s\sim\beta\cdot s$. If
$\mathcal{O}(i_{0},j_{0})\neq\mathcal{O}(j_{0},i_{0})$ then $\beta=1$, and the
statement of the lemma holds in this case. So suppose that
$\mathcal{O}(i_{0},j_{0})=\mathcal{O}(j_{0},i_{0})$. Then $\widetilde{u}\in
F_{i_{0},j_{0}}$, and for $c=1,\ldots,n_{1}-1$, $s^{c}(\widetilde{u})\in
F_{s^{c}(i_{0}),s^{c}(j_{0})}$. Thus the factors of
$\widetilde{u}M_{0}=\prod_{c=0}^{n_{1}-1}s^{c}(\widetilde{u})$ commute
pairwise, and hence
$M_{0}^{-1}s^{n_{1}}(\widetilde{u}^{-1})=\prod_{c=1}^{n_{1}}s^{c}(\widetilde{u}^{-1})$.
It follows that:

$\displaystyle\gamma_{1}\cdot s\sim\beta\cdot s$
$\displaystyle=\widetilde{u}s^{n_{1}}(\widetilde{u}^{-1})\cdot
s=\widetilde{u}M_{0}M_{0}^{-1}s^{n_{1}}(\widetilde{u}^{-1})\cdot
s=\left(\prod_{c=0}^{n_{1}-1}s^{c}(\widetilde{u})\right)\left(\prod_{c=1}^{n_{1}}s^{c}(\widetilde{u}^{-1})\right)\cdot
s$ $\displaystyle=\left(\prod_{c=0}^{n_{1}-1}s^{c}(\widetilde{u})\right)\cdot
s\cdot\left(\prod_{c=0}^{n_{1}-1}s^{c}(\widetilde{u}^{-1})\right).$

Now $\prod_{c=0}^{n_{1}-1}s^{c}(\widetilde{u})$ may be seen to be the inverse
of $\prod_{c=0}^{n_{1}-1}s^{c}(\widetilde{u}^{-1})$, and therefore
$\gamma_{1}\cdot s\sim s$ also in this case, which completes the proof of the
lemma. ∎

This enables us to prove Theorem 4.3.

###### Proof of Theorem 4.3.

Let $w\in UVB_{n}$ be a torsion element of order $r$. By Theorems 2.3 and 2.4
there exist (unique) $u\in UVP_{n}$ and $s\in S_{n}$ such that $w=us$. As in
(16), we see that $u\cdot s(u)\cdots s^{r-1}(u)=1$ in $UVP_{n}$ and $s^{r}=1$.
Further, from (14), $u=\gamma_{1}\cdots\gamma_{n(s)}$, where for
$l=1,\ldots,n(s)$, $\gamma_{l}\in F_{\mathcal{O}_{l}}$. Let
$l\in\\{1,\ldots,n(s)\\}$. Since $F_{\mathcal{O}_{l}}$ is invariant under
conjugation by $s$, it follows from (14) that $\gamma_{l}s(\gamma_{l})\cdots
s^{r-1}(\gamma_{l})=1$ in $F_{\mathcal{O}_{l}}$. Since
$\gamma_{l}s(\gamma_{l})\cdots s^{r-1}(\gamma_{l})$ may also be written as
$(\gamma_{l}s)^{r}$ using (16), we conclude from Lemma 4.5 that $\gamma_{l}s$
is conjugate to $s$ by an element of $F_{\mathcal{O}_{l}}$, or in other words,
there exists $\Lambda_{l}\in F_{\mathcal{O}_{l}}$ such that
$\gamma_{l}s=\Lambda_{l}\cdot s\cdot\Lambda_{l}^{-1}$. Let us prove by reverse
induction that for all $m=1,\ldots,n(s)+1$:

$w=\Lambda_{n(s)}\Lambda_{n(s)-1}\cdots\Lambda_{m}\gamma_{1}\cdots\gamma_{m-1}\cdot
s\cdot\Lambda_{m}^{-1}\cdots\Lambda_{n(s)-1}^{-1}\Lambda_{n(s)}^{-1}.$ (25)

If $m=n(s)+1$ then (25) follows directly from the fact that
$w=us=\gamma_{1}\cdots\gamma_{n(s)}\cdot s$. So suppose that (25) holds for
some $m\in\\{2,\ldots,n(s)+1\\}$. Then using (14) and the facts that
$\gamma_{1}\cdots\gamma_{m-2}\in\bigoplus_{i=1}^{m-2}F_{\mathcal{O}_{i}}$ and
$\gamma_{m-1}\in F_{\mathcal{O}_{m}-1}$, we obtain:

