Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists
We give a method to compute presentations of saturated cluster modular groups. Using this, we obtain finite presentations of the saturated cluster modular groups of finite mutation type X₆ and X₇. We verify that the cluster modular groups of finite mutation type Ẽ₆, Ẽ₇, Ẽ₈, G⁽*'*⁾₂, X₆ and X₇ a...
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| citation_txt | Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists. Tsukasa Ishibashi. SIGMA 16 (2020), 025, 22 pages |
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| description | We give a method to compute presentations of saturated cluster modular groups. Using this, we obtain finite presentations of the saturated cluster modular groups of finite mutation type X₆ and X₇. We verify that the cluster modular groups of finite mutation type Ẽ₆, Ẽ₇, Ẽ₈, G⁽*'*⁾₂, X₆ and X₇ are virtually generated by cluster Dehn twists.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 025, 22 pages
Presentations of Cluster Modular Groups
and Generation by Cluster Dehn Twists
Tsukasa ISHIBASHI
Graduate School of Mathematical Sciences, The University of Tokyo,
3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan
E-mail: ishiba@ms.u-tokyo.ac.jp
Received January 01, 2020, in final form March 27, 2020; Published online April 07, 2020
https://doi.org/10.3842/SIGMA.2020.025
Abstract. We give a method to compute presentations of saturated cluster modular groups.
Using this, we obtain finite presentations of the saturated cluster modular groups of finite
mutation type X6 and X7. We verify that the cluster modular groups of finite mutation
type Ẽ6, Ẽ7, Ẽ8, G
(∗,∗)
2 , X6 and X7 are virtually generated by cluster Dehn twists.
Key words: cluster algebras; cluster modular groups; mapping class groups; quivers of finite
mutation type
2020 Mathematics Subject Classification: 13F60; 05E15; 30F60
1 Introduction and main results
A cluster modular group, defined in [9], is a group Γ|s| associated with a combinatorial data s
called a seed. An element of the cluster modular group is a finite sequence of seed permutations
and mutations which preserve the exchange matrix, and two such sequences are identified if they
induce the same pair of cluster transformations. The cluster modular group acts on the cluster
modular complex M|s|. The quotient M|s|/Γ|s| is called the modular orbifold, which can be
considered as 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 [18]. Therefore the structure of the cluster
modular orbifold is of great interst, especially in the context of the higher Teichmüller theory [8].
For example, a fundamental problem is to compute the rational cohomology groups of the
modular orbifold, which coincide with those of its orbifold fundamental group. A presentation
of the cluster modular group will provide useful information for the latter.
Once trying to find a presentation of the cluster modular group, one immediately encounters
the difficulty which arises from the fact that a complete list of relations among the cluster
transformations is not known in general. In simple cases they are exhaused by standard (h+ 2)-
gon relations [10] such as the involutivity and the pentagon relation, while there are “non-
standard” relations in general, even for those associated with marked surfaces [11]. A nice
survey on this problem is found in [17]: not only an annoying thing is this, but also related to
certain “dualities” between supersymmetric gauge theories.
In order to isolate such a problem, we consider the saturated cluster modular group Γ̂s [10] in-
stead. It is defined by restricting the relations among cluster transformations to those generated
by standard ones, so that the cluster modular group is obtained as a quotient of the saturated
cluster modular group. The saturated cluster modular group is basically easier than the cluster
modular group to deal with, and already considered in several studies:
• The fundamental group of the modular orbifold M|s|/Γ|s| is actually isomorphic to the
saturated cluster modular group [9].
mailto:ishiba@ms.u-tokyo.ac.jp
https://doi.org/10.3842/SIGMA.2020.025
2 T. Ishibashi
1
2
3 4
5
6
0
1
2 3
4
56
Figure 1. Quivers of type X6 and X7.
• The Fock–Goncharov quantization provides a projective unitary representation of the sat-
urated cluster modular group [10, Theorem 5.5].
• The Kato–Terashima partition q-series [16] gives a map from the saturated cluster modular
group to the ring Z
[[
q1/∆
]]
of formal power series for some integer ∆.
