# Presentations of the saturated cluster modular groups of finite mutation
type $X_{6}$ and $X_{7}$

Tsukasa Ishibashi  Graduate School of Mathematical Sciences, the University of
Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan<EMAIL_ADDRESS>
###### Abstract.

We give finite presentations of the saturated cluster modular groups of type
$X_{6}$ and $X_{7}$. Using these presentations we compute the abelianizations
of these groups and compare with known examples. Further we verify that the
cluster modular group of type $X_{7}$ is generated by cluster Dehn twists.

###### Contents

  1. 0 Introduction and main results
  2. 1 Basic definitions
  3. 2 Brown’s algorithm for a group acting on a CW complex
  4. 3 Presentations of $\widehat{\Gamma}_{X_{7}}$ and $\widehat{\Gamma}_{X_{6}}$
  5. A Relations with the mapping class group of an annulus
  6. B Cluster Dehn twists in other examples of cluster modular groups

##  0\. Introduction and main results

A _cluster modular group_ , defined in [FG09a], is a discrete group
$\Gamma_{|{\mathbf{s}}|}$ associated with a combinatorial data ${\mathbf{s}}$
called a _seed_. An element of a cluster modular group is a finite sequence of
_seed permutations_ and _mutations_ , which preserve the _exchange matrix_ and
induce a pair of non-trivial _cluster transformations_. The cluster modular
group acts on the _cluster modular complex_ ${\mathcal{M}}_{|{\mathbf{s}}|}$.
The quotient ${\mathcal{M}}_{|{\mathbf{s}}|}/\Gamma_{|{\mathbf{s}}|}$ is
called the _modular orbifold_ , which is a combinatorial generalization of the
moduli space of Riemann surfaces. Indeed, the modular orbifold coincides with
the latter space when the seed is associated with an ideal triangulation of a
closed surface with one puncture [Pen12]. This relation justifies the study of
the $\Gamma_{|{\mathbf{s}}|}$-action on the cluster modular complex. It can be
thought as a study of _higher moduli spaces_ in the context of _higher
Teichmüller theory_ [FG06b].

However, there are only restricted examples of cluster modular groups whose
complete presentations are determined. Known examples of infinite cluster
modular groups are: (tagged) mapping class groups [BS15], Thompson’s group
$\mathbb{T}$ [FG09a], the braid group of type $G_{2}$ [FG06a], and so on. Our
goal is to provide new examples of such cluster modular groups. One difficulty
arises from the fact that the relations among cluster transformations are not
completely known in general. In many cases they are exhausted by _standard
relations_ [FG09a], while there are “non-standard” relations in some cases,
even for a seed associated to a marked surface. See Remark 9.19 of [FST08]. A
survey on this problem is also found in [KY16]. In this paper, we consider the
_saturated cluster modular group_ ${\widehat{\Gamma}}_{|{\varepsilon}|}$
[FG09a] instead. It is determined by an exchange matrix and relations are
given by the standard ones. The cluster modular group is a quotient of the
saturated cluster modular group. The saturated cluster modular group naturally
appears in several studies:

  * •

The fundamental group of the modular orbifold
${\mathcal{M}}_{|{\mathbf{s}}|}/\Gamma_{|{\mathbf{s}}|}$ is isomorphic to the
saturated cluster modular group [FG09a].

  * •

The Fock-Goncharov quantization [FG09b] provides a projective unitary
representation of the saturated cluster modular group.

  * •

The Kato-Terashima partition q-series [KT15] gives a map from the saturated
cluster modular group to the ring $\mathbb{Z}[[q^{1/\Delta}]]$ of formal power
series for some integer $\Delta$.

For any seed, we introduce the _saturated modular complex_ on which the
saturated cluster modular group acts, and show that it is simply-connected.
Therefore we can extract a presentation of the saturated cluster modular group
from the data of this action.

It is also difficult to find a generating set of a (saturated) cluster modular
group in general, but we can find a finite generating set (and a finite
presentation) when the seed is of _finite mutation type_. The mutation-
equivalence classes of quivers of finite mutation type are listed in [FST12b,
FST12a]. The list consists of the quivers associated with ideal triangulations
of marked surfaces, the quivers obtained from several generalized Dynkin
diagrams, and two mysterious quivers called $X_{6}$ and $X_{7}$. The quivers
of type $X_{6}$ and $X_{7}$ are shown in Figure 1. Our main theorems are
computations of finite presentations of the saturated cluster modular groups
associated with these seeds.

$1$$2$$3$$4$$5$$6$$0$$1$$2$$3$$4$$5$$6$

Figure 1. Quivers of type $X_{6}$ and $X_{7}$

######  Theorem 1.

The saturated cluster modular group ${\widehat{\Gamma}}_{X_{7}}$ of type
$X_{7}$ is generated by $\psi_{k}$, $\phi_{k}$ $(k=1,3,5)$ and the permutation
group $\mathcal{S}_{3}$ of numbers $\\{1,3,5\\}$, and the complete set of
relations among them is given as follows:

$\displaystyle\sigma\psi_{k}\sigma^{-1}$ $\displaystyle=\psi_{\sigma(k)},$
$\displaystyle\sigma\phi_{k}\sigma^{-1}$
$\displaystyle=\phi_{\sigma(k)}~{}\text{ for $\sigma\in\mathcal{S}_{3}$,
$k=1,3,5$},$ $\displaystyle\phi_{1}\phi_{3}$ $\displaystyle=\phi_{3}\phi_{1},$
$\displaystyle\phi_{1}$ $\displaystyle=\psi_{1}^{2}\sigma_{35},$
$\displaystyle 1$ $\displaystyle=(\psi_{3}^{-1}\psi_{1})^{2},$ $\displaystyle
1$ $\displaystyle=\sigma_{153}\psi_{5}\psi_{1}\psi_{3}\psi_{5}\psi_{1},$

and the usual relations of permutations. Here $\sigma_{ij}$ denotes the
tranposition of $i$ and $j$ and $\sigma_{135}$ denotes the cyclic permutation
$1\mapsto 3\mapsto 5\mapsto 1$.

We investigate a geometric origin of some of these relations in Appendix A.

######  Theorem 2.

The saturated cluster modular group ${\widehat{\Gamma}}_{X_{6}}$ of type
$X_{6}$ is generated by three elements $\alpha$, $\beta$ and $\sigma$, and the
complete set of relations among them is given as follows:

$\displaystyle\sigma^{2}$ $\displaystyle=1,$
$\displaystyle[\sigma,\beta\alpha]^{3}$ $\displaystyle=1,$
$\displaystyle[\sigma,\beta]^{2}$ $\displaystyle=1,$
$\displaystyle(\alpha^{-1}\sigma)^{2}$ $\displaystyle=[\sigma,\alpha],$
$\displaystyle[\beta,\alpha]$
$\displaystyle=[\sigma,\alpha]=[\sigma,\beta]\beta[(\beta\alpha)^{-1},\sigma]\alpha^{-1},$
$\displaystyle\beta$ $\displaystyle=\alpha[\sigma,\beta][\beta\alpha,\sigma].$

Felikson-Shapiro-Thomas-Tumarkin [FSTT14] constructed an embedding of the free
group $\mathbb{F}_{2}$ into $\Gamma_{X_{6}}$. It would be interesting to
describe it in terms of our generator.

Theorems 1 and 2 would provide useful test examples for a general theory of
cluster modular groups such as Nielsen-Thurston classification [Ish17].

###  0.1. First homology groups

Let $H_{1}(G;\mathbb{Z})=G/[G,G]$ be the first homology group
(=abelianization) of a group $G$.

######  Corollary 3.

We have
$H_{1}({\widehat{\Gamma}}_{X_{7}};\mathbb{Z})\cong\mathbb{Z}/5\times\mathbb{Z}/2$.
The generators are the images of $\psi_{1}$ and $\sigma_{13}$.

###### Proof.

It is well-known that the signature function gives an isomorphism
$H_{1}(\mathfrak{S}_{n};\mathbb{Z})\xrightarrow{\cong}\mathbb{Z}/2$. From the
first relation in Theorem 1 we get $\psi_{1}=\psi_{3}=\psi_{5}$ in the
abelianization, and the last relation implies $\psi_{1}^{5}=1$. ∎

######  Corollary 4.

We have
$H_{1}({\widehat{\Gamma}}_{X_{6}};\mathbb{Z})\cong\mathbb{Z}/2\times\mathbb{Z}/2$.
The generators are the images of $\alpha$ and $\sigma$.

###### Proof.

From the fourth and sixth relations in Theorem 2, we get $\alpha^{2}=1$ and
$\beta=\alpha$ in the abelianization. ∎

Corollary 3 implies that the saturated cluster modular group of type $X_{7}$
is different from mapping class groups of surfaces, Thompson’s group
$\mathbb{T}$, and Artin-Tits braid groups. Indeed, the pure mapping class
group of a marked surface $\Sigma$ is perfect (_i.e._ , the abelianization is
trivial) when the genus $>2$ (cf. [FM12, Theorem 5.2]). Hence from the exact
sequences

$\displaystyle 1\to PMC(\Sigma)\to MC(\Sigma)\to\mathfrak{S}(M)\to 1,$
$\displaystyle 1\to\\{\pm 1\\}^{p}\to MC^{\bowtie}(\Sigma)\to MC(\Sigma)\to
1,$

one can see that the abelianization of a (tagged) mapping class group is
either trivial or consists of $\mathbb{Z}/2$-components. See Example 1.3 for
notations. The group $\mathbb{T}$ is known to be simple (cf. [CFP96]) and
braid groups have no torsion in the abelianization. Hence
${\widehat{\Gamma}}_{X_{7}}$ is a new example of (saturated) cluster modular
group with a complete presentation. It requires a future research to
distinguish ${\widehat{\Gamma}}_{X_{6}}$ from tagged mapping class groups.

###  0.2. Cluster Dehn twists

_Cluster Dehn twists_ in cluster modular groups were introduced in the
author’s previous work [Ish17] and their dynamical behavior was investigated.
An element $\phi\in\Gamma_{|{\mathbf{s}}|}$ of infinite order is a _cluster
Dehn twist_ if there exists a seed ${\mathbf{t}}\in|{\mathbf{s}}|$ such that
$\phi^{n}=((i\ j)\mu_{j})^{\ell}$ as an element of $\Gamma_{\mathbf{t}}$ for
some integer $n,\ell$ and vertices $i,j\in I-I_{0}$. In fact the generator
$\phi_{k}$ (resp. $\psi_{k}$) is a cluster Dehn twist with $n=1$ (resp.
$n=4$). The following is a generalization of the classical theorem that the
mapping class group of a punctured surface is generated by (half-)Dehn twists
(cf. [FM12]).

######  Corollary 5.

The saturated cluster modular group ${\widehat{\Gamma}}_{X_{7}}$, and hence
the cluster modular group $\Gamma_{X_{7}}$, is generated by the six cluster
Dehn twists $\phi_{k}$ and $\psi_{k}$ $(k=1,3,5)$.

###### Proof.

We can delete the permutations using the relations
$\sigma_{35}=\psi_{1}^{-2}\phi_{1}$, $\sigma_{51}=\psi_{3}^{-2}\phi_{3}$ and
$\sigma_{13}=\psi_{5}^{-2}\phi_{5}$. Hence the saturated cluster modular group
is generated by $\psi_{k}$’s and $\phi_{k}$’s. ∎

For a geometric origin of these cluster Dehn twists, see Appendix A. We
believe that the following analogous statement for type $X_{6}$ holds.

