# On the structure of finitely presented Bestvina-Brady groups

Priyavrat Despande  Chennai Mathematical Institute, India
<EMAIL_ADDRESS>and  Mallika Roy  Chennai Mathematical Institute, India
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Right-angled Artin groups and their subgroups are of great interest because of
their geometric, combinatorial and algorithmic properties. It is convenient to
define these groups using finite simplicial graphs. The isomorphism type of
the group is uniquely determined by the graph. Moreover, many structural
properties of right angled Artin groups can be expressed in terms of their
defining graph.

In this article we address the question of understanding the structure of a
class of subgroups of right-angled Artin groups in terms of the graph.
Bestvina and Brady, in their seminal work, identified studied these subgroups
(now called Bestvina-Brady groups or Artin kernels) from finiteness conditions
viewpoint. Unlike the right-angled Artin groups the isomorphism type of
Bestvina-Brady groups is not uniquely determined by the defining graph. We
prove that certain finitely presented Bestvina-Brady groups can be expressed
as an iterated amalgamated product. Moreover, we show that this amalgamated
product can be read off from the graph defining the ambient right-angled Artin
group.

###### Key words and phrases:

right angled Artin group, amalgamated product, Bestvina-Brady group

###### 2010 Mathematics Subject Classification:

20F36, 20F65, 08B25

The authors are partially supported by a grant from the Infosys Foundation.

##  1\. Introduction

A right-angled Artin group (a RAAG, for short) is a finitely presented group
such that the commuting relations are the only relations. It is perhaps easier
to describe this group using finite simplicial graphs. Let $\Gamma$ be such a
graph; then associated RAAG, denoted by $A_{\Gamma}$, has generators
corresponding to vertices of $\Gamma$ and two generators commute whenever the
corresponding vertices are connected by an edge. We refer the reader to [4]
for an encyclopedic introduction to RAAGs. They have become central in group
theory, their study interweaves geometric group theory with other areas of
mathematics. This class interpolates between two of the most classical
families of groups, free and free abelian groups, and its study provides
uniform approaches and proofs, as well as rich generalisations of the results
for free and free abelian groups. The study of this class from different
perspectives has contributed to the development of new, rich theories such as
the theory of CAT(0) cube complexes and has been an essential ingredient in
Ian Agol’s solution to Thurston’s virtual fibering and virtual Haken
conjectures. RAAGs are important in geometric group theory for many reasons,
including the fact that they have interesting subgroups; for example,
Bestvina-Brady groups.

A Bestvina-Brady group (a B-B group, for short) is the kernel of the group
homomorphism $A_{\Gamma}\rightarrow\mathbb{Z}$ which takes all the generators
of $A_{\Gamma}$ to $1$. One of the reasons they are interesting is that they
provide an example of a group that satisfies the finiteness property
$\mathbf{FP_{n}}$ but not $\mathbf{FP_{n+1}}$. They also provide
counterexamples either to Eilenberg–Ganea Conjecture or Whitehead Conjecture;
see [2] for details.

Though RAAGs interpolate between free groups and free abelian groups their
structure is not always straightforward. However, many of their structural
properties can be read off from the underlying graph. For example, a RAAG
$A_{\Gamma}$ is a free product of two of its sub-RAAGs if and only if $\Gamma$
is disconnected; by a theorem of Clay [5] all nontrivial splittings of
$A_{\Gamma}$ over $\mathbb{Z}$ correspond to cut vertices of $\Gamma$, further
it was proved by Grove and Hull [6] that $A_{\Gamma}$ splits over an abelian
group if and only if $\Gamma$ is disconnected, or complete or contains a
separating clique.

It is easy to observe that two non-isomorphic graphs can give rise to the same
B-B group $H_{\Gamma}$ (see 3.8). Moreover, though the finiteness properties
are completely determined by the (topology of the) clique complex of $\Gamma$,
not much work has been done to understand structural properties of
$H_{\Gamma}$ in terms of graph theoretic input. We should mention here the
recent work of Chang [3] on abelian splittings of B-B groups and the work of
Barquinero-Ruffoni-Ye [1] decomposing Artin kernels (a class of subgroups
generalizing B-B groups) as graphs groups. In both the works the decomposition
of $H_{\Gamma}$ is expressed in terms of the underlying $\Gamma$; hence we
consider them as a motivation for our paper.

The aim of this article is to show that a certain class of finitely presented
B-B groups can be decomposed as an (iterated) amalgamated product of a RAAG
and a finitely many copies of $\mathbb{Z}^{2}$. We express this amalgamated
product completely in terms of the underlying graph. To be precise, we
decompose $\Gamma$ as a union of $\Gamma^{\prime}$ and some triangles (i.e.,
$3$-cliques); this union is with the help of a suitable spanning tree. The
subgraph $\Gamma^{\prime}$ is selected on the basis that the corresponding B-B
group $H_{\Gamma^{\prime}}$ is isomorphic to a RAAG; the triangles correspond
to copies of $\mathbb{Z}^{2}$. The classes of graphs for which this
decomposition works include $1$-skeleton to (extra)-special triangulation of
the $2$-disk and connected splittings with a separating clique. We should note
here that Papadima and Suciu [8, Proposition 9.4] showed that the B-B group
corresponding to an extra-special triangulation is not isomorphic to any RAAG.
Our main theorem implies that such B-B groups can be decomposed as an iterated
amalgamation of RAAGs.

##  2\. Preliminaries and notations

Below we present the necessary preliminaries on graph theory and B-B groups.

###  2.1. Graph theory

We recall and set up some basic notations and terminologies in graph theory.
Throughout this article, we assume finite graphs which have no loops and
multi-edges, i.e., all the graphs are finite simplicial. Given a graph
$\Gamma$, we denote the set of its vertices and edges by $V(\Gamma)$ and
$E(\Gamma)$, respectively. We denote $e=(v,w)$ to be an edge connecting
vertices $v$ and $w$. $\iota(e),\tau(e)$ respectively denotes the initial
point and the terminal point of the edge $e$, i.e., $e=(\iota(e),\tau(e))$.
Two vertices are adjacent if they are connected by an edge. We will denote any
tree of $\Gamma$ by $T$ and in most of the situations we will consider $T$ as
a spanning tree or maximal tree. As we are dealing with finite graphs it is
straightforward to see that $|E(T)|=|V(\Gamma)|-1$. Given any subset
$V^{\prime}$ of $V(\Gamma)$, the induced subgraph (in some literature, it is
also called as full subgraph) on $V^{\prime}$ is a graph $\Gamma^{\prime}$
whose vertex set is $V^{\prime}$, and two vertices are adjacent in
$\Gamma^{\prime}$ if and only if they are adjacent in $\Gamma$.

The star graph $S_{n}$ of order $n$, sometimes simply known as an n-star, is a
tree on $n$ vertices with one vertex having degree $n-1$ and the other $n-1$
vertices having degree $1$.

Given two graphs $\Gamma_{1}=(V(\Gamma_{1}),E(\Gamma_{1}))$ and
$\Gamma_{2}=(V(\Gamma_{2}),E(\Gamma_{2}))$, the union of $\Gamma_{1}$ and
$\Gamma_{2}$ is $\Gamma_{1}\cup\Gamma_{2}=(V(\Gamma_{1})\cup
V(\Gamma_{2}),E(\Gamma_{1})\cup E(\Gamma_{2}))$. The join of two graphs
$\Gamma_{1}$ and $\Gamma_{2}$, denoted by $\Gamma_{1}\vee\Gamma_{2}$, is
defined to the graph union $\Gamma_{1}\cup\Gamma_{2}$ and with every pair of
vertices $(v,w)\in V(\Gamma_{1})\times V(\Gamma_{2})$ being adjacent. The join
operation is commutative, that is,
$\Gamma_{1}\vee\Gamma_{2}=\Gamma_{2}\vee\Gamma_{1}$. When a graph $\Gamma$
decomposes as a join of a vertex $v$ and another graph $\Gamma^{\prime}$, the
vertex $v$ is called a dominating vertex, and $\Gamma$ is called a cone graph
or the cone on $\Gamma^{\prime}$.