In this paper, we introduce a simplicial complex called the saturated modular complex on which
the saturated cluster modular group faithfully acts, and show that it is simply-connected. Then
we can utilize a method established by Brown [3] to obtain a presentation of the saturated
cluster modular group from the data of this action. When the seed is of finite mutation type,
namely the mutation class of the exchange matrix is a finite set, this method works particularly
well. In this case the “fundamental domain” of the saturated modular complex is finite, and we
can obtain a finite presentation of the saturated cluster modular group. The mutation classes of
finite mutation type has been completely classified in [5, 6]: see Theorems 2.13 and 2.14. In the
case of skew-symmetric exchange matrices, the list consists of the mutation classes associated
with marked surfaces, several classes associated with generalized Dynkin diagrams, and two
mysterious classes called X6 and X7. The initial quivers of type X6 and X7 are shown in Fig. 1.
Our main result is a computation of finite presentations of the saturated cluster modular groups
of type X6 and X7.
Theorem 1.1. The saturated cluster modular group Γ̂X7 of type X7 is generated by elements
ψk, φk for k = 1, 3, 5 and the permutation group S3 of numbers {1, 3, 5}, and the complete set
of relations among them is given as follows:
σψkσ
−1 = ψσ(k), σφkσ
−1 = φσ(k) for σ ∈ S3, k = 1, 3, 5,
φ1φ3 = φ3φ1, φ1 = ψ2
1σ35, 1 =
(
ψ−1
3 ψ1
)2
, 1 = σ153ψ5ψ1ψ3ψ5ψ1,
and the usual relations of permutations. Here σij denotes the tranposition of i and j and σ153
denotes the cyclic permutation 1 7→ 5 7→ 3 7→ 1.
Some of these relations have geometric interpretations, which are studied in Appendix A.
Theorem 1.2. The saturated cluster modular group Γ̂X6 of type X6 is generated by five elements
α1, α2, β1, β2 and σ, and the complete set of relations among them is given as follows:
σ2 = 1, α2 = σα1σ
−1, β2 = σβ1σ
−1,
α1α2 = α2α1, β−1
1 α2β1 = α1,
(
β2β
−1
1
)2
= 1,(
α2β2β
−1
1 α−1
1
)3
= 1, β2(β1α1)−1β2 = Adα2β1 σ, β1 = α1
(
β1α2β
−1
1
)
α−1
2 σ−1.
Here Adx y := xyx−1 denotes the conjugation.
Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists 3
Remark 1.3 (the group Γ̂X7 as an amalgamated product1). From the presentation given in
Theorem 1.1, we can delete the generators φk for k = 1, 3, 5 using the fourth relation. Let G be
the group generated by the three elements ψk for k = 1, 3, 5 subject to the following relations:
ψiψ
−1
j = ψjψ
−1
i , (1.1)
ψ2
k[ψi, ψj ]ψ
−2
k = ψkψiψjψkψi, (1.2)
for {i, j, k} = {1, 3, 5}. Then we have an isomorphism S3 ∗Z/3G ∼= Γ̂X7 , where the amalgamation
data is given by
Z/3→ S3, 1 7→ σ153,
Z/3→ G, 1 7→ ψ5ψ1ψ3ψ5ψ1.
Indeed, using the first relation in Theorem 1.1 we can move all the generators from S3 to the left
and get Γ̂X7 = S3 · 〈ψ1, ψ3, ψ5〉. The fifth relation is equivalent to ψ1ψ
−1
3 = ψ3ψ
−1
1 , which holds
among the ψ’s and leads to the defining relation (1.1). The third relation (“the φ’s commute
with each other”) is rewritten as
σ153 = ψ2
5ψ
2
1ψ
−2
3 ψ−2
5 = ψ2
5ψ1ψ3ψ
−1
1 ψ−1
3 ψ−2
5 = ψ2
5[ψ1, ψ3]ψ−2
5 ,
which leads to (1.2) and describes the intersection of S3 and 〈ψ1, ψ3, ψ5〉, together with the sixth
relation.
First homology groups. As the first application, we compute the first homology groups
of Γ̂X6 and Γ̂X7 . Here the first homology group (= abelianization) of a group G is defined to be
H1(G;Z) = G/[G,G]. Here we present the results with proofs based on Theorems 1.1 and 1.2.
Corollary 1.4. We have H1
(
Γ̂X7 ;Z
) ∼= Z/5×Z/2. The generators are the images of ψ1 and σ13.
Proof. It is well-known that the signature function gives an isomorphism H1(Sn;Z)
∼−→ Z/2 for
the symmetric group Sn of degree n. From the first relation in Theorem 1.1 we get ψ1 = ψ3 = ψ5
in the abelianization, and the last relation implies ψ5
1 = 1. �
Corollary 1.5. We have H1
(
Γ̂X6 ;Z
) ∼= Z× Z/2. The generators are the images of α1 and β1.