######  Conjecture 6.

The saturated cluster modular group ${\widehat{\Gamma}}_{X_{6}}$, and hence
the cluster modular group $\Gamma_{X_{6}}$, is generated by cluster Dehn
twists.

Indeed, a computation using the software GAP shows that the normal closure of
a cluster Dehn twist coincides with the whole group
${\widehat{\Gamma}}_{X_{6}}$, but we don’t know explicit cluster Dehn twist
generators. We confirm in Appendix B that similar statements for some other
cluster modular groups hold.

###  0.3. Organization of the paper

In § 1, we recall the definition of the saturated cluster modular groups. We
introduce the saturated modular complexes and prove that they are simply-
connected. In § 2, we recall Brown’s algorithm [Bro84] which enables us to
compute a presentation of a group acting on a CW complex. We investigate
relevant properties of the action of the saturated cluster modular group on
the saturated modular complex in full generality. In § 3, we compute the
presentations of the saturated cluster modular groups of type $X_{7}$ and
$X_{6}$ in this order. In Appendix A, we discuss several relations between
these saturated cluster modular groups and the mapping class group of an
annulus. In Appendix B, we confirm that some of known cluster modular groups
are generated by cluster Dehn twists.

Acknowledgement. The author would like to express his gratitude to his
supervisor, Nariya Kawazumi for continuous encouragement during this work. He
is very grateful to Travis Scrimshaw for pointing out an error in the
presentation of ${\widehat{\Gamma}}_{X_{6}}$ in the first version of this
paper. Most of computations in this paper are done by using the _Java applet
for quiver mutations_ provided by Bernhard Keller, which is available at
http://www.math.lsa.umich.edu/~fomin/cluster.html. This work is partially
supported by JSPS KAKENHI Grant Number 18J13304 and the program for Leading
Graduate School, MEXT, Japan.

##  1\. Basic definitions

###  1.1. Seed mutations

In this section we review seed mutations in order to motivate the definitions
of saturated cluster modular group and saturated modular complex. We follow
the notations of [FG09a].

Let $I$ be a finite set. A _seed_ is a tuple
${\mathbf{s}}=({\varepsilon},(A_{i})_{i\in I},(X_{i})_{i\in I})$, where
${\varepsilon}=({\varepsilon}_{ij})_{i,j\in I}$ is a
$\frac{1}{2}\mathbb{Z}$-valued skew-symmetrizable matrix, $(A_{i})_{i\in I}$
and $(X_{i})_{i\in I}$ are two bunches of algebraically independent
commutative variables. Our convention of skew-symmetrization is that there
exist positive integers $d_{j}$ ($j\in I$) such that the matrix
$({\varepsilon}_{ij}d_{j}^{-1})_{i,j}$ is skew-symmetric. The matrix
${\varepsilon}$ is called the _exchange matrix_ and the variables $(A_{i})$
(resp. $(X_{i})$) are called the cluster ${\mathcal{A}}$\- (resp.
${\mathcal{X}}$-)variables. We fix a subset $I_{0}\subset I$ called the
_frozen subset_ , and assume ${\varepsilon}_{ij}\in\mathbb{Z}$ unless $i,j\in
I_{0}$.

The exchange matrix ${\varepsilon}$ can be represented by a weighted quiver
without loops and 2-cycles. The corresponding quiver has the set of vertices
$I$ and each vertex $i\in I$ is assigned a weight $d_{i}$, and there exist
$|d_{j}^{-1}{\varepsilon}_{ij}|\mathrm{gcd}(d_{i},d_{j})$ arrows from the
vertex $i$ to the vertex $j$ (resp. $j$ to $i$) if ${\varepsilon}_{ij}>0$
(resp. ${\varepsilon}_{ij}<0$). See [Zic16] for details. We tacitly use this
correspondence in the sequel. In this sense we call an element of $I$ a
_vertex_ and elements of $I_{0}$ are called _frozen vertices_.

######  Definition 1.1 (seed mutations).

For a seed ${\mathbf{s}}=({\varepsilon},(A_{i})_{i\in I},(X_{i})_{i\in I})$
and a vertex $k\in I\setminus I_{0}$, we define a new seed
${\mathbf{s}}^{\prime}=({\varepsilon}^{\prime},(A^{\prime}_{i})_{i\in
I},(X^{\prime}_{i})_{i\in I})$ as follows:

$\displaystyle{\varepsilon}^{\prime}_{ij}$
$\displaystyle:=\begin{cases}-{\varepsilon}_{ij}&\text{if
$k\in\\{i,j\\}$},\vspace{2mm}\\\
{\varepsilon}_{ij}+\displaystyle\frac{|{\varepsilon}_{ik}|{\varepsilon}_{kj}+{\varepsilon}_{ik}|{\varepsilon}_{kj}|}{2}&\text{
otherwise}.\end{cases}$ (1.1) $\displaystyle A_{i}^{\prime}$
$\displaystyle:=\begin{cases}A_{i}^{-1}(\prod_{{\varepsilon}_{kj}>0}A_{j}^{{\varepsilon}_{kj}}+\prod_{{\varepsilon}_{kj}<0}A_{j}^{-{\varepsilon}_{kj}})&\text{if
$i=k$},\\\ A_{i}&\text{otherwise}.\end{cases}$ (1.2) $\displaystyle
X_{i}^{\prime}$ $\displaystyle:=\begin{cases}X_{k}^{-1}&\text{if $i=k$},\\\
X_{i}(1+X_{k}^{-\mathrm{sign}({\varepsilon}_{ik}}))^{-{\varepsilon}_{ik}}&\text{otherwise}.\end{cases}$
(1.3)

We write ${\mathbf{s}}^{\prime}=\mu_{k}({\mathbf{s}})$ and refer to this
transformation of seeds as the _mutation directed to the vertex $k$_. The
transformation (1.1) is independent of the other data and called the _matrix
mutation_ (or _quiver mutation_). The rational transformations (1.2) and (1.3)
are called the _${\mathcal{A}}$ \- and ${\mathcal{X}}$-cluster
transformations_.

A _seed permutation_ is a permutation $\sigma$ of $I$ which preserves $I_{0}$
setwise. It acts on a seed as
$\sigma({\varepsilon},(A_{i}),(X_{i}))=({\varepsilon}^{\prime},(A^{\prime}_{i}),(X^{\prime}_{i}))$,
where

${\varepsilon}^{\prime}_{ij}:={\varepsilon}_{\sigma^{-1}(i),\sigma^{-1}(j)},~{}A^{\prime}_{i}:=A_{\sigma^{-1}(i)},~{}X^{\prime}_{i}:=X_{\sigma^{-1}(i)}.$

It is called a _seed isomorphism_ if it satisfies
${\varepsilon}^{\prime}={\varepsilon}$. A _mutation sequence_ is a finite
composition of mutations and seed permutations. A mutation sequence is called
a _mutation loop_ if it preserves the initial exchange matrix. A mutation loop
is said to be _trivial_ if it also preserves the initial cluster variables.
The set $|{\mathbf{s}}|:=\\{\phi({\mathbf{s}})\mid\text{$\phi$: mutation
sequence with the initial seed ${\mathbf{s}}$}\\}$ is called the _mutation
class_ of ${\mathbf{s}}$. The set $|{\varepsilon}|$ of exchange matrices (or
weighted quivers) appearing in $|{\mathbf{s}}|$ is called the mutation class
of ${\varepsilon}$.

The _cluster modular group_ $\Gamma_{\mathbf{s}}$ based at a seed
${\mathbf{s}}$ is the group of mutation loops with the initial seed
${\mathbf{s}}$, modulo trivial ones. Since the group structure of the cluster
modular group only depends on the mutation class $|{\mathbf{s}}|$, we will use
the notation $\Gamma_{|{\mathbf{s}}|}$ when no confusion can occur.

######  Remark 1.2.

The cluster modular group acts on some geometric objects
${\mathcal{A}}_{|{\mathbf{s}}|}$ and ${\mathcal{X}}_{|{\mathbf{s}}|}$ called
the _cluster ${\mathcal{A}}-$ and ${\mathcal{X}}-$varieties_, preserving their
geometric structures. See [FG09a] for details.

######  Example 1.3.

Here is a geometric example. Let $\Sigma$ be a marked surface. It is a
connected oriented compact 2-dimensional manifold equipped with a finite
subset $M\subset\Sigma$ of marked points, satisfying some conditions. See
[FST12a] for details. An ideal triangulation of $\Sigma$ is the isotopy class
of a collection $\Delta=\\{\alpha_{i}\\}_{i=1}^{n(\Sigma)}$ of simple arcs
whose endpoints are marked points, and the complement
$\Sigma\setminus\bigcup_{i=1}^{n(\Sigma)}\alpha_{i}$ consists of triangles.
Here $n(\Sigma)$ only depends on $\Sigma$.

Given such an ideal triangulation $\Delta$, we draw a quiver on each triangle
as shown in Figure 2. Gluing them along the edges, we get a quiver drawn on a
surface. The vertices on $\partial\Sigma$ are declared to be frozen. For
example, a torus with one marked point and its ideal triangulation yields a
quiver whose exchange matrix is given by

${\varepsilon}=\begin{pmatrix}0&2&-2\\\ -2&0&2\\\ 2&-2&0\end{pmatrix}.$

Hence we can form a seed ${\mathbf{s}}_{\Delta}$. Here cluster variables can
be interpreted as coordinate function on Teichmüller spaces of $\Sigma$: the
${\mathcal{A}}$-variables are $\lambda$-length coordinates on the _decorated
Teichmüller space_ and the ${\mathcal{X}}$-variables are cross-ratio
coordinates on the _enhanced Teichmüller space_. See [Pen12] for details. We
have

$\Gamma_{{\mathbf{s}}_{\Delta}}\cong\begin{cases}MC(\Sigma)&\text{if $\Sigma$
is a closed surface with one marked point},\\\ MC(\Sigma)\ltimes\\{\pm
1\\}^{p}&\text{otherwise}.\end{cases}$

Here $p$ is the number of interior marked points (punctures) and
$MC(\Sigma):=\pi_{0}\mathrm{Homeo}^{+}(\Sigma,M)$ denotes the _mapping class
group_ of $\Sigma$ whose action on $\\{\pm 1\\}^{p}$ is induced by the
permutation of punctures. See [BS15, Ish17]. The _pure mapping class group_
$PMC(\Sigma)$ consists of elements of $MC(\Sigma)$ which fix each marked
point. We write $MC^{\bowtie}(\Sigma):=MC(\Sigma)\ltimes\\{\pm 1\\}^{p}$ and
call it the _tagged mapping class group_.

Figure 2. The quiver associated with a triangle

The definition of the cluster modular group contains an equivalence relation
which comes from those among mutations. In general it is difficult to list up
the generators of relations, while we know the following “standard” ones:

######  Lemma 1.4 (standard relations [FG09a]).

Let ${\mathbf{s}}=({\varepsilon},(A_{i})_{i\in I},(X_{i})_{i\in I})$ be a seed
and $(p,h)=(0,2)$, $(1,3)$, $(2,4)$ or $(3,6)$. Then for each $i,j\in I$ such
that ${\varepsilon}_{ij}=-p{\varepsilon}_{ji}=p$, the mutation sequence
$r_{ij}:=((i\ j)\mu_{i})^{h+2}=((i\ j)\mu_{j})^{h+2}$ is trivial. Here $(i\
j)$ denotes the seed permutation given by the transposition of $i$ and $j$.