Two graphs $\Gamma_{1}$ and $\Gamma_{2}$ are said to be isomorphic, denoted by
$\Gamma_{1}\simeq\Gamma_{2}$, if there is a bijection
$\phi:V(\Gamma_{1})\rightarrow V(\Gamma_{2})$ such that two vertices $v,w$ are
adjacent in $\Gamma_{1}$ if and only if $\phi(v),\phi(w)$ are adjacent in
$\Gamma_{2}$. In this case, we call $\phi$ a graph isomorphism between
$\Gamma_{1}$ and $\Gamma_{2}$. The star graph $S_{n}$ is isomorphic to the
complete bipartite graph $K_{(1,n-1)}$. Also let $\Gamma_{1}$, $\Gamma_{2}$ be
cone graphs on $\Gamma^{\prime}_{1}\text{ and }\Gamma^{\prime}_{2}$
respectively, then $\Gamma_{1}\simeq\Gamma_{2}$ if and only if
$\Gamma^{\prime}_{1}\simeq\Gamma^{\prime}_{2}$.

Given a graph $\Gamma$ we construct a simplicial complex $\triangle_{\Gamma}$,
called the _flag complex_ , as follows: the vertex set is the ground vertex
set $V(\Gamma)$ and a subset of cardinality $k$ is a $(k-1)$-simplex if and
only if the induced subgraph is a $k$-clique. We do not differentiate between
an abstract simplicial complex and its geometric realization, any topological
statement about the clique complex is about its geometric realization. Our
main focus is on those graphs whose clique complex is simply connected. Such
graphs form a fairly large class, for example, a connected chordal graph has
contractible clique complex.

###  2.2. Bestvina–Brady groups

######  Definition 2.1.

Let $\Gamma$ be a finite simplicial graph with the vertex set $V(\Gamma)$ and
the edge set $E(\Gamma)$. The right-angled Artin group $A_{\Gamma}$ associated
to $\Gamma$ has the following finite presentation:

$A_{\Gamma}=\bigl{\langle}V(\Gamma)\mid[v,w]=1\text{ for each edge }(v,w)\in
E(\Gamma)\bigr{\rangle}.$

Let $\phi\colon A_{\Gamma}\rightarrow\mathbb{Z}$ be the group homomorphism
sending all generators of $A_{\Gamma}$ to $1$. The Bestvina–Brady group
$H_{\Gamma}$ associated to $\Gamma$ is the kernel $\ker\phi$.

Some algebraic properties of $H_{\Gamma}$ can be seen directly from the
defining graph $\Gamma$ and $\triangle_{\Gamma}$, the flag complex on
$\Gamma$.

######  Theorem 2.2 ([2], Main Theorem).

Let $\Gamma$ be a finite simplicial graph.

  * (1)

$H_{\Gamma}$ is finitely generated if and only if $\Gamma$ is connected.

  * (2)

$H_{\Gamma}$ is finitely presented if and only if $\triangle_{\Gamma}$ is
simply-connected.

$u$$v$$w$$g$$f$$e$ Figure 1. A directed triangle.

Let us fix a linear order on the vertices, and orient the edges increasingly.
A triplet of edges $(e,f,g)$ forms a directed triangle if
$e=(u,v),f=(v,w),g=(u,w)$, and $u<v<w$; see Figure 1.

When $\Gamma$ is connected and $\triangle_{\Gamma}$ is simply connected, we
can write down a presentation for $H_{\Gamma}$, called the Dicks–Leary
presentation [7, Theorem 1, Corollary 3].

######  Theorem 2.3.

Suppose the flag complex $\triangle_{\Gamma}$ is simply connected. Then
$H_{\Gamma}$ has presentation

$H_{\Gamma}=\langle e\in E(\Gamma)\mid ef=fe,ef=g\text{ if
}\triangle(e,f,g)\text{ is a directed triangle }\rangle.$

Moreover, the inclusion $\iota\colon H_{\Gamma}\hookrightarrow A_{\Gamma}$ is
given by $\iota({e})=uv^{-1}$ for every edge ${e}=(u,v)$ of $\Gamma$.

The Dicks-Leary presentation is not necessarily a minimal presentation, i.e.,
there are some redundant generators. Dicks-Leary considered all the edges of
$\Gamma$ as the generators. The simpler presentation was given by Papadima-
Suciu in [8]. The authors proved that for the generators of $H_{\Gamma}$ it is
enough to consider the edges of a spanning tree of $\Gamma$.

######  Theorem 2.4 ([8, Corollary 2.3]).

If $\triangle_{\Gamma}$ is simply-connected, then $H_{\Gamma}$ has a
presentation $H_{\Gamma}=F/R$, where $F$ is the free group generated by the
edges of a spanning tree of $\Gamma$, and $R$ is a finitely generated normal
subgroup of the commutator group $[F,F]$.

Here are some examples of B-B groups.

######  Example 2.5.

If $\Gamma$ is a complete graph on $n$ vertices, then any spanning tree has
$n-1$ edges. Moreover, any two edges in a spanning tree form $2$ sides of a
triangle in $\Gamma$. The corresponding Bestvina–Brady group has $n-1$
generators and any two of them commute, hence it is $\mathbb{Z}^{n-1}$.

######  Example 2.6.

Now let $\Gamma$ be a tree on $n$ vertices. The spanning tree is the graph
$\Gamma$ itself and there are no triangles. The corresponding Bestvina–Brady
group has $n-1$ generators and none of them commute, hence it is $F_{n-1}$ the
free group on $n-1$ generators.

######  Example 2.7.

Let $\Gamma$ be the graph in Figure 2. Choosing a maximal tree
$T=\\{e_{1},\ldots,e_{5}\\}$ as indicated, the presentation of the B-B group
reads as follows:

$H_{\Gamma}=\langle
e_{1},\ldots,e_{5}\mid[e_{1},e_{2}],[e_{2},e_{3}],[e_{3},e_{4}],e_{5}{e_{2}}^{-1}e_{3}={e_{2}}^{-1}e_{3}e_{5}\rangle.$

Unlike the previous two examples, this particular $H_{\Gamma}$ is not
isomorphic to any right-angled Artin group (see [8, Proposition 9.4] for
details).

$v_{6}$$v_{4}$$v_{5}$$v_{3}$$v_{2}$$v_{1}$$e_{1}$$e_{2}$$e_{3}$$e_{4}$$e_{5}$
Figure 2.

Here we also recall the definition of an amalgamated product.

######  Definition 2.8.

Let $G_{1}$ and $G_{2}$ be groups with distinguished isomorphic subgroups
$H\leq G_{1}$ and $K\leq G_{2}$. Fix an isomorphism $\phi\colon H\rightarrow
K$. The free product of $G_{1}$ and $G_{2}$ with amalgamation of $H$ and $K$
by the isomorphism $\phi$ is the quotient of $G_{1}*G_{2}$ by the normal
closure of the set $\\{\phi(h)h^{-1}\mid h\in H\\}$. We will refer to this
factor group briefly as the amalgamated product and have the following
notations:

$\langle G_{1}*G_{2}\mid h=\phi(h),h\in
H\rangle,\phantom{aa}G_{1}*_{H=K}G_{2},\phantom{aa}G_{1}*_{H}G_{2}.$

##  3\. On the structure of B-B groups

As stated earlier the aim of this article is to describe the structure of
Bestvina-Brady groups and we do this, up to some extent, in this section.
However, we start by focusing at a class of B-B groups that have are in fact
isomorphic to some RAAG. These type of B–B groups can be recognized from the
graph defining the ambient RAAG. Then we move to the iterated amalgamated
product structure having the very first factor group isomorphic to a RAAG.
Most importantly, the iterated amalgamated product structure of the B–B group
can also be derived from the graph defining its ambient RAAG.