Proof. From the second and third relations in Theorem 1.2, we get α1 = α2 and β1 = β2 in
the abelianization. The last two relations together with σ2 = 1 imply that σ = α−1
1 β1 and
α2
1 = β2
1 . �
Cluster Dehn twists. Next let us turn our attention to the problem finding a good genera-
tors of cluster modular groups. As a candidate for an appropriate class of generators, the cluster
Dehn twists has been introduced in the author’s previous work [15]. He proved that the cluster
Dehn twists have a similar dynamical behavior to that of Dehn twists in mapping class groups.
In the case of mapping class groups, Dehn twists and half-twists are cluster Dehn twists.
It is a classical theorem in the Teichmüller theory that mapping class groups of marked
surfaces are generated by Dehn twists and half-twists. See, for instance, [4]. The following is
a cluster algebraic generalization of this theorem, for several cases of finite mutation type:
Theorem 1.6. The cluster modular groups of finite mutation type Ẽ6, Ẽ7, Ẽ8, G
(∗,∗)
2 and X7
are generated by finitely many cluster Dehn twists. The cluster modular group of type X6 is
virtually generated by four cluster Dehn twists.
1The author thank an anonymous referee for their suggestion to get this kind of presentation.
4 T. Ishibashi
For the former three cases, the theorem follows from the computation of the cluster modular
groups given by Assem–Schiffler–Shramchenko [1] using the cluster categories. The saturated
cluster modular group of type G
(∗,∗)
2 is computed by Fock–Goncharov [7]. For the last two cases
we use Theorems 1.1 and 1.2.
We can also find cluster Dehn twists in the remaining cluster modular groups of finite muta-
tion type, at least for skew-symmetric cases. Our general expectation is that any cluster modular
group of finite mutation type is virtually generated by cluster Dehn twists. It will be especially
interesting to study the cases E
(1,1)
7 and E
(1,1)
8 , since they appear as the unfrozen parts of the
quivers defining the higher Teichmüller theory [8] for certain polygon (as a marked surface) and
a Lie algebra of type A. See Table 1. Their mutation classes consist of 506 and 5739 quivers,
respectively.
3-gon 4-gon 5-gon 6-gon 7-gon
sl2 ∅ A1 A2 A3 A4
sl3 A1 D4 E7 E
(1,1)
8 ∞13
sl4 A3 E
(1,1)
7 ∞15 ∞21 ∞27
sl5 D6 ∞16 ∞26 ∞36 ∞46
sl6 E
(1,1)
8 ∞25 ∞40 ∞55 ∞70
sl7 ∞15 ∞36 ∞57 ∞78 ∞99
Table 1. Type of the unfrozen part of the quiver defining the higher Teichmüller theory for the (m+ 2)-
gon and the Lie algebra sln+1 with m = 1, . . . , 5 and n = 1, . . . , 6. Here ∅ denotes the empty quiver,
and ∞N denotes a mutation class of a quiver of infinite mutation type with N vertices.
Organization of the paper. In Section 2, we recall the definition of the saturated clus-
ter modular groups. We introduce the saturated modular complexes and prove that they are
simply-connected. In Section 3, we recall Brown’s algorithm [3] 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 gen-
erality. In Section 4, we compute the presentations of the saturated cluster modular groups of
type X7 and X6. In Section 5, we give a proof of Theorem 1.6.
2 Basic definitions
2.1 Seed mutations
In this section we review seed mutations, in order to motivate the definitions of the saturated
cluster modular group and the saturated modular complex. We follow the notations of [9].
Let I be a finite set. A seed is a tuple s = (ε, (Ai)i∈I , (Xi)i∈I), where ε = (εij)i,j∈I is
a 1
2Z-valued skew-symmetrizable matrix, (Ai)i∈I and (Xi)i∈I are two bunches of algebraically
independent commutative variables. Our convention of skew-symmetrization is that there exist
positive integers dj (j ∈ I) such that the matrix
(
εijd
−1
j
)
i,j
is skew-symmetric. The matrix ε
is called the exchange matrix and the variables (Ai) (resp. (Xi)) are called the cluster A-
(resp. X -)variables. We fix a subset I0 ⊂ I called the frozen subset, and assume εij ∈ Z unless
(i, j) ∈ I0 × I0.
