Paolo Bellingeri Luis Paris Anne-Laure Thiel primarymsc200020F36

# Virtual Artin groups

Paolo Bellingeri  Paolo Bellingeri, Normandie Univ, UNICAEN, CNRS, LMNO, 14000
Caen, France<EMAIL_ADDRESS>Luis Paris  Luis Paris, IMB, UMR
5584, CNRS, Université Bourgogne Franche-Comté, 21000 Dijon, France
<EMAIL_ADDRESS>Anne-Laure Thiel  Anne-Laure Thiel, Normandie Univ,
UNICAEN, CNRS, LMNO, 14000 Caen, France<EMAIL_ADDRESS>
###### Abstract

Starting from the observation that the standard presentation of a virtual
braid group mixes the standard presentation of the corresponding braid group
with the standard presentation of the corresponding symmetric group and some
mixed relations that mimic the action of the symmetric group on its root
system, we define a virtual Artin group ${\rm VA}[\Gamma]$ of a Coxeter graph
$\Gamma$ mixing the standard presentation of the Artin group $A[\Gamma]$ with
the standard presentation of the Coxeter group $W[\Gamma]$ and some mixed
relations that mimic the action of $W[\Gamma]$ on its root system. By
definition we have two epimorphisms $\pi_{K}:{\rm VA}[\Gamma]\to W[\Gamma]$
and $\pi_{P}:{\rm VA}[\Gamma]\to W[\Gamma]$ whose kernels are denoted by ${\rm
KVA}[\Gamma]$ and ${\rm PVA}[\Gamma]$ respectively. We calculate presentations
for these two subgroups. In particular ${\rm KVA}[\Gamma]$ is an Artin group.
We prove that the center of any virtual Artin group is trivial. In the case
where $\Gamma$ is of spherical type or of affine type, we show that each free
of infinity parabolic subgroup of ${\rm KVA}[\Gamma]$ is also of spherical
type or of affine type, and we show that ${\rm VA}[\Gamma]$ has a solution to
the word problem. In the case where $\Gamma$ is of spherical type we show that
${\rm KVA}[\Gamma]$ satisfies the $K(\pi,1)$ conjecture and we infer the
cohomological dimension of ${\rm KVA}[\Gamma]$ and the virtual cohomological
dimension of ${\rm VA}[\Gamma]$. In the case where $\Gamma$ is of affine type
we determine upper bounds for the cohomological dimension of ${\rm
KVA}[\Gamma]$ and for the virtual cohomological dimension of ${\rm
VA}[\Gamma]$.

##  1 Introduction

Virtual braid groups were introduced by Kauffman [25] in his founding paper on
the theory of virtual knots and links. They play the same role for virtual
links as the braid groups for classical links. We know by Kamada [24] and
Vershinin [35] that the virtual braid group on $n$ strands, denoted ${\rm
VB}_{n}$, admits a presentation with generators
$\sigma_{1},\dots,\sigma_{n-1},\tau_{1},\dots,\tau_{n-1}$, and relations:

  * (v1)

$\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i}$ for $|i-j|\geq 2$,
$\sigma_{i}\sigma_{j}\sigma_{i}=\sigma_{j}\sigma_{i}\sigma_{j}$ for $|i-j|=1$,

  * (v2)

$\tau_{i}\tau_{j}=\tau_{j}\tau_{i}$ for $|i-j|\geq 2$,
$\tau_{i}\tau_{j}\tau_{i}=\tau_{j}\tau_{i}\tau_{j}$ for $|i-j|=1$,
$\tau_{i}^{2}=1$ for $1\leq i\leq n-1$,

  * (v3)

$\tau_{j}\sigma_{i}=\sigma_{i}\tau_{j}$ for $|i-j|\geq 2$,
$\tau_{i}\tau_{j}\sigma_{i}=\sigma_{j}\tau_{i}\tau_{j}$ for $|i-j|=1$.

The relations (v1) are the relations of Artin’s presentation for the braid
group $B_{n}$, and they determine an embedding of $B_{n}$ into ${\rm VB}_{n}$.
The relations (v2) are the relations of the Coxeter type presentation of the
symmetric group $S_{n}$, and they determine an embedding of $S_{n}$ into ${\rm
VB}_{n}$. As for the relations (v3), called _mixed relations_ , they mimic the
action of the symmetric group on its associated root system. This situation
extends naturally to all Artin groups and Coxeter groups, which explains the
forthcoming definition of virtual Artin group.

A _Coxeter matrix_ on a countable set $S$ is a symmetric square matrix
$M=(m_{s,t})_{s,t\in S}$, indexed by the elements of $S$, with entries in
$\mathbb{N}\cup\\{\infty\\}$, satisfying $m_{s,s}=1$ for all $s\in S$ and
$m_{s,t}=m_{t,s}\geq 2$ for all $s,t\in S$, $s\neq t$. Such a matrix is
usually represented by a labeled graph $\Gamma$, called _Coxeter graph_ ,
defined as follows. The set $S$ is the set of vertices of $\Gamma$. Two
vertices $s,t$ are joined by an edge if $m_{s,t}\geq 3$, and this edge is
labeled with $m_{s,t}$ if $m_{s,t}\geq 4$. We say that $\Gamma$ (or $M$) is
_finite_ if $S$ is finite. In most of the studies on Coxeter groups and/or
Artin groups the Coxeter graph is assumed to be finite. Here we will not
assume that $\Gamma$ is necessarily finite since, from a given Coxeter graph
$\Gamma$, we will construct another Coxeter graph $\hat{\Gamma}$ which is not
finite in general, even when $\Gamma$ is. However, we will always assume the
set of vertices to be countable.

If $a,b$ are two letters and $m$ is an integer $\geq 2$, then we denote by
${\rm Prod}_{L}(a,b,m)$ the word $aba\cdots$ of length $m$ and by ${\rm
Prod}_{R}(b,a,m)$ the word $\cdots aba$ of length $m$. If $a,b$ are two
elements of a group $G$, then we denote again by ${\rm Prod}_{L}(a,b,m)$
(resp. ${\rm Prod}_{R}(b,a,m)$) the element of $G$ represented by ${\rm
Prod}_{L}(a,b,m)$ (resp. ${\rm Prod}_{R}(b,a,m)$). Let $\Gamma$ be a Coxeter
graph and let $M=(m_{s,t})_{s,t\in S}$ be its Coxeter matrix. The _Artin
group_ associated with $\Gamma$, denoted $A[\Gamma]$, is the group defined by
the following presentation:

$A[\Gamma]=\langle S\mid{\rm Prod}_{R}(t,s,m_{s,t})={\rm
Prod}_{R}(s,t,m_{s,t})\text{ for }s,t\in S\,,\ s\neq t\,,\
m_{s,t}\neq\infty\rangle\,.$

The _Coxeter group_ associated with $\Gamma$, denoted $W[\Gamma]$, is the
quotient of $A[\Gamma]$ by the relations $s^{2}=1$, $s\in S$.

######  Definition.

Let $\Gamma$ be a Coxeter graph and let $M=(m_{s,t})_{s,t\in S}$ be its
Coxeter matrix. Let $\SS=\\{\sigma_{s}\mid s\in S\\}$ and
$\mathcal{T}=\\{\tau_{s}\mid s\in S\\}$ be two abstract sets in one-to-one
correspondence with $S$. The _virtual Artin group_ associated with $\Gamma$,
denoted ${\rm VA}[\Gamma]$, is the group defined by the presentation with
generating set $\SS\sqcup\mathcal{T}$ and relations:

  * (v1)

${\rm Prod}_{R}(\sigma_{t},\sigma_{s},m_{s,t})={\rm
Prod}_{R}(\sigma_{s},\sigma_{t},m_{s,t})$ for $s,t\in S$, $s\neq t$ and
$m_{s,t}\neq\infty$;

  * (v2)

${\rm Prod}_{R}(\tau_{t},\tau_{s},m_{s,t})={\rm
Prod}_{R}(\tau_{s},\tau_{t},m_{s,t})$ for $s,t\in S$, $s\neq t$ and
$m_{s,t}\neq\infty$; $\tau_{s}^{2}=1$ for $s\in S$;

  * (v3)

${\rm Prod}_{R}(\tau_{s},\tau_{t},m_{s,t}-1)\,\sigma_{s}=\sigma_{r}\,{\rm
Prod}_{R}(\tau_{s},\tau_{t},m_{s,t}-1)$, where $r=s$ if $m_{s,t}$ is even and
$r=t$ if $m_{s,t}$ is odd, for $s,t\in S$, $s\neq t$ and $m_{s,t}\neq\infty$.

Like for virtual braid groups, the relations (v1) are the relations that
define the Artin group $A[\Gamma]$, and we have a natural homomorphism
$\iota_{A}:A[\Gamma]\to{\rm VA}[\Gamma]$ which sends $s$ to $\sigma_{s}$ for
all $s\in S$. On the other hand, the relations (v2) are the relations that
define the Coxeter group $W[\Gamma]$, and we have a natural homomorphism
$\iota_{W}:W[\Gamma]\to{\rm VA}[\Gamma]$ which sends $s$ to $\tau_{s}$ for all
$s\in S$. We will see in Section 2 that these two homomorphisms are injective.
Finally, the relations (v3) mimic the action of $W[\Gamma]$ on its root system
(see Lemma 2.2).

Artin groups were introduced in the early 1970s by Brieskorn [9],
Brieskorn–Saito [10] and Deligne [17] as natural generalizations of braid
groups. This generalization goes beyond the presentations themselves since a
large part of the theory of braid groups extends to some Artin groups,
especially those of spherical type, and a large part of the study of braid
groups is now made in this more general framework. We expect that the same
phenomenon will hold for virtual Artin groups viewed as generalizations of
virtual braid groups.

Notice that the third author [33] proposed a notion of virtual braid group of
type $B_{n}$ by extending the diagrammatic description provided by tom Dieck
for Artin groups of type $B$ and which turns out to be a quotient of the
virtual Artin group of type $B_{n}$ defined above.

The virtual braid group on $n$ strands ${\rm VB}_{n}$ admits two surjective
homomorphisms onto the symmetric group $S_{n}$, $\pi_{K}:{\rm VB}_{n}\to
S_{n}$ and $\pi_{P}:{\rm VB}_{n}\to S_{n}$. Actually, when $n\geq 5$, by
Bellingeri–Paris [6], these are the only surjective homomorphisms from ${\rm
VB}_{n}$ onto $S_{n}$ up to automorphism of $S_{n}$. The kernel of $\pi_{P}$
is called the _pure virtual braid group_ and it is denoted by ${\rm PVB}_{n}$,
and the kernel of $\pi_{K}$ is called the _kure virtual braid group_ and it is
denoted by ${\rm KVB}_{n}$. The homomorphisms $\pi_{K}$ and $\pi_{P}$ admit a
common section $\iota_{n}:S_{n}\to{\rm VB}_{n}$, hence we have the semi-direct
product decompositions ${\rm VB}_{n}={\rm KVB}_{n}\rtimes S_{n}$ and ${\rm
VB}_{n}={\rm PVB}_{n}\rtimes S_{n}$. A presentation for ${\rm PVB}_{n}$ was
calculated by Bardakov [1]. Furthermore, ${\rm PVB}_{n}$ coincides with a
group studied by Bartholdi–Enriquez–Etingof–Rains [3] in relation with the
Yang–Baxter equations. By Bartholdi–Enriquez–Etingof–Rains [3], ${\rm
PVB}_{n}$ admits a finite classifying space, that is, ${\rm PVB}_{n}$ is of
type F, which implies also that it is torsion free. On the other hand, the
group ${\rm KVB}_{n}$ is an Artin group (see Rabenda [31] and
Bardakov–Bellingeri [2]) and it is quite well-understood thanks to
Godelle–Paris [23]. In particular ${\rm KVB}_{n}$ is also of type F, therefore
it is torsion free and it has a solution to the word problem. Since ${\rm
KVB}_{n}$ is of finite index in ${\rm VB}_{n}$, we can therefore use ${\rm
KVB}_{n}$ to solve the word problem in ${\rm VB}_{n}$ (see
Bellingeri–Cisneros-de-la-Cruz–Paris [5]).

Let $\Gamma$ be a Coxeter graph. Like for virtual braid groups, we have two
surjective homomorphisms $\pi_{P}:{\rm VA}[\Gamma]\to W[\Gamma]$ and
$\pi_{K}:{\rm VA}[\Gamma]\to W[\Gamma]$ that admit a common section
$\iota_{W}:W[\Gamma]\to{\rm VA}[\Gamma]$. Let ${\rm PVA}[\Gamma]$ denote the
kernel of $\pi_{P}$ and ${\rm KVA}[\Gamma]$ denote the kernel of $\pi_{K}$.
Then we have semi-direct product decompositions ${\rm VA}[\Gamma]={\rm
PVA}[\Gamma]\rtimes W[\Gamma]$ and ${\rm VA}[\Gamma]={\rm KVA}[\Gamma]\rtimes
W[\Gamma]$. In a first step we compute presentations for ${\rm KVA}[\Gamma]$
and ${\rm PVA}[\Gamma]$ (see Theorem 2.3 and Theorem 2.6). These presentations
are generalizations of the presentations of ${\rm PVB}_{n}$ and ${\rm
KVB}_{n}$ found by Bardakov [1], Rabenda [31] and Bardakov–Bellingeri [2], and
provide a new insight into these groups. For example ${\rm KVA}[\Gamma]$ as
well as ${\rm PVA}[\Gamma]$ has a generating set naturally in one-to-one
correspondence with the root system of $W[\Gamma]$, and the action of
$W[\Gamma]$ on ${\rm KVA}[\Gamma]$ and/or ${\rm PVA}[\Gamma]$ is given by the
action of $W[\Gamma]$ on its root system. As a first application we show that
the natural homomorphism $\iota_{A}:A[\Gamma]\to{\rm VA}[\Gamma]$ which sends
$s$ to $\sigma_{s}$ for all $s\in S$ is injective (see Corollary 2.4). Note
that the homomorphism $\iota_{W}:W[\Gamma]\to{\rm VA}[\Gamma]$ is necessarily
injective since it is a section of both $\pi_{P}$ and $\pi_{K}$ (see
Proposition 2.1).

From Theorem 2.3 it follows that ${\rm KVA}[\Gamma]$ is an Artin group whose
Coxeter graph, denoted $\hat{\Gamma}$, is determined by the root system of
$W[\Gamma]$. We use this in the rest of the paper. In Section 3 we prove that
the center of ${\rm VA}[\Gamma]$ is trivial whatever $\Gamma$ is (see
Corollary 3.4). This result is rather surprising since determining the center
of all Artin groups is an open question. In Sections 4, 5 and 6 we assume that
the Coxeter graph $\Gamma$ is of spherical type or of affine type. There is a
general principle which says that, if one understands the free of infinity
parabolic subgroups of a given Artin group, then we can understand the Artin
group itself (see Godelle–Paris [22]). The aim of Section 4 is to show that we
can apply this principle to ${\rm KVA}[\Gamma]$ when $\Gamma$ is of spherical
type or of affine type (see Theorem 4.1). In particular we show that each free
of infinity parabolic subgroup of ${\rm KVA}[\Gamma]$ is of spherical type or
of affine type and its spherical dimension is bounded above by the spherical
dimension of $A[\Gamma]$. The definition of “spherical dimension” of an Artin
group is given in the beginning of Section 4. In the good cases, that include
Artin groups of spherical type and of affine type, it coincides with the
cohomological dimension of the group. In Section 5 we show that, if $\Gamma$
is a Coxeter graph of spherical type or of affine type, then ${\rm
VA}[\Gamma]$ has a solution to the word problem (see Theorem 5.1). Section 6
concerns some (co)homological properties of ${\rm VA}[\Gamma]$ and ${\rm
KVA}[\Gamma]$ when $\Gamma$ is of spherical type or of affine type. We show
that, if $\Gamma$ is of spherical type, then ${\rm KVA}[\Gamma]$ satisfies the
so-called $K(\pi,1)$ conjecture (see Theorem 6.3), hence it is of type F, its
cohomological dimension is finite, and it is torsion free (see Corollary 6.5).
It also follows that the virtual cohomological dimension of ${\rm VA}[\Gamma]$
is equal to the number of vertices of $\Gamma$. For the case where $\Gamma$ is
of affine type we show that the cohomological dimension of ${\rm KVA}[\Gamma]$
and the virtual cohomological dimension of ${\rm VA}[\Gamma]$ are bounded
above (see Theorem 6.6).