###  3.1. Isomorphism between RAAGs and B-B groups

Since the only relations in B–B groups are the commuting relations, for a
given $\Gamma$ it is natural to ask whether $H_{\Gamma}$ isomorphic to
$A_{\Gamma^{\prime}}$ for some finite simplicial graph $\Gamma^{\prime}$. This
question was first considered by Papadima and Suciu in [8]. The authors also
constructed a family of B-B groups not isomorphic to any RAAGs.

We briefly review their result; first we recall the definition of special
(Figure 5) and extra-special triangulation (Figure 3).

######  Definition 3.1.

A triangulation of the $2$-disk $D^{2}$ is said to be special if it can be
obtained from a triangle by adding one triangle at a time, along a unique
boundary edge. A triangulation of $D^{2}$ is called extra-special if it is
obtained from a special triangulation, by adding one triangle along each
boundary edge (see Figure 3).

Figure 3. Building an extra-special triangulation of the disk.

######  Proposition 3.2 ([8, Proposition 9.4]).

Let $\Gamma$ be the $1$-skeleton of an extra-special triangulation of $D^{2}$.
Then the corresponding Bestvina–Brady group $H_{\Gamma}$ is not isomorphic to
any Artin group.

Papadima and Suciu [8] also describe an explicit presentation of $H_{\Gamma}$,
where $\Gamma$ is the $1$-skeleton of a special triangulation of $D^{2}$.

######  Lemma 3.3.

Let $\Gamma=(V,E)$ be the $1$-skeleton of a special triangulation of $D^{2}$,
then:

  * (i)

$2|V|-|E|=3$,

  * (ii)

$H_{\Gamma}$ admits a presentation with $|V|-1$ generators and $|V|-2$
commutator relators.

We now identify a class of graphs such that the corresponding B-B group is
isomorphic to some RAAG. The proof is due to Chang, for the benefit of the
reader we also provide it here.

######  Theorem 3.4.

Let $\Gamma$ be a finite simplicial graph such that $\triangle_{\Gamma}$ is
simply-connected and $\Gamma$ has a spanning tree $T$ such that each triangle
of $\Gamma$ has either $0$ or $2$ edges in $E(T)$. Then $H_{\Gamma}\cong
A_{\Gamma^{\prime}}$ for some finite simplicial graph $\Gamma^{\prime}$.

###### Proof.

Let $T$ be a spanning tree of $\Gamma$ such that each triangle of $\Gamma$ has
either $0$ or $2$ edges in $E(T)$ and we denote
$E(T)=\\{e_{1},e_{2},\ldots,e_{n}\\}$. From Theorem 2.4, $E(T)$ is a
generating set for $H_{\Gamma}$. Since $H_{\Gamma}$ is finitely presented and
$E(T)$ contains $0$ or $2$ edges from each triangle, the relators of
$H_{\Gamma}$ are of the form $[e_{i},e_{j}]$. We construct the graph
$\Gamma^{\prime}$ as follows: the vertex set $V(\Gamma^{\prime})$ is $E(T)$;
and two vertices in $V(\Gamma^{\prime})$ are adjacent whenever the
corresponding edges form a triangle in $\Gamma$, i.e., they commute in
$H_{\Gamma}$. Let us denote
$V(\Gamma^{\prime})=\\{v_{1},v_{2},\ldots,v_{n}\\}$, and let $\phi\colon
E(T)\rightarrow V(\Gamma^{\prime})$ be a map sending each $e_{i}$ to $v_{i}$
for $i=1,2,\ldots,n$. Clearly, $\phi$ is a bijection. By the construction of
$\Gamma^{\prime}$, we have $\phi([e_{i},e_{j}])=[v_{i},v_{j}]$ and
$\phi^{-1}([v_{i},v_{j}])=[e_{i},e_{j}]$. Thus, $\phi$ is our desired
isomorphism between $H_{\Gamma}$ and $A_{\Gamma^{\prime}}$. ∎

######  Example 3.5.

Let $\Gamma=v\vee\Gamma^{\prime}$ be a cone graph with $v$ as its cone vertex.
A spanning tree can be chosen to be the star graph consisting of $v$,
$V(\Gamma^{\prime})$ and all the edges joining $v$ to each vertex of
$\Gamma^{\prime}$. Every edge in $\Gamma^{\prime}$ and the two edges that
connect each of its boundary vertices to $v$ form a triangle. Hence,
generators of $H_{\Gamma}$ correspond to vertices of $\Gamma^{\prime}$ and
commuting relations are given by the edges of $\Gamma^{\prime}$. Consequently,
$H_{\Gamma}\cong A_{\Gamma^{\prime}}$.

######  Example 3.6.

Let $\Gamma$ be the graph as shown in Figure 4. Let us choose $T$ to be the
spanning tree of $\Gamma$ that consists of edges $e_{1}$, $e_{2}$, $e_{3}$,
$e_{4}$ and $e_{5}$. By Theorem 3.4, $H_{\Gamma}$ is isomorphic to
$A_{\Gamma^{\prime}}$, where $\Gamma^{\prime}$ is a line graph with vertices
$w_{1},w_{2},w_{3},w_{4}\text{ and }w_{5}$ as shown on the right hand side of
Figure 4. Note that, $\Gamma$ is not a cone graph. But $H_{\Gamma}$ is still
isomorphic to $A_{\Gamma^{\prime}}$.

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A graph whose corresponding B-B group is a RAAG.

######  Example 3.7.

Let $\Gamma$ be the graph as shown in Figure 5. In [8], the authors showed
that $H_{\Gamma}$ is not isomorphic to any right-angled Artin group. Observe
that for any spanning tree $T$ of $\Gamma$, there is a triangle that has only
one edge in $E(T)$.

$v_{6}$$v_{4}$$v_{5}$$v_{3}$$v_{2}$$v_{1}$$e_{1}$$e_{2}$$e_{3}$$e_{5}$$e_{4}$
Figure 5. A graph whose corresponding B-B group is not a RAAG.

######  Note 3.8.

Recall that two RAAGs $A_{\Gamma_{1}}$ and $A_{\Gamma_{2}}$ are isomorphic if
and only if their underlying finite simplicial graphs $\Gamma_{1}$ and
$\Gamma_{2}$ are isomorphic. However, this is not the case for B-B groups. For
example, consider any two non-isomorphic trees on $n$ vertices for $n\geq 2$.
In both the cases the corresponding B-B group is $F_{n-1}$. Let $\Gamma_{1}$,
$\Gamma_{2}$ be cone graphs on $\Gamma^{\prime}_{1}\text{ and
}\Gamma^{\prime}_{2}$ respectively, then $\Gamma_{1}\cong\Gamma_{2}$ if and
only if $\Gamma^{\prime}_{1}\cong\Gamma^{\prime}_{2}$. So if we restrict
ourselves to cone graphs, then $\Gamma_{1}\cong\Gamma_{2}$ if and only if
$H_{\Gamma_{1}}\cong H_{\Gamma_{2}}$.

###  3.2. B-B groups as an iterated amalgamated product

In this section we will mainly focus on the triangles of $\Gamma$. More
precisely, we will concentrate on how a particular triangle $\triangle$ of
$\Gamma$ intersects its edge-set complement (see Def. 3.9). We introduce the
notion of favourable and unfavourable triangles with respect to the chosen
spanning tree $T$. Accordingly we also introduce the notion of favourable and
unfavourable graphs.

We start this subsection defining the aforementioned notions.

######  Definition 3.9.

Let $\Gamma$ be a connected graph and $\triangle$ be a triangle of $\Gamma$.
The edge-set complement of $\triangle$ is the induced subgraph of $\Gamma$
generated by $\big{(}\Gamma\setminus
E(\triangle)\big{)}~{}\setminus~{}\\{\text{isolated vertices}\\}$. We will
denote the edge-set complement of $\triangle$ in $\Gamma$ by
$\triangle_{\Gamma}^{c}$.