We call these sequences $r_{ij}(=r_{ji})$ the _standard sequences_. Note that
if the exchange matrix is skew-symmetric, which is the case we treat mainly in
this paper, we have only two of the standard relations:

  * •

$\mu_{i}\mu_{j}=\mu_{j}\mu_{i}$ if ${\varepsilon}_{ij}=0$. (Square relation)

  * •

$\mu_{i}\mu_{j}\mu_{i}\mu_{j}\mu_{i}=(i\ j)$ if ${\varepsilon}_{ij}=\pm 1$.
(Pentagon relation)

######  Example 1.5.

Here is a basic example of the pentagon relation in cluster modular group. Let
$I:=\\{0,1\\}$, $I_{0}:=\emptyset$ and a skew-symmetric exchange matrix

${\varepsilon}:=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix},$

which is called the exchange matrix of type $A_{2}$. The mutation sequence
$\phi:=(0\ 1)\mu_{0}$ gives an element of the cluster modular group. It turns
out that it is the generator of the cluster modular group. The associated
cluster transformations are given by:

$\displaystyle\phi(A_{0},A_{1})$
$\displaystyle=\left(A_{1},\frac{1+A_{1}}{A_{0}}\right),$
$\displaystyle\phi(X_{0},X_{1})$ $\displaystyle=(X_{1}(1+X_{0}),X_{0}^{-1}).$

Then the pentagon relation implies that $\phi$ has order $5$, or one can check
it directly. In particular we have $\Gamma_{A_{2}}\cong\mathbb{Z}/5$.

###  1.2. Saturated cluster modular groups and saturated modular complexes

In this section we recall the definition of the _saturated cluster modular
group_ , which is called the _special cluster modular group_ in [FG09a]. Also
we introduce the _saturated cluster complex_ on which the saturated cluster
modular group acts.

######  Definition 1.6.

The _saturated cluster modular group_ $\widehat{\Gamma}_{\varepsilon}$ based
at an exchange matrix ${\varepsilon}$ is the group of (matrix) mutation loops
with the initial exchange matrix ${\varepsilon}$, modulo the equivalence
relation generated by:

  1. (1)

Involutivity: $\mu_{k}\mu_{k}=1$ for each $k\in I$,

  2. (2)

Naturality: $\sigma\mu_{k}\sigma^{-1}=\mu_{\sigma(k)}$ for each $k\in I$ and a
seed permutation $\sigma$, and

  3. (3)

Standard relations: $r_{ij}=r_{ji}=1$ under the situation of Lemma 1.4.

Take a seed ${\mathbf{s}}=({\varepsilon},(A_{i})_{i\in I},(X_{i})_{i\in I})$
with the exchange matrix ${\varepsilon}$. Then Lemma 1.4 implies that we have
a surjective homomorphism
$\widehat{\Gamma}_{\varepsilon}\to\Gamma_{\mathbf{s}}$.

To investigate the saturated cluster modular group, we define a 2-complex on
which the saturated cluster modular group acts simplicially. Let
${\varepsilon}=({\varepsilon}_{ij})_{i,j\in I}$ be an exchange matrix and set
$n:=|I\setminus I_{0}|$. A labeled $n$-regular tree $T$ is an (infinite)
$n$-valent tree with a labeling $\\{\text{edges of $T$}\\}\to\\{1,\dots,n\\}$
which induces a bijection $\\{\text{edges incident to
$v$}\\}\xrightarrow{\sim}\\{1,\dots,n\\}$ for each vertex $v$ of $T$. Choose a
vertex $v_{0}$ of $T$ and assign a matrix to each vertex of $T$ by the
following rules:

  * •

The matrix assigned to $v_{0}$ is the initial exchange matrix ${\varepsilon}$.

  * •

The matrices assigned to a pair of vertices connected by an edge labeled by
$k$ are related by the matrix mutation $\mu_{k}$.

Let $T_{|{\varepsilon}|}$ denotes the $n$-regular tree equipped with such an
assignment of exchange matrices on the vertices. Let $D$ be the subgroup of
$\mathrm{Aut}(T_{|{\varepsilon}|})$ which consists of elements which preserve
the assigned matrices.

######  Lemma 1.7.

_We have a natural surjective homomorphism
$\Phi_{v_{0}}:D\to\widehat{\Gamma}_{{\varepsilon}}$_.

###### Proof.

For an element $\gamma\in D$, the two matrices assigned to $v_{0}$ and
$\gamma^{-1}(v_{0})$ are the same. Let $p(\gamma)=e_{i_{k}}\dots e_{i_{1}}$ be
the unique edge path in $T_{|{\varepsilon}|}$ from $v_{0}$ to
$\gamma^{-1}(v_{0})$, where an edge $e_{i}$ has the labeling $i$. Then there
exists a seed isomorphism $\sigma=\sigma(\gamma)$ from ${\varepsilon}$ to
$\mu_{i_{k}}\dots\mu_{i_{1}}({\varepsilon})$ such that
$\gamma(e_{i})=e_{\sigma(i)}$ for all $i\in I$. For another element
$\gamma^{\prime}\in D$, let $p(\gamma^{\prime})=e_{j_{l}}\dots e_{j_{1}}$ be
the edge path from $v_{0}$ to $\gamma^{\prime-1}(v_{0})$. Then
$\gamma^{-1}(p(\gamma^{\prime}))=e_{\sigma^{-1}(j_{l})}\dots
e_{\sigma^{-1}(j_{1})}$ is the edge path from $\gamma^{-1}(v_{0})$ to
$(\gamma^{\prime}\gamma)^{-1}(v_{0})$. Hence we have

$\displaystyle\Phi_{v_{0}}(\gamma^{\prime})\Phi_{v_{0}}(\gamma)$
$\displaystyle=\sigma(\gamma^{\prime})\mu_{j_{l}}\dots\mu_{j_{1}}\sigma(\gamma)\mu_{i_{k}}\dots\mu_{i_{1}}$
$\displaystyle=\sigma(\gamma^{\prime})\sigma(\gamma)\mu_{\sigma^{-1}(j_{l})}\dots\mu_{\sigma^{-1}(j_{1})}\mu_{i_{k}}\dots\mu_{i_{1}}$
$\displaystyle=\sigma(\gamma^{\prime}\gamma)\mu_{\sigma^{-1}(j_{l})}\dots\mu_{\sigma^{-1}(j_{1})}\mu_{i_{k}}\dots\mu_{i_{1}}$
$\displaystyle=\Phi_{v_{0}}(\gamma^{\prime}\gamma).$

Thus the map $\Phi_{v_{0}}:D\to\widehat{\Gamma}_{{\varepsilon}}$ given by
$\gamma\mapsto{\sigma\mu_{i_{k}}\dots\mu_{i_{1}}}$ is a group homomorphism. It
is clearly surjective. ∎

The graph
${\widehat{\mathbb{E}}}_{|{\varepsilon}|}:=T_{|{\varepsilon}|}/\mathrm{ker}\Psi_{v_{0}}$
is called the _saturated exchange graph_. The saturated cluster modular group
acts on ${\widehat{\mathbb{E}}}_{|{\varepsilon}|}$ as graph automorphisms via
$\Phi_{v_{0}}$. We write
$\widehat{\Gamma}_{v_{0}}:=\Phi_{v_{0}}^{-1}(\widehat{\Gamma}_{{\varepsilon}})\subset\mathrm{Aut}({\widehat{\mathbb{E}}}_{|{\varepsilon}|})$.

######  Remark 1.8.

  1. (1)

An element $\gamma$ of $D$ belongs to the subgroup $\mathrm{ker}\Psi_{v_{0}}$
if and only if there exists a vertex $v$ equipped with ${\varepsilon}$ and the
edge path from $v$ to $\gamma(v)$ is given by a concatenation of standard
sequences.

  2. (2)

If we change the basepoint of $T$ to a vertex $v_{1}$ which is connected with
$v_{0}$ by an edge $e_{k}$, then we have
$\Psi_{v_{1}}=\mathrm{Ad}_{\mu_{k}}\circ\Psi_{v_{0}}:D\to\widehat{\Gamma}_{\mu_{k}({\varepsilon})}$.
Here $\mathrm{Ad}_{\mu_{k}}$ denotes the conjugation by $\mu_{k}$.

The _exchange graph_ of the mutation class $|{\mathbf{s}}|$ is defined as
follows [FZ07]. Starting with a $n$-regular tree $T$ with a fixed vertex
$v_{0}$, we assign a seed to each vertex of $T$ by the following rules:

  * •

The seed assigned to $v_{0}$ is the initial seed ${\mathbf{s}}$.

  * •

The seeds assigned to a pair of vertices connected by an edge labeled by $k$
are related by the seed mutation $\mu_{k}$.

We identify two vertices with the same seeds, and the resulting graph
$\mathbb{E}_{|{\mathbf{s}}|}$ is the exchange graph. From Lemmas 1.4 and 1.8
we have a natural covering map
${\widehat{\mathbb{E}}}_{|{\varepsilon}|}\to\mathbb{E}_{|{\mathbf{s}}|}$.

######  Lemma 1.9.

The group homomorphism
$\widehat{\Gamma}_{|{\varepsilon}|}\to\Gamma_{|{\mathbf{s}}|}$ is an
isomorphism if and only if the covering map
${\widehat{\mathbb{E}}}_{|{\varepsilon}|}\to\mathbb{E}_{|{\mathbf{s}}|}$ is a
homeomorphism.

###### Proof.

Both conditions are equivalent to the condition that relations among mutations
are generated by standard ones. ∎

######  Definition 1.10 (saturated modular complex).

To each cycle in ${\widehat{\mathbb{E}}}_{|{\varepsilon}|}$ which is given by
a standard sequence, we attach a 2-cell given by the corresponding standard
polygon. The resulting 2-complex $\widehat{\mathcal{M}}_{|{\varepsilon}|}$ is
called the _saturated modular complex_.

Here is a simple but important lemma:

######  Lemma 1.11.

The saturated modular complex $\widehat{\mathcal{M}}_{|{\varepsilon}|}$ is
simply-connected.

###### Proof.

From Remark 1.8, each combinatorial cycle $\alpha$ in
$\widehat{\mathcal{M}}_{|{\varepsilon}|}$ is a concatenation of standard
cycles. Since each standard cycle spans a 2-cell given by the corresponding
standard polygon, $\alpha$ can be contracted to a point along these 2-cells.
Therefore $\widehat{\mathcal{M}}_{|{\varepsilon}|}$ is simply-connected. ∎

#### Seeds of finite mutation type

An exchange matrix ${\varepsilon}$ is said to be of _finite mutation type_ if
the mutation class $|{\varepsilon}|$ is a finite set. If ${\varepsilon}$ is
mutation-equivalent to a Dynkin quiver, then it is said to be of _finite
type_. Otherwise it is of _infinite type_.