$\displaystyle w$
$\displaystyle=\Lambda_{n(s)}\Lambda_{n(s)-1}\cdots\Lambda_{m}\gamma_{1}\cdots\gamma_{m-2}(\gamma_{m-1}\cdot
s)\cdot\Lambda_{m}^{-1}\cdots\Lambda_{n(s)-1}^{-1}\Lambda_{n(s)}^{-1}$
$\displaystyle=\Lambda_{n(s)}\Lambda_{n(s)-1}\cdots\Lambda_{m}\gamma_{1}\cdots\gamma_{m-2}\Lambda_{m-1}\cdot
s\cdot\Lambda_{m-1}^{-1}\Lambda_{m}^{-1}\cdots\Lambda_{n(s)-1}^{-1}\Lambda_{n(s)}^{-1}$
$\displaystyle=\Lambda_{n(s)}\Lambda_{n(s)-1}\cdots\Lambda_{m}\Lambda_{m-1}\gamma_{1}\cdots\gamma_{m-2}\cdot
s\cdot\Lambda_{m-1}^{-1}\Lambda_{m}^{-1}\cdots\Lambda_{n(s)-1}^{-1}\Lambda_{n(s)}^{-1},$

which proves (25) for $m-1$. Taking $m=1$, we see that $w=\Lambda\cdot
s\cdot\Lambda^{-1}$, where $\Lambda=\Lambda_{n(s)}\cdots\Lambda_{1}\in
UVP_{n}$, and this completes the proof of the theorem. ∎

######  Remark 4.6.

It is an open question whether a result similar to that of Theorem 4.3 holds
for $VB_{n}$ and for $WB_{n}$.

######  Acknowledgements.

The first and third authors are supported by the French project ‘AlMaRe’
(ANR-19-CE40-0001-01).

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11institutetext:  Minerva Schools at KGI, San Francisco, California 94103,
USA. 22institutetext: Maulana Azad National Urdu University, Gachibowli,
Hyderabad 500 032, India.

# Membership of Stars in Open Clusters using Random Forest with Gaia Data

Md Mahmudunnobe 11 Priya Hasan 22<EMAIL_ADDRESS>Mudasir Raja 22 S N
Hasan 22

###### Abstract

Membership of stars in open clusters is one of the most crucial parameters in
studies of star clusters. Gaia opened a new window in the estimation of
membership because of its unprecedented 6-D data. In the present study, we
used published membership data of nine open star clusters as a training set to
find new members from Gaia DR2 data using a supervised random forest model
with a precision of around 90%. The number of new members found is often
double the published number. Membership probability of a larger sample of
stars in clusters is a major benefit in determination of cluster parameters
like distance, extinction and mass functions. We also found members in the
outer regions of the cluster and found sub-structures in the clusters studied.
The color magnitude diagrams are more populated and enriched by the addition
of new members making their study more promising.

##  1 Introduction

Star clusters are the building blocks of galaxies and are the key to
understanding the formation and evolution of stars and galaxies lynga82
janes94  khar05  friel95  bon06 . Star clusters are seen as an over-density
and are a sample of stars in the same region of the sky, gravitationally bound
and formed from the same molecular cloud. Hence the member stars are
approximately at the same distance, of the same age and only differ in mass.
These stars move with a common velocity, which is an imprint of their
formation process. The well-known open cluster catalogues are
2002A&A…389..871D  and 2013A&A…558A..53K  and they list about 3000 open
clusters of the Milky Way.

In the study of star clusters, one of the most crucial parameters is cluster
membership. To identify members, we look for stars in the same region of the
sky, at the same distance and with a common velocity. However, such samples
may be contaminated as there can be field stars with similar velocities or at
similar distances. Also, photometric and spectroscopic (kinematic) data of
cluster stars are generally not available for all the stars in a cluster. For
most stars, we determine a membership probability, which is a non-trivial
problem. In the presence of only photometric data, the standard method used is
a photometric envelope in the color magnitude diagram (CMD) due to which
variable stars, outliers, etc. are difficult or impossible to be identified as
members. Spectroscopic data is generally available only for a few selected
bright stars of the cluster putting limitations on membership determination.
Various methods have been used for membership determination using stellar
positions, proper motions, parallaxes, radial velocities, photometry and their
combinations 1958AJ…..63..387V ; 1977A&AS…27…89S ; 1990A&A…237…54Z ;
2020MNRAS.492.5811B .