######  Definition 3.10.

A triangle $\triangle$ of $\Gamma$ is said to be a internal triangle if its
intersection with the edge-set complement $\triangle_{\Gamma}^{c}$ is neither
one vertex nor one edge.

######  Definition 3.11.

Let $T$ be a spanning tree of $\Gamma$. A triangle $\triangle$ of $\Gamma$ is
a favourable triangle with respect to $T$ if it has either $0$ or $2$ edges in
$E(T)$. Otherwise, we will say $\triangle$ is unfavourable.

######  Note 3.12.

Any graph $\Gamma$ can have several spanning trees. So, it may very well
happen that a triangle $\triangle$ is favourable in one spanning tree $T_{1}$
of $\Gamma$ but not favourable in another spanning tree $T_{2}$ of $\Gamma$.
Hence to talk about favourable triangles we have to fix a spanning tree $T$ of
$\Gamma$. The canonical (choice of a) spanning tree is the one with the
_maximal_ number of favourable triangles or equivalently, the the number of
unfavourable triangles is minimal.

######  Definition 3.13.

Let $\Gamma$ be a finite simplicial graph such that the flag complex on
$\Gamma$ is simply-connected. $\Gamma$ is said to be a favourable graph if
$\Gamma$ has a spanning tree $T$ such that each triangle of $\Gamma$ is
favourable triangle with respect to $T$. And if for every spanning tree $T$ of
$\Gamma$, there exists at least one unfavourable triangle of $\Gamma$ then
$\Gamma$ is called a unfavourable graph.

The graph on the left in Figure 4 is a favourable graph and the graph in
Figure 5 is a unfavourable graph.

In light of 3.4 if $\Gamma$ is a favourable graph, then the corresponding B-B
group is always isomorphic to a RAAG. It is important to note that a
“unfavourable” graph doesn’t mean that the corresponding B-B group is not
isomorphic to any RAAG.

Before going to our main result we will first prove a couple of Lemmas and
Propositions we need regarding internal triangles and unfavourable triangles.

######  Lemma 3.14.

If a triangle $\triangle$ of $\Gamma$ intersects the edge-set complement
$\triangle_{\Gamma}^{c}$ in one vertex, then it is a favourable triangle
irrespective of the choice of a spanning tree.

###### Proof.

Let the triangle $\triangle$ intersects $\triangle_{\Gamma}^{c}$ in one
vertex, say at the vertex $v$. Then, $\triangle$ is hanging from
$\triangle_{\Gamma}^{c}$ at $v$ like a pendant. In this set up, it is straight
forward to see that for any spanning tree $T$, $\triangle$ has exactly two
edges in $E(T)$. Hence from the definition, $\triangle$ is a favourable
triangle. ∎

For the iterated amalgamated product structure, we are mainly interested in
unfavourable graphs. We want to have an iterated amalgamation structure such
that the factor groups are RAAGs. Such a decomposition is helpful in
understanding various properties that are known for RAAGs and can pass through
the iteration of amalgamations. The class of graphs for which such an iterated
amalagamation exists is the following:

######  Definition 3.15.

Let $\mathcal{G}$ be a family of finite, simplicial, unfavourable graphs
$\Gamma$ with simply connected flag complex which in addition has a spanning
tree $T$ such that all the internal triangles are favourable with respect to
$T$.

To identify the class of graphs which belong to the family $\mathcal{G}$ we
use the notion of connected splitting from the literature (see [1] for more
details).

######  Definition 3.16.

Let $\Gamma_{1},\Gamma_{2},\Gamma_{3}$ be pairwise different induced subgraphs
of $\Gamma$ such that $\Gamma=\Gamma_{1}\cup\Gamma_{2}$ and
$\Gamma_{3}=\Gamma_{1}\cap\Gamma_{2}$. Then we say that
$(\Gamma_{1},\Gamma_{2},\Gamma_{3})$ is a splitting of $\Gamma$. We say that
$(\Gamma_{1},\Gamma_{2},\Gamma_{3})$ is a connected splitting of $\Gamma$ if
$\Gamma_{3}$ is connected.

The condition $\Gamma=\Gamma_{1}\cup\Gamma_{2}$ means the both
$V(\Gamma)=V(\Gamma_{1})\cup V(\Gamma_{2})$ and
$E(\Gamma)=~{}E(\Gamma_{1})\cup E(\Gamma_{2})$ (which we already discussed in
section 2). Also the condition $\Gamma_{3}=\Gamma_{1}\cap\Gamma_{2}$ means the
both $V(\Gamma_{3})=V(\Gamma_{1})\cap V(\Gamma_{2})$ and
$E(\Gamma_{3})=E(\Gamma_{1})\cap E(\Gamma_{2})$.

######  Lemma 3.17 ([1, Lemma 2.5]).

Let $(\Gamma_{1},\Gamma_{2},\Gamma_{3})$ be a splitting of $\Gamma$. Then we
have:

  * (i)

There is no edge between $V(\Gamma_{1})\setminus V(\Gamma_{3})$ and
$V(\Gamma_{2})\setminus V(\Gamma_{3})$;

  * (ii)

$\Gamma_{3}$ is separating;

  * (iii)

If $\Gamma$ and $\Gamma_{3}$ are connected, then $\Gamma_{1}$ and $\Gamma_{2}$
are connected too.

Here is a characterization of unfavourable graphs in $\mathcal{G}$ that we
use.

######  Theorem 3.18.

Let $\Gamma$ is a finite simplicial connected unfavourable graph such that
$\triangle_{\Gamma}$ is simply connected. Then $\Gamma\in\mathcal{G}$ if one
of the following conditions hold:

  1. (1)

$\Gamma$ is a $1$-skeleton of special triangulation (or extra-special
triangulation) of the $2$-disk.

  2. (2)

$\Gamma$ has a connected splitting $(\Gamma_{1},\Gamma_{2},\Gamma_{3})$ and
$\Gamma_{3}$ is a clique $K_{n}$, $n\geq 3$.

###### Proof.

Proof of (1): It is straightforward from the definition of special
triangulation or extra-special triangulation of $D^{2}$. If we consider an
unfavorable graph $\Gamma$ which is a $1$-skeleton of a special triangulation
or extra-special triangulation, we can always choose a spanning tree $T$ of
$\Gamma$ for which none of the internal triangle is unfavourable. This is
possible as we attach the triangles along the boundary for both the special
and extra-special triangulation.

Proof of (2): Let us suppose that $\Gamma$ has $\ell\geq 1$ many internal
triangles, say $\triangle_{1},\triangle_{2},\ldots,\triangle_{\ell}$. Our
claim is that each $\triangle_{i}$ is a favourable triangle, for
$i=1,2,\ldots,\ell$. We will prove our claim by induction on $\ell$.

First consider the case $\ell=1$, i.e., only one internal triangle, say
$\triangle$. Since, there is only one triangle like this, we can choose a
spanning tree $T$ that touches the vertices of $\triangle$ but not its edges.
Thus, $\triangle$ has $0$ edge in $E(T)$ and so favourable.