######  Theorem 1.12 (Fomin-Zelevinsky [FZ03]).

For a seed ${\mathbf{s}}=({\varepsilon},(A_{i})_{i\in I},(X_{i})_{i\in I})$,
the underlying exchange matrix ${\varepsilon}$ is of finite type if and only
if the mutation class $|{\mathbf{s}}|$ is a finite set.

Indeed, the latter condition is the original definition of a seed of finite
type. An important point is that whether a seed is of finite type is
determined by its exchange matrix. The following theorems give the
classification of exchange matrices of finite mutation type.

######  Theorem 1.13 (Felikson-Shapiro-Tumarkin [FST12b]).

Any skew-symmetric exchange matrix with size $\geq 3$ of finite mutation type
is either obtained by an ideal triangulation of a marked surface or contained
in one of the eleven mutation classes: $E_{6}$, $E_{7}$, $E_{8}$,
$\tilde{E}_{6}$, $\tilde{E}_{7}$, $\tilde{E}_{8}$, $E_{6}^{(1,1)}$,
$E_{7}^{(1,1)}$, $E_{8}^{(1,1)}$, $X_{6}$, $X_{7}$.

The quivers representing the eleven mutation classes are given in Figure 6.1
of [FST12b]. The first three are of finite type.

######  Theorem 1.14 (Felikson-Shapiro-Tumarkin [FST12a]).

An exchange matrix with size $\geq 3$ which is not skew-symmetric, is of
finite mutation type if and only if it is s-decomposable or contained in one
of the seven mutation classes: $\tilde{G}_{2}$, $F_{4}$, $\tilde{F}_{4}$,
$G_{2}^{(*,+)}$, $G_{2}^{(*,*)}$, $F_{4}^{(*,+)}$, $F_{4}^{(*,*)}$

The quivers representing the seven mutation classes are given in Figure 1.1 of
[FST12a]. Only $F_{4}$ is of finite type. Here an exchange matrix is said to
be _s-decomposable_ if it is obtained by gluing certain “blocks”, see
[FST12a]. We say that a seed ${\mathbf{s}}=({\varepsilon},(A_{i})_{i\in
I},(X_{i})_{i\in I})$ is of finite mutation type if the underlying exchange
matrix is.

######  Lemma 1.15.

If a seed ${\mathbf{s}}$ is of finite mutation type and infinite type, then
the cluster modular group $\Gamma_{|{\mathbf{s}}|}$ is a finitely generated
infinite group.

###### Proof.

In this case we have infinitely many seeds whose underlying exchange matrix
coincides with the initial one ${\varepsilon}$. Each such seed gives rise to a
distinct element of the cluster modular group, hence the cluster modular group
is infinite. The cluster modular group is finitely generated by Lemma 3.6. ∎

##  2\. Brown’s algorithm for a group acting on a CW complex

Let us recall Brown’s algorithm [Bro84] with a suitable specialization for our
purpose. Let $G$ be a group acting on a simply-connected CW complex $X$
preserving the cell structure. It is known that if the action has no fixed
points, then $G\cong\pi_{1}(X)$. It is not true under existence of a fixed
point (which corresponds to existence of a quiver isomorphism in our cluster
setting). Brown’s algorithm enables us to compute a presentation of $G$ even
in this situation.

We prepare some notations and terminology. The set of $k$-cells of $X$ is
denoted by $C_{k}(X)$. A 1-cell $\sigma\in C_{1}(X)$ is said to be _inverted_
if there exists an element $g\in G$ which fixes $\sigma$ and reverses the
orientation of $\sigma$. If $\sigma$ is not a loop in $X$, which is the case
we will deal with, the condition means that $g$ transposes two endpoints of
$\sigma$. In the following, we will assume the following condition:

Each 1-cell of $X$ has distinct endpoints and is not inverted. (2.1)

A subtree $T$ of $X$ is called a _tree of representatives_ for $X$ mod $G$ if
the vertices of $T$ form a set of representatives for the vertices of $X$ mod
$G$. Such a tree always exists and the 1-cells of $T$ are inequivalent mod
$G$. By an _edge_ of $X$ we mean an oriented 1-cell of $X$. Let us denote the
set of edges of $X$ by $E(X)$, which is again a $G$-set.

Let $\pi:E(X)\to C_{1}(X)$ be the projection forgetting the orientation. A
$G$-equivariant section $P:C_{1}(X)\to E(X)$ is called an _orientation_ of
$X$. Such an orientation exists under the assumption (2.1). The initial (resp.
terminal) vertex of an edge $e\in E(X)$ is denoted by $o(e)$ (resp. $t(e)$).
We represent an edge $e$ by the symbol $(o(e)\xrightarrow{e}t(e))$.

For a 2-cell in $X$, the attaching map $\partial\Delta^{2}\to X^{(1)}$ is
represented by a combinatorial cycle $\alpha=e_{1}\dots e_{n}$ such that
$o(e_{1})=t(e_{n})\in C_{0}(T)$. Here $\Delta^{2}$ denotes the standard
2-simplex, and we read the combinatorial cycle from the left to the right by
convention.

######  Definition 2.1.

A choice of the following data $(T,P,E^{+},F)$ is called a _data of
representatives_ for the action of $G$ on $X$:

  1. (1)

$T\subseteq X^{(1)}$ is a tree of representatives for $X$ mod $G$.

  2. (2)

$P:C_{1}(X)\to E(X)$ is an orientation of $X$.

  3. (3)

$E^{+}\subseteq P(C_{1}(X))$ is a set of representatives mod $G$ such that
$P(C_{1}(T))\subseteq E^{+}$ and $o(e)\in V$ for each edge $e$ in $E^{+}$.

  4. (4)

$F\subseteq C_{2}(X)$ is a set of representatives mod $G$ together with a
choice of a combinatorial cycle representing the attaching map for each
2-cell.

Fixing a data of representatives, we can compute a presentation of the group
$G$. For each edge $e\in E^{+}$, there is a unique vertex $w_{e}\in C_{0}(T)$
which is $G$-equivalent to the terminal vertex $t(e)$. Choose an element
$g_{e}\in G$ such that

$t(e)=g_{e}(w_{e}).$ (2.2)

Here we choose $g_{e}:=1\in G$ if $e$ is contained in $T$ by convention. These
elements together with the elements of the isotropy group $G_{v}:=\\{g\in
G|g(v)=v\\}$ ($v\in C_{0}(T)$) form a generating set of $G$. We call them
_Brown generators_. We shall describe the relations among the Brown
generators. Note that each edge $e\in E(X)$ such that $v:=o(e)\in V$ has one
of the following forms:

  1. (a)

$e=(v\xrightarrow{he}hg_{e}w_{e})$, where $e\in E^{+}$ and $h\in G_{v}$.

  2. (b)

$e=(w_{e}\xrightarrow{hg_{e}^{-1}\bar{e}}hg_{e}^{-1}v)$, where $e\in E^{+}$,
$\bar{e}$ is the opposite edge and $h\in G_{w_{e}}$.

Then we set

$g:=\begin{cases}hg_{e}&\text{(case(a))}\\\
hg_{e}^{-1}&\text{(case(b)).}\end{cases}$

For a combinatorial cycle $\alpha=e_{1}\dots e_{n}$ such that
$o(e_{1})=t(e_{n})\in C_{0}(T)$, the above rule provides us with elements
$g_{1}^{\tau},\dots,g_{n}^{\tau}\in G$ such that $t(e_{i})\in
g_{1}^{\tau}\dots g_{i}^{\tau}(C_{0}(T))$. Let $g^{\tau}:=g_{1}^{\tau}\dots
g_{n}^{\tau}$. Note that $g^{\tau}\in G_{o(e_{1})}$.

######  Theorem 2.2 (Brown, [Bro84]).

Let $G$ be a group acting on a simply-connected CW complex $X$ preserving the
cell structure. Assume the condition (2.1). Then $G$ is generated by the
elements $g_{e}$ ($e\in E^{+}$) and the isotropy groups $G_{v}$ ($v\in
C_{0}(T)$), subject to the following relations:

  1. (1)

$g_{e}=1$ if $e$ is contained in $T$.

  2. (2)

$g_{e}^{-1}i_{e}(g)g_{e}=c_{e}(g)$ for any $e\in E^{+}$ and $g\in G_{e}$,
where $i_{e}:G_{e}\to G_{o(e)}$ is the inclusion and $c_{e}:G_{e}\to
G_{w_{e}}$ is defined by $g\mapsto g_{e}^{-1}gg_{e}$. $[$Isotropy relations
$]$

  3. (3)

$g^{\tau}=g_{1}^{\tau}\dots g_{n}^{\tau}$ for any $\tau\in F$. $[$Face
relations $]$

##  3\. Presentations of $\widehat{\Gamma}_{X_{7}}$ and
$\widehat{\Gamma}_{X_{6}}$

Let ${\varepsilon}=({\varepsilon}_{ij})_{i,j\in I}$ be an exchange matrix. Let
$V:=C_{0}({\widehat{\mathcal{M}}}_{|{\varepsilon}|})$ be the set of vertices
of the saturated modular complex. The following several lemmas will be
employed in the computations of presentations. First note that we have a
natural surjective map $[\ ]:V\to|{\varepsilon}|$ which extracts the exchange
matrix assigned to each vertex.

######  Lemma 3.1.

The natural map $[\
]:V/{\widehat{\Gamma}}_{|{\varepsilon}|}\to|{\varepsilon}|$ gives a bijection.

###### Proof.

The surjective map $[\ ]:V\to|{\varepsilon}|$ descends to a map $[\
]:V/{\widehat{\Gamma}}_{|{\varepsilon}|}\to|{\varepsilon}|$, since the action
of the saturated cluster modular group preserves the assigned exchange
matrices. If two vertices $v_{0}$ and $v_{1}$ are assigned the same matrices,
then there exists an element $\gamma\in\mathrm{Aut}(D)$ such that
$\gamma(v_{0})=v_{1}$. Hence $v_{0}$ is sent to $v_{1}$ by an element of
${\widehat{\Gamma}}_{|{\varepsilon}|}$. Thus the map $[\
]:V/{\widehat{\Gamma}}_{|{\varepsilon}|}\to|{\varepsilon}|$ is bijective. ∎

######  Lemma 3.2.

The action of the saturated cluster modular group on the saturated modular
complex $\widehat{\mathcal{M}}_{|{\varepsilon}|}$ satisfies the condition
(2.1).

###### Proof.

First note that there are no loops in the saturated cluster modular complex,
since we have no standard relation of length $1$. Assume that an element
$\phi\in{\widehat{\Gamma}}_{|{\varepsilon}|}$, $\phi\neq 1$, fixes an edge $e$
of $\widehat{\mathcal{M}}_{|{\varepsilon}|}$. Let $x$ and $y$ be endpoints and
$i\in I$ the label of $e$. Then $\phi$ is of the form $\phi=\sigma\mu_{i}$ as
an element of ${\widehat{\Gamma}}_{[x]}$, where $\sigma$ is an involutive
quiver isomorphism. Now suppose we have $\phi(x)=y$ and $\phi(y)=x$. Then we
have $\phi^{2}\in\mathrm{Aut}([x])$, namely, $\phi^{2}$ must be given by a
seed isomorphism $\tau$ as an element of ${\widehat{\Gamma}}_{[x]}$. Hence we
have a non-trivial relation $\tau=\mu_{\sigma^{-1}(i)}\mu_{i}$, which cannot
be written as a composition of the standard relations. Thus we have a
contradiction. ∎

If an edge $e$ of the saturated modular complex has endpoints $v_{0}$ and
$v_{1}$, the above lemma provides an inclusion of isotropy groups
$(\widehat{\Gamma}_{|{\varepsilon}|})_{e}\subseteq(\widehat{\Gamma}_{|{\varepsilon}|})_{v_{0}}\cap(\widehat{\Gamma}_{|{\varepsilon}|})_{v_{1}}$.
An element $\sigma$ of $(\widehat{\Gamma}_{|{\varepsilon}|})_{e}$ is called a
_simultaneous seed isomorphism_ of the pair $(v_{0},v_{1})$. We also say that
$\sigma$ is a simultaneous seed isomorphism of the pair of underlying exchange
matrices $([v_{0}],[v_{1}])$.