In order to avoid possible misunderstandings let us finish this section with a
caveat. The term ”virtual” has different meanings in this text: a virtual
Artin group is called in such way because it is a natural extension of the
virtual braid group ${\rm VB}_{n}$, while ”virtual cohomogical dimension” and
”virtually torsion free” are referred to combinatorial properties of finite
index subgroups of virtual Artin groups. However, when $\Gamma$ is of
spherical type, the fact that ${\rm KVA}[\Gamma]$ is an Artin group and that
${\rm VA}[\Gamma]={\rm KVA}[\Gamma]\rtimes W[\Gamma]$ can be rephrased saying
that the virtual Artin group ${\rm VA}[\Gamma]$ is virtually an Artin group.

######  Acknowledgments.

We thank Conchita Martínez-Pérez for all the information and references on
cohomological dimension of groups that she gave us. Her help was important for
achieving the results of Section 6. The first and second authors are supported
by the French project “AlMaRe” (ANR-19-CE40-0001-01) of the ANR. The first and
third authors were also supported by the project ARTIQ (ERDF/RIN).

##  2 The groups ${\rm KVA}[\Gamma]$ and ${\rm PVA}[\Gamma]$

Let $\Gamma$ be a Coxeter graph and let $M=(m_{s,t})_{s,t\in S}$ be its
Coxeter matrix. Recall that $\Gamma$ is not assumed to be finite in the sense
that the set of vertices of $\Gamma$ can be a countable infinite set. For
$X\subset S$ we set $M_{X}=(m_{s,t})_{s,t\in X}$ and we denote by $\Gamma_{X}$
the Coxeter graph of $M_{X}$. Note that $\Gamma_{X}$ is the full Coxeter
subgraph of $\Gamma$ spanned by $X$. We denote by $\mathcal{P}_{\rm fin}(S)$
the set of finite subsets of $S$. If $X,Y\in\mathcal{P}_{\rm fin}(S)$ are such
that $X\subset Y$, then, by Van der Lek [27, Lemma 4.11] and Bourbaki [8,
Corollaire 1, Page 19], the homomorphisms $A[\Gamma_{X}]\to A[\Gamma_{Y}]$ and
$W[\Gamma_{X}]\to W[\Gamma_{Y}]$ induced by the inclusion are injective. It
follows that $A[\Gamma]$ and $W[\Gamma]$ are the following direct limits:

$A[\Gamma]=\varinjlim_{X\in\mathcal{P}_{\rm fin}(S)}A[\Gamma_{X}]\text{ and
}W[\Gamma]=\varinjlim_{X\in\mathcal{P}_{\rm fin}(S)}W[\Gamma_{X}]\,.$

So, most of the results on Artin and Coxeter groups that we will use in this
paper, although proved within the framework of finite Coxeter graphs, extend
immediately to Coxeter graphs with countable many vertices. We will point out
when and where this is not the case.

We see in the presentations of ${\rm VA}[\Gamma]$ and $W[\Gamma]$ that there
are surjective homomorphisms $\pi_{K}:{\rm VA}[\Gamma]\to W[\Gamma]$ and
$\pi_{P}:{\rm VA}[\Gamma]\to W[\Gamma]$ defined by:

$\displaystyle\pi_{K}(\sigma_{s})=1\text{ and }\pi_{K}(\tau_{s})=s\text{ for
all }s\in S\,,$ $\displaystyle\pi_{P}(\sigma_{s})=\pi_{P}(\tau_{s})=s\text{
for all }s\in S\,.$

We set ${\rm KVA}[\Gamma]={\rm Ker}(\pi_{K})$ and ${\rm PVA}[\Gamma]={\rm
Ker}(\pi_{P})$. Recall the homomorphism $\iota_{W}:W[\Gamma]\to{\rm
VA}[\Gamma]$ which sends $s$ to $\tau_{s}$ for all $s\in S$. Observe that
$\pi_{K}\circ\iota_{W}={\rm id}$ and $\pi_{P}\circ\iota_{W}={\rm id}$, hence
$\iota_{W}$ is a section of both epimorphisms. So:

######  Proposition 2.1.

Let $\Gamma$ be a Coxeter graph. The homomorphism $\iota_{W}:W[\Gamma]\to{\rm
VA}[\Gamma]$ is injective and we have the following semi-direct product
decompositions:

${\rm VA}[\Gamma]={\rm KVA}[\Gamma]\rtimes W[\Gamma]\text{ and }{\rm
VA}[\Gamma]={\rm PVA}[\Gamma]\rtimes W[\Gamma]\,.$

Let $\Gamma$ be a Coxeter graph and let $M=(m_{s,t})_{s,t\in S}$ be its
Coxeter matrix. Let $\Pi=\\{\alpha_{s}\mid s\in S\\}$ be an abstract set in
one-to-one correspondence with $S$. The elements of $\Pi$ are called _simple
roots_. Let $V$ be the real vector space having $\Pi$ as a basis and let
$\langle\cdot,\cdot\rangle:V\times V\to\mathbb{R}$ be the symmetric bilinear
form, called the _canonical bilinear form_ , defined by:

$\langle\alpha_{s},\alpha_{t}\rangle=\left\\{\begin{array}[]{ll}-2\cos(\pi/m_{s,t})&\text{if
}m_{s,t}\neq\infty\,,\\\ -2&\text{if }m_{s,t}=\infty\,.\end{array}\right.$

There is a faithful linear representation $W[\Gamma]\hookrightarrow{\rm
GL}(V)$, called the _canonical linear representation_ of $W[\Gamma]$,
preserving the bilinear form $\langle\cdot,\cdot\rangle$, and defined by:

$s(v)=v-\langle v,\alpha_{s}\rangle\alpha_{s}\,,\quad\text{for }v\in V\,,\
s\in S\,.$

We will assume $W[\Gamma]$ to be embedded into ${\rm GL}(V)$ via this linear
representation. We set $\Phi[\Gamma]=\\{w(\alpha_{s})\mid s\in S\text{ and
}w\in W[\Gamma]\\}$, and we denote by $\Phi^{+}[\Gamma]$ (resp.
$\Phi^{-}[\Gamma]$) the set of elements $\beta\in\Phi[\Gamma]$ that can be
written $\beta=\sum_{s\in S}\lambda_{s}\alpha_{s}$ with $\lambda_{s}\geq 0$
(resp. $\lambda_{s}\leq 0$) for all $s\in S$. By Deodhar [16] we have the
disjoint union $\Phi[\Gamma]=\Phi^{+}[\Gamma]\sqcup\Phi^{-}[\Gamma]$ and
$\Phi^{-}[\Gamma]=\\{-\beta\mid\beta\in\Phi^{+}[\Gamma]\\}$. The set
$\Phi[\Gamma]$ is called the _root system_ of $\Gamma$ and the elements of
$\Phi^{+}[\Gamma]$ (resp. $\Phi^{-}[\Gamma]$) are called _positive roots_
(resp. _negative roots_).

For each $\beta\in\Phi[\Gamma]$ we denote by $r_{\beta}:V\to V$ the linear
reflection defined by $r_{\beta}(v)=v-\langle v,\beta\rangle\beta$. Note that,
if $s\in S$ and $w\in W[\Gamma]$ are such that $\beta=w(\alpha_{s})$, then
$r_{\beta}=wsw^{-1}$. So $r_{\beta}\in W[\Gamma]$ for all
$\beta\in\Phi[\Gamma]$ and $r_{\alpha_{s}}=s$ for all $s\in S$. Note also that
$r_{\beta}=r_{-\beta}$ for all $\beta\in\Phi[\Gamma]$.

We denote by $\cdot$ the action of $W[\Gamma]$ on ${\rm KVA}[\Gamma]$ (resp.
${\rm PVA}[\Gamma]$). In other words, for $w\in W[\Gamma]$ and $g\in{\rm
KVA}[\Gamma]$ (resp. $g\in{\rm PVA}[\Gamma]$), we set $w\cdot
g=\iota_{W}(w)\,g\,\iota_{W}(w)^{-1}$. The following lemma gives its real
meaning to the remark “the relations (v3) mimic the action of $W[\Gamma]$ on
the root system of $W[\Gamma]$” made in the introduction.

######  Lemma 2.2.

Let $\Gamma$ be a Coxeter graph and let $M=(m_{s,t})_{s,t\in S}$ be its
Coxeter matrix. Let $u,v\in W[\Gamma]$ and $s,t\in S$. If
$u(\alpha_{s})=v(\alpha_{t})$, then $u\cdot\sigma_{s}=v\cdot\sigma_{t}$ (in
${\rm KVA}[\Gamma]$) and
$u\cdot(\tau_{s}\sigma_{s})=v\cdot(\tau_{t}\sigma_{t})$ (in ${\rm
PVA}[\Gamma]$).

###### Proof.

We prove the equality $u\cdot\sigma_{s}=v\cdot\sigma_{t}$. The other equality
can be proved in the same way. We denote by $\lg:W[\Gamma]\to\mathbb{N}$ the
word length with respect to $S$. The proof is based on the following result
which can be found in Deodhar [16].

Claim 1.

  * (1)

Let $w\in W[\Gamma]$ and $s\in S$. We have $w(\alpha_{s})\in\Phi^{+}[\Gamma]$
if and only if $\lg(ws)=\lg(w)+1$.

  * (2)

Let $w\in W[\Gamma]$ and $s,t\in S$, $s\neq t$, such that
$\lg(ws)=\lg(wt)=\lg(w)-1$. Then $m_{s,t}\neq\infty$ and $\lg(w\,{\rm
Prod}_{R}(t,s,m_{s,t})^{-1})=\lg(w)-m_{s,t}$.

  * (3)

Let $s,t\in S$, $s\neq t$, such that $m_{s,t}\neq\infty$. Then ${\rm
Prod}_{R}(s,t,m_{s,t}-1)(\alpha_{s})=\alpha_{r}$, where $r=s$ if $m_{s,t}$ is
even and $r=t$ if $m_{s,t}$ is odd.

Let $u,v\in W[\Gamma]$ and $s,t\in S$ such that $u(\alpha_{s})=v(\alpha_{t})$.
Upon replacing $u$ with $v^{-1}u$, we may assume that $v=1$, that is,
$u(\alpha_{s})=\alpha_{t}$. We show that $u\cdot\sigma_{s}=\sigma_{t}$ by
induction on $\lg(u)$. If $\lg(u)=0$, then $\alpha_{s}=\alpha_{t}$, hence
$s=t$, and therefore $\sigma_{s}=\sigma_{t}$. So, we can assume that
$\lg(u)\geq 1$ and that the induction hypothesis holds. Since
$u(\alpha_{s})=\alpha_{t}$, we have $usu^{-1}=t$, hence $us=tu$. Moreover,
$\lg(us)=\lg(u)+1$, since $u(\alpha_{s})=\alpha_{t}\in\Phi^{+}[\Gamma]$ (by
Claim 1). Set $w=us=tu$. Choose $r\in S$ such that $\lg(ur)<\lg(u)$. We have
$r\neq s$, since $\lg(us)=\lg(u)+1$ and $\lg(ur)=\lg(u)-1$. Moreover,
$\lg(ws)=\lg(wr)=\lg(w)-1$ hence, by Claim 1, $m_{r,s}\neq\infty$ and
$\lg(w^{\prime})=\lg(w)-m_{r,s}$, where $w^{\prime}=w\,{\rm
Prod}_{R}(s,r,m_{r,s})^{-1}$. We set $x=s$ if $m_{r,s}$ is even and $x=r$ if
$m_{r,s}$ is odd. Since $w=us=w^{\prime}\,{\rm Prod}_{R}(s,r,m_{r,s}-1)\,s$,
we have $u=w^{\prime}\,{\rm Prod}_{R}(s,r,m_{r,s}-1)$, hence, by Claim 1,

$\alpha_{t}=u(\alpha_{s})=w^{\prime}\,{\rm
Prod}_{R}(s,r,m_{r,s}-1)(\alpha_{s})=w^{\prime}(\alpha_{x})\,.$

Since $\lg(w^{\prime})=\lg(w)-m_{r,s}\leq\lg(w)-2<\lg(w)-1=\lg(u)$, by the
induction hypothesis, $w^{\prime}\cdot\sigma_{x}=\sigma_{t}$. On the other
hand,

$\displaystyle{\rm Prod}_{R}(s,r,m_{r,s}-1)\cdot\sigma_{s}={\rm
Prod}_{R}(\tau_{s},\tau_{r},m_{r,s}-1)\,\sigma_{s}\,{\rm
Prod}_{R}(\tau_{s},\tau_{r},m_{r,s}-1)^{-1}=$ $\displaystyle\sigma_{x}\,{\rm
Prod}_{R}(\tau_{s},\tau_{r},m_{r,s}-1)\,{\rm
Prod}_{R}(\tau_{s},\tau_{r},m_{r,s}-1)^{-1}=\sigma_{x}\,,$

hence:

$u\cdot\sigma_{s}=w^{\prime}\,{\rm
Prod}_{R}(s,r,m_{r,s}-1)\cdot\sigma_{s}=w^{\prime}\cdot\sigma_{x}=\sigma_{t}\,.\proved$

∎

Let $\Gamma$ be a Coxeter graph and let $M=(m_{s,t})_{s,t\in S}$ be its
Coxeter matrix, we now define some important elements of our two kernels. Let
$\beta\in\Phi[\Gamma]$, we choose $w\in W[\Gamma]$ and $s\in S$ such that
$w(\alpha_{s})=\beta$ and we set $\delta_{\beta}=w\cdot\sigma_{s}\in{\rm
KVA}[\Gamma]$ and $\zeta_{\beta}=w\cdot(\tau_{s}\sigma_{s})\in{\rm
PVA}[\Gamma]$. By Lemma 2.2 these definitions do not depend on the choice of
$w$ and $s$. We will see in the proofs of Theorem 2.3 and Theorem 2.6 that
$\\{\delta_{\beta}\mid\beta\in\Phi[\Gamma]\\}$ generates ${\rm KVA}[\Gamma]$
and $\\{\zeta_{\beta}\mid\beta\in\Phi[\Gamma]\\}$ generates ${\rm
PVA}[\Gamma]$.

We define a new Coxeter matrix
$\hat{M}=(\hat{m}_{\beta,\gamma})_{\beta,\gamma\in\Phi[\Gamma]}$ indexed by
the elements of $\Phi[\Gamma]$ as follows.

  * (a)

We set $\hat{m}_{\beta,\beta}=1$ for all $\beta\in\Phi[\Gamma]$.

  * (b)

Let $\beta,\gamma\in\Phi[\Gamma]$, $\beta\neq\gamma$. If there exist $w\in
W[\Gamma]$ and $s,t\in S$ such that $\beta=w(\alpha_{s})$,
$\gamma=w(\alpha_{t})$ and $m_{s,t}\neq\infty$, then we set
$\hat{m}_{\beta,\gamma}=m_{s,t}$. We set $\hat{m}_{\beta,\gamma}=\infty$
otherwise.

Note that the definition of $\hat{m}_{\beta,\gamma}$ in Case (b) does not
depend on the choice of $w$, $s$ and $t$. Indeed, suppose there exist
$w,w^{\prime}\in W[\Gamma]$ and $s,t,s^{\prime},t^{\prime}\in S$ such that
$m_{s,t}\neq\infty$, $m_{s^{\prime},t^{\prime}}\neq\infty$,
$\beta=w(\alpha_{s})=w^{\prime}(\alpha_{s^{\prime}})$ and
$\gamma=w(\alpha_{t})=w^{\prime}(\alpha_{t^{\prime}})$. Then:

$\langle\beta,\gamma\rangle=-2\cos(\pi/m_{s,t})=-2\cos(\pi/m_{s^{\prime},t^{\prime}})\,,$

hence $m_{s,t}=m_{s^{\prime},t^{\prime}}$. We denote by $\hat{\Gamma}$ the
Coxeter graph of $\hat{M}$. Moreover, for convenience, we will denote by
$\\{\hat{\delta}_{\beta}\mid\beta\in\Phi[\Gamma]\\}$ the standard generating
set of $A[\hat{\Gamma}]$.