The exchange matrix ε 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 ∈ I is assigned a weight di,
and there exist
∣∣d−1
j εij
∣∣ gcd(di, dj) arrows from the vertex i to the vertex j (resp. j to i) if εij > 0
(resp. εij < 0). See [19] for details. We tacitly use this correspondence in the sequel. In this
sense we call an element of I a vertex and elements of I0 are called frozen vertices.
Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists 5
Definition 2.1 (seed mutations). For a seed s = (ε, (Ai)i∈I , (Xi)i∈I) and a vertex k ∈ I \ I0,
we define a new seed s′ = (ε′, (A′i)i∈I , (X
′
i)i∈I) as follows:
ε′ij :=
−εij if k ∈ {i, j},
εij +
|εik|εkj + εik|εkj |
2
otherwise,
(2.1)
A′i :=
A−1
i
( ∏
εkj>0
A
εkj
j +
∏
εkj<0
A
−εkj
j
)
if i = k,
Ai otherwise,
(2.2)
X ′i :=
{
X−1
k if i = k,
Xi
(
1 +X
− sgn(εik)
k
)−εik otherwise.
(2.3)
We write s′ = µk(s) and refer to this transformation of seeds as the mutation directed to the
vertex k. The transformation (2.1) is independent of the other data and called the matrix
mutation (or quiver mutation). The rational transformations (2.2) and (2.3) are called the
cluster A- and X -transformations.
A seed permutation is a permutation σ of I which preserves I0 setwise. It acts on a seed as
σ(ε, (Ai), (Xi)) = (ε′, (A′i), (X
′
i)), where
ε′ij := εσ−1(i),σ−1(j), A′i := Aσ−1(i), X ′i := Xσ−1(i).
It is called a seed isomorphism if it satisfies ε′ = ε. 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. It is said to be trivial if it also preserves the initial cluster variables.
The set
|s| := {φ(s) |φ: mutation sequence with the initial seed s}
is called the mutation class of s. The set |ε| of exchange matrices (or weighted quivers) appearing
in |s| is called the mutation class of ε.
The cluster modular group Γs based at a seed s is the group of mutation loops with the initial
seed s, modulo trivial ones. Since the group structure of the cluster modular group only depends
on the mutation class |s|, we will use the notation Γ|s| when no confusion can occur.
Remark 2.2. The cluster modular group acts on some geometric objects A|s| and X|s| called
the cluster A- and X -varieties, preserving their geometric structures. See [9] for details.
Example 2.3. Here is a geometric example. Let Σ be a marked surface. It is a connected
oriented compact 2-dimensional manifold with boundary equipped with a finite subset M ⊂ Σ
of marked points, satisfying some conditions. See [5] for details. An ideal triangulation of Σ is
the isotopy class of a collection ∆ = {αi}n(Σ)
i=1 of simple arcs whose endpoints are marked points,
and the complement Σ \
⋃n(Σ)
i=1 αi consists of triangles. Here n(Σ) only depends on Σ.
Given such an ideal triangulation ∆, we draw a quiver on each triangle as shown in Fig. 2.
Gluing them along the edges, we get a quiver drawn on a surface. The vertices on ∂Σ 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
ε =
0 2 −2
−2 0 2
2 −2 0
.
6 T. Ishibashi
Figure 2. The quiver associated with a triangle.
Hence we can form a seed s∆. Here cluster variables can be interpreted as coordinate functions
on certain extensions of the Teichmüller space of Σ. See [18] for details. Except for a few small
surfaces, we have
Γs∆
∼=
{
MC(Σ) if Σ is a closed surface with one marked point,
MC(Σ) n {±1}p otherwise.
Here p is the number of interior marked points (punctures) and MC(Σ) denotes the mapping
class group of Σ, which is the group of isotopy classes of orientation-preserving diffeomorphisms
on Σ that fix the subset M setwise. Its action on {±1}p is induced by the permutation of
punctures. See [2, 15].
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 2.4 (standard (h + 2)-gon relations [9]). Let s = (ε, (Ai)i∈I , (Xi)i∈I) be a seed and
(p, h) = (0, 2), (1, 3), (2, 4) or (3, 6). Then for each i, j ∈ I such that εij = −pεji = p, the
mutation sequence rij := ((i j)µi)
h+2 = ((i j)µj)
h+2 is trivial. Here (i j) denotes the seed
permutation given by the transposition of i and j.
We call these sequences rij(= rji) 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:
• µiµj = µjµi if εij = 0 (square relation).
• µiµjµiµjµi = (i j) if εij = ±1 (pentagon relation).