Assume that the claim holds for all graphs which have connected splittings
(with a separating clique) and have fewer than $\ell$ internal triangles. We
will consider the induced subgraphs $\Gamma_{1}^{\prime}\text{ and
}\Gamma_{2}^{\prime}$ of $\Gamma$ spanned by the vertices
$V(\Gamma_{1}\setminus\Gamma_{3})$ and $V(\Gamma_{2}\setminus\Gamma_{3})$
respectively. Let $V(\Gamma_{3})=V(K_{n})=\\{u_{1},u_{2},\ldots,u_{n}\\}$.
There exists an $u_{i}$ which is adjacent to a vertex of
$\Gamma_{1}^{\prime}$. Without loss of generality we can assume that it to be
$u_{1}$. Similarly, without loss of generality we can assume that $u_{2}$ is
adjacent to a vertex of $\Gamma_{2}^{\prime}$. Now let
$\Gamma_{1}^{\prime\prime}$ be the induced subgraph spanned by
$V(\Gamma_{1}^{\prime})\cup\\{u_{1}\\}$ and $\Gamma_{2}^{\prime\prime}$ be the
induced subgraph spanned by $V(\Gamma_{2}^{\prime})\cup\\{u_{2}\\}$. Then both
$\Gamma_{1}^{\prime\prime}$ and $\Gamma_{2}^{\prime\prime}$ have less than
$\ell$ internal triangles. By induction hypothesis,
$\Gamma_{1}^{\prime\prime}$ and $\Gamma_{2}^{\prime\prime}$ have spanning
trees $T_{1}^{\prime\prime}$ and $T_{2}^{\prime\prime}$ respectively for which
all the internal triangles are favourable. Since $\Gamma_{3}$ is a complete
graph, we can choose the spanning tree of $\Gamma_{3}$ as a star graph. Let
$T_{3}$ be the spanning tree of $\Gamma_{3}$. Hence for $T_{3}$ every triangle
of $\Gamma_{3}$ is favourable. Let us consider $T=T_{1}^{\prime\prime}\cup
T_{3}\cup T_{2}^{\prime\prime}$. Then $T$ is the desired spanning tree of
$\Gamma$ for which all the internal triangles are favourable. ∎

######  Lemma 3.19.

Let $\Gamma$ be a finite simplicial graph with simply-connected clique
complex. Then there is a group element in $H_{\Gamma}for$each edge $e\in
E(\Gamma)$.

###### Proof.

By abusing the notation we let $e$ denote the edge as well as the
corresponding group element. Let us first fix a spanning tree $T$ of $\Gamma$,
and $\iota(e)=u\text{ and }\tau(e)=v$. If $e\in E(T)$, then there is nothing
to prove. Let us suppose that $e$ is an edge not in $T$. Also, let
$A_{\Gamma}$ be the RAAG defined on $\Gamma$. Picking a path
$e_{1}^{\epsilon_{1}},\ldots,e_{r}^{\epsilon_{r}}$ in $T$ connecting $u$ to
$v$, we see that
$\iota(e)=\iota({e_{1}}^{\epsilon_{1}}\ldots{e_{r}}^{\epsilon_{r}})\in
A_{\Gamma}$, $\epsilon_{i}=\pm 1$. Clearly,
$e={e_{1}}^{\epsilon_{1}}\ldots{e_{r}}^{\epsilon_{r}}\in H_{\Gamma}$. ∎

######  Proposition 3.20.

Let $T$ be a spanning tree of $\Gamma$ and let $\triangle$ be an unfavourable
triangle of $\Gamma$ with respect to $T$ such that $\triangle$ intersects the
edge-set complement $\triangle_{\Gamma}^{c}$ exactly in one edge. Then
$H_{\Gamma}\cong H_{\triangle}*_{\mathbb{Z}}H_{\Gamma^{\prime}}$.

###### Proof.

The Bestvina–Brady group $H_{\Gamma}$ is finitely presented and its generators
are the edges of the spanning tree $T$. Let
$E(\triangle)=\\{e_{1},e_{2},e_{3}\\}$, since it is unfavourable we may assume
that $e_{1}\in E(T)$ and $e_{2},e_{3}\notin E(T)$. We know that
$H_{\triangle}$, the Bestvina–Brady group associated to $\triangle$ is a copy
of $\mathbb{Z}^{2}$ with the following presentation:

$H_{\triangle}=\bigl{\langle}e_{1},e_{2}\mid
e_{1}e_{2}=e_{2}e_{1}\bigr{\rangle}.$

Let $e_{2}$ be the edge common to both $\triangle$ and
$\triangle_{\Gamma}^{c}$. To avoid any confusion, we will denote $e_{2}$ by
$\overline{e_{2}}$ when seen as an edge of $\triangle_{\Gamma}^{c}$. Note that
the restriction of $T$ to $\triangle_{\Gamma}^{c}$ denoted by $T^{\prime}$ is
a spanning tree of $\triangle_{\Gamma}^{c}$. As, $\overline{e_{2}}\notin
E(T^{\prime})$, it is not a generator of $H_{\triangle_{\Gamma}^{c}}$.
However, $\langle\overline{e_{2}}\rangle$ is a subgroup of
$H_{\triangle_{\Gamma}^{c}}$. On the other hand $e_{2}$ is a generator of
$H_{\triangle}$ and of course, $\langle e_{2}\rangle$ is a subgroup of
$H_{\triangle}$.

Note that all generators and relators of $H_{\Gamma}$ are present in
$H_{\triangle_{\Gamma}^{c}}$, except the generator $e_{1}$ and the relations
involving $e_{1}$. So our first step is to include $e_{1}$ in some extension
of $H_{\triangle_{\Gamma}^{c}}$; which can be done by taking the free product
of $H_{\triangle_{\Gamma}^{c}}$ and $H_{\triangle}$. Thus
$H_{\triangle}*H_{\triangle_{\Gamma}^{c}}$ is generated by all the generators
of $H_{\Gamma}$ and $e_{2}$. So our next step is to get rid of $e_{2}$. Let us
consider the following amalgamated product:

$H_{\triangle}*_{\langle
e_{2}\rangle=\langle\overline{e_{2}}\rangle}H_{\triangle_{\Gamma}^{c}}=H_{\triangle}*_{\langle
e_{2}\rangle}H_{\triangle_{\Gamma}^{c}}.$

Thus we include $e_{1}$ in the generating set and at the same time we discard
the generator $e_{2}$. In the amalgamated free product one of the relations is
$e_{2}=\overline{e_{2}}$ and $\overline{e_{2}}$ can be expressed as a word in
terms of the other generators of $H_{\triangle_{\Gamma}^{c}}$; leaving $e_{2}$
redundant. Now comparing the presentations of $H_{\Gamma}$ and
$H_{\triangle}*_{\langle e_{2}\rangle}H_{\triangle_{\Gamma}^{c}}$, it is clear
that, $H_{\Gamma}\cong H_{\triangle}*_{\langle
e_{2}\rangle}H_{\triangle_{\Gamma}^{c}}\cong
H_{\triangle}*_{\mathbb{Z}}H_{\triangle_{\Gamma}^{c}}$. This completes the
proof. ∎

Now we describe the structure of B-B groups associated to graphs in
$\mathcal{G}$.

######  Theorem 3.21.

Let $\Gamma\in\mathcal{G}$. Then $H_{\Gamma}$ splits as an iterated
amalgamated product and in each iteration the amalgamated group is an infinite
cyclic group.

###### Proof.

As $\Gamma$ is in $\mathcal{G}$, we can choose a spanning tree $T$ with
respect to which $\Gamma$ has minimal number of unfavourable triangles and
none of them is internal. Let us enumerate these unfavourable triangles of
$\Gamma$ as $\\{\triangle_{1},\triangle_{2},\ldots,\triangle_{n}\\}$ and
denote by $e_{i}$ that edge of $\triangle_{i}$ which is in $E(T)$. We denote
the other two edges $\triangle_{i}$ by $f_{i}\text{ and }f^{\prime}_{i}$ and
they are not in $E(T)$. It follows from 3.14 that $\triangle_{i}$ intersects
the edge-set complement ${\triangle_{i}}_{\Gamma}^{c}$ in exactly one edge,
for each $i=1,2,\dots,n$.