######  Remark.

We know by Deodhar [16] that $\Phi[\Gamma]$ is finite if and only if
$W[\Gamma]$ is finite. So, $\hat{\Gamma}$ is finite if and only if $W[\Gamma]$
is finite.

######  Example 1.

Suppose $\Gamma$ is the graph $A_{n-1}$ of Figure 2.1. Then $W[\Gamma]$ is the
symmetric group $S_{n}$, $A[\Gamma]$ is the braid group $B_{n}$ and ${\rm
VA}[\Gamma]$ is the virtual braid group ${\rm VB}_{n}$. We set
$U=\mathbb{R}^{n}$ that we assume to be endowed with the standard scalar
product, $\langle\cdot,\cdot\rangle$, and we denote by
$\\{e_{1},\dots,e_{n}\\}$ its canonical basis. We consider the action of
$S_{n}$ on $U$ by permutations of the coordinates and we denote by $V$ the
hyperplane of $U$ defined by the equation $x_{1}+\cdots+x_{n}=0$. Then $V$ is
invariant under the action of $S_{n}$ and the induced representation
$W[\Gamma]=S_{n}\to{\rm GL}(V)$ is the canonical representation. For
$i,j\in\\{1,\dots,n\\}$, $i\neq j$, we set $\alpha_{i,j}=e_{j}-e_{i}$. Then
$\Pi=\\{\alpha_{1,2},\dots,\alpha_{n-1,n}\\}$ is the set of simple roots, and
$\Phi[\Gamma]=\\{\alpha_{i,j}\mid 1\leq i\neq j\leq n\\}$ is the root system
of $W[\Gamma]=S_{n}$. The set $\\{\delta_{\alpha_{i,j}}\mid 1\leq i\neq j\leq
n\\}$ is the generating set of ${\rm KVB}_{n}={\rm KVA}[\Gamma]$ given by
Rabenda [31] and Bardakov–Bellingeri [2] and the set
$\\{\zeta_{\alpha_{i,j}}\mid 1\leq i\neq j\leq n\\}$ is the generating set of
${\rm PVB}_{n}={\rm PVA}[\Gamma]$ given by Bardakov [1]. The Coxeter matrix
$\hat{M}$ is defined by:

  * •

$\hat{m}_{\alpha_{i,j},\alpha_{i,j}}=1$ for $1\leq i\neq j\leq n$.

  * •

$\hat{m}_{\alpha_{i,j},\alpha_{k,\ell}}=2$ for $i,j,k,\ell$ pairwise distinct.

  * •

$\hat{m}_{\alpha_{i,j},\alpha_{j,k}}=3$ for $i,j,k$ pairwise distinct.

  * •

$\hat{m}_{\alpha_{i,j},\alpha_{k,\ell}}=\infty$ otherwise.

Figure 2.1: Coxeter graph $A_{n-1}$

######  Example 2.

We say that $A[\Gamma]$ (or $\Gamma$) is _simply laced_ if
$m_{s,t}\in\\{2,3\\}$ for all $s,t\in S$, $s\neq t$. Suppose $\Gamma$ is
simply laced. Then each root $\beta\in\Phi[\Gamma]$ can be written
$\beta=\sum_{s\in S}\lambda_{s}\alpha_{s}$ with $\lambda_{s}\in\mathbb{Z}$ for
all $s\in S$. Moreover, we have $\langle\beta,\gamma\rangle\in\mathbb{Z}$ for
all $\beta,\gamma\in\Phi[\Gamma]$. In this case $\hat{M}$ is defined by:

  * •

$\hat{m}_{\beta,\beta}=1$ for $\beta\in\Phi[\Gamma]$.

  * •

$\hat{m}_{\beta,\gamma}=2$ for $\beta,\gamma\in\Phi[\Gamma]$,
$\beta\neq\gamma$ and $\langle\beta,\gamma\rangle=0$.

  * •

$\hat{m}_{\beta,\gamma}=3$ for $\beta,\gamma\in\Phi[\Gamma]$,
$\beta\neq\gamma$ and $\langle\beta,\gamma\rangle=-1$.

  * •

$\hat{m}_{\beta,\gamma}=\infty$ for $\beta,\gamma\in\Phi[\Gamma]$,
$\beta\neq\gamma$ and $\langle\beta,\gamma\rangle\not\in\\{0,-1\\}$.

######  Theorem 2.3.

Let $\Gamma$ be a Coxeter graph. Then the map
$\\{\hat{\delta}_{\beta}\mid\beta\in\Phi[\Gamma]\\}\to\\{\delta_{\beta}\mid\beta\in\Phi[\Gamma]\\}$,
$\hat{\delta}_{\beta}\mapsto\delta_{\beta}$, induces an isomorphism
$\varphi:A[\hat{\Gamma}]\to{\rm KVA}[\Gamma]$.

######  Remark.

When $\Gamma$ is the graph $A_{n-1}$ of Figure 2.1 we get from Theorem 2.3 the
presentation of ${\rm KVB}_{n}={\rm KVA}[\Gamma]$ given by Rabenda [31] and
Bardakov–Bellingeri [2].

###### Proof.

We start by showing that $\\{\delta_{\beta}\mid\beta\in\Phi[\Gamma]\\}$
generates ${\rm KVA}[\Gamma]$. Let $g\in{\rm KVA}[\Gamma]$. There exist
$w_{0},w_{1},\dots,w_{p}\in W[\Gamma]$, $s_{1},\dots,s_{p}\in S$ and
$\varepsilon_{1},\dots,\varepsilon_{p}\in\\{\pm 1\\}$ such that:

$g=\iota_{W}(w_{0})\,\sigma_{s_{1}}^{\varepsilon_{1}}\,\iota_{W}(w_{1})\cdots\sigma_{s_{p}}^{\varepsilon_{p}}\,\iota_{W}(w_{p})\,.$

For $i\in\\{1,\dots,p\\}$ we set $\beta_{i}=(w_{0}w_{1}\cdots
w_{i-1})(\alpha_{s_{i}})$, and we set $w=w_{0}w_{1}\cdots w_{p}$. Then
$g=\delta_{\beta_{1}}^{\varepsilon_{1}}\cdots\delta_{\beta_{p}}^{\varepsilon_{p}}\,\iota_{W}(w)$.
Since $g\in{\rm KVA}[\Gamma]$, we have $1=\pi_{K}(g)=w$, hence
$g=\delta_{\beta_{1}}^{\varepsilon_{1}}\cdots\delta_{\beta_{p}}^{\varepsilon_{p}}$.

Now we show that there is a homomorphism $\varphi:A[\hat{\Gamma}]\to{\rm
KVA}[\Gamma]$ which sends $\hat{\delta}_{\beta}$ to $\delta_{\beta}$ for all
$\beta\in\Phi[\Gamma]$. Let $\beta,\gamma\in\Phi[\Gamma]$ such that
$\beta\neq\gamma$ and $\hat{m}_{\beta,\gamma}\neq\infty$. There exist $w\in
W[\Gamma]$ and $s,t\in S$ such that $\beta=w(\alpha_{s})$,
$\gamma=w(\alpha_{t})$ and $m_{s,t}=\hat{m}_{\beta,\gamma}$. Then:

$\displaystyle{\rm
Prod}_{R}(\delta_{\gamma},\delta_{\beta},\hat{m}_{\beta,\gamma})=\iota_{W}(w)\,{\rm
Prod}_{R}(\sigma_{t},\sigma_{s},m_{s,t})\,\iota_{W}(w)^{-1}=$
$\displaystyle\iota_{W}(w)\,{\rm
Prod}_{R}(\sigma_{s},\sigma_{t},m_{s,t})\,\iota_{W}(w)^{-1}={\rm
Prod}_{R}(\delta_{\beta},\delta_{\gamma},\hat{m}_{\beta,\gamma})\,.$

Note that this homomorphism is surjective since we know that
$\\{\delta_{\beta}\mid\beta\in\Phi[\Gamma]\\}$ generates ${\rm KVA}[\Gamma]$.

The action of $W[\Gamma]$ on
$\\{\hat{\delta}_{\beta}\mid\beta\in\Phi[\Gamma]\\}$ defined by
$w\cdot\hat{\delta}_{\beta}=\hat{\delta}_{w(\beta)}$ extends to an action of
$W[\Gamma]$ on $A[\hat{\Gamma}]$. We can therefore consider the semi-direct
product $G=A[\hat{\Gamma}]\rtimes W[\Gamma]$ determined by this action. Now,
we show that the homomorphisms $\varphi:A[\hat{\Gamma}]\to{\rm
KVA}[\Gamma]\subset{\rm VA}[\Gamma]$ and $\iota_{W}:W[\Gamma]\to{\rm
VA}[\Gamma]$ induce a homomorphism $\tilde{\varphi}:G\to{\rm VA}[\Gamma]$. In
order to do this, we just need to check that
$\varphi(w\cdot\hat{\delta}_{\beta})=\iota_{W}(w)\,\varphi(\hat{\delta}_{\beta})\,\iota_{W}(w)^{-1}$
for all $w\in W[\Gamma]$ and all $\beta\in\Phi[\Gamma]$, and this equality is
easily verified since both terms are by definition equal to
$\delta_{w(\beta)}$.

Let $\psi:\SS\sqcup\mathcal{T}\to G$ be the map defined by
$\psi(\sigma_{s})=\hat{\delta}_{\alpha_{s}}$ and $\psi(\tau_{s})=s$ for all
$s\in S$. Now we show that this map induces a homomorphism $\psi:{\rm
VA}[\Gamma]\to G$. Let $s,t\in S$, $s\neq t$, such that $m_{s,t}\neq\infty$.
Then:

$\displaystyle{\rm Prod}_{R}(\psi(\sigma_{t}),\psi(\sigma_{s}),m_{s,t})={\rm
Prod}_{R}(\hat{\delta}_{\alpha_{t}},\hat{\delta}_{\alpha_{s}},\hat{m}_{\alpha_{s},\alpha_{t}})=$
$\displaystyle{\rm
Prod}_{R}(\hat{\delta}_{\alpha_{s}},\hat{\delta}_{\alpha_{t}},\hat{m}_{\alpha_{s},\alpha_{t}})={\rm
Prod}_{R}(\psi(\sigma_{s}),\psi(\sigma_{t}),m_{s,t})\,.$

Let $s,t\in S$, $s\neq t$, such that $m_{s,t}\neq\infty$. Then:

${\rm Prod}_{R}(\psi(\tau_{t}),\psi(\tau_{s}),m_{s,t})={\rm
Prod}_{R}(t,s,m_{s,t})={\rm Prod}_{R}(s,t,m_{s,t})={\rm
Prod}_{R}(\psi(\tau_{s}),\psi(\tau_{t}),m_{s,t})\,.$

Let $s\in S$. Then:

$\psi(\tau_{s})^{2}=s^{2}=1\,.$

Let $s,t\in S$, $s\neq t$, such that $m_{s,t}\neq\infty$. Set $r=s$ if
$m_{s,t}$ is even and $r=t$ if $m_{s,t}$ is odd. Let $w={\rm
Prod}_{R}(s,t,m_{s,t}-1)\in W[\Gamma]$. Recall that, by Deodhar [16],
$w(\alpha_{s})=\alpha_{r}$. Then:

$\displaystyle{\rm
Prod}_{R}(\psi(\tau_{s}),\psi(\tau_{t}),m_{s,t}-1)\,\psi(\sigma_{s})={\rm
Prod}_{R}(s,t,m_{s,t}-1)\,\hat{\delta}_{\alpha_{s}}=w\hat{\delta}_{\alpha_{s}}=\hat{\delta}_{w(\alpha_{s})}w=$
$\displaystyle\hat{\delta}_{\alpha_{r}}\,{\rm
Prod}_{R}(s,t,m_{s,t}-1)=\psi(\sigma_{r})\,{\rm
Prod}_{R}(\psi(\tau_{s}),\psi(\tau_{t}),m_{s,t}-1)\,.$

Now we prove that $\psi\circ\tilde{\varphi}={\rm id}$. This implies that
$\tilde{\varphi}$ is injective, and therefore that $\varphi$ is injective,
finishing the proof of the theorem. Let $s\in S$. Then
$(\psi\circ\tilde{\varphi})(s)=\psi(\tau_{s})=s$. Let $\beta\in\Phi[\Gamma]$.
Let $w\in W[\Gamma]$ and $s\in S$ such that $\beta=w(\alpha_{s})$. Then:

$\displaystyle(\psi\circ\tilde{\varphi})(\hat{\delta}_{\beta})=\psi(\delta_{\beta})=\psi(\delta_{w(\alpha_{s})})=\psi(w\cdot\delta_{\alpha_{s}})=\psi(\iota_{W}(w)\,\sigma_{s}\,\iota_{W}(w)^{-1})=w\hat{\delta}_{\alpha_{s}}w^{-1}=$
$\displaystyle
w\cdot\hat{\delta}_{\alpha_{s}}=\hat{\delta}_{w(\alpha_{s})}=\hat{\delta}_{\beta}\,.$

Since $S\cup\\{\hat{\delta}_{\beta}\mid\beta\in\Phi[\Gamma]\\}$ generates $G$,
it follows that $\psi\circ\tilde{\varphi}={\rm id}$. ∎

From now on we identify ${\rm KVA}[\Gamma]$ with $A[\hat{\Gamma}]$. In
particular, we assume that $\\{\delta_{\beta}\mid\beta\in\Phi[\Gamma]\\}$ is
the standard generating set of $A[\hat{\Gamma}]$.

######  Corollary 2.4.

Let $\Gamma$ be a Coxeter graph. Then $\iota_{A}:A[\Gamma]\to{\rm VA}[\Gamma]$
is injective.

###### Proof.

Let $\Gamma$ be a Coxeter graph and let $M=(m_{s,t})_{s,t\in S}$ be its
Coxeter matrix. Recall that, for $X\subset S$, we set $M_{X}=(m_{s,t})_{s,t\in
X}$ and we denote by $\Gamma_{X}$ the Coxeter graph of $M_{X}$. Recall also
that, by Van der Lek [27], the homomorphism $\iota_{X}:A[\Gamma_{X}]\to
A[\Gamma]$ induced by the inclusion is injective.

Let $\Pi=\\{\alpha_{s}\mid s\in S\\}$ be the set of simple roots. Observe that
the map $S\to\Pi$, $s\mapsto\alpha_{s}$, induces an isomorphism from $\Gamma$
to $\hat{\Gamma}_{\Pi}$, and therefore an isomorphism from $A[\Gamma]$ to
$A[\hat{\Gamma}_{\Pi}]$. By composing this isomorphism with the embedding
$\iota_{\Pi}:A[\hat{\Gamma}_{\Pi}]\hookrightarrow A[\hat{\Gamma}]$, we get an
injective homomorphism from $A[\Gamma]$ into $A[\hat{\Gamma}]={\rm
KVA}[\Gamma]$ which sends $s$ to $\delta_{\alpha_{s}}$ for all $s\in S$. Since
$\delta_{\alpha_{s}}=\sigma_{s}$ for all $s$, this homomorphism is nothing
else than $\iota_{A}$. ∎

We turn now to determine a presentation for ${\rm PVA}[\Gamma]$. Take an
abstract set $\\{\hat{\zeta}_{\beta}\mid\beta\in\Phi[\Gamma]\\}$ in one-to-one
correspondence with $\Phi[\Gamma]$. Let $\beta,\gamma\in\Phi[\Gamma]$ such
that $\hat{m}_{\beta,\gamma}\neq\infty$. Let $m=\hat{m}_{\beta,\gamma}$. We
define roots $\beta_{1},\dots,\beta_{m}\in\Phi[\Gamma]$ by $\beta_{1}=\beta$
and, for $k\geq 2$,

$\beta_{k}=\left\\{\begin{array}[]{ll}{\rm
Prod}_{R}(r_{\gamma},r_{\beta},k-1)(\gamma)&\text{if }k\text{ is even}\,,\\\
{\rm Prod}_{R}(r_{\beta},r_{\gamma},k-1)(\beta)&\text{if }k\text{ is
odd}\,.\end{array}\right.$

We set

$Z(\gamma,\beta,\hat{m}_{\beta,\gamma})=\hat{\zeta}_{\beta_{m}}\cdots\hat{\zeta}_{\beta_{2}}\hat{\zeta}_{\beta_{1}}$

that we consider as a word over
$\\{\hat{\zeta}_{\beta}\mid\beta\in\Phi[\Gamma]\\}$. We denote by
$\widehat{{\rm PVA}}[\Gamma]$ the group defined by the presentation with
generating set $\\{\hat{\zeta}_{\beta}\mid\beta\in\Phi[\Gamma]\\}$ and
relations:

$Z(\gamma,\beta,\hat{m}_{\beta,\gamma})=Z(\beta,\gamma,\hat{m}_{\beta,\gamma})\text{
for }\beta,\gamma\in\Phi[\Gamma],\ \beta\neq\gamma,\
\hat{m}_{\beta,\gamma}\neq\infty.$

######  Example 1.

Suppose that $\Gamma$ is the graph $A_{n-1}$ of Figure 2.1. Recall that
$W[\Gamma]$ is the symmetric group $S_{n}$, $A[\Gamma]$ is the braid group
$B_{n}$, and ${\rm VA}[\Gamma]$ is the virtual braid group ${\rm VB}_{n}$.
Recall also that $\Pi=\\{\alpha_{1,2},\dots,\alpha_{n-1,n}\\}$ is the set of
simple roots and $\Phi[\Gamma]=\\{\alpha_{i,j}\mid 1\leq i\neq j\leq n\\}$ is
the root system of $W[\Gamma]=S_{n}$. Then $\widehat{{\rm PVA}}[A_{n-1}]$ is
the group defined by the presentation with generating set
$\\{\hat{\zeta}_{\alpha_{i,j}}\mid 1\leq i\neq j\leq n\\}$ and relations:

  * •

$\hat{\zeta}_{\alpha_{i,j}}\hat{\zeta}_{\alpha_{k,\ell}}=\hat{\zeta}_{\alpha_{k,\ell}}\hat{\zeta}_{\alpha_{i,j}}$
for $i,j,k,\ell$ pairwise distinct,

  * •

$\hat{\zeta}_{\alpha_{i,j}}\hat{\zeta}_{\alpha_{i,k}}\hat{\zeta}_{\alpha_{j,k}}=\hat{\zeta}_{\alpha_{j,k}}\hat{\zeta}_{\alpha_{i,k}}\hat{\zeta}_{\alpha_{i,j}}$
for $i,j,k$ pairwise distinct.