Example 2.5. Here is a basic example of the pentagon relation in cluster modular group. Let
I := {0, 1}, I0 := ∅ and consider a skew-symmetric exchange matrix
ε :=
(
0 1
−1 0
)
,
which is called the exchange matrix of type A2. The mutation sequence φ := (0 1)µ0 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
φ(A0, A1) =
(
A1,
1 +A1
A0
)
, φ(X0, X1) =
(
X1(1 +X0), X−1
0
)
.
Then the pentagon relation implies that φ has order 5, or one can check it by seeing the cluster
transformations. In particular we have ΓA2
∼= Z/5.
Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists 7
Cluster Dehn twists. An element φ ∈ Γ|s| of infinite order is called a cluster Dehn twist
if there exists a seed t ∈ |s| such that φn = ((i j)µj)
` as an element of Γt for some non-
zero integers n, ` and vertices i, j ∈ I − I0. For example, I := {0, 1}, I0 := ∅ and consider
a skew-symmetric exchange matrix
ε :=
(
0 k
−k 0
)
for an integer k ≥ 2. Then the mutation sequence φ := (0 1)µ0 gives an element of the cluster
modular group, and it is a cluster Dehn twist.
2.2 Saturated cluster modular groups and saturated modular complexes
In this section we recall the definition of the saturated cluster modular group [10] (or the spe-
cial cluster modular group [9]). Also we introduce the saturated cluster complex on which the
saturated cluster modular group acts.
Definition 2.6. The saturated cluster modular group Γ̂ε based at an exchange matrix ε is the
group of (matrix) mutation loops with the initial exchange matrix ε, modulo the equivalence
relation generated by:
1. Involutivity: µkµk = 1 for each k ∈ I,
2. Naturality: σµkσ
−1 = µσ(k) for each k ∈ I and a seed permutation σ, and
3. Standard relations: rij = rji = 1 under the situation of Lemma 2.4.
Take a seed s = (ε, (Ai)i∈I , (Xi)i∈I) with the exchange matrix ε. Then Lemma 2.4 implies
that we have a surjective homomorphism Γ̂ε → Γs.
To investigate the saturated cluster modular group, we define a 2-complex on which the
saturated cluster modular group acts simplicially. Let ε = (εij)i,j∈I be an exchange matrix
and set n := |I \ I0|. A labeled n-regular tree T is an (infinite) n-valent tree with a labeling
{edges of T} → {1, . . . , n} which induces a bijection {edges incident to v} ∼−→ {1, . . . , n} for each
vertex v of T . Choose a vertex v0 of T and assign a matrix to each vertex of T by the following
rules:
• The matrix assigned to v0 is the initial exchange matrix ε.
• The matrices assigned to a pair of vertices connected by an edge labeled by k are related
by the matrix mutation µk.
Let T|ε| denotes the n-regular tree equipped with such an assignment of exchange matrices on
the vertices. Let D be the subgroup of Aut(T|ε|) which consists of elements which preserve the
assigned matrices.
Lemma 2.7. We have a natural surjective homomorphism Φv0 : D → Γ̂ε.
Proof. For an element γ ∈ D, the two matrices assigned to v0 and γ−1(v0) are the same. Let
p(γ) = eik . . . ei1 be the unique edge path in T|ε| from v0 to γ−1(v0), where an edge ei has the
labeling i. Then there exists a seed isomorphism σ = σ(γ) from ε to µik . . . µi1(ε) such that
γ(ei) = eσ(i) for all i ∈ I. For another element γ′ ∈ D, let p(γ′) = ejl . . . ej1 be the edge path
from v0 to γ′−1(v0). Then γ−1(p(γ′)) = eσ−1(jl) . . . eσ−1(j1) is the edge path from γ−1(v0) to
(γ′γ)−1(v0). Hence we have
Φv0(γ′)Φv0(γ) = σ(γ′)µjl · · ·µj1σ(γ)µik · · ·µi1 = σ(γ′)σ(γ)µσ−1(jl) · · ·µσ−1(j1)µik · · ·µi1
= σ(γ′γ)µσ−1(jl) · · ·µσ−1(j1)µik · · ·µi1 = Φv0(γ′γ).
Thus the map Φv0 : D → Γ̂ε given by γ 7→ σµik · · ·µi1 is a group homomorphism. It is clearly
surjective. �
8 T. Ishibashi
The graph Ê|ε| := T|ε|/ ker Φv0 is called the saturated exchange graph. The saturated cluster
modular group acts on Ê|ε| as graph automorphisms via Φv0 . We write Γ̂v0 := Φ−1
v0
(
Γ̂ε
)
⊂
Aut
(
Ê|ε|
)
.