The second Gaia data release DR2 2016AA…595A…1G 2018AA…616A…1G  (and
references therein) contains precise astrometry at the sub-milliarcsecond
level and homogeneous three-band photometry for about 1.3 billion sources.
This can be used to characterize a large number of clusters over the entire
sky. Using Gaia data DR2, 2018AA…618A..93C  provided a list of 1229 clusters
with membership data and derived parameters, in particular, mean distances and
proper motions. They applied an unsupervised membership assignment code,
UPMASK 2014A&A…561A..57K , which is primarily based on the $K$-means
clustering algorithm to detect the cluster members and then using random
sampling to assign the membership probability. This work was extended by the
same authors for a final list of members for 1481 clusters refId0 .

In this paper, we used membership data from 2018AA…618A..93C  as our training
data to find new members in a sample of nine open star clusters. We discuss
the results obtained and present the advantages of applying such a technique
in membership determination of star clusters.

The paper is structured as follows: Section 1 is the Introduction, Section 2
describes the Gaia DR2 Data and Sample Selection, Section 3 has the details of
the Machine Learning technique applied in the problem, Section 4 presents the
Results and in Section 5 we discuss our Conclusions.

##  2 Data and Sample Selection

Figure 1: WISE images of the cluster sample: NGC 581, NGC 1893, IC 1805, NGC
6231, NGC 6823, NGC 3293, NGC 6913, NGC 2264, NGC 2244. In all images, North
is up, East is left.

We selected a sample of nine young clusters viz. NGC 581, NGC 1893, IC 1805,
NGC 6231, NGC 6823, NGC 3293, NGC 6913, NGC 2264 and NGC 2244 (Fig. 1). The
sample has clusters with ages ranging from 1.3–20 Myr, at galactocentric
distance $R_{GC}$ ranging from 7.3–14.5 kpc and at varying galactic latitudes
$l$ and longitudes $b$ which have been listed in Table 1. This sample will
also be useful to study the effectiveness of this procedure as the clusters
vary in their position in the galaxy and their individual structures and
parameters.

Table 1: Basic cluster parameters Cluster | $l$ | $b$ | Ang.Dia | Distance | $E(B-V)$ | log(age) | $R_{GC}$
---|---|---|---|---|---|---|---
| deg | deg | arc min | pc | mag | log(yr) | kpc
NGC 581 | 128.05 | -01.80 | 5.0 | 2194 | 0.38 | 7.3 | 10.0
NGC 1893 | 173.59 | -1.68 | 25 | 6000 | 0.45 | 6.5 | 14.5
IC 1805 | 134.73 | 0.92 | 20 | 2344 | 0.87 | 6.1 | 10.3
NGC 6231 | 343.46 | +01.18 | 14.0 | 1243 | 0.85 | 6.5 | 7.4
NGC 3293 | 285.86 | +00.07 | 6.0 | 2327 | 0.26 | 7.0 | 8.2
NGC 6913 | 76.91 | +00.59 | 10.0 | 1148 | 0.74 | 7.1 | 8.3
NGC 2264 | 202.94 | +02.2 | 39.0 | 667 | 0.05 | 6.9 | 9.1
NGC 2244 | 206.31 | -02.07 | 29.0 | 1445 | 0.46 | 6.9 | 9.8

##  3 Machine Learning and the RF Method

Machine Learning (ML) has been used very effectively in various astrophysical
problems, including the problem of membership in star clusters. The Random
Forest (RF) method was used to identify reliable members of the old (4 Gyr)
open cluster M67 based on the high-precision astrometry and photometry taken
from the Gaia DR2 and was found to be very effective for membership
determination of open clusters Gao_2018 . Once cluster members are determined,
cluster parameters like distance, proper motion, radial velocity and spatial
structure of M67 were found. The method found not only likely cluster members
but also possible escaped members, which may lie outside the tidal radius of
the cluster.

Membership of stars in NGC 188 was studied using a machine-learning-based
method which combined two widely used algorithms: spectral clustering (SC) and
random forest (RF) 2014RAA….14..159G . The $K$-means clustering algorithm was
used to study membership of NGC 188 and NGC 2266 2016ExA….42…49E .