By Bardakov [1], this group is isomorphic to ${\rm PVB}_{n}$.

######  Example 2.

Suppose $\Gamma$ is simply laced. Let $\beta,\gamma\in\Phi[\Gamma]$. If
$\hat{m}_{\beta,\gamma}=2$, then $\langle\beta,\gamma\rangle=0$ and
$r_{\beta}(\gamma)=\gamma$, hence
$Z(\gamma,\beta,\hat{m}_{\beta,\gamma})=\hat{\zeta}_{\gamma}\hat{\zeta}_{\beta}$.
If $\hat{m}_{\beta,\gamma}=3$, then $\langle\beta,\gamma\rangle=-1$,
$r_{\beta}(\gamma)=\beta+\gamma$ and $(r_{\beta}r_{\gamma})(\beta)=\gamma$,
hence
$Z(\gamma,\beta,\hat{m}_{\beta,\gamma})=\hat{\zeta}_{\gamma}\hat{\zeta}_{\beta+\gamma}\hat{\zeta}_{\beta}$.
So, $\widehat{{\rm PVA}}[\Gamma]$ is the group defined by the presentation
with generating set $\\{\hat{\zeta}_{\beta}\mid\beta\in\Phi[\Gamma]\\}$ and
relations:

  * •

$\hat{\zeta}_{\gamma}\hat{\zeta}_{\beta}=\hat{\zeta}_{\beta}\hat{\zeta}_{\gamma}$
for $\beta,\gamma\in\Phi[\Gamma]$, $\beta\neq\gamma$ and
$\langle\beta,\gamma\rangle=0$,

  * •

$\hat{\zeta}_{\gamma}\hat{\zeta}_{\beta+\gamma}\hat{\zeta}_{\beta}=\hat{\zeta}_{\beta}\hat{\zeta}_{\beta+\gamma}\hat{\zeta}_{\gamma}$
for $\beta,\gamma\in\Phi[\Gamma]$, $\beta\neq\gamma$ and
$\langle\beta,\gamma\rangle=-1$.

The following lemma will not be used in the paper, but it is useful for
understanding the presentation of ${\rm PVA}[\Gamma]$ given in Theorem 2.6.

######  Lemma 2.5.

Let $\Gamma$ be a Coxeter graph. Let $\beta,\gamma\in\Phi[\Gamma]$ such that
$\hat{m}_{\beta,\gamma}\neq\infty$. Let $m=\hat{m}_{\beta,\gamma}$ and let
$\beta_{1},\dots,\beta_{m}\in\Phi[\Gamma]$ be the roots such that
$Z(\gamma,\beta,\hat{m}_{\beta,\gamma})=\hat{\zeta}_{\beta_{m}}\cdots\hat{\zeta}_{\beta_{2}}\hat{\zeta}_{\beta_{1}}$.
Then $Z(\beta,\gamma,\hat{m}_{\beta,\gamma})$ is the reverse word of
$Z(\gamma,\beta,\hat{m}_{\beta,\gamma})$, that is,
$Z(\beta,\gamma,\hat{m}_{\beta,\gamma})=\hat{\zeta}_{\beta_{1}}\hat{\zeta}_{\beta_{2}}\cdots\hat{\zeta}_{\beta_{m}}$.

###### Proof.

We set $\beta_{1}=\beta$, $\gamma_{1}=\gamma$ and, for $k\geq 2$,

$\displaystyle\beta_{k}=\left\\{\begin{array}[]{ll}{\rm
Prod}_{R}(r_{\gamma},r_{\beta},k-1)(\gamma)&\text{if }k\text{ is even}\,,\\\
{\rm Prod}_{R}(r_{\beta},r_{\gamma},k-1)(\beta)&\text{if }k\text{ is
odd}\,,\end{array}\right.$
$\displaystyle\gamma_{k}=\left\\{\begin{array}[]{ll}{\rm
Prod}_{R}(r_{\beta},r_{\gamma},k-1)(\beta)&\text{if }k\text{ is even}\,,\\\
{\rm Prod}_{R}(r_{\gamma},r_{\beta},k-1)(\gamma)&\text{if }k\text{ is
odd}\,.\end{array}\right.$

We have:

$Z(\gamma,\beta,\hat{m}_{\beta,\gamma})=\hat{\zeta}_{\beta_{m}}\cdots\hat{\zeta}_{\beta_{2}}\hat{\zeta}_{\beta_{1}}\text{
and
}Z(\beta,\gamma,\hat{m}_{\beta,\gamma})=\hat{\zeta}_{\gamma_{m}}\cdots\hat{\zeta}_{\gamma_{2}}\hat{\zeta}_{\gamma_{1}}\,.$

We need to show that $\beta_{k}=\gamma_{m+1-k}$ for all $k\in\\{1,\dots,m\\}$.
We assume that $m$ is even. The case where $m$ is odd can be proved in the
same way. Recall that, by Deodhar [16], we have:

$\gamma={\rm Prod}_{R}(r_{\gamma},r_{\beta},m-1)(\gamma)\text{ and }\beta={\rm
Prod}_{R}(r_{\beta},r_{\gamma},m-1)(\beta)\,.$

If $k$ is even, then $m+1-k$ is odd, hence:

$\displaystyle\gamma_{m+1-k}={\rm
Prod}_{R}(r_{\gamma},r_{\beta},m-k)(\gamma)={\rm
Prod}_{R}(r_{\gamma},r_{\beta},m-k)\,{\rm
Prod}_{R}(r_{\gamma},r_{\beta},m-1)(\gamma)=$ $\displaystyle{\rm
Prod}_{R}(r_{\gamma},r_{\beta},m-k)\,{\rm
Prod}_{L}(r_{\beta},r_{\gamma},m-1)(\gamma)={\rm
Prod}_{L}(r_{\beta},r_{\gamma},k-1)(\gamma)=$ $\displaystyle{\rm
Prod}_{R}(r_{\gamma},r_{\beta},k-1)(\gamma)=\beta_{k}\,.$

If $k$ is odd, then $m+1-k$ is even, hence:

$\displaystyle\gamma_{m+1-k}={\rm
Prod}_{R}(r_{\beta},r_{\gamma},m-k)(\beta)={\rm
Prod}_{R}(r_{\beta},r_{\gamma},m-k)\,{\rm
Prod}_{R}(r_{\beta},r_{\gamma},m-1)(\beta)=$ $\displaystyle{\rm
Prod}_{R}(r_{\beta},r_{\gamma},m-k)\,{\rm
Prod}_{L}(r_{\gamma},r_{\beta},m-1)(\beta)={\rm
Prod}_{L}(r_{\beta},r_{\gamma},k-1)(\beta)=$ $\displaystyle{\rm
Prod}_{R}(r_{\beta},r_{\gamma},k-1)(\beta)=\beta_{k}\,.\qquad\proved$

∎

######  Theorem 2.6.

Let $\Gamma$ be a Coxeter graph. Then the map
$\\{\hat{\zeta}_{\beta}\mid\beta\in\Phi[\Gamma]\\}\to\\{\zeta_{\beta}\mid\beta\in\Phi[\Gamma]\\}$,
$\hat{\zeta}_{\beta}\mapsto\zeta_{\beta}$, induces an isomorphism
$\varphi:\widehat{{\rm PVA}}[\Gamma]\to{\rm PVA}[\Gamma]$.

###### Proof.

The proof is similar to that of Theorem 2.3. One can easily show that
$\\{\zeta_{\beta}\mid\beta\in\Phi[\Gamma]\\}$ generates ${\rm PVA}[\Gamma]$ in
the same way as we shown that $\\{\delta_{\beta}\mid\beta\in\Phi[\Gamma]\\}$
generates ${\rm KVA}[\Gamma]$ in the proof of Theorem 2.3.

Now, we show that we have a homomorphism $\varphi:\widehat{{\rm
PVA}}[\Gamma]\to{\rm PVA}[\Gamma]$ which sends $\hat{\zeta}_{\beta}$ to
$\zeta_{\beta}$ for all $\beta\in\Phi[\Gamma]$. Let
$\beta,\gamma\in\Phi[\Gamma]$ such that $\hat{m}_{\beta,\gamma}\neq\infty$.
There exist $w\in W$ and $s,t\in S$ such that $\beta=w(\alpha_{s})$,
$\gamma=w(\alpha_{t})$, and $m_{s,t}=\hat{m}_{\beta,\gamma}$. Recall the
definition of $Z(\gamma,\beta,\hat{m}_{\beta,\gamma})$. Let
$m=\hat{m}_{\beta,\gamma}=m_{s,t}$ and let
$\beta_{1},\dots,\beta_{m}\in\Phi[\Gamma]$ be the roots defined by
$\beta_{1}=\beta$ and, for $k\geq 2$,

$\beta_{k}=\left\\{\begin{array}[]{ll}{\rm
Prod}_{R}(r_{\gamma},r_{\beta},k-1)(\gamma)&\text{if }k\text{ is even}\,,\\\
{\rm Prod}_{R}(r_{\beta},r_{\gamma},k-1)(\beta)&\text{if }k\text{ is
odd}\,.\end{array}\right.$

Then
$Z(\gamma,\beta,\hat{m}_{\beta,\gamma})=\hat{\zeta}_{\beta_{m}}\cdots\hat{\zeta}_{\beta_{2}}\hat{\zeta}_{\beta_{1}}$.
Let $k\in\\{1,\dots,m\\}$. Suppose that $k$ is odd. Then:

$\beta_{k}={\rm Prod}_{R}(r_{\beta},r_{\gamma},k-1)(\beta)=(w\,{\rm
Prod}_{R}(s,t,k-1)\,w^{-1}w)(\alpha_{s})=(w\,{\rm
Prod}_{R}(s,t,k-1))(\alpha_{s})\,,$

hence:

$\displaystyle\zeta_{\beta_{k}}=w\,{\rm
Prod}_{R}(s,t,k-1)\cdot\zeta_{\alpha_{s}}=$ $\displaystyle\iota_{W}(w)\,{\rm
Prod}_{R}(\tau_{s},\tau_{t},k-1)\,(\tau_{s}\sigma_{s})\,{\rm
Prod}_{R}(\tau_{s},\tau_{t},k-1)^{-1}\iota_{W}(w)^{-1}=$
$\displaystyle\iota_{W}(w)\,{\rm
Prod}_{R}(\tau_{t},\tau_{s},k)\,\sigma_{s}\,{\rm
Prod}_{R}(\tau_{s},\tau_{t},k-1)^{-1}\iota_{W}(w)^{-1}\,.$

We show in the same way that, if $k$ is even, then:

$\zeta_{\beta_{k}}=\iota_{W}(w)\,{\rm
Prod}_{R}(\tau_{s},\tau_{t},k)\,\sigma_{t}\,{\rm
Prod}_{R}(\tau_{t},\tau_{s},k-1)^{-1}\iota_{W}(w)^{-1}\,.$

Thus:

$\zeta_{\beta_{m}}\cdots\zeta_{\beta_{2}}\zeta_{\beta_{1}}=\iota_{W}(w)\,{\rm
Prod}_{L}(\tau_{s},\tau_{t},m)\,{\rm
Prod}_{R}(\sigma_{t},\sigma_{s},m)\,\iota_{W}(w)^{-1}\,.$

Similarly, if $\gamma_{1},\dots,\gamma_{m}$ are the roots such that
$Z(\beta,\gamma,\hat{m}_{\beta,\gamma})=\hat{\zeta}_{\gamma_{m}}\cdots\hat{\zeta}_{\gamma_{2}}\hat{\zeta}_{\gamma_{1}}$,
then:

$\zeta_{\gamma_{m}}\cdots\zeta_{\gamma_{2}}\zeta_{\gamma_{1}}=\iota_{W}(w)\,{\rm
Prod}_{L}(\tau_{t},\tau_{s},m)\,{\rm
Prod}_{R}(\sigma_{s},\sigma_{t},m)\,\iota_{W}(w)^{-1}\,.$

So,
$\zeta_{\beta_{m}}\cdots\zeta_{\beta_{2}}\zeta_{\beta_{1}}=\zeta_{\gamma_{m}}\cdots\zeta_{\gamma_{2}}\zeta_{\gamma_{1}}$.

The action of $W[\Gamma]$ on
$\\{\hat{\zeta}_{\beta}\mid\beta\in\Phi[\Gamma]\\}$ defined by
$w\cdot\hat{\zeta}_{\beta}=\hat{\zeta}_{w(\beta)}$ extends to an action of
$W[\Gamma]$ on $\widehat{{\rm PVA}}[\Gamma]$. Thus, we can consider the semi-
direct product $G=\widehat{{\rm PVA}}[\Gamma]\rtimes W[\Gamma]$ determined by
this action. Now, we show that the homomorphisms $\varphi:\widehat{{\rm
PVA}}[\Gamma]\to{\rm PVA}[\Gamma]\subset{\rm VA}[\Gamma]$ and
$\iota_{W}:W[\Gamma]\to{\rm VA}[\Gamma]$ induce a homomorphism
$\tilde{\varphi}:G\to{\rm VA}[\Gamma]$. To do this, we just need to check that
$\varphi(w\cdot\hat{\zeta}_{\beta})=\iota_{W}(w)\,\varphi(\hat{\zeta}_{\beta})\,\iota_{W}(w)^{-1}$
for all $w\in W[\Gamma]$ and all $\beta\in\Phi[\Gamma]$, and this equality is
easily verified since both terms are by definition equal to
$\zeta_{w(\beta)}$.