We already know that the B–B group $H_{\triangle_{i}}$ associated to
$\triangle_{i}$ is finitely presented and the presentation is the following:

(3.1)
$H_{\triangle_{i}}=\bigl{\langle}e_{i},f_{i}|e_{i}f_{i}=f^{\prime}_{i}=f_{i}e_{i}\bigr{\rangle}\cong\mathbb{Z}^{2};\phantom{aaa}i=1,2,\ldots,n.$

Consider the subgraphs defined as follows:

(3.2) $\Gamma_{1}:={\triangle_{1}}_{\Gamma}^{c}\text{ and
}\Gamma_{i}:={\triangle_{i}}_{\Gamma_{i-1}}^{c},\phantom{aa}i=2,3,\ldots,n.$

Also note that, $\Gamma_{1}$ is the induced subgraph of $\Gamma$ and
$\Gamma_{i}$ is the induced subgraph of $\Gamma_{i-1}$.

By 3.20 we have,

(3.3) $H_{\Gamma}=H_{\Gamma_{1}}*_{\langle f_{1}\rangle}H_{\triangle_{1}}$

and

(3.4) $H_{\Gamma_{i-1}}=H_{\Gamma_{i}}*_{\langle
f_{i}\rangle}H_{\triangle_{i}}\phantom{aa}\hbox{for~{}}i=2,3,\ldots,n.$

Note that we choose the the restriction of $T$ to $\Gamma_{i}$ as the spanning
tree for $\Gamma_{i}$, for $i=1,2,3,\ldots,n$. From the construction and
Equation 3.2 it is clear that the graph
$\Gamma_{n}:={\triangle_{n}}_{\Gamma_{n-1}}^{c}$ is a favourable graph. So,
$H_{\Gamma_{n}}$ is isomorphic to $A_{\overline{\Gamma}}$ for some finite
simplicial graph $\overline{\Gamma}$, see 3.4. Recall that, the edges of
$T_{n}$ are the vertices of $\overline{\Gamma}$ and two such vertices are
adjacent whenever the corresponding edges form a triangle in $\Gamma_{n}$.

We can now express the B-B group $H_{\Gamma}$ as the following (iterated)
amalgamated product.

$H_{\Gamma}=\Bigg{(}\Big{(}(A_{\overline{\Gamma}}*_{\langle
f_{n}\rangle}H_{\triangle_{n}})*_{\langle
f_{n-1}\rangle}H_{\triangle_{n-1}}\Big{)}*_{\langle
f_{n-2}\rangle}H_{\triangle_{n-2}}\bigg{)}\cdots*_{\langle
f_{n-i}\rangle}H_{\triangle_{n-i}}\cdots\Bigg{)}*_{\langle
f_{1}\rangle}H_{\triangle_{1}}.$

This completes the proof. ∎

We now look at two examples. First we consider the graph in Fig. 5 which is an
extra-special triangulation. Recall that, from [8, Proposition 9.4] it follows
that the corresponding B-B group is not isomorphic to a RAAG.

######  Example 3.22.

Our choice of spanning tree is $T=\\{e_{1},\dots,e_{5}\\}$. Hence the
presentation for the B-B groups is

$H_{\Gamma}=\langle
e_{1},\dots,e_{5}\mid[e_{2},e_{3}],[e_{2},e_{4}],[e_{4},e_{5}],e_{1}e_{2}e_{4}^{-1}=e_{2}e_{4}^{-1}e_{1}\rangle.$

Note that the triangle $\triangle_{1}$ spanned by $\\{v_{1},v_{2},v_{5}\\}$ is
unfavourable with respect to $T$. Let $\Gamma_{1}$ be the edge-set complement
of $\triangle_{1}$. It is not hard to see that $\Gamma_{1}$ is favourable
graph; in fact, $\\{e_{2},e_{3},e_{4},e_{5}\\}$ is a spanning tree with
respect to which all the three triangles are favourable. Consequently,

$H_{\Gamma}=A_{\Gamma_{1}}\ast_{\langle f_{1}\rangle}H_{\triangle_{1}},$

where $f_{1}$ denotes the (edge and the) group element corresponding to
$(v_{2},v_{5})$. The group $A_{\Gamma_{1}}$ is an example of RAAG which is
free-by-cyclic and $H_{\triangle_{1}}$ is isomorphic to $\mathbb{Z}^{2}$.

Now we consider an example where the underlying graph is not an extra-special
triangulation.

######  Example 3.23.

Let us consider $\Gamma$ in Figure 6. We choose the following spanning tree
$T=\\{e_{1},e_{2},\ldots,e_{11}\\}$. Note that there are two non-favourable
triangles $\triangle(v_{1},v_{2},v_{5})=~{}\triangle_{1}$ and
$\triangle(v_{10},v_{11},v_{9})=\triangle_{2}$. In fact, the reader can verify
that for any choice of spanning tree, there will be at least two unfavourable
triangles.

We have, $E(\triangle_{1})=\\{e_{1},f_{1},f_{1}^{\prime}\\}\text{ where
}e_{1}\in~{}E(T)\text{ and }f_{1},f_{1}^{\prime}\notin E(T)$ and
$E(\triangle_{2})=\\{e_{10},f_{2},f_{2}^{\prime}\\}\text{ where
}e_{10}\in~{}E(T)\text{ and }f_{2},f_{2}^{\prime}\notin~{}E(T)$. Let us denote
$\Gamma_{1}:={\triangle_{1}}_{\Gamma}^{c}$ (see Figure 7) and
$\Gamma_{2}:={\triangle_{2}}_{\Gamma_{1}}^{c}$ (see Figure 8). Hence,
$\Gamma_{1}$ is the induced subgraph of $\Gamma$ and $\Gamma_{2}$ is the
induced subgraph of $\Gamma_{1}$. We have:

$\displaystyle H_{\triangle_{1}}$
$\displaystyle=\bigl{\langle}e_{1},f_{1}|e_{1}f_{1}=f^{\prime}_{1}=f_{1}e_{1}\bigr{\rangle}\cong\mathbb{Z}^{2}.$
$\displaystyle H_{\triangle_{2}}$
$\displaystyle=\bigl{\langle}e_{10},f_{2}|e_{10}f_{2}=f^{\prime}_{2}=f_{2}e_{10}\bigr{\rangle}\cong\mathbb{Z}^{2}.$

Thus

(3.5) $H_{\Gamma}=H_{\Gamma_{1}}*_{\langle f_{1}\rangle}H_{\triangle_{1}},$

and,

(3.6) $H_{\Gamma_{1}}=H_{\Gamma_{2}}*_{\langle
f_{2}\rangle}H_{\triangle_{2}}.$

Note that $\Gamma_{2}$ is a favourable graph. So, $H_{\Gamma_{2}}\cong
A_{\overline{\Gamma}}$ where

$A_{\overline{\Gamma}}=\langle
w_{2},\ldots,w_{9},w_{11}\mid[w_{2},w_{3}],[w_{2},w_{4}],[w_{4},w_{5}],[w_{7},w_{8}],[w_{8},w_{9}],[w_{9},w_{11}]\rangle.$

Thus finally we have the desired iterated amalgamation structure of
$H_{\Gamma}\colon$

$H_{\Gamma}=(A_{\overline{\Gamma}}*_{\langle
f_{2}\rangle}H_{\triangle_{2}})*_{\langle
f_{1}\rangle}H_{\triangle_{1}}\cong(A_{\overline{\Gamma}}*_{\mathbb{Z}}\mathbb{Z}^{2})*_{\mathbb{Z}}\mathbb{Z}^{2}.$

$v_{6}$$v_{4}$$v_{5}$$v_{3}$$v_{2}$$v_{1}$$v_{13}$$v_{14}$$v_{15}$$e_{1}$$f_{1}$$e_{2}$$e_{3}$$e_{5}$$f_{1}^{\prime}$$e_{4}$$v_{7}$$v_{8}$$v_{9}$$v_{12}$$v_{11}$$v_{10}$$e_{10}$$f_{2}$$e_{9}$$e_{11}$$e_{7}$$f_{2}^{\prime}$$e_{8}$$e_{6}$
Figure 6. The graph $\Gamma$.
$v_{6}$$v_{4}$$v_{5}$$v_{3}$$v_{2}$$v_{1}$$v_{13}$$v_{14}$$v_{15}$$f_{1}$$e_{2}$$e_{3}$$e_{5}$$e_{4}$$v_{7}$$v_{8}$$v_{9}$$v_{12}$$v_{11}$$v_{10}$$e_{10}$$f_{2}$$e_{9}$$e_{11}$$e_{7}$$f_{2}^{\prime}$$e_{8}$$e_{6}$
Figure 7. The graph $\Gamma_{1}$.
$v_{6}$$v_{4}$$v_{5}$$v_{3}$$v_{2}$$v_{1}$$v_{13}$$v_{14}$$v_{15}$$f_{1}$$e_{2}$$e_{3}$$e_{5}$$e_{4}$$v_{7}$$v_{8}$$v_{9}$$v_{12}$$v_{11}$$v_{10}$$f_{2}$$e_{9}$$e_{11}$$e_{7}$$e_{8}$$e_{6}$
Figure 8. The graph $\Gamma_{2}$.