Reliable memberships and fundamental astrophysical properties of the clusters
NGC 6791 and NGC 6405 were found using the astrometric and photometric data of
the Gaia DR2 2020Ap&SS.365…24G  2018AJ….156..121G . ML based membership
determination algorithms for open clusters in the absence of a priori
information of cluster parameters was done by 2021MNRAS.502.2582A  using
k-nearest neighbour ($kNN$) and Gaussian Mixture Model ($GMM$).

RF is an ensemble machine-learning technique mainly used for classification
and regression tasks. In recent years, RF has been widely used to investigate
a variety of astronomical problems. RF can easily handle high-dimensional data
without data normalization and obtain class membership probabilities that
allow us to isolate the most likely cluster members. RF has very short
execution time which allows us to effectively handle large, high-dimensional
datasets. However, as a supervised ML technique, the effectiveness of RF
strongly depends on a reliable training set; a poor training set will yield
unreliable results and hence is a major caveat.

###  3.1 Training Data

In order to train our model, we need samples of both member stars and non-
member stars. For each of the clusters, we take all the members provided by
2018AA…618A..93C  as our training set. The above authors also provided a
membership probability for each of the member stars. In their method, any star
with a membership probability $>0$ is a possible member.

We added additional criteria to filter bad data, and only included stars whose
parameters from the Gaia DR2 satisfied $parallax\\_over\\_error>3$,
$pmra\\_error<0.3$ and $pmdec\\_error<0.3$111Right Ascension (RA) is denoted
by $\alpha$ and declination (Dec) by $\delta$. Proper motion in RA is pmra
($\mu_{\alpha}$) and proper motion in declination is pmdec ($\mu_{\delta}$)
and both are in units of milli-arcseconds per year (mas/yr). Parallax is
denoted by $\omega$ and has units of (mas).. Henceforth we will refer to these
member stars of the training set as CG members and we reported the total CG
members in each clusters in Table 3.

The radius of open clusters is a difficult parameter to define. The primary
reason behind this is the lack of precise data to identify members. Earlier
observations detected only a few members closer to the cluster center, thus
reporting a smaller radius. Improved observations lead to detection of members
in outer regions, thus deriving a larger radius of clusters. For example,
2018AA…618A..93C  reported a larger radius for many clusters using the precise
Gaia data. In order to find new members in outer regions, we used a search
radius around each cluster which was double the maximum radius found in CG
members, where the maximum radius is defined as the distance of the farthest
members from the center. The only exception in this case was NGC 2264, where
the presence of an additional sub-clump required us to increase our search
radius to thrice the cluster radius.

For the sample of non-members, we took stars from a concentric ring outside
the search radius as shown in Table 3. To balance our training data and avoid
any over-fitting, we took a random sample of an equal number of non-members
and members for the cluster. Thus the total number of stars in the training
set for each cluster is twice the CG members of the cluster.

Similar to 2018AA…618A..93C , we used $\mu_{\alpha}$, $\mu_{\delta}$ and
$\omega$ to build our model since all the members of a cluster should be
concentrated in a small region in the 3D space of $\mu_{\alpha}$,
$\mu_{\delta}$ and $\omega$, while the non-members should lie outside this
region. For an open cluster, of course, the members lie in the same region of
the sky ($\alpha$, $\delta$). But we do not use $\alpha$, $\delta$ as training
features as we have already selected our search radius around the cluster
center. We also did not use the distance from the cluster center as a feature
because the non-members are chosen from an outer ring area and any supervised
ML method will lead to an over-fitting to get a higher accuracy by setting a
cutoff with this distance parameter.

Figure 2: Proper motions of the CG (orange) and new (blue) members for our
sample of clusters.  Figure 3: Parallax distribution of CG (orange), new
(blue) and combined (green) members for all nine clusters which shows that the
new members are very similar in parallax to the CG members. Figure 4:
Distribution of the CG members (orange) and new members (blue) in sky
coordinates, showing a large number of identified new members on the boundary
of the clusters. Figure 5: Color-Magnitude Diagram of CG (orange) and new
members (blue) for all nine clusters. We can see that this method is effective
in deriving variable stars, pre-main sequence stars and non-main sequence
stars and enriches the CMD. Figure 6: Sky plot of CG (orange), new members
(blue) and combined (green) members of the sample clusters with the
probability density of the stars. Note: NGC 1893, 2244, 2264 and 6913 show
sub-cluster regions. Figure 7: Sky plot of CG (orange), new members (blue) and
combined (green) members of the sample clusters (contd) Figure 8: Proper
Motion Plot of CG (orange), new members (blue) and combined (green) members of
the sample clusters with the probability density of the stars. These clusters
show smaller additional clumps in the proper motion space. Figure 9: Proper
Motion Plot of CG (orange), new members (blue) and combined (green) members of
the sample clusters (contd) Figure 10: Stellar distribution of the clusters
with CG (orange) and combined (green) members of the sample.