Let $\psi:\SS\sqcup\mathcal{T}\to G$ be the map defined by
$\psi(\sigma_{s})=s\hat{\zeta}_{\alpha_{s}}$ and $\psi(\tau_{s})=s$ for all
$s\in S$. Now, we show that this map induces a homomorphism $\psi:{\rm
VA}[\Gamma]\to G$. Let $s,t\in S$, $s\neq t$, such that $m_{s,t}\neq\infty$.
Then:

${\rm Prod}_{R}(\psi(\tau_{t}),\psi(\tau_{s}),m_{s,t})={\rm
Prod}_{R}(t,s,m_{s,t})={\rm Prod}_{R}(s,t,m_{s,t})={\rm
Prod}_{R}(\psi(\tau_{s}),\psi(\tau_{t}),m_{s,t})\,.$

Let $s\in S$. Then:

$\psi(\tau_{s})^{2}=s^{2}=1\,.$

Let $s,t\in S$, $s\neq t$, such that $m_{s,t}\neq\infty$. Let $m=m_{s,t}$ and
let $\beta_{1},\dots,\beta_{m}\in\Phi[\Gamma]$ be the roots defined by
$\beta_{1}=\alpha_{s}$ and, for $k\geq 2$,

$\beta_{k}=\left\\{\begin{array}[]{ll}{\rm
Prod}_{R}(t,s,k-1)(\alpha_{t})&\text{if }k\text{ is even}\,,\\\ {\rm
Prod}_{R}(s,t,k-1)(\alpha_{s})&\text{if }k\text{ is odd}\,.\end{array}\right.$

Then:

$\displaystyle{\rm Prod}_{R}(\psi(\sigma_{t}),\psi(\sigma_{s}),m_{s,t})=\cdots
s\hat{\zeta}_{\alpha_{s}}t\hat{\zeta}_{\alpha_{t}}s\hat{\zeta}_{\alpha_{s}}={\rm
Prod}_{R}(t,s,m_{s,t})\,\hat{\zeta}_{\beta_{m}}\cdots\hat{\zeta}_{\beta_{2}}\hat{\zeta}_{\beta_{1}}=$
$\displaystyle{\rm
Prod}_{R}(t,s,m_{s,t})\,Z(\alpha_{t},\alpha_{s},\hat{m}_{\alpha_{s},\alpha_{t}})\,.$

Similarly,

${\rm Prod}_{R}(\psi(\sigma_{s}),\psi(\sigma_{t}),m_{s,t})={\rm
Prod}_{R}(s,t,m_{s,t})\,Z(\alpha_{s},\alpha_{t},\hat{m}_{\alpha_{s},\alpha_{t}})\,.$

Hence:

${\rm Prod}_{R}(\psi(\sigma_{t}),\psi(\sigma_{s}),m_{s,t})={\rm
Prod}_{R}(\psi(\sigma_{s}),\psi(\sigma_{t}),m_{s,t})\,.$

Let $s,t\in S$, $s\neq t$, such that $m_{s,t}\neq\infty$. Set $r=s$ if
$m_{s,t}$ is even and $r=t$ if $m_{s,t}$ is odd. Then the following equality
can be proved in the same way as the corresponding equality in the proof of
Theorem 2.3:

${\rm
Prod}_{R}(\psi(\tau_{s}),\psi(\tau_{t}),m_{s,t}-1)\,\psi(\sigma_{s})=\psi(\sigma_{r})\,{\rm
Prod}_{R}(\psi(\tau_{s}),\psi(\tau_{t}),m_{s,t}-1)\,.$

Finally, we prove that $\psi\circ\tilde{\varphi}={\rm id}$ in the same way as
in the proof of Theorem 2.3, which implies the injectivity of
$\tilde{\varphi}$, hence of $\varphi$, and concludes the proof of Theorem 2.6.
∎

##  3 Center of ${\rm VA}[\Gamma]$

We start this section by recalling some results on centers of Coxeter groups.
If $\Gamma$ is a Coxeter graph and $M=(m_{s,t})_{s,t\in S}$ is its Coxeter
matrix, then we denote by $\lg:W[\Gamma]\to\mathbb{N}$ the word length in
$W[\Gamma]$ with respect to $S$. The following can be found in Bourbaki [8,
Exercice 22, Page 22].

######  Proposition 3.1 (Bourbaki [8]).

Let $\Gamma$ be a Coxeter graph and let $M=(m_{s,t})_{s,t\in S}$ be its
Coxeter matrix. The following conditions on an element $w_{0}\in W[\Gamma]$
are equivalent:

  * (a)

For all $u\in W[\Gamma]$, $\lg(w_{0})=\lg(u)+\lg(u^{-1}w_{0})$.

  * (b)

For all $s\in S$, $\lg(w_{0})>\lg(sw_{0})$.

Moreover, $w_{0}$ exists if and only if $W[\Gamma]$ is finite. If $w_{0}$
satisfies (a) and/or (b), then $w_{0}$ is unique, $w_{0}$ is an involution,
and $w_{0}Sw_{0}=S$.

The element $w_{0}$ of Proposition 3.1, if it exists, is called _the longest
element_ of $W[\Gamma]$. The center of a group $G$ will be denoted by $Z(G)$.
The following can be found in Bourbaki [8, Exercice 3, Page 130].

######  Proposition 3.2 (Bourbaki [8]).

Let $\Gamma$ be a connected Coxeter graph.

  * (1)

If $W[\Gamma]$ is infinite, then $Z(W[\Gamma])=\\{1\\}$.

  * (2)

Suppose that $W[\Gamma]$ is finite. Let $w_{0}$ be the longest element of
$W[\Gamma]$. Then $Z(W[\Gamma])=\\{1,w_{0}\\}$ if $w_{0}$ is central, and
$Z(W[\Gamma])=\\{1\\}$ otherwise.

Now, we return to the study of virtual Artin groups.

######  Theorem 3.3.

Let $\Gamma$ be a Coxeter graph. Assume that $W[\Gamma]$ is a subgroup of
${\rm VA}[\Gamma]$ via the embedding $\iota_{W}:W[\Gamma]\hookrightarrow{\rm
VA}[\Gamma]$. Then the centralizer of $W[\Gamma]$ in ${\rm VA}[\Gamma]$ is
$Z(W[\Gamma])$.

###### Proof.

If $\mathcal{X}$ is a subset of $\Phi[\Gamma]$, then we denote by
$\hat{\Gamma}_{\mathcal{X}}$ the full subgraph of $\hat{\Gamma}$ spanned by
$\mathcal{X}$. By Van der Lek [27, Lemma 4.11] we can identify
$A[\hat{\Gamma}_{\mathcal{X}}]$ with the subgroup of $A[\hat{\Gamma}]={\rm
KVA}[\Gamma]$ generated by $\\{\delta_{\beta}\mid\beta\in\mathcal{X}\\}$.
Moreover, again by Van der Lek [27, Lemma 4.12], if $\\{\mathcal{X}_{i}\mid
i\in I\\}$ is a family of subsets of $\Phi[\Gamma]$, then:

(3.1) $\bigcap_{i\in
I}A[\hat{\Gamma}_{\mathcal{X}_{i}}]=A[\hat{\Gamma}_{\cap_{i\in
I}\mathcal{X}_{i}}]\,.$

It is clear that $Z(W[\Gamma])$ is contained in the centralizer of $W[\Gamma]$
in ${\rm VA}[\Gamma]$. So, we just have to prove that the centralizer of
$W[\Gamma]$ in ${\rm VA}[\Gamma]$ is contained in $Z(W[\Gamma])$. Let $g$ be
an element of the centralizer of $W[\Gamma]$ in ${\rm VA}[\Gamma]$. We write
$g$ in the form $g=hw$ with $h\in{\rm KVA}[\Gamma]$ and $w\in W[\Gamma]$. The
element $\pi_{K}(g)=w$ must lie in the center of $W[\Gamma]$, hence $h$ also
lies in the centralizer of $W[\Gamma]$ in ${\rm VA}[\Gamma]$. Let
$\beta\in\Phi^{+}[\Gamma]$. We set
$\mathcal{X}^{+}_{\beta}=\Phi[\Gamma]\setminus\\{-\beta\\}$,
$\mathcal{X}^{-}_{\beta}=\Phi[\Gamma]\setminus\\{\beta\\}$ and
$\mathcal{Y}_{\beta}=\Phi[\Gamma]\setminus\\{\beta,-\beta\\}$. Since
$\hat{m}_{\beta,-\beta}=\infty$,
$\mathcal{X}^{+}_{\beta}\cup\mathcal{X}^{-}_{\beta}=\Phi[\Gamma]$ and
$\mathcal{X}^{+}_{\beta}\cap\mathcal{X}^{-}_{\beta}=\mathcal{Y}_{\beta}$, by
Bellingeri–Paris [6, Lemma 3.3],

${\rm
KVA}[\Gamma]=A[\hat{\Gamma}]=A[\hat{\Gamma}_{\mathcal{X}^{+}_{\beta}}]*_{A[\hat{\Gamma}_{\mathcal{Y}_{\beta}}]}A[\hat{\Gamma}_{\mathcal{X}^{-}_{\beta}}]\,.$

Since $r_{\beta}(\beta)=-\beta$ and $r_{\beta}(-\beta)=\beta$, we have
$r_{\beta}(\mathcal{X}^{+}_{\beta})=\mathcal{X}^{-}_{\beta}$,
$r_{\beta}(\mathcal{X}^{-}_{\beta})=\mathcal{X}^{+}_{\beta}$ and
$r_{\beta}(\mathcal{Y}_{\beta})=\mathcal{Y}_{\beta}$, hence $r_{\beta}\cdot
A[\hat{\Gamma}_{\mathcal{X}^{+}_{\beta}}]=A[\hat{\Gamma}_{\mathcal{X}^{-}_{\beta}}]$,
$r_{\beta}\cdot
A[\hat{\Gamma}_{\mathcal{X}^{-}_{\beta}}]=A[\hat{\Gamma}_{\mathcal{X}^{+}_{\beta}}]$
and $r_{\beta}\cdot
A[\hat{\Gamma}_{\mathcal{Y}_{\beta}}]=A[\hat{\Gamma}_{\mathcal{Y}_{\beta}}]$.
Since $r_{\beta}\in W[\Gamma]$ and $h$ lies in the centralizer of $W[\Gamma]$
in ${\rm VA}[\Gamma]$, $r_{\beta}$ and $h$ commute, which means in terms of
actions that $r_{\beta}\cdot h=h$. By Bellingeri–Paris [6, Lemma 3.6], it
follows that $h\in A[\hat{\Gamma}_{\mathcal{Y}_{\beta}}]$. We have
$\cap_{\beta\in\Phi^{+}[\Gamma]}\mathcal{Y}_{\beta}=\emptyset$, hence, by
Equation (3.1),

$h\in\bigcap_{\beta\in\Phi^{+}[\Gamma]}A[\hat{\Gamma}_{\mathcal{Y}_{\beta}}]=A[\hat{\Gamma}_{\emptyset}]=\\{1\\}\,.$

Thus, $h=1$ and $g=w\in Z(W[\Gamma])$. ∎

######  Corollary 3.4.

Let $\Gamma$ be a Coxeter graph. Then $Z({\rm VA}[\Gamma])=\\{1\\}$.

###### Proof.

Let $\Gamma_{1},\dots,\Gamma_{\ell}$ be the connected components of $\Gamma$.
We have ${\rm VA}[\Gamma]={\rm VA}[\Gamma_{1}]\times\cdots\times{\rm
VA}[\Gamma_{\ell}]$, hence:

$Z({\rm VA}[\Gamma])=Z({\rm VA}[\Gamma_{1}])\times\cdots\times Z({\rm
VA}[\Gamma_{\ell}])\,.$

So, we can assume that $\Gamma$ is connected. Note that the center of ${\rm
VA}[\Gamma]$ is contained in the centralizer of $W[\Gamma]$ in ${\rm
VA}[\Gamma]$. If either $W[\Gamma]$ is infinite or $W[\Gamma]$ is finite and
the longest element $w_{0}$ is not central, then $Z({\rm VA}[\Gamma])=\\{1\\}$
by an easy combination of Proposition 3.2 and Theorem 3.3. Suppose $W[\Gamma]$
is finite and $w_{0}$ is central. Then, by Proposition 3.2 and Theorem 3.3,
$Z({\rm VA}[\Gamma])\subset\\{1,w_{0}\\}$. Let $s\in S$. By Deodhar [16] we
have $w_{0}(\alpha_{s})=-\alpha_{s}$, hence
$w_{0}\cdot\delta_{\alpha_{s}}=\delta_{-\alpha_{s}}\neq\delta_{\alpha_{s}}$,
thus $w_{0}$ and $\delta_{\alpha_{s}}=\sigma_{s}$ do not commute. So,
$w_{0}\not\in Z({\rm VA}[\Gamma])$, hence $Z({\rm VA}[\Gamma])=\\{1\\}$. ∎

######  Remark.

Determining the center of $A[\Gamma]$ for any Coxeter graph is an open
problem.

######  Remark.

The fact that $Z({\rm VB}_{n})=\\{1\\}$ has been proved in [18] using the
construction by Bartholdi–Enriquez–Etingof–Rains [3] of a classifying space
for ${\rm PVB}_{n}$.

We can use the same techniques to prove the following which is also a direct
consequence of Charney–Morris-Wright [13].

######  Proposition 3.5.

Let $\Gamma$ be a Coxeter graph. Then $Z({\rm KVA}[\Gamma])=\\{1\\}$.

###### Proof.

Let $\beta\in\Phi^{+}[\Gamma]$. We set
$\mathcal{X}^{+}_{\beta}=\Phi[\Gamma]\setminus\\{-\beta\\}$,
$\mathcal{X}^{-}_{\beta}=\Phi[\Gamma]\setminus\\{\beta\\}$ and
$\mathcal{Y}_{\beta}=\Phi[\Gamma]\setminus\\{\beta,-\beta\\}$. As in the proof
of Theorem 3.3, by Bellingeri–Paris [6, Lemma 3.3], from the equalities
$\hat{m}_{\beta,-\beta}=\infty$,
$\mathcal{X}^{+}_{\beta}\cup\mathcal{X}^{-}_{\beta}=\Phi[\Gamma]$ and
$\mathcal{X}^{+}_{\beta}\cap\mathcal{X}^{-}_{\beta}=\mathcal{Y}_{\beta}$ it
follows that:

${\rm
KVA}[\Gamma]=A[\hat{\Gamma}]=A[\hat{\Gamma}_{\mathcal{X}^{+}_{\beta}}]*_{A[\hat{\Gamma}_{\mathcal{Y}_{\beta}}]}A[\hat{\Gamma}_{\mathcal{X}^{-}_{\beta}}]\,.$

It is well-known that the center of an amalgamated product is a subgroup of
the amalgam, hence $Z({\rm KVA}[\Gamma])\subset
Z(A[\hat{\Gamma}_{\mathcal{Y}_{\beta}}])$. Since we know from the proof of
Theorem 3.3 that
$\cap_{\beta\in\Phi^{+}[\Gamma]}A[\hat{\Gamma}_{\mathcal{Y}_{\beta}}]=\\{1\\}$,
we conclude that $Z({\rm KVA}[\Gamma])=\\{1\\}$. ∎

##  4 Virtual Artin groups of spherical type and of affine type

Let $\Gamma$ be a finite Coxeter graph. We say that $\Gamma$ (or $A[\Gamma]$
or $W[\Gamma]$ or ${\rm VA}[\Gamma]$) is of _spherical type_ if $W[\Gamma]$ is
finite. If $\Gamma_{1},\dots,\Gamma_{p}$ are the connected components of
$\Gamma$, then $W[\Gamma]=W[\Gamma_{1}]\times\cdots\times W[\Gamma_{p}]$,
hence $\Gamma$ is of spherical type if and only if all its connected
components are of spherical type. The connected Coxeter graphs of spherical
type have been classified by Coxeter himself [15]. These are the Coxeter
graphs $A_{n}$ ($n\geq 1$), $B_{n}$ ($n\geq 2$), $D_{n}$ ($n\geq 4$), $E_{n}$
($n\in\\{6,7,8\\}$), $F_{4}$, $H_{3}$, $H_{4}$ and $I_{2}(m)$ $(m\geq 5$)
shown in Figure 4.1.