######  Lemma 3.3.

The ${\widehat{\Gamma}}_{|{\varepsilon}|}$-orbit of an oriented edge in
${\widehat{\mathcal{M}}}_{|{\varepsilon}|}$ is determined by a pair of
exchange matrices $({\varepsilon},{\varepsilon}^{\prime})$ and an index $k\in
I$ such that ${\varepsilon}^{\prime}=\mu_{k}({\varepsilon})$, modulo
simultaneous seed isomorphism of the pair
$({\varepsilon},{\varepsilon}^{\prime})$.

###### Proof.

Fix a tree $T$ of representative. Then each oriented edge is translated by an
element of ${\widehat{\Gamma}}_{|{\varepsilon}|}$ so that its origin belongs
to $V(T)$. If we have two oriented edges with the same origin, they are
translated each other by an element of ${\widehat{\Gamma}}_{|{\varepsilon}|}$
if and only if there exists a simultaneous seed isomorphism of the two
endpoints of one of them. ∎

######  Lemma 3.4.

The ${\widehat{\Gamma}}_{|{\varepsilon}|}$-orbit of a standard $(h+2)$-gon
cycle $C$ in ${\widehat{\mathcal{M}}}_{|{\varepsilon}|}$ is determined by an
exchage matrix ${\varepsilon}^{\prime}\in|{\varepsilon}|$ and the
$\mathrm{Aut}({\varepsilon}^{\prime})$-orbit of a pair of indices $k,l\in
I-I_{0}$ such that
${\varepsilon}^{\prime}_{kl}=-p{\varepsilon}^{\prime}_{lk}=p$. Here
$(p,h)=(0,2)$, $(1,3)$, $(2,4)$ or $(3,6)$.

###### Proof.

The base point of each standard cycle is translated by an element of
${\widehat{\Gamma}}_{|{\varepsilon}|}$ to a point belonging to a fixed tree of
representative. Then the assertion is clear. ∎

A standard $(h+2)$-gon cycle which path through a vertex $v\in V$ and
determined by a pair of indices $k,l$ is denoted by $C_{h+2}(k,l)_{v}$. A
cycle may have several expressions.

######  Definition 3.5 (the saturated modular graph).

We define the _saturated modular graph_ $\widehat{G}_{|{\varepsilon}|}$ to be
the graph whose vertices are exchange matrices in $|{\varepsilon}|$ and edges
are ${\widehat{\Gamma}}_{|{\varepsilon}|}$-orbits of edges in
${\widehat{\mathcal{M}}}_{|{\varepsilon}|}$.

From Lemma 3.4, each ${\widehat{\Gamma}}_{|{\varepsilon}|}$-orbit of a
standard cycle is represented by a circuit in the saturated modular graph
$\widehat{G}_{|{\varepsilon}|}$. The following lemma is clear from the
definition.

######  Lemma 3.6.

If ${\varepsilon}$ is of finite mutation type, then the saturated modular
graph is a finite graph.

###  3.1. The saturated cluster modular group of type $X_{7}$

$0$$1$$2$$3$$4$$5$$6$$Q_{0}$$0$$1$$2$$3$$4$$5$$6$$Q_{1}$

Figure 3. Two quivers in $|X_{7}|$

The mutation class $X_{7}$ consists of two quivers $Q_{0}$ and $Q_{1}$. They
are shown in Figure 3. Let $v_{0}\in V$ be a vertex of the saturated modular
complex such that $[v_{0}]=Q_{0}$. Let us fix a data of representatives as
follows:

  1. (1)

The tree $T$ consists of two vertices $v_{0}$ and $v_{1}:=\mu_{0}(v_{0})$
together with an edge $e_{0}:=(v_{0}\xrightarrow{\mu_{0}}v_{1})$.

  2. (2)

The set $E^{+}$ of representative of oriented edges consists of three oriented
edges $e_{0}$, $e_{1}:=(v_{0}\xrightarrow{\mu_{1}}\mu_{1}(v_{0}))$ and
$e_{2}:=(v_{1}\xrightarrow{\mu_{1}}\mu_{1}(v_{1}))$. There is an orientation
of ${\widehat{\mathcal{M}}}_{X_{7}}$ which extends them.

  3. (3)

The set $F:=\\{\tau_{1},\tau_{2},\tau_{3},\tau_{4}\\}$ of representative of
2-cells consists of two square cycles $\tau_{1}:=C_{4}(1,4)_{v_{0}}$,
$\tau_{2}:=C_{4}(1,3)_{v_{1}}$ and two pentagon cycles
$\tau_{3}:=C_{5}(0,1)_{v_{0}}$, $\tau_{4}:=C_{5}(1,4)_{v_{1}}$.

$Q_{0}$$Q_{1}$$[e_{0}]$$[e_{1}]$$[e_{2}]$ Figure 4. The modular graph
$\widehat{G}_{X_{7}}$. The tree $T$ is shown by a thick line. The set $E^{+}$
is shown by dashed lines.

The tree $T$ and the set $E^{+}$ are shown in Figure 4. Then the following
proposition can be easily verified using Lemmas 3.1, 3.2, 3.3 and 3.4.

######  Proposition 3.7.

The data $(T,P,E^{+},F)$ determines a data of representatives for the action
of ${\widehat{\Gamma}}_{X_{7}}$ on ${\widehat{\mathcal{M}}}_{X_{7}}$.

Then we choose the Brown generators as follows:

  1. (1)

The isotropy group of the vertex $v_{0}$ is the image of the group
homomorphism

$\iota:\mathfrak{S}(\\{1,3,5\\})\to\mathfrak{S}(\\{0,1,2,3,4,5,6\\})$

given by $\iota(\sigma):0\mapsto
0,~{}k\mapsto\sigma(k),~{}k+1\mapsto\sigma(k)+1$ for $k\in\\{1,3,5\\}$. Here
$\mathfrak{S}(X)$ denotes the symmetric group of a finite set $X$. We write a
cyclic permutation $k_{1}\mapsto\dots\mapsto k_{n}\mapsto k_{1}$ of
$\\{1,3,5\\}$ as $\sigma_{k_{1}\dots k_{n}}$. It turns out that the isotropy
group of $v_{1}$ is the same, hence
$\mathcal{S}_{3}:=\iota(\mathfrak{S}(\\{1,3,5\\}))$ consists of simultaneous
seed isomorphisms.

  2. (2)

Set $g_{e_{0}}:=1$, $g_{e_{1}}:=(1\ 2)\mu_{1}\in{\widehat{\Gamma}}_{v_{0}}$
and $g_{e_{2}}:=(0\ 1\ 2)(3\ 4\ 5\ 6)\mu_{1}\in{\widehat{\Gamma}}_{v_{1}}$.
Then they satisfy the condition (2.2). Changing their basepoints to the vertex
$v_{0}$ using the edge $e_{0}$ (see Remark 1.8), we get $\phi_{1}:=(1\
2)\mu_{1}$ and $\psi_{1}:=(0\ 1\ 2)(3\ 4\ 5\ 6)\mu_{2}\mu_{1}\mu_{0}$, which
we regard as elements in ${\widehat{\Gamma}}_{Q_{0}}$.

From Theorem 2.2, the elements in $\mathcal{S}_{3}$ together with $\phi_{1}$
and $\phi_{2}$ generate the saturated cluster modular group. Let us
investigate the relations among them. To simplify the computations, let us
introduce auxiliary elements $\phi_{3}:=\sigma_{135}\phi_{1}\sigma_{135}^{-1}$
and $\phi_{5}:=\sigma_{153}\phi_{1}\sigma_{153}^{-1}$. Similarly define
$\psi_{3}$ and $\psi_{5}$. Then the relations are determined as follows.

#### Isotropy relations.

The isotropy group of the edge $e_{1}$ is generated by $\sigma_{35}$. Then the
isotropy relation implies $\phi_{1}^{-1}\sigma_{35}\phi_{1}=\sigma_{35}$.
Similarly, the isotropy group of the edge $e_{2}$ is generated by
$\sigma_{35}$. Then the isotropy relation implies
$\psi_{1}^{-1}\sigma_{35}\psi_{1}=\sigma_{35}$.

#### Face relations.

From the cycle $\tau_{1}$ we get:

$\phi_{3}\phi_{1}^{-1}\phi_{3}^{-1}\phi_{1}=(3\ 4)\mu_{3}(1\ 2)\mu_{2}(3\
4)\mu_{4}(1\ 2)\mu_{1}=\mu_{4}\mu_{1}\mu_{4}\mu_{1}=1.$

From the cycle $\tau_{3}$ we get:

$\displaystyle\sigma_{35}\phi_{1}^{-1}\psi_{1}^{2}$ $\displaystyle=(3\ 5)(4\
6)(1\ 2)\mu_{2}(0\ 1\ 2)(3\ 4\ 5\ 6)\mu_{2}\mu_{1}\mu_{0}(0\ 1\ 2)(3\ 4\ 5\
6)\mu_{2}\mu_{1}\mu_{0}$ $\displaystyle=(0\
1)\mu_{0}\mu_{1}\mu_{0}\mu_{1}\mu_{0}=1.$

In order to investigate $\tau_{2}$ and $\tau_{4}$, we represent each element
in ${\widehat{\Gamma}}_{v_{1}}$: $\psi_{1}=(0\ 1\ 2)(3\ 4\ 5\ 6)\mu_{1}$,
$\psi_{3}=(0\ 3\ 4)(5\ 6\ 1\ 2)\mu_{3}$ and $\psi_{5}=(0\ 5\ 6)(1\ 2\ 3\
4)\mu_{5}$. Then from the cycle $\tau_{2}$ we get:

$\displaystyle\psi_{3}^{-1}\psi_{1}\psi_{3}^{-1}\psi_{1}$ $\displaystyle=(0\
4\ 3)(6\ 5\ 2\ 1)\mu_{4}(0\ 1\ 2)(3\ 4\ 5\ 6)\mu_{1}(0\ 4\ 3)(6\ 5\ 2\
1)\mu_{4}(0\ 1\ 2)(3\ 4\ 5\ 6)\mu_{1}$
$\displaystyle=\mu_{3}\mu_{1}\mu_{3}\mu_{1}=1.$

From the cycle $\tau_{4}$ we get:

$\displaystyle\sigma_{153}\psi_{5}\psi_{1}\psi_{3}\psi_{5}\psi_{1}$
$\displaystyle=(1\ 5\ 3)(2\ 6\ 4)(0\ 5\ 6)(1\ 2\ 3\ 4)\mu_{5}(0\ 1\ 2)(3\ 4\
5\ 6)\mu_{1}$ $\displaystyle\qquad\ (0\ 3\ 4)(5\ 6\ 1\ 2)\mu_{3}(0\ 5\ 6)(1\
2\ 3\ 4)\mu_{5}(0\ 1\ 2)(3\ 4\ 5\ 6)\mu_{1}$ $\displaystyle=(1\
4)\mu_{1}\mu_{4}\mu_{1}\mu_{4}\mu_{1}=1.$

Summarizing these relations, we obtain Theorem 1.