We end the article with a non-example.

######  Example 3.24.

Consider the graph in Fig. 9, which is the $3$-fold join of two isolated
vertices. The RAAG corresponding to this graph is the direct product of three
copies of $F_{2}$ and the B-B group is the same group that appeared in the
seminal paper of Stallings. It is a finitely presented group which is not of
type $\mathbf{FP}_{3}$. The reader can easily verify that no matter which
spanning tree is chosen there are at least two internal triangles that are
unfavourable. In fact, any $k$-fold join of two isolated vertices has this
property; so these graphs do not belong to the class $\mathcal{G}$.

Figure 9. A graph not in the class $\mathcal{G}$.

## References

  * [1] E. M. Barquinero, L. Ruffoni, K. Ye, “Graphical splittings of Artin kernels”, Journal of Group Theory vol. 24, no. 4, (2021), pp. 711-735.
  * [2] M. Bestvina and N. Brady, “Morse theory and finiteness properties of groups”, Invent. Math. 129(3) (1997), 445–470.
  * [3] Y. Chan-Chang, “Abelian splittings and JSJ-Decompositions of Bestvina–Brady groups”, preprint _arXiv:2003.09927_.
  * [4] R. Charney, “An introduction to right-angled Artin groups”, Geom. Dedicata 125 (2007), 141–158.
  * [5] M. Clay, “When does a right-angled Artin group split over Z?”, Internat. J. Algebra Comput., 24(6):815–825, 2014.
  * [6] D. Groves, M. Hull, “Abelian splittings of right-angled Artin groups”, In Hyperbolic geometry and geometric group theory, volume 73 of Adv. Stud. Pure Math., pages 159–165. Math. Soc. Japan, Tokyo, 2017.
  * [7] W. Dicks and I. J. Leary, “Presentations for subgroups of Artin groups”, Proc. Amer. Math. Soc. 127(2) (1999), 343–348.
  * [8] S. Papadima and A. Suciu, “Algebraic invariants for Bestvina-Brady groups”, J. Lond. Math. Soc. (2), 76(2) (2007), 273–292.


xv,\mathsf{H}(v)\rangle=-\frac{1}{2\pi}\int_{0}^{\infty}\mathrm{Re}[\overline{\hat{v}}(\xi)\partial_{\xi}\hat{v}(\xi)]\,d\xi+\frac{1}{2\pi}\int_{-\infty}^{0}\mathrm{Re}[\overline{\hat{v}}(\xi)\partial_{\xi}\hat{v}(\xi)]\,d\xi=\frac{1}{2\pi}|\hat{v}(0)|^{2}\
,$

which is (C.1).
Since $\mathsf{H}$ preserves real valued functions, we have

$\mathrm{Re}\,\langle\partial_{x}v,\mathsf{H}(|v|^{2})v\rangle=\langle\mathrm{Re}[\overline{v}\partial_{x}v],\mathsf{H}(|v|^{2}\rangle=\frac{1}{2}\langle\partial_{x}(|v|^{2}),\mathsf{H}(|v|^{2}\rangle=-\frac{1}{2}\langle|v|^{2},\mathsf{H}\partial_{x}(|v|^{2}\rangle\
,$

which leads to (C.2) since $\mathsf{H}\partial_{x}=|D|$.
Finally, setting $\varrho:=|v|^{2}$, we have
$\varrho=\Pi_{+}\varrho+\overline{\Pi_{+}\varrho}$ and

$\|v\|_{L^{6}}^{6}=\int_{\mathbb{R}}\varrho^{3}\,dx=\int_{\mathbb{R}}(\Pi_{+}\varrho+\overline{\Pi_{+}\varrho})^{3}\,dx=3\int_{\mathbb{R}}[(\Pi_{+}\varrho)^{2}\overline{\Pi_{+}\varrho}+\Pi_{+}\varrho(\overline{\Pi_{+}\varrho})^{2}]\,dx\
,$

because $H^{s}_{\pm}$ is preserved by the product (if $s$ is large enough) and
the ranges of $\Pi_{+}$ and of $\Pi_{-}$ are orthogonal. We obtain

$\int_{\mathbb{R}}\varrho^{3}\,dx=3\int_{\mathbb{R}}|\Pi_{+}\varrho|^{2}\varrho\,dx=\frac{3}{4}\int_{\mathbb{R}}(\varrho^{2}+(H\varrho)^{2})\varrho\,dx\
,$

whence

$\int_{\mathbb{R}}\varrho^{3}\,dx=3\int\varrho(H\varrho)^{2}\,dx\ ,$

which is precisely (C.3).

###  C.1. Derivation of Pohozaev Identity

In this paragraph we provide the details for the proof of Proposition 4.1 when
deriving the Pohozaev type identities. We integrate (4.11) against
$x\partial_{x}\overline{S}$ over the compact interval $[-R,R]$. By taking the
real part, we find

$\mathrm{Re}\int_{-R}^{R}\left(-x\overline{S}^{\prime}S^{\prime\prime}-x\overline{S}^{\prime}(|D||S|^{2})S+\frac{1}{4}x\overline{S}^{\prime}|S|^{4}S\right)dx=\tilde{\omega}\mathrm{Re}\int_{-R}^{R}x\overline{S}^{\prime}S\,dx.$

For the first term on the left-hand side, we find

$\displaystyle\mathrm{Re}\int_{-R}^{R}x\overline{S}^{\prime}S^{\prime\prime}\,dx$
$\displaystyle=\frac{1}{2}\int_{-R}^{R}x\partial_{x}|S^{\prime}|^{2}\,dx=\frac{1}{2}x|S^{\prime}(x)|^{2}\Big{|}_{-R}^{R}-\frac{1}{2}\int_{-R}^{R}|S^{\prime}(x)|^{2}\,dx.$

In view of $|D|=\mathsf{H}\partial_{x}$ and integrating by parts, we obtain

$\displaystyle\mathrm{Re}\int_{-R}^{R}x\overline{S}^{\prime}(|D||S|^{2})S\,dx=\frac{1}{2}\int_{-R}^{R}x\partial_{x}|S|^{2}(\mathsf{H}\partial_{x}|S|^{2})\,dx$
$\displaystyle\to\frac{1}{2\pi}\left(\int_{-\infty}^{\infty}\partial_{x}|S|^{2}\,dx\right)^{2}=0\quad\mbox{as}\quad
R\to+\infty,$

where we also used (C.1). Next, we notice

$\mathrm{Re}\,\int_{-R}^{R}x\partial_{x}\overline{S}|S|^{4}S\,dx=\frac{1}{6}x|S(x)|^{6}\Big{|}_{-R}^{R}-\frac{1}{6}\int_{-R}^{R}|S|^{6}\,dx$
$\tilde{\omega}\mathrm{Re}\int_{-R}^{R}x\partial_{x}\overline{S}S\,dx=\frac{\tilde{\omega}}{2}x|S(x)|^{2}\Big{|}_{-R}^{R}-\frac{\tilde{\omega}}{2}\int_{-R}^{R}|S(x)|^{2}\,dx.$