$A_{n}$:  $B_{n}$:

$D_{n}$:  $E_{6}$:

$E_{7}$:  $E_{8}$:

$F_{4}$:  $H_{3}$:  $H_{4}$:

$I_{2}(m)$:

Figure 4.1: Connected Coxeter graphs of spherical type

Let $\Gamma$ be a finite Coxeter graph and let $M=(m_{s,t})_{s,t\in S}$ be its
Coxeter matrix. Let $\Pi=\\{\alpha_{s}\mid s\in S\\}$ be the set of simple
roots, let $V$ be the real vector space having $\Pi$ as a basis and let
$\langle\cdot,\cdot\rangle:V\times V\to\mathbb{R}$ be the canonical bilinear
form of $W[\Gamma]$. We know by Coxeter [14, 15] that $W[\Gamma]$ is finite if
and only if $\langle\cdot,\cdot\rangle$ is positive definite. We say that
$\Gamma$ (or $A[\Gamma]$ or $W[\Gamma]$ or ${\rm VA}[\Gamma]$) is of _affine
type_ if $\langle\cdot,\cdot\rangle$ is positive but not positive definite. It
is easily seen that $\Gamma$ is of affine type if and only if all its
connected components are of affine type or of spherical type and at least one
of the connected components is of affine type. On the other hand a
classification of the connected Coxeter graphs of affine type can be found in
Bourbaki [8]. These are the Coxeter graphs $\tilde{A}_{n}$ ($n\geq 1$),
$\tilde{B}_{n}$ ($n\geq 3$), $\tilde{C}_{n}$ ($n\geq 2$), $\tilde{D}_{n}$
($n\geq 4$), $\tilde{E}_{n}$ ($n\in\\{6,7,8\\}$), $\tilde{F}_{4}$ and
$\tilde{G}_{2}$ shown in Figure 4.2.

$\tilde{A}_{1}$:  $\tilde{A}_{n}$:  $\tilde{B}_{n}$:

$\tilde{C}_{n}$:  $\tilde{D}_{n}$:

$\tilde{E}_{6}$:  $\tilde{E}_{7}$:

$\tilde{E}_{8}$:  $\tilde{F}_{4}$:

$\tilde{G}_{2}$:

Figure 4.2: Connected Coxeter graphs of affine type

Artin groups of spherical type are well understood since Artin groups
themselves were introduced by Brieskorn [9], Deligne [17] and Brieskorn–Saito
[10] in the early 1970s. One would have expected that the next class of Artin
groups to be well understood would be the class of Artin groups of affine
type, but this was not the case. It was not until the last decade, with the
development of the dual structures and the work of Digne [19, 20], McCammond
[28], McCammond–Sulway [29] and Paolini–Salvetti [30], that elementary
problems on these groups such as determining the center, determining the
torsion part, or solving the word problem, have been solved, as well as the
famous $K(\pi,1)$ conjecture (see Section 6). In contrast, virtual Artin
groups of affine type are treated more or less in the same way as those of
spherical type. Their most important common point is the following Theorem 4.1
which allows us to understand the free of infinity subsets of $\Phi[\Gamma]$,
viewed as the set of vertices of $\hat{\Gamma}$.

Let $\Gamma$ be a Coxeter graph and let $M=(m_{s,t})_{s,t\in S}$ be its
Coxeter matrix. We denote by $\mathcal{P}_{\rm sph}(S)$ the set of subsets
$X\subset S$ such that $\Gamma_{X}$ is of spherical type. Then the _spherical
dimension_ of $A[\Gamma]$ is defined to be $n_{\rm
sph}(A[\Gamma])=\max\\{|X|\mid X\in\mathcal{P}_{\rm sph}(S)\\}$. We know by
Charney–Davis [12] that, if $A[\Gamma]$ satisfies the $K(\pi,1)$ conjecture,
then $n_{\rm sph}(A[\Gamma])$ is equal to the cohomological dimension of
$A[\Gamma]$, and it is conjectured that all Artin groups satisfy the
$K(\pi,1)$ conjecture. This conjecture will be recalled in the beginning of
Section 6. Note however that, by Deligne [17] and Paolini–Salvetti [30],
$A[\Gamma]$ satisfies the $K(\pi,1)$ conjecture if $\Gamma$ is of spherical
type or of affine type.

######  Example.

Suppose that $\Gamma$ is of spherical type or of affine type. Let
$\langle\cdot,\cdot\rangle:V\times V\to\mathbb{R}$ be the canonical bilinear
form of $W[\Gamma]$. Then $n_{\rm sph}(A[\Gamma])$ is equal to the rank of
$\langle\cdot,\cdot\rangle$. In particular, if $\Gamma$ is of spherical type,
then $n_{\rm sph}(A[\Gamma])$ is equal to the number of vertices of $\Gamma$,
and if $\Gamma$ is connected of affine type, then $n_{\rm sph}(A[\Gamma])$ is
equal to the number of vertices of $\Gamma$ minus $1$.

Let $\Gamma$ be a Coxeter graph and let $M=(m_{s,t})_{s,t\in S}$ be its
Coxeter matrix. A subset $X\subset S$ is called _free of infinity_ if
$m_{s,t}\neq\infty$ for all $s,t\in X$. We denote by
$\mathcal{P}_{<\infty}(S)$ the set of free of infinity subsets of $S$. Note
that $\mathcal{P}_{\rm sph}(S)\subset\mathcal{P}_{<\infty}(S)$, hence:

$n_{\rm sph}(A[\Gamma])=\max\\{n_{\rm sph}(A[\Gamma_{X}])\mid
X\in\mathcal{P}_{<\infty}(S)\\}\,.$

The following theorem is the main result of the present section. Recall the
general principle which says that, if we understand $A[\Gamma_{X}]$ for all
$X\in\mathcal{P}_{<\infty}(S)$, then we can understand $A[\Gamma]$ (see
Godelle–Paris [22]). Since we understand the Artin groups of spherical type
and the Artin groups of affine type, this theorem enables to apply this
principle and therefore to understand $A[\hat{\Gamma}]={\rm KVA}[\Gamma]$ when
$\Gamma$ is of spherical type or of affine type. Since ${\rm VA}[\Gamma]={\rm
KVA}[\Gamma]\rtimes W[\Gamma]$ and Coxeter groups are well-understood, we can
therefore deduce results on ${\rm VA}[\Gamma]$ itself. We will apply this
strategy in Section 5 and Section 6 to prove that ${\rm VA}[\Gamma]$ has a
solution to the word problem and to compute the virtual cohomological
dimension of ${\rm VA}[\Gamma]$ when $\Gamma$ is of spherical type or of
affine type.

######  Theorem 4.1.

Let $\Gamma$ be a Coxeter graph of spherical type or of affine type, and let
$\mathcal{X}$ be a free of infinity subset of $\Phi[\Gamma]$. Then
$\mathcal{X}$ is finite, $\hat{\Gamma}_{\mathcal{X}}$ is of spherical type or
of affine type, and $n_{\rm sph}(A[\hat{\Gamma}_{\mathcal{X}}])\leq n_{\rm
sph}(A[\Gamma])$.

###### Proof.

A symmetric matrix $A=(a_{i,j})_{1\leq i,j\leq n}$ is called _decomposable_ if
there exists a non-trivial partition of the set of indices,
$\\{1,\dots,n\\}=I\sqcup J$, such that $a_{i,j}=a_{j,i}=0$ for all $i\in I$
and $j\in J$. We say that $A$ is _indecomposable_ otherwise. The following
claim is well-known and can be found for example in Bourbaki [8, Lemme 4, Page
78].

Claim 1. Let $A=(a_{i,j})_{1\leq i,j\leq n}$ be a positive symmetric matrix
but not positive definite. Suppose that $A$ is indecomposable, that
$a_{i,i}>0$ for all $i\in\\{1,\dots,n\\}$, and that $a_{i,j}\leq 0$ for all
$i,j\in\\{1,\dots,n\\}$, $i\neq j$.

  * (1)

The rank of $A$ is $n-1$.

  * (2)

If $X$ is a proper subset of $\\{1,\dots,n\\}$, then the matrix
$A_{X}=(a_{i,j})_{i,j\in X}$ is positive definite.

Now take a Coxeter graph $\Gamma$ of spherical type or of affine type, and
denote by $M=(m_{s,t})_{s,t\in S}$ its Coxeter matrix. Let
$\hat{\Pi}=\\{\hat{E}_{\beta}\mid\beta\in\Phi[\Gamma]\\}$ be a set in one-to-
one correspondence with $\Phi[\Gamma]$, and let
$\hat{V}=\oplus_{\beta\in\Phi[\Gamma]}\mathbb{R}\,\hat{E}_{\beta}$ be a real
vector space having $\hat{\Pi}$ as a basis. Define the symmetric bilinear form
$\langle\\!\langle\cdot,\cdot\rangle\\!\rangle:\hat{V}\times\hat{V}\to\mathbb{R}$
by
$\langle\\!\langle\hat{E}_{\beta},\hat{E}_{\gamma}\rangle\\!\rangle=-2\cos(\pi/\hat{m}_{\beta,\gamma})$
if $\hat{m}_{\beta,\gamma}\neq\infty$ and
$\langle\\!\langle\hat{E}_{\beta},\hat{E}_{\gamma}\rangle\\!\rangle=-2$ if
$\hat{m}_{\beta,\gamma}=\infty$.

Claim 2. Let $\beta,\gamma\in\Phi[\Gamma]$ such that
$\hat{m}_{\beta,\gamma}\neq\infty$. Then
$\langle\\!\langle\hat{E}_{\beta},\hat{E}_{\gamma}\rangle\\!\rangle=\langle\beta,\gamma\rangle$.

Proof of Claim 2. If $\beta=\gamma$, then
$\langle\\!\langle\hat{E}_{\beta},\hat{E}_{\gamma}\rangle\\!\rangle=\langle\beta,\gamma\rangle=2$.
Suppose $\beta\neq\gamma$. By definition there exist $w\in W$ and $s,t\in S$
such that $m_{s,t}\neq\infty$, $\beta=w(\alpha_{s})$, $\gamma=w(\alpha_{t})$
and $\hat{m}_{\beta,\gamma}=m_{s,t}$. Then:

$\langle\beta,\gamma\rangle=\langle
w(\alpha_{s}),w(\alpha_{t})\rangle=\langle\alpha_{s},\alpha_{t}\rangle=-2\cos(\pi/m_{s,t})=-2\cos(\pi/\hat{m}_{\beta,\gamma})=\langle\\!\langle\hat{E}_{\beta},\hat{E}_{\gamma}\rangle\\!\rangle\,.$

This completes the proof of Claim 2.

Now take a free of infinity finite subset $\mathcal{X}$ of $\Phi[\Gamma]$. We
turn to show that $A[\hat{\Gamma}_{\mathcal{X}}]$ is of spherical type or of
affine type, that $|\mathcal{X}|\leq 2\,n_{\rm sph}(A[\Gamma])$, and that
$n_{\rm sph}(A[\hat{\Gamma}_{\mathcal{X}}])\leq n_{\rm sph}(A[\Gamma])$. The
existence of this uniform bound on the cardinality of $\mathcal{X}$ in
particular implies that there exists no free of infinity infinite subset of
$\Phi[\Gamma]$. This will prove the theorem. We denote by
$\hat{V}_{\mathcal{X}}$ the linear subspace of $\hat{V}$ spanned by
$\\{\hat{E}_{\beta}\mid\beta\in\mathcal{X}\\}$ and by
$\langle\\!\langle\cdot,\cdot\rangle\\!\rangle_{\mathcal{X}}:\hat{V}_{\mathcal{X}}\times\hat{V}_{\mathcal{X}}\to\mathbb{R}$
the restriction of $\langle\\!\langle\cdot,\cdot\rangle\\!\rangle$ to
$\hat{V}_{\mathcal{X}}$.

Let $\hat{u}\in\hat{V}_{\mathcal{X}}$. We write
$\hat{u}=\sum_{\beta\in\mathcal{X}}\lambda_{\beta}\hat{E}_{\beta}$ with
$\lambda_{\beta}\in\mathbb{R}$ for all $\beta\in\mathcal{X}$. Let
$u=\sum_{\beta\in\mathcal{X}}\lambda_{\beta}\beta\in V$. Then, by Claim 2,

$\langle\\!\langle\hat{u},\hat{u}\rangle\\!\rangle_{\mathcal{X}}=\sum_{\beta\in\mathcal{X}}\sum_{\gamma\in\mathcal{X}}\lambda_{\beta}\lambda_{\gamma}\langle\\!\langle\hat{E}_{\beta},\hat{E}_{\gamma}\rangle\\!\rangle=\sum_{\beta\in\mathcal{X}}\sum_{\gamma\in\mathcal{X}}\lambda_{\beta}\lambda_{\gamma}\langle\beta,\gamma\rangle=\langle
u,u\rangle\geq 0\,.$

This shows that $\langle\\!\langle\cdot,\cdot\rangle\\!\rangle_{\mathcal{X}}$
is positive, hence $A[\hat{\Gamma}_{\mathcal{X}}]$ is of spherical type or of
affine type.

Let $\hat{\Gamma}_{1},\dots,\hat{\Gamma}_{m}$ be the connected components of
$\hat{\Gamma}_{\mathcal{X}}$. For each $k\in\\{1,\dots,m\\}$ we denote by
$\mathcal{X}_{k}$ the set of vertices of $\hat{\Gamma}_{k}$ and we consider
the matrix $A_{k}=(\hat{m}_{\beta,\gamma})_{\beta,\gamma\in\mathcal{X}_{k}}$.
The matrix $A_{k}$ is symmetric, positive and indecomposable, hence, by Claim
1, the rank of $A_{k}$ is either $|\mathcal{X}_{k}|$ or $|\mathcal{X}_{k}|-1$.
In the second case we necessarily have $|\mathcal{X}_{k}|-1\geq 1$. Thus, we
have $m\leq{\rm
rank}(\langle\\!\langle\cdot,\cdot\rangle\\!\rangle_{\mathcal{X}})$, and
therefore:

$|\mathcal{X}|=\sum_{k=1}^{m}|\mathcal{X}_{k}|=m+\sum_{k=1}^{m}(|\mathcal{X}_{k}|-1)\leq
m+{\rm rank}(\langle\\!\langle\cdot,\cdot\rangle\\!\rangle_{\mathcal{X}})\leq
2\,{\rm rank}(\langle\\!\langle\cdot,\cdot\rangle\\!\rangle_{\mathcal{X}})\,.$

Since both $\hat{\Gamma}_{\mathcal{X}}$ and $\Gamma$ are of spherical type or
of affine type, we have $n_{\rm sph}(A[\hat{\Gamma}_{\mathcal{X}}])={\rm
rank}(\langle\\!\langle\cdot,\cdot\rangle\\!\rangle_{\mathcal{X}})$ and
$n_{\rm sph}(A[\Gamma])={\rm rank}(\langle\cdot,\cdot\rangle)$, hence it
remains to show that ${\rm
rank}(\langle\\!\langle\cdot,\cdot\rangle\\!\rangle_{\mathcal{X}})\leq{\rm
rank}(\langle\cdot,\cdot\rangle)$.

By applying again Claim 1 to the matrices $A_{k}$ we see that there exists a
subset $\mathcal{Y}\subset\mathcal{X}$ such that $|\mathcal{Y}|={\rm
rank}(\langle\\!\langle\cdot,\cdot\rangle\\!\rangle_{\mathcal{X}})$ and
$\langle\\!\langle\cdot,\cdot\rangle\\!\rangle_{\mathcal{Y}}$ is positive
definite. Let $\\{\lambda_{\beta}\\}_{\beta\in\mathcal{Y}}$ be a collection of
real numbers such that $\sum_{\beta\in\mathcal{Y}}\lambda_{\beta}\beta=0$ (in
$V$). Then, by Claim 2,

$\displaystyle 0=\langle
0,0\rangle=\langle\sum_{\beta\in\mathcal{Y}}\lambda_{\beta}\beta,\sum_{\beta\in\mathcal{Y}}\lambda_{\beta}\beta\rangle=\sum_{\beta\in\mathcal{Y}}\sum_{\gamma\in\mathcal{Y}}\lambda_{\beta}\lambda_{\gamma}\langle\beta,\gamma\rangle=$
$\displaystyle\sum_{\beta\in\mathcal{Y}}\sum_{\gamma\in\mathcal{Y}}\lambda_{\beta}\lambda_{\gamma}\langle\\!\langle\hat{E}_{\beta},\hat{E}_{\gamma}\rangle\\!\rangle=\langle\\!\langle\sum_{\beta\in\mathcal{Y}}\lambda_{\beta}\hat{E}_{\beta},\sum_{\beta\in\mathcal{Y}}\lambda_{\beta}\hat{E}_{\beta}\rangle\\!\rangle_{\mathcal{Y}}\,.$

Since $\langle\\!\langle\cdot,\cdot\rangle\\!\rangle_{\mathcal{Y}}$ is
positive definite and $\\{\hat{E}_{\beta}\mid\beta\in\mathcal{Y}\\}$ is a
basis of $\hat{V}_{\mathcal{Y}}$, it follows that $\lambda_{\beta}=0$ for all
$\beta\in\mathcal{Y}$, hence $\mathcal{Y}$ is a linearly independent subset of
$V$. Let $V_{\mathcal{Y}}$ be the linear subspace of $V$ spanned by
$\mathcal{Y}$ and let
$\langle\cdot,\cdot\rangle_{\mathcal{Y}}:V_{\mathcal{Y}}\times
V_{\mathcal{Y}}\to\mathbb{R}$ be the restriction of
$\langle\cdot,\cdot\rangle$ to $V_{\mathcal{Y}}$. The matrix of
$\langle\cdot,\cdot\rangle_{\mathcal{Y}}$ in the basis $\mathcal{Y}$ is the
same as the matrix of
$\langle\\!\langle\cdot,\cdot\rangle\\!\rangle_{\mathcal{Y}}$ in the basis
$\\{\hat{E}_{\beta}\mid\beta\in\mathcal{Y}\\}$ and
$\langle\\!\langle\cdot,\cdot\rangle\\!\rangle_{\mathcal{Y}}$ is positive
definite, hence $\langle\cdot,\cdot\rangle_{\mathcal{Y}}$ is positive
definite. So, ${\rm
rank}(\langle\\!\langle\cdot,\cdot\rangle\\!\rangle_{\mathcal{X}})=|\mathcal{Y}|=\dim(V_{\mathcal{Y}})={\rm
rank}(\langle\cdot,\cdot\rangle_{\mathcal{Y}})\leq{\rm
rank}(\langle\cdot,\cdot\rangle)$. ∎

##  5 Word problem

######  Theorem 5.1.

Let $\Gamma$ be a Coxeter graph of spherical type or of affine type. Then
${\rm VA}[\Gamma]$ has a solution to the word problem.