###  3.2 Evaluation Metric

The members of open clusters can be used to study the initial mass function,
stellar evolution and dynamics and the star formation process in clusters.
Having a large number of members helps to reduce errors in determination of
parameters, while wrongly identified members can lead to misleading results.
Thus it is a common practice to apply strict criteria to avoid any non-members
being selected as members. As we would like to avoid false positives as
cluster members, we will use precision as our metric to evaluate the
performance of the model, which is defined as

$\frac{\rm True\ Positive}{\rm True\ Positive+False\ Positive}$

We will use this to hypertune the model parameters.

###  3.3 Hyper-tuning Model Parameters

Cluster members occupy a closely bound region in the 3D space of
($\omega,\mu_{\alpha},\mu_{\delta}$) in a spherical or normal distribution,
while non-members are randomly distributed outside this region. If we analyze
and compare the possible supervised classification models for this problem,
then firstly we can say that any linear parametric model (i.e. linear support
vector machine (SVM)) would not be suitable as the separation is not linear
but spherical. The other possible options we could explore are SVM with a
suitable kernel, Naive Bayes with an Gaussian distribution and $kNN$ after
scaling the features to reduce over-fitting. Another suitable model is the
decision tree method as we have smaller number of features and the members are
quite separated. But using a single decision tree can increase the over-
fitting which results in a less stable model. Therefore in this paper we
explore with RF method which can decrease the bias and increase the variance
by using a large number of trees.

Table 2: The random selection grid with the chosen range of values for important RF model parameters Model Parameter | Chosen Values to Select From
---|---
$bootstrap$ | True, False
(Whether bootstrapping the samples) |
$ccp\\_alpha$ |  $2^{-10}$, $2^{-9}$, $2^{-8}$, $2^{-7}$, $2^{-6}$, $2^{-5}$, $2^{-4}$, $2^{-3}$, $2^{-2}$, $2^{-1}$, 0
(Complexity parameter) |
$max\\_depth$ | 10, 20, 30, 40, 50, 60,70, 80, 90, 100, 110, None
(Maximum depth of the tree) |
$max\\_features$ | ‘auto’, ‘sqrt’
(Number of features in each tree) |
$min\\_samples\\_leaf$ | 1, 2, 4
(Minimum samples required for a leaf node) |
$min\\_samples\\_split$ | 2, 5, 10
(Minimum samples required to split a node) |
$n\\_estimators$ | 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000
(Number of decision trees) |

For each cluster we chose the RF model parameters in a way such that the model
has the maximum precision. We first divided the training data for each cluster
in a test and train subset in a ratio of $30:70$. Then we used the train
subset to fine-tune the parameters of the RF model using cross validation. We
made a grid with the possible range of values for important model parameters
(i.e. number of trees in RF, maximum depth of a tree, minimum samples needed
for a split, minimum sample for a leaf node etc), which are reported in Table
2. Then we applied a randomized search 5-fold cross validation in the train
subset with 100 iteration which in total builds 500 models with randomly
chosen parameters from the grid and select the model which resulted in maximum
precision. Finally, we estimated the precision of our selected model using the
test subset, which is shown in Table 3.

###  3.4 Prediction

After extracting the Gaia DR2 data within the search radius as specified in
Table 3, we subtracted the identified members as we already have them in the
training data. In our training data for each cluster we defined membership
probability as 1 for the member stars and 0 for the non-member stars. We then
ran the chosen RF classification model on the data to classify them as either
members or non-members. We also assigned a membership probability for the
stars by measuring the percentage of trees that classify them as members.