Since $|S|^{2}+|S^{\prime}|^{2}\in L^{1}(\mathbb{R})$, there exists a sequence
$R_{n}\to+\infty$ such that $x(|S(x)|^{2}+|S^{\prime}(x)|^{2})\to 0$ with
$x=\pm R_{n}$ as $n\to\infty$ and we obtain

$\frac{1}{2}\int_{\mathbb{R}}|\partial_{x}S|^{2}\,dx-\frac{1}{24}\int_{\mathbb{R}}|S|^{6}\,dx=-\frac{\tilde{\omega}}{2}\int_{\mathbb{R}}|S|^{2}\,dx.$

###  C.2. Gauge Transformation and Pseudo-Conformal Law

Let $s\geq 0$ be given. We consider the nonlinear map

$\Phi:H^{s}(\mathbb{R})\to H^{s}(\mathbb{R}),\quad u(x)\mapsto
u(x)\mathrm{e}^{-\frac{i}{2}\int_{-\infty}^{x}|u(y)|^{2}\,dy}.$ (C.4)

Clearly, we have mass preservation property $M[\Phi(u)]=M(u)$ and it can be
shown that $\Phi:H^{s}(\mathbb{R})\to H^{s}(\mathbb{R})$ is a diffeomorphism.
We refer to the map $\Phi$ as the gauge transform. We remark that the map
$\Phi$ (with different numerical factors in the exponential term) also plays
an important role for the derivative NLS (DNLS).

Suppose $u(t)\in H^{s}(\mathbb{R})$ (not necessarily restricted to
$H^{s}_{+}(\mathbb{R})$) solves (CM-DNLS) on some time interval
$I\subset\mathbb{R}$. Then $v(t)=\Phi(u(t))\in H^{s}(\mathbb{R})$ is found to
solve the derivative type NLS equation:

$i\partial_{t}v=-\partial_{xx}v-(|D||v|^{2})v+\frac{1}{4}|v|^{4}v,$ (C.5)

using that $\Pi_{+}=\frac{1}{2}(1+i\mathsf{H})$ and
$|D|=\mathsf{H}\partial_{x}$. We readily check that (C.5) has the conserved
energy

$\widetilde{E}(v)=\frac{1}{2}\int_{\mathbb{R}}|\partial_{x}v|^{2}-\frac{1}{4}\int_{\mathbb{R}}|v|^{2}(|D||v|^{2})+\frac{1}{24}\int_{\mathbb{R}}|v|^{6}.$
(C.6)

From identities (C.2) and (C.3), we have

$\widetilde{E}(v)=\frac{1}{2}\int_{\mathbb{R}}|\partial_{x}v+\frac{1}{2}\mathsf{H}(|v|^{2})v|^{2}=\frac{1}{2}\int_{\mathbb{R}}|\partial_{x}u-i\Pi_{+}(|u|^{2})u|^{2}=E(u)$
(C.7)

since $v=\Phi(u)$ and $\Pi_{+}=\frac{1}{2}(1+i\mathsf{H})$.

Now, let $\Sigma=\\{u\in H^{1}(\mathbb{R}):xu\in L^{2}(\mathbb{R})\\}$ denote
the space of solutions with finite variance and energy (not necessarily
restricted to the Hardy space).

######  Lemma C.1 (Pseudo-Conformal Law).

Suppose that $u\in C([-T,T];H^{1}(\mathbb{R}))$ solves (CM-DNLS) with
$u(0)=u_{0}\in\Sigma$. Then $u(t)\in\Sigma$ for all $t\in[-T,T]$ and we have

$\frac{d^{2}}{dt^{2}}\int_{\mathbb{R}}|x|^{2}|u(t,x)|^{2}\,dx=16E(u_{0})\quad\mbox{for
$t\in[-T,T]$}.$

As a consequence, it holds that

$8t^{2}E(\mathrm{e}^{i|x|^{2}/4t}u_{0})=\int_{\mathbb{R}}|x|^{2}|u(t,x)|^{2}\,dx$

for all $t\in[-T,T]$ with $t\neq 0$.

######  Remarks.

1) Recall that the ground state $\mathcal{R}(x)=\frac{\sqrt{2}}{x+i}\in
H^{1}_{+}(\mathbb{R})$ does not belong to $\Sigma$. In fact, it can be shown
that all multi-solitons for (CM-DNLS) fail to have finite variance as well.

2) In view of the non-negativity $E(u)\geq 0$, we see that the classical
Zakharov–Glassey argument to prove existence for negative energy data (with
finite variance) cannot be invoked for (CM-DNLS).

###### Proof.

Let $v(t)=\Phi(u(t))$ for $t\in[-T,T]$. We readily check that
$\Phi:\Sigma\to\Sigma$ holds and, in particular, we have
$v_{0}=v(0)\in\Sigma$. Recall that $v\in C([-T,T];H^{1}(\mathbb{R}))$ solves
(C.5). By following arguments for standard-type NLS, see, e. g., [10][Section
7.6], we can deduce that $v(t)\in\Sigma$ for all $t\in[-T,T]$. Moreover, by
calculations analogous to $L^{2}$-critical NLS (see [10] again), we find the
variance-virial identities

$\frac{d}{dt}\int_{\mathbb{R}}|x|^{2}|v(t,x)|^{2}\,dx=4\int_{\mathbb{R}}x\mathrm{Im}(\overline{v}\partial_{x}v)\,dx,$
(C.8)
$\quad\frac{d^{2}}{dt^{2}}\int_{\mathbb{R}}|x|^{2}|v(t,x)|^{2}\,dx=16\widetilde{E}_{0},$
(C.9)

with $v_{0}=\Phi(u_{0})$ and $\widetilde{E}_{0}=\widetilde{E}(v_{0})$. By
integration in $t\in[-T,T]$, we obtain that the variance
$V(t)=\int_{\mathbb{R}}|x|^{2}|v(t,x)|^{2}\,dx$ is given by

$V(t)=8\widetilde{E}_{0}t^{2}+A_{0}t+V_{0}$ (C.10)

with the constants
$A_{0}=4\int_{\mathbb{R}}x\mathrm{Im}(\overline{v}_{0}\partial_{x}v_{0})\,dx$
and $V_{0}=V(0)$. For $t\in[-T,T]$ with $t\neq 0$, we observe

$\displaystyle 8t^{2}\widetilde{E}(\mathrm{e}^{i|x|^{2}/4t}v_{0})$
$\displaystyle=8t^{2}\left(\frac{1}{2}\int_{\mathbb{R}}|\partial_{x}(\mathrm{e}^{i|x|^{2}/4t}v_{0})|^{2}-\frac{1}{4}\int_{\mathbb{R}}|v_{0}|^{2}(|D||v_{0}|^{2})+\frac{1}{24}\int_{\mathbb{R}}|v_{0}|^{6}\right)$
$\displaystyle=8t^{2}\widetilde{E}_{0}+A_{0}t+V_{0}=V(t).$

Finally, we go back to the function $u=u(t,x)$. Here we note that
$|u(t,x)|=|v(t,x)|$ and $\widetilde{E}(v_{0})=E(u_{0})$ since
$v(t)=\Phi(u(t))$. This proves this first claim in Lemma C.1. For the second
statement, we remark that $\Phi$ commutes with multiplication by
$\mathrm{e}^{i|x|^{2}/4t}$ for $t\neq 0$, i. e., we have
$\Phi(\mathrm{e}^{i|x|^{2}/4t}u_{0})=\mathrm{e}^{i|x|^{2}/4t}\Phi(u_{0})=\mathrm{e}^{i|x|^{2}/4t}v_{0}$.
Therefore
$\widetilde{E}(\mathrm{e}^{i|x|^{2}|4t}v_{0})=E(\mathrm{e}^{i|x|^{2}/4t}u_{0})$.
This completes the proof. ∎

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