The proof of Theorem 5.1 relies on the following proposition which we state
and prove in a general framework, without assuming that the Coxeter graph is
of spherical type or of affine type.

######  Proposition 5.2.

Let $\Gamma$ be a finite Coxeter graph. Let $\SS\sqcup\mathcal{T}$ be the
standard generating set of ${\rm VA}[\Gamma]$, where $\SS=\\{\sigma_{s}\mid
s\in S\\}$ and $\mathcal{T}=\\{\tau_{s}\mid s\in S\\}$. There exists an
algorithm which, given a word $w$ on $(\SS\sqcup\mathcal{T})^{\pm 1}$ which
represents an element $g\in{\rm KVA}[\Gamma]$, determines:

  * (i)

a finite subset $\mathcal{X}\subset\Phi[\Gamma]$ such that $g\in
A[\hat{\Gamma}_{\mathcal{X}}]$,

  * (ii)

the Coxeter matrix $\hat{M}_{\mathcal{X}}$ of $\hat{\Gamma}_{\mathcal{X}}$,

  * (iii)

a word $\hat{w}$ over $\\{\delta_{\beta}\mid\beta\in\mathcal{X}\\}^{\pm 1}$
which represents $g$.

###### Proof.

Let $M=(m_{s,t})_{s,t\in S}$ be the Coxeter matrix of $\Gamma$. For $s,t\in S$
we set $a_{s,t}=-2\cos(\pi/m_{s,t})$ if $m_{s,t}\neq\infty$ and $a_{s,t}=-2$
if $m_{s,t}=\infty$. The number $a_{s,t}$ is an algebraic number for all
$s,t\in S$, hence we can consider a number field $\mathbb{K}$ containing all
$a_{s,t}$ and make all linear calculations with $\mathbb{K}$ in place of
$\mathbb{R}$. This makes them algorithmically computable. Take an abstract set
$\Pi=\\{\alpha_{s}\mid s\in S\\}$ in one-to-one correspondence with $S$ and
denote by $V$ the vector space over $\mathbb{K}$ having $\Pi$ as a basis.
Moreover, consider the symmetric bilinear form
$\langle\cdot,\cdot\rangle:V\times V\to\mathbb{K}$ defined by
$\langle\alpha_{s},\alpha_{t}\rangle=a_{s,t}$ for all $s,t\in S$ and the
embedding $W[\Gamma]\hookrightarrow{\rm GL}(V)$ defined by $s(v)=v-\langle
v,\alpha_{s}\rangle\alpha_{s}$ for $s\in S$ and $v\in V$. Set
$\Phi[\Gamma]=\\{w(\alpha_{s})\mid s\in S\text{ and }w\in W[\Gamma]\\}$.

In this context a root $\beta$ is given by a word $w=s_{1}\cdots s_{\ell}\in
S^{*}$ and an element $s\in S$ such that $\bar{w}(\alpha_{s})=\beta$, where
$\bar{w}$ is the element of $W[\Gamma]$ represented by $w$. The word problem
in $W[\Gamma]$ is solvable (see Tits [34]) and the linear operations in $V$
can be down with algorithmic means, hence we can decide whether two pairs
$(w,s)$ and $(w^{\prime},s^{\prime})$ in $S^{*}\times S$ represent the same
root or not.

Claim 1. Let $\beta,\gamma\in\Phi[\Gamma]$ such that
$\langle\beta,\gamma\rangle\leq 0$ and let $s,t\in S$ such that
$m_{s,t}\neq\infty$. There exists $u\in W[\Gamma]$ such that
$\\{u(\alpha_{s}),u(\alpha_{t})\\}=\\{\beta,\gamma\\}$ if and only if $st$ and
$r_{\beta}r_{\gamma}$ are conjugate in $W[\Gamma]$.

Proof of Claim 1. Note that $ts$ and $st$ are conjugate, since $ts=s\,st\,s$,
hence we can replace $st$ with $ts$ if necessary. Suppose first there exists
$u\in W[\Gamma]$ such that
$\\{u(\alpha_{s}),u(\alpha_{t})\\}=\\{\beta,\gamma\\}$, say
$u(\alpha_{s})=\beta$ and $u(\alpha_{t})=\gamma$. Then
$r_{\beta}=ur_{\alpha_{s}}u^{-1}=usu^{-1}$ and
$r_{\gamma}=ur_{\alpha_{t}}u^{-1}=utu^{-1}$, hence
$r_{\beta}r_{\gamma}=u\,st\,u^{-1}$.

Now, suppose that $r_{\beta}r_{\gamma}$ and $st$ are conjugate in $W[\Gamma]$.
Let $w\in W[\Gamma]$ such that $wr_{\beta}r_{\gamma}w^{-1}=st$. Upon replacing
$\beta$ with $w(\beta)$ and $\gamma$ with $w(\gamma)$, we can assume that
$r_{\beta}r_{\gamma}=st$. By Baumeister–Dyer–Stump–Wegener [4, Theorem 1.4],
$r_{\beta}$ and $r_{\gamma}$ lie in the subgroup $W[\Gamma_{\\{s,t\\}}]$ of
$W[\Gamma]$ generated by $\\{s,t\\}$, and they are conjugate in
$W[\Gamma_{\\{s,t\\}}]$ to elements of $\\{s,t\\}$. Since, furthermore,
$r_{\beta}r_{\gamma}=st$, there exists $k\in\mathbb{N}$ such that
$r_{\beta}=(st)^{k+1}s$ and $r_{\gamma}=(st)^{k}s$. If $k$ is even, say
$k=2\ell$, then, upon conjugating by $s(ts)^{\ell}$, we may assume that
$r_{\beta}=t$ and $r_{\gamma}=s$. If $k$ is odd, say $k=2\ell-1$, then, upon
conjugating by $(ts)^{\ell}$, we may assume that $r_{\beta}=s$ and
$r_{\gamma}=t$. So, since we can swap $s$ and $t$, we may assume that
$r_{\beta}=s$ and $r_{\gamma}=t$. This means in terms of roots that
$\beta\in\\{\alpha_{s},-\alpha_{s}\\}$ and
$\gamma\in\\{\alpha_{t},-\alpha_{t}\\}$. If $\langle\beta,\gamma\rangle=0$,
then $\langle\alpha_{s},\alpha_{t}\rangle=0$. In this case
$s(\alpha_{s})=-\alpha_{s}$, $s(\alpha_{t})=\alpha_{t}$,
$t(\alpha_{s})=\alpha_{s}$ and $t(\alpha_{t})=-\alpha_{t}$. By using these
equalities we easily reduce to the case where $\beta=\alpha_{s}$ and
$\gamma=\alpha_{t}$, which concludes the proof in this case. Suppose
$\langle\beta,\gamma\rangle<0$. Then $m_{s,t}\geq 3$,
$\langle\alpha_{s},\alpha_{t}\rangle=\langle\beta,\gamma\rangle$, and we have
two possibilities: either $\beta=\alpha_{s}$ and $\gamma=\alpha_{t}$, or
$\beta=-\alpha_{s}$ and $\gamma=-\alpha_{t}$. If $\beta=\alpha_{s}$ and
$\gamma=\alpha_{t}$, then there is nothing to prove. So, we can assume
$\beta=-\alpha_{s}$ and $\gamma=-\alpha_{t}$. We know by Deodhar [16] that:

$\displaystyle{\rm Prod}_{R}(s,t,m_{s,t})(\alpha_{s})=-\alpha_{s}\text{ and
}{\rm Prod}_{R}(s,t,m_{s,t})(\alpha_{t})=-\alpha_{t}\text{ if }m_{s,t}\text{
is even}\,,$ $\displaystyle{\rm
Prod}_{R}(s,t,m_{s,t})(\alpha_{s})=-\alpha_{t}\text{ and }{\rm
Prod}_{R}(s,t,m_{s,t})(\alpha_{t})=-\alpha_{s}\text{ if }m_{s,t}\text{ is
odd}\,,$

hence, upon applying ${\rm Prod}_{R}(s,t,m_{s,t})$, we may assume that
$\beta=\alpha_{s}$ and $\gamma=\alpha_{t}$, which finishes the proof of Claim
1.

Claim 2. There exists an algorithm which, for given
$\beta,\gamma\in\Phi[\Gamma]$, computes $\hat{m}_{\beta,\gamma}$.

Proof of Claim 2. If $\langle\beta,\gamma\rangle\not\in]-2,0]$, then
$\hat{m}_{\beta,\gamma}=\infty$. Thus, we can assume
$\langle\beta,\gamma\rangle\in]-2,0]$. As shown in the beginning of the proof
of Proposition 1.4.7 in Krammer [26], this implies that $r_{\beta}r_{\gamma}$
is of finite order. Let $s,t\in S$ such that $m_{s,t}\neq\infty$. Then the
algorithm of Krammer [26, Proposition 3.3.2] determines whether
$r_{\beta}r_{\gamma}$ and $st$ are conjugate. By Claim 1 this algorithm also
decides whether there exists $u\in W[\Gamma]$ such that
$\\{u(\alpha_{s}),u(\alpha_{t})\\}=\\{\beta,\gamma\\}$. Then we compute
$\hat{m}_{\beta,\gamma}$ by applying this algorithm to all the pairs $(s,t)\in
S\times S$ such that $s\neq t$ and $m_{s,t}\neq\infty$. This concludes the
proof of Claim 2.

End of the proof of Proposition 5.2. Let $g\in{\rm KVA}[\Gamma]$ given by a
word over $(\SS\sqcup\mathcal{T})^{\pm 1}$. From this word we can easily write
$g$ in the form:

$g=\iota_{W}(w_{0})\sigma_{s_{1}}^{\varepsilon_{1}}\iota_{W}(w_{1})\cdots\sigma_{s_{p}}^{\varepsilon_{p}}\iota_{W}(w_{p})\,,$

where $w_{0},w_{1},\dots,w_{p}\in W[\Gamma]$, $s_{1},\dots,s_{p}\in S$ and
$\varepsilon_{1},\dots,\varepsilon_{p}\in\\{\pm 1\\}$. For $i=1,\dots,p$ set
$\beta_{i}=(w_{0}\cdots w_{i-1})(\alpha_{s_{i}})$. Set also
$w=w_{0}w_{1}\cdots w_{p}$. Then
$g=\delta_{\beta_{1}}^{\varepsilon_{1}}\delta_{\beta_{2}}^{\varepsilon_{2}}\cdots\delta_{\beta_{p}}^{\varepsilon_{p}}\iota_{W}(w)$.
But $w=\pi_{K}(g)=1$, since $g\in{\rm KVA}[\Gamma]$, hence
$g=\delta_{\beta_{1}}^{\varepsilon_{1}}\delta_{\beta_{2}}^{\varepsilon_{2}}\cdots\delta_{\beta_{p}}^{\varepsilon_{p}}$.
This proves Part (i) and Part (iii) if we set
$\mathcal{X}=\\{\beta_{1},\dots,\beta_{p}\\}$. Then Part (ii) immediately
follows from Claim 2. ∎

###### Proof of Theorem 5.1.

Let $\Gamma$ be a Coxeter graph of spherical type or of affine type. Let
$g\in{\rm VA}[\Gamma]$ given by a word over $(\SS\sqcup\mathcal{T})^{\pm 1}$.
We can calculate $\pi_{K}(g)\in W[\Gamma]$ and the word problem in $W[\Gamma]$
is solvable (see Tits [34]), hence we can decide whether $\pi_{K}(g)$ is
trivial or not. If $\pi_{K}(g)\neq 1$, then $g\neq 1$ and the algorithm stops
here. So, we can assume $\pi_{K}(g)=1$, that is, $g\in{\rm KVA}[\Gamma]$. The
algorithm of Proposition 5.2 gives a finite subset
$\mathcal{X}\subset\Phi[\Gamma]$ and a word $\hat{w}$ over
$\\{\delta_{\beta}\mid\beta\in\mathcal{X}\\}^{\pm 1}$ such that $g\in
A[\hat{\Gamma}_{\mathcal{X}}]$ and $\hat{w}$ represents $g$. It also gives the
matrix $\hat{M}_{\mathcal{X}}$ of $\hat{\Gamma}_{\mathcal{X}}$. We know by
Godelle–Paris [22] that, if, for each free of infinity subset
$\mathcal{Y}\subset\mathcal{X}$ the group $A[\hat{\Gamma}_{\mathcal{Y}}]$ has
a solution to the word problem, then $A[\hat{\Gamma}_{\mathcal{X}}]$ itself
has a solution to the word problem. On the other hand, by Theorem 4.1,
$A[\hat{\Gamma}_{\mathcal{Y}}]$ is of spherical type or of affine type for
each free of infinity subset $\mathcal{Y}\subset\mathcal{X}$, by Deligne [17]
and Brieskorn–Saito [10] the Artin groups of spherical type have a solvable
word problem, and by McCammond–Sulway [29] the Artin groups of affine type
have a solvable word problem. So, $A[\hat{\Gamma}_{\mathcal{X}}]$ has a
solution to the word problem. This means that there is an algorithm which
decides whether $g$ is trivial or not. ∎

##  6 $K(\pi,1)$ conjecture and cohomological dimension

We start by recalling the $K(\pi,1)$ conjecture for Artin groups. Let $\Gamma$
be a Coxeter graph and let $M=(m_{s,t})_{s,t\in S}$ be its Coxeter matrix.
Recall that $\Pi=\\{\alpha_{s}\mid s\in S\\}$ is the set of simple roots, that
$V$ is the real vector space having $\Pi$ as a basis, and that
$\langle\cdot,\cdot\rangle:V\times V\to\mathbb{R}$ is the canonical bilinear
form. We have an embedding $W[\Gamma]\hookrightarrow{\rm GL}(V)$ defined by
$s(v)=v-\langle v,\alpha_{s}\rangle\alpha_{s}$ for $s\in S$ and $v\in V$, and
$W[\Gamma]$, viewed as a subgroup of ${\rm GL}(V)$, is generated by linear
reflections.

Suppose $\Gamma$ is finite. There exists a non-empty open cone $I\subset V$,
called _Tits cone_ , on which $W[\Gamma]$ acts properly discontinuously. The
set of reflections in $W[\Gamma]$ is $R=\\{wsw^{-1}\mid w\in W[\Gamma]\text{
and }s\in S\\}$ and $W[\Gamma]$ acts freely and properly discontinuously on
$I\setminus\cup_{r\in R}H_{r}$, where, for $r\in R$, $H_{r}$ denotes the
hyperplane of $V$ fixed by $r$ (see Bourbaki [8]). Set:

$\mathcal{M}[\Gamma]=(I\times I)\setminus\left(\bigcup_{r\in R}(H_{r}\times
H_{r})\right)\,.$

This is a connected manifold of dimension $2\,|S|$ on which $W[\Gamma]$ acts
freely and properly discontinuously. A key result in the domain is the
following.