###  3.2. The saturated cluster modular group of type $X_{6}$

$1$$2$$3$$4$$5$$6$$Q_{0}$$1$$2$$3$$4$$5$$6$$Q_{1}$$1$$6$$2$$4$$3$$5$$Q_{2}$$1$$2$$3$$4$$5$$6$$Q_{3}$$1$$2$$3$$4$$5$$6$$Q_{4}$

Figure 5. Five quivers in $|X_{6}|$

Next we investigate the mutation class $X_{6}$. Although the computation is a
little more complicated in this case, the strategy is the same as before. The
mutation class $X_{6}$ consists of five quivers $Q_{0}$,…, $Q_{4}$. They are
shown in Figure 5. Let $v_{0}\in V$ be a vertex of the saturated modular
complex such that $[v_{0}]=Q_{0}$. Let us fix a data of representatives as
follows, see Figure 6.

  1. (1)

The tree $T$ consists of five vertices $v_{0}$, $v_{1}:=(2\ 3)\mu_{1}(v_{0})$,
$v_{2}:=(3\ 5)\mu_{6}(v_{1})$, $v_{3}:=(2\ 4)(1\ 6)\mu_{6}(v_{2})$ and
$v_{4}:=(4\ 5)\mu_{1}(v_{3})$ together with four edges

$\displaystyle e^{0}:=(v_{0}\xrightarrow{\mu_{1}}v_{1}),$ $\displaystyle\
{}e^{1}:=(v_{1}\xrightarrow{\mu_{6}}v_{2}),$ $\displaystyle
e^{2}:=(v_{2}\xrightarrow{\mu_{6}}v_{3}),$ $\displaystyle\
{}e^{3}:=(v_{3}\xrightarrow{\mu_{1}}v_{4}).$

  2. (2)

The set $E^{+}:=\\{e^{0},\dots,e^{3},e_{0},\dots,e_{6}\\}$ of representatives
of oriented edges consists of eleven oriented edges, where

$\displaystyle e_{0}$ $\displaystyle:=(v_{0}\xrightarrow{\mu_{6}}(v_{4})),$
$\displaystyle\ {}e_{1}$
$\displaystyle:=(v_{0}\xrightarrow{\mu_{2}}\mu_{2}(v_{0})),$ $\displaystyle\
{}e_{2}$ $\displaystyle:=(v_{4}\xrightarrow{\mu_{2}}\mu_{2}(v_{4})),$
$\displaystyle e_{3}$
$\displaystyle:=(v_{1}\xrightarrow{\mu_{2}}\mu_{2}(v_{1})),$ $\displaystyle\
{}e_{4}$ $\displaystyle:=(v_{1}\xrightarrow{\mu_{3}}\mu_{3}(v_{1})),$
$\displaystyle\ {}e_{5}$
$\displaystyle:=(v_{2}\xrightarrow{\mu_{4}}\mu_{4}(v_{2})),$ $\displaystyle
e_{6}$ $\displaystyle:=(v_{2}\xrightarrow{\mu_{3}}\mu_{3}(v_{2})).$

There is an orientation of ${\widehat{\mathcal{M}}}_{X_{6}}$ which extends
them.

  3. (3)

The set $F:=\\{\tau_{1},\dots,\tau_{11}\\}$ of representatives of 2-cells
consists of six square cycles

$\displaystyle\tau_{1}$ $\displaystyle:=C_{4}(2,6)_{v_{0}},$ $\displaystyle\
{}\tau_{2}$ $\displaystyle:=C_{4}(2,5)_{v_{0}},$ $\displaystyle\ {}\tau_{3}$
$\displaystyle:=C_{4}(2,5)_{v_{4}},$ $\displaystyle\tau_{4}$
$\displaystyle:=C_{4}(3,4)_{v_{1}},$ $\displaystyle\ {}\tau_{5}$
$\displaystyle:=C_{4}(2,5)_{v_{0}},$ $\displaystyle\ {}\tau_{6}$
$\displaystyle:=C_{4}(4,6)_{v_{1}}$

and five pentagon cycles

$\displaystyle\tau_{7}$ $\displaystyle:=C_{5}(1,6)_{v_{0}},$ $\displaystyle\
{}\tau_{8}$ $\displaystyle:=C_{5}(1,2)_{v_{0}},$ $\displaystyle\ {}\tau_{9}$
$\displaystyle:=C_{5}(1,5)_{v_{4}},$ $\displaystyle\tau_{10}$
$\displaystyle:=C_{5}(2,4)_{v_{1}},$ $\displaystyle\ {}\tau_{11}$
$\displaystyle:=C_{5}(2,4)_{v_{3}}.$

$Q_{0}$$Q_{1}$$Q_{2}$$Q_{3}$$Q_{4}$$e^{0}$$e^{1}$$e^{2}$$e^{3}$$e_{0}$$e_{1}$$e_{2}$$e_{3}$$e_{4}$$e_{5}$$e_{6}$
Figure 6. The modular graph $\widehat{G}_{X_{6}}$. The tree $T$ is shown by
thick lines. The set $E^{+}$ is shown by dashed lines.

Then the following proposition can be easily verified using Lemmas 3.1, 3.2,
3.3 and 3.4.

######  Proposition 3.8.

The data $(T,E^{+},F)$ determines a data of representatives for the action of
${\widehat{\Gamma}}_{X_{6}}$ on ${\widehat{\mathcal{M}}}_{X_{6}}$.

Then we get the Brown generators as follows:

  1. (1)

The isotropy group of the vertices $v_{0}$, $v_{1}$, $v_{3}$ and $v_{4}$ are
the same, and generated by an involution $\sigma$. It is written as
$\sigma=(2\ 4)(3\ 5)$ in ${\widehat{\Gamma}}_{Q_{0}}$ and
${\widehat{\Gamma}}_{Q_{4}}$, $\sigma=(2\ 5)(3\ 4)$ in
${\widehat{\Gamma}}_{Q_{1}}$ and ${\widehat{\Gamma}}_{Q_{3}}$. The isotropy
group of the vertex $v_{2}$ is generated by a permutation $\rho:=(1\ 2\ 5)(3\
6\ 4)\in{\widehat{\Gamma}}_{Q_{2}}$ of order 3.

  2. (2)

Set $g_{e^{i}}:=1$ for $i=0,\dots,3$, and

$\displaystyle g_{e_{0}}$ $\displaystyle:=(1\
6)\mu_{1}\mu_{6}\mu_{1}\mu_{6}\mu_{1}\in{\widehat{\Gamma}}_{v_{0}},$
$\displaystyle g_{e_{1}}$ $\displaystyle:=(2\
3)\mu_{2}\in{\widehat{\Gamma}}_{v_{0}},$ $\displaystyle g_{e_{2}}$
$\displaystyle:=(2\ 3)\mu_{2}\in{\widehat{\Gamma}}_{v_{4}},$ $\displaystyle
g_{e_{3}}$ $\displaystyle:=(4\ 6)(1\ 3)\mu_{2}(3\ 5)\mu_{6}(2\ 4)(1\
6)\mu_{6}\in{\widehat{\Gamma}}_{v_{3}},$ $\displaystyle g_{e_{4}}$
$\displaystyle:=(4\ 6)(1\ 2)\mu_{3}(3\ 5)\mu_{6}(2\ 4)(1\
6)\mu_{6}\in{\widehat{\Gamma}}_{v_{3}},$ $\displaystyle g_{e_{5}}$
$\displaystyle:=(3\ 5)(2\ 1\ 5\ 4\ 6\ 3)\mu_{4}\in{\widehat{\Gamma}}_{v_{2}},$
$\displaystyle g_{e_{6}}$ $\displaystyle:=(2\ 4)(1\ 6)\mu_{6}(1\ 3\ 6\ 5)(2\
4)\mu_{3}\in{\widehat{\Gamma}}_{v_{2}}.$

Then they satisfy the condition (2.2). Changing their basepoints to the vertex
$v_{0}$ _along_ the tree $T$, we get the corresponding elements in
${\widehat{\Gamma}}_{Q_{0}}$:

$\displaystyle\phi_{0}$ $\displaystyle:=(1\
6)\mu_{1}\mu_{6}\mu_{1}\mu_{6}\mu_{1},$ $\displaystyle\ {}\phi_{1}$
$\displaystyle:=(2\ 3)\mu_{2},$ $\displaystyle\phi_{2}$ $\displaystyle:=(2\
3)\mu_{6}\mu_{2}\mu_{6},$ $\displaystyle\ {}\phi_{3}$ $\displaystyle:=(1\ 5\
2\ 6\ 3\ 4)\mu_{4}\mu_{2}\mu_{4}\mu_{3}\mu_{1},$ $\displaystyle\phi_{4}$
$\displaystyle:=(1\ 4)(2\ 5)(3\ 6)\mu_{4}\mu_{3}\mu_{4}\mu_{2}\mu_{1},$
$\displaystyle\ {}\phi_{5}$ $\displaystyle:=(1\ 5\ 3)(2\ 4\
6)\mu_{3}\mu_{4}\mu_{6}\mu_{1},$ $\displaystyle\phi_{6}$ $\displaystyle:=(1\
5)(2\ 6)\mu_{5}\mu_{2}\mu_{6}\mu_{1}.$

For example, $\phi_{3}=((2\ 4)(1\ 6)\mu_{6}(3\ 5)\mu_{6}(2\
3)\mu_{1})^{-1}g_{e_{3}}((2\ 4)(1\ 6)\mu_{6}(3\ 5)\mu_{6}(2\ 3)\mu_{1})$. From
Theorem 2.2, these elements generate the saturated cluster modular group of
type $X_{6}$. Let us investigate the relations among them.

#### Isotropy relations.

We have the relations $\sigma^{2}=1$, $\rho^{3}=1$ and $\rho=\sigma\phi_{6}$.

#### Face relations.

The verification of the following face relations is tedious but
straightforward.

  * •

$\tau_{1}$: $\phi_{2}=\phi_{1}$. This relation implies that $\tau_{2}$ and
$\tau_{3}$ induces the same relation.

  * •

$\tau_{2}(=\tau_{3})$: $[\sigma,\phi_{3}]^{2}=1$.

  * •

$\tau_{4}$: $\sigma^{-1}\phi_{4}^{-1}\phi_{6}\phi_{4}=1$.

  * •

$\tau_{5}$: $(\phi_{1}\sigma)^{2}(\phi_{1}^{-1}\sigma)^{2}=1$.

  * •

$\tau_{6}$: $(\sigma\phi_{4}^{-1}\sigma^{-1})\phi_{4}\phi_{5}=1$.

  * •

$\tau_{7}$: $\phi_{0}=1$.

  * •

$\tau_{8}$: $\phi_{1}^{-1}\phi_{3}^{-1}\phi_{4}=1$.