Table 3: Prediction from the Random Forest Model |  | Members | Members | Non-Member | Search | New | Precision | Ratio of
---|---|---|---|---|---|---|---|---
Cluster | Radius | before filter | after filter | radius | radius | Members |  | new to CG
| deg |  |  | deg | deg |  | % |
NGC 581 | 0.17 | 306 | 290 | 0.7-0.8 | 0.34 | 525 | 86 | 1.81
NGC 1893 | 0.41 | 494 | 218 | 1.0-1.1 | 0.82 | 774 | 93 | 3.55
NGC 2244 | 0.67 | 1701 | 1192 | 1.4-1.5 | 1.33 | 3043 | 88 | 2.55
NGC 2264 | 0.19 | 186 | 179 | 1.0-1.1 | 0.60 | 514 | 99 | 2.87
NGC 3293 | 0.20 | 657 | 617 | 0.7-0.8 | 0.40 | 1089 | 94 | 1.76
NGC 6231 | 0.47 | 1580 | 1354 | 0.95-1.0 | 0.94 | 2710 | 92 | 2.00
NGC 6823 | 0.2 | 236 | 220 | 0.7-0.8 | 0.40 | 304 | 93 | 1.38
NGC 6913 | 0.3 | 170 | 170 | 0.7-0.8 | 0.60 | 536 | 95 | 3.15
IC 1805 | 0.33 | 456 | 430 | 0.7-0.8 | 0.66 | 1104 | 90 | 2.57

##  4 Results and Discussions

Using the RF method we have almost doubled the number of members in our sample
of nine clusters as shown in Table 3. The Figures 2 to 10 show the plots
obtained from our analysis.

Figures 2 and 3 show the proper motion and parallax distribution for the CG
members (orange) and the new members (blue). We can see that the new members
are distributed very closely to the CG (training) members in all of these 3
parameters ($\mu_{\alpha},\mu_{\delta},\omega$). Figure 4 shows the
distribution of the CG members and new members in sky coordinates. We can see
that there are many new members found in the outer regions of the clusters.
Figure 5 shows the CMDs obtained for all the nine clusters with the CG members
in orange and the new members in blue. Though we did not use any photometric
information while building the model and predicting new stars, all new members
agree with the CMD of CG members for all nine clusters, which further
validates the accuracy of our member classification. Moreover, as we did not
use the photometric information while identifying new members, we can find
non-main sequence stars in our sample of clusters from the CMDs.

In order to visualize the substructures, instead of making a radial profile,
we made a kernel density estimation (KDE) contour plot for both
($\alpha,\delta$) space and ($\mu_{\alpha},\mu_{\delta}$) space. The color
bars in these figures represent the probability density, which is the
normalized density of the stars for a given range of ($\alpha,\delta$) or
($\mu_{\alpha},\mu_{\delta}$). We can get the probability (i.e. frequency) of
finding stars within a given range of ($\alpha,\delta$) or
($\mu_{\alpha},\mu_{\delta}$) by taking the product of the area with their
probability density. Integration of the probability density over all the space
will be exactly 1. If we multiply the color bar with the total number of stars
for a given plot, we will get the density of stars (count/sq.deg).

Figure 6 shows the 4 clusters for which we found additional clump(s) besides
the cluster center. For all other clusters, the combined KDE plots show a
Gaussian distribution as expected (Fig. 7). Some of this sub-structure were
evident in the training members, but got further enhanced in the combined
sample with the addition of new members. As the clumps are overlapped with
each other, it not not very straightforward to find the size or radius for
each clump. So we found the center of the clusters and the half radius $R50$,
defined as the distance from the center within which, 50% of the member stars
lie. We reported the center and R50 of the central clump for GC members and
combined members along with the center of the additional clumps for combined
members in Table 4. These clumps show the substructures and fractions in the
cluster and they can be further studied to understand the mass segregation
across the clusters.

Similarly, in proper motion space, we also found some additional clumps as
seen in Fig. 8. The presence of more than one clump in the proper motion space
indicates the effects of dynamic interactions inside a open star cluster.
These clumps can be further studied to understand the underlying dynamics of
clusters. Figure 9 shows the KDE plot in proper motion space for the remaining
clusters where we see the expected Gaussian distributions.

The stellar distribution of the stars in the clusters is shown in Fig. 10. We
can see that many clusters show a bimodal distribution for the combined
members, some of them can be due to the presence of clumps outside the central
region. For example, For example, for NGC 2264, we can see that the second
peak is near 0.4 deg, which is consistent with the finding that the lower
clump center is around 0.4 deg from the central clump as shown in Fig. 6 and
reported in Table 4. But we can also see bimodal distribution for the clusters
where we did not find any major clump (i.e. NGC 581, NGC 6231), which might be
due to a possible asymmetry in the stellar distribution in the clusters.