######  Theorem 6.1 (Van der Lek [27]).

Let $\Gamma$ be a finite Coxeter graph. Then the fundamental group of
$\mathcal{M}[\Gamma]/W[\Gamma]$ is isomorphic to $A[\Gamma]$.

Recall that a CW-complex $X$ is a _classifying space_ for a (discrete) group
$G$ if $\pi_{1}(X)=G$ and the universal covering of $X$ is contractible. These
spaces play a major role in the computation of cohomology of groups (see Brown
[11]). The following is one of the main questions in the theory of Artin
groups.

######  Conjecture 6.2 ($K(\pi,1)$ Conjecture).

Let $\Gamma$ be a finite Coxeter graph. Then $\mathcal{M}[\Gamma]/W[\Gamma]$
is a classifying space for $A[\Gamma]$.

We say that an Artin group $A[\Gamma]$ with $\Gamma$ finite _satisfies the
$K(\pi,1)$ conjecture_ if $\mathcal{M}[\Gamma]/W[\Gamma]$ is a classifying
space for $A[\Gamma]$. So, the above conjecture says that any Artin group
$A[\Gamma]$ with $\Gamma$ finite satisfies the $K(\pi,1)$ conjecture. Note
also that, by Deligne [17] and Paolini–Salvetti [30], $A[\Gamma]$ satisfies
the $K(\pi,1)$ conjecture if $\Gamma$ is of spherical type or of affine type.

The following makes sense only when $\Gamma$ is of spherical type since this
is the only case where $\hat{\Gamma}$ is finite.

######  Theorem 6.3.

Let $\Gamma$ be a Coxeter graph of spherical type. Then $A[\hat{\Gamma}]$
satisfies the $K(\pi,1)$ conjecture.

Theorem 6.3 is a direct consequence of the following proposition combined with
Theorem 4.1.

######  Proposition 6.4.

Let $\Gamma$ be a finite Coxeter graph and let $M=(m_{s,t})_{s,t\in S}$ be its
Coxeter matrix. Suppose that, for each free of infinity subset $X\subset S$,
the Coxeter graph $\Gamma_{X}$ is of spherical type or of affine type. Then
$A[\Gamma]$ satisfies the $K(\pi,1)$ conjecture.

###### Proof.

By Deligne [17], if $\Gamma$ is of spherical type, then $A[\Gamma]$ satisfies
the $K(\pi,1)$ conjecture, and by Paolini–Salvetti [30], if $\Gamma$ is of
affine type, then $A[\Gamma]$ satisfies the $K(\pi,1)$ conjecture. On the
other hand, by Ellis–Sköldberg [21] (see also Godelle–Paris [23]), if
$A[\Gamma_{X}]$ satisfies the $K(\pi,1)$ conjecture for every free of infinity
subset $X\subset S$, then $A[\Gamma]$ satisfies the $K(\pi,1)$ conjecture. So,
if for each free of infinity subset $X\subset S$ the Coxeter graph
$\Gamma_{X}$ is of spherical type or of affine type, then $A[\Gamma]$
satisfies the $K(\pi,1)$ conjecture. ∎

###### Proof of Theorem 6.3.

It is a straightforward consequence of Proposition 6.4 and Theorem 4.1. ∎

For a group $G$ we denote by ${\rm cd}(G)$ the cohomological dimension of $G$
and by ${\rm vcd}(G)$ the virtual cohomological dimension of $G$.

######  Corollary 6.5.

Let $\Gamma$ be a Coxeter graph of spherical type and let $n$ be the number of
vertices of $\Gamma$. Then ${\rm cd}({\rm KVA}[\Gamma])={\rm vcd}({\rm
VA}[\Gamma])=n$. In particular, ${\rm KVA}[\Gamma]$ is torsion free and ${\rm
VA}[\Gamma]$ is virtually torsion free.

###### Proof.

We have $n=n_{\rm sph}(A[\Gamma])$ by definition. By Theorem 6.3,
$A[\hat{\Gamma}]$ satisfies the $K(\pi,1)$ conjecture, hence, by Charney–Davis
[12], ${\rm cd}({\rm KVA}[\Gamma])={\rm cd}(A[\hat{\Gamma}])=n_{\rm
sph}(A[\hat{\Gamma}])$. By Theorem 4.1 it follows that ${\rm cd}({\rm
KVA}[\Gamma])\leq n$. On the other hand, $A[\Gamma]$ is a subgroup of ${\rm
KVA}[\Gamma]$ (see Corollary 2.4) and, by Deligne [17], ${\rm
cd}(A[\Gamma])=n$, hence $n\leq{\rm cd}({\rm KVA}[\Gamma])$, and therefore
${\rm cd}({\rm KVA}[\Gamma])=n$. Finally, since ${\rm VA}[\Gamma]={\rm
KVA}[\Gamma]\rtimes W[\Gamma]$ and $W[\Gamma]$ is finite, ${\rm vcd}({\rm
VA}[\Gamma])={\rm cd}({\rm KVA}[\Gamma])=n$. ∎

######  Remark.

It follows from Bartholdi–Enriquez–Etingof–Rains [3] that the cohomological
dimension of ${\rm PVB}_{n}$ is $n-1$, which implies also that ${\rm PVB}_{n}$
is torsion free. We think that this extends to all virtual Artin groups of
spherical type in the sense that, for a Coxeter graph $\Gamma$ of spherical
type, we think that the cohomological dimension of ${\rm PVA}[\Gamma]$ is
equal to the number of vertices of $\Gamma$, and therefore that ${\rm
PVA}[\Gamma]$ is torsion free as well. Note that, since in this case ${\rm
PVA}[\Gamma]$ is of finite index in ${\rm VA}[\Gamma]$ and by Corollary 6.5
the virtual cohomological dimension of ${\rm VA}[\Gamma]$ is equal to the
number of vertices of $\Gamma$, by a theorem of Serre [32] it would suffice to
prove that ${\rm PVA}[\Gamma]$ is torsion free.

We are not able to give precise values for the cohomological dimension of
${\rm KVA}[\Gamma]$ and for the virtual cohomological dimension of ${\rm
VA}[\Gamma]$ when $\Gamma$ is of affine type, but we can give bounds:

######  Theorem 6.6.

Let $\Gamma$ be a Coxeter graph of affine type.

  * (1)

${\rm KVA}[\Gamma]$ is torsion free and ${\rm cd}({\rm KVA}([\Gamma])\leq
n_{\rm sph}(A[\Gamma])+1$.

  * (2)

${\rm VA}[\Gamma]$ is virtually torsion free and ${\rm vcd}({\rm
VA}[\Gamma])\leq 2\,n_{\rm sph}(A[\Gamma])+1$.

###### Proof.

Let $\mathcal{X}$ be a finite subset of $\Phi[\Gamma]$. We denote by
$\mathcal{P}_{<\infty}(\mathcal{X})$ the set of free of infinity subsets of
$\mathcal{X}$. Recall that $n_{\rm
sph}(A[\hat{\Gamma}_{\mathcal{X}}])=\max\\{n_{\rm
sph}(A[\hat{\Gamma}_{\mathcal{Y}}])\mid\mathcal{Y}\in\mathcal{P}_{<\infty}(\mathcal{X})\\}$
and, by Theorem 4.1, $n_{\rm sph}(A[\hat{\Gamma}_{\mathcal{Y}}])\leq n_{\rm
sph}(A[\Gamma])$ for all $\mathcal{Y}\in\mathcal{P}_{<\infty}(\mathcal{X})$,
hence $n_{\rm sph}(A[\hat{\Gamma}_{\mathcal{X}}])\leq n_{\rm sph}(A[\Gamma])$.
Furthermore, by Proposition 6.4 and Theorem 4.1,
$A[\hat{\Gamma}_{\mathcal{X}}]$ satisfies the $K(\pi,1)$ conjecture, hence, by
Charney–Davis [12], ${\rm cd}(A[\hat{\Gamma}_{\mathcal{X}}])=n_{\rm
sph}(A[\hat{\Gamma}_{\mathcal{X}}])\leq n_{\rm sph}(A[\Gamma])$. Recall that
$\mathcal{P}_{\rm fin}(\Phi[\Gamma])$ denotes the set of finite subsets of
$\Phi[\Gamma]$. Since ${\rm
KVA}[\Gamma]=A[\hat{\Gamma}]=\varinjlim_{\mathcal{X}\in\mathcal{P}_{\rm
fin}(\Phi[\Gamma])}A[\hat{\Gamma}_{\mathcal{X}}]$ and ${\rm
cd}(A[\hat{\Gamma}_{\mathcal{X}}])\leq n_{\rm sph}(A[\Gamma])$ for all
$\mathcal{X}\in\mathcal{P}_{\rm fin}(\Phi[\Gamma])$, by Bieri [7, Theorem 4.7]
we conclude that ${\rm cd}({\rm KVA}[\Gamma])\leq n_{\rm sph}(A[\Gamma])+1$.
This also implies that ${\rm KVA}[\Gamma]$ is torsion free.

We know by Bourbaki [8, Chapitre VI] that $W[\Gamma]$ admits a semi-direct
product decomposition $W[\Gamma]=\mathbb{Z}^{m}\rtimes W_{0}$ where $m\leq
n_{\rm sph}(A[\Gamma])$ and $W_{0}$ is a finite Coxeter group. Let
$H=\pi_{K}^{-1}(\mathbb{Z}^{m})$. Then $H$ is a finite index subgroup of ${\rm
VA}[\Gamma]$ which admits a semi-direct product decomposition $H={\rm
KVA}[\Gamma]\rtimes\mathbb{Z}^{m}$. By Brown [11, Proposition 2.4] it follows
that:

${\rm cd}(H)\leq{\rm cd}({\rm KVA}[\Gamma])+{\rm cd}(\mathbb{Z}^{m})\leq
n_{\rm sph}(A[\Gamma])+1+m\leq 2\,n_{\rm sph}(A[\Gamma])+1\,,$

hence ${\rm VA}[\Gamma]$ is virtually torsion free and ${\rm vcd}({\rm
VA}[\Gamma])={\rm cd}(H)\leq 2\,n_{\rm sph}(A[\Gamma])+1$. ∎

## References

  * [1] V G Bardakov, The virtual and universal braids, Fund. Math. 184 (2004), 1–18.
  * [2] V G Bardakov, P Bellingeri, Combinatorial properties of virtual braids, Topology Appl. 156 (2009), no. 6, 1071–1082.
  * [3] L Bartholdi, B Enriquez, P Etingof, E Rains, Groups and Lie algebras corresponding to the Yang-Baxter equations, J. Algebra 305 (2006), no. 2, 742–764.
  * [4] B Baumeister, M Dyer, C Stump, P Wegener, A note on the transitive Hurwitz action on decompositions of parabolic Coxeter elements, Proc. Amer. Math. Soc. Ser. B 1 (2014), 149–154.
  * [5] P Bellingeri, B A Cisneros de la Cruz, L Paris, A simple solution to the word problem for virtual braid groups, Pacific J. Math. 283 (2016), no. 2, 271–287.
  * [6] P Bellingeri, L Paris, Virtual braids and permutations, Ann. Inst. Fourier (Grenoble) 70 (2020), no. 3, 1341–1362.
  * [7] R Bieri, Homological dimension of discrete groups, Second edition. Queen Mary College Mathematics Notes. Queen Mary College, Department of Pure Mathematics, London, 1981.
  * [8] N Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles No 1337. Hermann, Paris, 1968.
  * [9] E Brieskorn, Sur les groupes de tresses [d’après V. I. Arnol’d], Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 401, pp. 21–44. Lecture Notes in Math., Vol. 317, Springer, Berlin, 1973.
  * [10] E Brieskorn, K Saito, Artin-Gruppen und Coxeter-Gruppen, Invent. Math. 17 (1972), 245–271.
  * [11] K S Brown, Cohomology of groups. Corrected reprint of the 1982 original, Graduate Texts in Mathematics, 87. Springer-Verlag, New York, 1994.
  * [12] R Charney, M W Davis, Finite $K(\pi,1)$s for Artin groups, Prospects in topology (Princeton, NJ, 1994), 110–124, Ann. of Math. Stud., 138, Princeton Univ. Press, Princeton, NJ, 1995.
  * [13] R Charney, R Morris-Wright, Artin groups of infinite type: trivial centers and acylindrical hyperbolicity, Proc. Amer. Math. Soc. 147 (2019), no. 9, 3675–3689.
  * [14] H S M Coxeter, Discrete groups generated by reflections, Ann. of Math. (2) 35 (1934), no. 3, 588–621.
  * [15] H S M Coxeter, The complete enumeration of finite groups of the form $R_{i}^{2}=(R_{i}R_{j})^{k_{ij}}=1$, J. London Math. Soc. 10 (1935), 21–25.
  * [16] V V Deodhar, On the root system of a Coxeter group, Comm. Algebra 10 (1982), no. 6, 611–630.
  * [17] P Deligne, Les immeubles des groupes de tresses généralisés, Invent. Math. 17 (1972), 273–302.
  * [18] E. Dies, A. Nicas, The center of the virtual braid group is trivial. J. Knot Theory Ramifications 23 (2014), no. 8, 1450042, 6 pp.
  * [19] F Digne, Présentations duales des groupes de tresses de type affine $\tilde{A}$. Comment. Math. Helv. 81 (2006), no. 1, 23–47.
  * [20] F Digne, A Garside presentation for Artin-Tits groups of type $\tilde{C}_{n}$, Ann. Inst. Fourier (Grenoble) 62 (2012), no. 2, 641–666.
  * [21] G Ellis, E Sköldberg, The $K(\pi,1)$ conjecture for a class of Artin groups, Comment. Math. Helv. 85 (2010), no. 2, 409–415.
  * [22] E Godelle, L Paris, Basic questions on Artin-Tits groups, Configuration spaces, 299–311, CRM Series, 14, Ed. Norm., Pisa, 2012.
  * [23] E Godelle, L Paris, $K(\pi,1)$ and word problems for infinite type Artin-Tits groups, and applications to virtual braid groups, Math. Z. 272 (2012), no. 3-4, 1339–1364.
  * [24] S Kamada, Braid presentation of virtual knots and welded knots, Osaka J. Math. 44 (2007), no. 2, 441–458.
  * [25] L H Kauffman, Virtual knot theory, European J. Combin. 20 (1999), no. 7, 663–690.
  * [26] D Krammer, The conjugacy problem for Coxeter groups, Groups Geom. Dyn. 3 (2009), no. 1, 71–171.
  * [27] H van der Lek, The homotopy type of complex hyperplane complements, Ph. D. Thesis, Nijmegen, 1983.
  * [28] J McCammond, Dual euclidean Artin groups and the failure of the lattice property, J. Algebra 437 (2015), 308–343.
  * [29] J McCammond, R Sulway, Artin groups of Euclidean type, Invent. Math. 210 (2017), no. 1, 231–282.
  * [30] G Paolini, M Salvetti, Proof of the $K(\pi,1)$ conjecture for affine Artin groups, Invent. Math. 224 (2021), no. 2, 487–572.
  * [31] L Rabenda, Mémoire de DEA, Université de Bourgogne, 2003.
  * [32] J-P Serre, Cohomologie des groupes discrets, Prospects in mathematics (Proc. Sympos., Princeton Univ., Princeton, N.J., 1970), pp. 77–169. Ann. of Math. Studies, No. 70, Princeton Univ. Press, Princeton, N.J., 1971.
  * [33] A-L Thiel, Virtual braid groups of type B and weak categorification, J. Knot Theory Ramifications 21 (2012), no. 2, 1250020, 21 pp.
  * [34] J Tits, Le problème des mots dans les groupes de Coxeter, 1969 Symposia Mathematica (INDAM, Rome, 1967/68), Vol. 1 pp. 175–185 Academic Press, London.
  * [35] V V Vershinin, On homology of virtual braids and Burau representation, Knots in Hellas ’98, Vol. 3 (Delphi), J. Knot Theory Ramifications 10 (2001), no. 5, 795–812.