  * •

$\tau_{9}$: $(\sigma\phi_{1}\sigma^{-1})\phi_{3}\phi_{4}^{-1}=1$.

  * •

$\tau_{10}$: $\phi_{4}^{-1}(\sigma\phi_{3}\sigma^{-1})\phi_{5}\phi_{3}=1$.

  * •

$\tau_{11}$:
$(\sigma\phi_{4}\sigma^{-1})\phi_{3}^{-1}(\sigma\phi_{6}\sigma^{-1})\phi_{3}^{-1}=1$.

For example, the relation induced by $\tau_{4}$ is verified as follows. First
we change the basepoint to $v_{1}$ using the path $(2\ 3)\mu_{1}$ from $v_{0}$
to $v_{1}$. Then we have $\phi_{4}=(1\ 4)(2\ 6)(3\ 5)\mu_{2}\mu_{4}\mu_{3}$
and $\phi_{6}=(1\ 5)(3\ 6)\mu_{3}\mu_{6}$, represented as elements of
${\widehat{\Gamma}}_{v_{1}}$. Then we compute

$\displaystyle\phi_{4}^{-1}\phi_{6}\phi_{4}$ $\displaystyle=(1\ 4)(2\ 6)(3\
5)\mu_{2}\mu_{4}\mu_{3}(1\ 5)(3\ 6)\mu_{3}\mu_{6}(1\ 4)(2\ 6)(3\
5)\mu_{2}\mu_{4}\mu_{3}$ $\displaystyle=(2\ 5)(3\
4)\mu_{4}\mu_{3}\mu_{4}\mu_{3}=\sigma.$

Now we prove Theorem 2.

###### Proof of Theorem 2.

From the face relations $\tau_{1}$, $\tau_{4}$, $\tau_{6}$, $\tau_{7}$ and
$\tau_{8}$, we can reduce the number of generators as $\phi_{2}=\phi_{1}$,
$\phi_{6}=\phi_{4}\sigma\phi_{4}^{-1}$,
$\phi_{5}=\phi_{4}^{-1}\sigma\phi_{4}\sigma=[\phi_{4}^{-1},\sigma]$,
$\phi_{0}=1$ and $\phi_{4}=\phi_{3}\phi_{1}$. Then the remaining relations are
summarized as

$\displaystyle\sigma^{2}=1,$ $\displaystyle[\sigma,\phi_{3}\phi_{1}]^{3}=1$
$\displaystyle(\text{From $\rho^{3}=1$}),$
$\displaystyle[\sigma,\phi_{3}]^{2}=1$ $\displaystyle(\text{From
$\tau_{2}$}),$ $\displaystyle(\phi_{1}\sigma)^{2}=[\sigma,\phi_{1}]$
$\displaystyle(\text{From $\tau_{5}$}),$
$\displaystyle\phi_{3}\phi_{1}\phi_{3}^{-1}=\sigma\phi_{1}\sigma^{-1}=\sigma\phi_{3}\sigma^{-1}[(\phi_{3}\phi_{1})^{-1},\sigma]$
$\displaystyle(\text{From $\tau_{9}$ and $\tau_{10}$}),$
$\displaystyle\phi_{3}=\sigma\phi_{3}\phi_{1}\sigma^{-1}\phi_{3}^{-1}\sigma[\phi_{3}\phi_{1},\sigma]$
$\displaystyle(\text{From $\tau_{11}$}).$

Using the first and fifth relaion
($\phi_{3}\phi_{1}=\sigma\phi_{1}\sigma^{-1}\phi_{3}$), the last relation is
slightly simplified as

$\displaystyle\phi_{3}=\phi_{1}[\sigma,\phi_{3}][\phi_{3}\phi_{1},\sigma].$

The fifth relaion can be rewritten as

$[\phi_{3},\phi_{1}]=[\sigma,\phi_{1}]=[\sigma,\phi_{3}]\phi_{3}[(\phi_{3}\phi_{1})^{-1},\sigma]\phi_{1}^{-1}$

by multiplying $\phi_{1}^{-1}$ from the right. Setting $\alpha:=\phi_{1}$ and
$\beta:=\phi_{3}$, we get the desired presentation. ∎

##  Appendix A Relations with the mapping class group of an annulus

In this section, we investigate several relations between the cluster modular
groups of type $X_{6}$ and $X_{7}$ with the mapping class group of an annulus.
Recall from Examples 1.3 and 1.13 that an ideal triangulation $\Delta$ of a
marked surface $\Sigma$ gives rise to the seed ${\mathbf{s}}_{\Delta}$ of
finite mutation type. The following lemma is repeatedly used in this section.

######  Proposition A.1.

For a seed ${\mathbf{s}}=({\varepsilon},(A_{i})_{i\in I},(X_{i})_{i\in I})$,
write its non-frozen part as ${\mathbf{s}}_{0}:=({\varepsilon}|_{I\setminus
I_{0}},(A_{i})_{i\in I\setminus I_{0}},(X_{i})_{i\in I\setminus I_{0}})$. Then
we have a group homomorphism
$e:\Gamma_{{\mathbf{s}}}\to\Gamma_{{\mathbf{s}}_{0}}$ satisfying the following
exact sequence:

$1\to\mathrm{Aut}_{f}({\mathbf{s}})\to\Gamma_{{\mathbf{s}}}\xrightarrow{e}\Gamma_{{\mathbf{s}}_{0}},$

where
$\mathrm{Aut}_{f}({\mathbf{s}}):=\\{\sigma\in\mathrm{Aut}({\mathbf{s}})|\text{$\sigma(i)=i$
for all $i\in I-I_{0}$}\\}$.

We call the homomorphism $e:\Gamma_{{\mathbf{s}}}\to\Gamma_{{\mathbf{s}}_{0}}$
the _elimination homomorphism_.

###### Proof.

Each element $\phi\in\Gamma_{\mathbf{s}}$ can be written as
$\phi={\sigma\mu_{i_{k}}\dots\mu_{i_{1}}}$, where $i_{1},\dots,i_{k}\in
I-I_{0}$ and $\sigma$ is a seed isomorphism. Here we can write
$\sigma=\sigma_{f}\sigma_{m}$, where $\sigma_{f}$ is a permutation of $I_{0}$
and $\sigma_{m}$ is a permutation of $I-I_{0}$. Then we define
$e(\phi):=\sigma_{m}\mu_{i_{l}}\dots\mu_{i_{1}}\in\Gamma_{{\mathbf{s}}_{0}}$.
The map $e$ is a group homomorphism, since the permutation $\sigma_{f}$
commutes with mutations. Moreover $e(\phi)=1$ if and only if
$\phi=\sigma_{f}\in\mathrm{Aut}_{f}({\mathbf{s}})$. Thus we get the exact
sequence. ∎

Similarly we can define the elimination homomorphism between the corresponding
saturated cluster modular groups, and we have a similar exact sequence.

$0$$1$$2$
Figure 7. The seed ${\mathbf{t}}$ 021 Figure 8. The ideal triangulation
$\Delta$ of the annulus with (2+1) marked points

Let ${\mathbf{t}}$ be the quiver shown in Figure 7. Note that it gives a
subquiver of both $X_{6}$ and $X_{7}$.

######  Lemma A.2.

The quiver ${\mathbf{t}}$ coincides with the non-frozen part of the quiver
associated with the ideal triangulation $\Delta$ of the annulus $S$ with
$(1+2)$ marked points shown in Figure 8. Moreover, the elimination
homomorphism induces an isomorphism
$e:\Gamma_{{\mathbf{s}}_{\Delta}}=MC(S)\xrightarrow{\cong}\Gamma_{|{\mathbf{t}}|}$.
Under this isomorphism, the generator $s\in MC(S)\cong\mathbb{Z}$ which
rotates the two marked points on one boundary component (called a “half
twist”) corresponds to the element $(0\ 1\
2)\mu_{2}\mu_{1}\mu_{0}\in\Gamma_{|{\mathbf{t}}|}$. The Dehn twist
$t:=s^{2}\in MC(S)$ corresponds to the element $(1\
2)\mu_{1}\in\Gamma_{|{\mathbf{t}}|}$.

###### Proof.

The injectivity is easily verified using the exact sequence in Proposition
A.1. It can be verified that $\Gamma_{|{\mathbf{t}}|}$ is generated by the
element $(0\ 1\ 2)\mu_{2}\mu_{1}\mu_{0}$, by observing that the mutation class
$|{\mathbf{t}}|$ consists of two seeds and the element acts on the set $[\
]^{-1}({\mathbf{t}})\subset V({\widehat{\mathcal{M}}}_{|{\mathbf{t}}|})$
transitively. Then the surjectivity follows from the third statement. The
verifications of the third and fourth statements are straightforward. ∎

######  Remark A.3.

We have
${\widehat{\Gamma}}_{{\mathbf{s}}_{\Delta}}\cong\Gamma_{{\mathbf{s}}_{\Delta}}$
in this case, see Theorem 9.17 in [FST08]. Hence we also have
${\widehat{\Gamma}}_{{\mathbf{t}}}\cong\Gamma_{{\mathbf{t}}}$.

Let $\Gamma_{X_{7}}^{(12)}$ be the subgroup of $\Gamma_{X_{7}}$ which consists
of elements of the form $\phi={\sigma\mu_{i_{k}}\dots\mu_{i_{1}}}$, where
$i_{1},\dots i_{k}\in\\{0,1,2\\}$ and $\sigma$ is a permutation such that
$\sigma(\\{0,1,2\\})=\\{0,1,2\\}$. Note that the group $\Gamma_{X_{7}}^{(12)}$
is naturally identified with the cluster modular group of the seed obtained
from the seed $X_{7}$ by freezing the vertices $\\{3,4,5,6\\}$.

######  Lemma A.4.

We have a split exact sequence

$1\to\mathbb{Z}/2\xrightarrow{i^{(12)}}\Gamma_{X_{7}}^{(12)}\xrightarrow{e^{(12)}}\Gamma_{|{\mathbf{t}}|}=MC(S)\to
1.$

Here $i^{(12)}$ is given by $1\mapsto\sigma_{35}=(3\ 5)(4\ 6)$ and $e^{(12)}$
is the elimination homomorphism.

###### Proof.

The exactness of the sequence follows from Proposition A.1 and
$s=e^{(12)}(\psi_{1})$. The commutativity relations
$\psi_{1}\sigma_{35}=\sigma_{35}\psi_{1}$ and
$\phi_{1}\sigma_{35}=\sigma_{35}\phi_{1}$ in Theorem 1 show that the image of
$i^{(12)}$ is central. Since $MC(S)\cong\mathbb{Z}$, there exists a section of
$e^{(12)}$. ∎

Having in mind this lemma, we rewrite
$MC(S)^{(12)}:=\Gamma_{X_{7}}^{(12)}\cong MC(S)\times\mathbb{Z}/2$.

######  Remark A.5.

The relation $t=s^{2}$ in $MC(S)$ corresponds to the relation
$\phi_{1}=\psi_{1}^{2}\sigma_{35}$ in $MC(S)^{(12)}$.