Table 4: Cluster and sub-clump centers and half radius, $R50$. $R50$ is defined as the radius within which 50% of the members are situated. For each cluster, the first row corresponds to the central clump and the following rows corresponds to the additional subclumps. Cluster | CG Members |  | Combined members |
---|---|---|---|---
|  Center ($\alpha$, $\delta$) | $R50$ |  Center ($\alpha$, $\delta$) | $R50$
NGC 1893 | (80.72, 33.44) | 0.17 | (80.75, 33.47) | 0.61
|  |  | (80.86, 33.96) |
NGC 2244 | (98.02, 4.90) | 0.37 | (98.03, 4.90) | 0.90
|  |  | (98.74, 5.40) |
|  |  | (98.37, 5.55) |
NGC 2264 | (100.26, 9.88) | 0.08 | (100.27, 9.49) | 0.33
|  |  | (100.23, 9.87) |
NGC 6913 | (305.95, 38.50) | 0.17 | (305.94, 38.50) | 0.45
|  |  | (305.39, 39.60) |
NGC 581 | (23.36, 60.66) | 0.11 | (23.35, 60.65) | 0.27
NGC 3293 | (158.96, -58.24) | 0.11 | (158.96, -58.24) | 0.34
NGC 6231 | (253.56, -41.83) | 0.24 | (253.56, -41.82) | 0.66
NGC 6823 | (295.79, 23.29) | 0.10 | (295.79, 23.30) | 0.24
IC 1805 | (38.22, 61.47) | 0.27 | (38.22, 61.51) | 0.60

As seen in Table 3, the number of new members found are more than three times
the number (after filtering) of CG members for NGC 1893 and NGC 6913, while
for NGC 6823 the ratio was less than 1.5.

Figure 11: Correlations between parameters in the study

For our sample of nine clusters, we found that the precision appears to be
mildly anti-correlated to the search radius, number of stars used to train the
model and the number of new members (with the outlier being NGC 581).
Precision appears to be correlated to the ratio of the new members to the CG
members. For the ratio of new to CG members there appears to be a anti-
correlation to the number of stars used to train the model and a positive
correlation to the search radius. The correlation between stars used to train
the model and the number of new members is very tight and has a value of 0.97.
The detailed correlation values are described in Fig. 11.

As our sample size is small, these correlations are mere indicators and are
subject to change with larger samples222The combined membership data for all
the nine clusters can be requested from PH..

##  5 Conclusion

We have used RF to estimate membership of a larger sample of stars in Gaia DR2
data for nine open clusters. This kind of analysis has the following major
advantages:

  * •

Our results indicate that this machine-learning-based method is highly
suitable for membership determination of open clusters in high dimensional
feature space.

  * •

The sample of stars in clusters can be increased by a large factor, almost 2–3
times. This improves our accuracy in determining various parameters of a star
cluster ranging from distance, extinction and mass function. The sizes of the
studied clusters also increased with the increase in membership and we can
study the outer regions of clusters. The method may find not only likely
cluster members but also likely escaped members, which may lie outside the
tidal radius of the cluster.

  * •

This method makes it possible to find sub-structure in position space as well
as velocity space of the cluster as the members lie in regions beyond the
reported sizes of clusters and hence we can identify sub-clumps as in NGC
2244, NGC 2264 and NGC 6913.

  * •

As we have not used photometric data while estimating the membership, we can
identify variables, pre-main sequence stars (in NGC 1893, NGC 3293),
unresolved binary sequences (in NGC 6231) as well as all other possible non
main-sequence members of the cluster from its CMD.

This work strongly indicates that the RF method has very good potential
applications in star cluster studies.

##  6 Acknowledgments

This work has made use of data from the European Space Agency (ESA) mission
Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing
and Analysis Consortium (DPAC, https://www.cosmos.esa.int/
web/gaia/dpac/consortium). Funding for the DPAC institutions, in particular
the institutions participating in the Gaia Multilateral Agreement. This work
has made use of data from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC, https:
//www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC
institutions, in particular the institutions participating in the Gaia
Multilateral Agreement. Virtual observatory tools like Aladin, Vizier and
TOPCAT have been used in the analysis. Astropy, Scikit Learn, Seaborn, Numpy
and Pandas packages in python have been used in the analysis and
visualizations. MM is very thankful to Dr. Rohan Sekhar, Associate Professor
of Computational Sciences in Minerva Schools at KGI for providing necessary
support and insights on the RF model.

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