Similarly we define the subgroups $MC(S)^{(34)}$, $MC(S)^{(56)}$ of the
cluster modular group $\Gamma_{X_{7}}$. They are isomorphic to
$MC(S)\times\mathbb{Z}/2$. Summarizing, we have proved the following geometric
interpretation of Corollary 5 in terms of these extended mapping class groups:

######  Theorem A.6.

The extended mapping class groups $MC(S)^{(12)}$, $MC(S)^{(34)}$ and
$MC(S)^{(56)}$ generate the cluster modular group $\Gamma_{X_{7}}$. The six
cluster Dehn twists given in Corollary 5 corresponds to the half twist and
Dehn twist in $MC(S)$ via the elimination homomorphisms.

Similarly we consider the subgroup $\Gamma_{X_{6}}^{(12)}$ of $\Gamma_{X_{6}}$
which consists of elements of the form
$\phi={\sigma\mu_{i_{k}}\dots\mu_{i_{1}}}$, where $i_{1},\dots
i_{k}\in\\{0,1,2\\}$ and $\sigma$ is a permutation such that
$\sigma(\\{0,1,2\\})=\\{0,1,2\\}$.

######  Lemma A.7.

The elimination homomorphism
$e^{(12)}:\Gamma_{X_{6}}^{(12)}\to\Gamma_{|{\mathbf{t}}|}$ is injective. The
image is a subgroup of index $2$.

###### Proof.

The injectivity follows from Proposition A.1, noting that in this case the
term $\mathrm{Aut}_{f}({\mathbf{s}})$ is trivial. We have
$t=e^{(12)}(\phi_{1})$, but the half twist $s$ is not extendable to an element
of $\Gamma_{X_{6}}^{(12)}$. Hence the image is generated by the Dehn twist
$t$, and it has index 2. ∎

Similarly we define the subgroup $\Gamma_{X_{6}}^{(34)}$ by replacing
$\\{1,2\\}$ by $\\{3,4\\}$, which is embedded into $MC(S)$ by
$\sigma\phi_{1}\sigma^{-1}\mapsto t$. Thus the cluster modular group
$\Gamma_{X_{6}}$ contains the subgroup of $MC(S)\times MC(S)$ generated by two
Dehn twists. However they do not generate the cluster modular group
$\Gamma_{X_{6}}$ in this case.

##  Appendix B Cluster Dehn twists in other examples of cluster modular groups

Here we confirm that some other examples of (saturated) cluster modular groups
are generated by cluster Dehn twists. Our interest is the case when the
cluster modular group is infinite. We will use the notation of [FST12b] and
[FST12a] for quivers of finite mutation type.

#### Type $G_{2}^{(*,*)}$.

The weighted quiver representing the mutation class $G_{2}^{(*,*)}$ is shown
in Figure 9.

######  Theorem B.1 (Fock-Goncharov [FG06a, Corollary 4.2]).

The saturated cluster modular group of type $G_{2}^{(*,*)}$ is isomorphic to
the braid group of type $G_{2}$:

${\widehat{\Gamma}}_{G_{2}^{(*,*)}}\cong\langle a,b|(ab)^{3}=(ba)^{3}\rangle.$

This theorem can be verified by a similar computation given in § 3 of this
paper. A data of representative (a “spanning tree” they call) is fixed in
[FG06a], though they determine the presentation by a more topological
observation _loc. cit_.

######  Proposition B.2.

The generators $a$ and $b$ are given by $a:=p_{a}((3\ 4)\mu_{4})p_{a}^{-1}$
and $b:=\phi b^{\prime}\phi^{-1}$, which are cluster Dehn twists. Here
$b^{\prime}:=p_{b}((1\ 2)\mu_{2})p_{b}^{-1}$ and $\phi:=(3\
4)\mu_{4}\mu_{3}\mu_{4}\mu_{2}\mu_{1}$ are elements of
$\Gamma_{G_{2}^{(*,*)}}$, $p_{a}:=(1\ 2)(3\ 4)\mu_{1}$ and $p_{b}:=(3\
4)\mu_{4}\mu_{2}\mu_{1}$ are mutation sequences. In particular the cluster
modular group of type $G_{2}^{(*,*)}$ is generated by cluster Dehn twists.

###### Proof.

The elements $a$ and $b^{\prime}$ are two of the five Brown generators
associated with the data of representatives fixed in [FG06a]. The mutation
sequences $p_{a}^{-1}$ and $p_{b}^{-1}$ are the ones which connect
$G_{2}^{(*,*)}$ with the quivers ${\mathbf{s}}_{a}$ and ${\mathbf{s}}_{b}$
shown in Figure 10, respectively. Then observe that the vertices 3 and 4 are
connected by two arrows in the quiver ${\mathbf{s}}_{a}$. Hence the element
$(3\ 4)\mu_{4}\in\Gamma_{{\mathbf{s}}_{a}}$ is a cluster Dehn twist, so is
$a\in\Gamma_{G_{2}^{(*,*)}}$. Similarly $b^{\prime}$ is a cluster Dehn twist,
so is $b=\phi b^{\prime}\phi^{-1}$. ∎

#### Type $\tilde{E}_{6}$, $\tilde{E}_{7}$ and $\tilde{E}_{8}$.

In [ASS12], they introduce the _cluster automorphism groups_ and compute them
for quivers of finite or euclidean types using the cluster categories. It is
known that if a quiver has no frozen vertices, then the cluster modular group
is isomorphic to the _group of direct cluster automorphisms_
$\mathrm{Aut}^{+}({\mathcal{A}})$ (here ${\mathcal{A}}$ stands for the
_cluster algebra_), which is a subgroup of index $\leq 2$ of the cluster
automorphism group. See [Fra16]. Hence we can restate the results described in
Table 1 of [ASS12] as follows, restricting our attention to the quivers of
affine and finite mutation type.

######  Theorem B.3 (Assem-Schiffler-Shramchenko [ASS12]).

The cluster modular groups of affine type $\tilde{E}_{6}$, $\tilde{E}_{7}$ and
$\tilde{E}_{8}$ are given as follows:

  * •

$\Gamma_{\tilde{E}_{6}}\cong\mathbb{Z}\times\mathfrak{S}_{3}$.

  * •

$\Gamma_{\tilde{E}_{7}}\cong\mathbb{Z}\times\mathbb{Z}/2$.

  * •

$\Gamma_{\tilde{E}_{8}}\cong\mathbb{Z}$.

######  Proposition B.4.

The cluster modular groups of type $\tilde{E}_{6}$, $\tilde{E}_{7}$ and
$\tilde{E}_{8}$ are generated by cluster Dehn twists.

###### Proof.

Since each group is virtually cyclic, it suffices to find one cluster Dehn
twist. For example, it is directly verified that the seed of type
$\tilde{E}_{6}$ is mutation-equivalent to the seed shown in Figure 11. Then
the vertices 3 and 4 are connected by two arrows in ${\mathbf{s}}$, hence
$\phi^{\prime}:=(3\ 4)\mu_{3}\in\Gamma_{\mathbf{s}}$ is a cluster Dehn twist.
Similarly one can find a quiver with a pair of vertices and two arrows between
them in each mutation class $\tilde{E}_{7}$ and $\tilde{E}_{8}$. Hence we get
a cluster Dehn twist. ∎

$3$$3$$1$$2$$3$$4$

Figure 9. The seed of type $G_{2}^{(*,*)}$

$3$$3$$1$$2$$3$$4$$Q_{a}$$3$$3$$1$$2$$3$$4$$Q_{b}$

Figure 10. Two seeds equivalent to $G_{2}^{(*,*)}$ $1$$2$$3$$4$$5$$6$$7$
Figure 11. A seed equivalent to $\tilde{E}_{6}$

## References

  * [ASS12] I. Assem, R. Schiffler, and V. Shramchenko.  Cluster automorphisms.  Proc. Lond. Math. Soc. (3), 104(6):1271–1302, 2012.
  * [Bro84] K. S. Brown.  Presentations for groups acting on simply-connected complexes.  J. Pure Appl. Algebra, 32(1):1–10, 1984.
  * [BS15] T. Bridgeland and I. Smith.  Quadratic differentials as stability conditions.  Publ. Math. Inst. Hautes Études Sci., 121:155–278, 2015.
  * [CFP96] J. W. Cannon, W. J. Floyd, and W. R. Parry.  Introductory notes on Richard Thompson’s groups.  Enseign. Math. (2), 42(3-4):215–256, 1996.
  * [FG06a] V. V. Fock and A. B. Goncharov.  Cluster X-varieties, amalgamation, and Poisson-Lie groups.  In Algebraic geometry and number theory, volume 253 of Progr. Math., pages 27–68. Birkhäuser Boston, Boston, MA, 2006.
  * [FG06b] V. V. Fock and A. B. Goncharov.  Moduli spaces of local systems and higher Teichmüller theory.  Publ. Math. Inst. Hautes Études Sci., (103):1–211, 2006.
  * [FG09a] V. V. Fock and A. B. Goncharov.  Cluster ensembles, quantization and the dilogarithm.  Ann. Sci. Éc. Norm. Supér. (4), 42(6):865–930, 2009.
  * [FG09b] V. V. Fock and A. B. Goncharov.  The quantum dilogarithm and representations of quantum cluster varieties.  Invent. Math., 175(2):223–286, 2009.
  * [FM12] B. Farb and D. Margalit.  A primer on mapping class groups, volume 49 of Princeton Mathematical Series.  Princeton University Press, Princeton, NJ, 2012.
  * [Fra16] C. Fraser.  Quasi-homomorphisms of cluster algebras.  Adv. in Appl. Math., 81:40–77, 2016.
  * [FST08] S. Fomin, M. Shapiro, and D. Thurston.  Cluster algebras and triangulated surfaces. I. Cluster complexes.  Acta Math., 201(1):83–146, 2008.
  * [FST12a] A. Felikson, M. Shapiro, and P. Tumarkin.  Cluster algebras of finite mutation type via unfoldings.  Int. Math. Res. Not. IMRN, (8):1768–1804, 2012.
  * [FST12b] A. Felikson, M. Shapiro, and P. Tumarkin.  Skew-symmetric cluster algebras of finite mutation type.  J. Eur. Math. Soc. (JEMS), 14(4):1135–1180, 2012.
  * [FSTT14] A. Felikson, M. Shapiro, H. Thomas, and P. Tumarkin.  Growth rate of cluster algebras.  Proc. Lond. Math. Soc. (3), 109(3):653–675, 2014.
  * [FZ03] S. Fomin and A. Zelevinsky.  Cluster algebras. II. Finite type classification.  Invent. Math., 154(1):63–121, 2003.
  * [FZ07] S. Fomin and A. Zelevinsky.  Cluster algebras. IV. Coefficients.  Compos. Math., 143(1):112–164, 2007.
  * [Ish17] T. Ishibashi.  On a Nielsen-Thurston classification theory on cluster modular groups.  To Appear in Annales de l’Institut de Fourier, 2017.
  * [KT15] A. Kato and Y. Terashima.  Quiver mutation loops and partition $q$-series.  Comm. Math. Phys., 336(2):811–830, 2015.
  * [KY16] H. K. Kim and M. Yamazaki.  Comments on exchange graphs in cluster algebras, 2016.
  * [Pen12] R. C. Penner.  Decorated Teichmüller theory.  QGM Master Class Series. European Mathematical Society (EMS), Zürich, 2012.  With a foreword by Yuri I. Manin.
  * [Zic16] C. K. Zickert.  Fock-Goncharov coordinates for rank two Lie groups, 2016.