Remark 2.8.
1. An element γ of D belongs to the subgroup ker Φv0 if and only if there exists a vertex v
equipped with ε and the edge path from v to γ(v) is given by a concatenation of standard
sequences.
2. If we change the basepoint of T to a vertex v1 which is connected with v0 by an edge ek,
then we have Φv1 = Adµk ◦Φv0 : D → Γ̂µk(ε). Here Adµk denotes the conjugation by µk.
The exchange graph of the mutation class |s| is defined as follows [13]. Starting with a n-
regular tree T with a fixed vertex v0, we assign a seed to each vertex of T by the following
rules:
• The seed assigned to v0 is the initial seed s.
• The seeds assigned to a pair of vertices connected by an edge labeled by k are related by
the seed mutation µk.
We identify two vertices with the same seeds, and the resulting graph E|s| is the exchange graph.
From Lemma 2.4 and Remark 2.8 we have a natural covering map Ê|ε| → E|s|.
Lemma 2.9. The group homomorphism Γ̂|ε| → Γ|s| is an isomorphism if and only if the covering
map Ê|ε| → E|s| is a homeomorphism.
Proof. Both conditions are equivalent to the condition that relations among mutations are
generated by standard ones. �
Definition 2.10 (saturated modular complex). To each cycle in Ê|ε| which is given by a standard
sequence, we attach a 2-cell given by the corresponding standard polygon. The resulting 2-
complex M̂|ε| is called the saturated modular complex.
Here is a simple but important lemma:
Lemma 2.11. The saturated modular complex M̂|ε| is simply-connected.
Proof. From Remark 2.8, each combinatorial cycle α in M̂|ε| is a concatenation of standard
cycles. Since each standard cycle spans a 2-cell given by the corresponding standard polygon,
α can be contracted to a point along these 2-cells. Therefore M̂|ε| is simply-connected. �
2.3 Seeds of finite mutation type
An exchange matrix ε is said to be of finite mutation type if the mutation class |ε| is a finite set.
If ε is mutation-equivalent to a Dynkin quiver, then it is said to be of finite type. Otherwise it
is of infinite type.
Theorem 2.12 (Fomin–Zelevinsky [12]). For a seed s = (ε, (Ai)i∈I , (Xi)i∈I), the underlying
exchange matrix ε is of finite type if and only if the mutation class |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.
Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists 9
Theorem 2.13 (Felikson–Shapiro–Tumarkin [6]). Any skew-symmetric exchange matrix with
size ≥ 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: E6, E7, E8, Ẽ6, Ẽ7, Ẽ8, E
(1,1)
6 , E
(1,1)
7 , E
(1,1)
8 ,
X6, X7.
The quivers representing the eleven mutation classes are given in Fig. 6.1 of [6]. The first
three are of finite type.
Theorem 2.14 (Felikson–Shapiro–Tumarkin [5]). An exchange matrix with size ≥ 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: G̃2, F4, F̃4, G
(∗,+)
2 , G
(∗,∗)
2 , F
(∗,+)
4 , F
(∗,∗)
4
The quivers representing the seven mutation classes are given in Fig. 1.1 of [5]. Only F4 is
of finite type. Here an exchange matrix is said to be s-decomposable if it is obtained by gluing
certain “blocks”, see [5]. We say that a seed s = (ε, (Ai)i∈I , (Xi)i∈I) is of finite mutation type
if the underlying exchange matrix is.
Lemma 2.15. If a seed s is of finite mutation type and infinite type, then the cluster modular
group Γ|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 ε. 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 3.8. �
3 Brown’s algorithm for a group acting on a CW complex
Let us recall Brown’s algorithm [3] 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 ∼= π1(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.
3.1 General setting
We prepare some notations and terminology. The set of k-cells of a CW complex X is denoted
by Ck(X). A 1-cell σ ∈ C1(X) is said to be inverted if there exists an element g ∈ G which
fixes σ and reverses the orientation of σ. If σ is not a loop in X, which is the case we will deal
with, the condition means that g transposes two endpoints of σ. In the following, we will assume
the following condition:
each 1-cell of X has distinct endpoints and is not inverted. (3.1)
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 π : E(X) → C1(X) be the projection forgetting the orientation of each 1-cell. A G-
equivariant section P : C1(X)→ E(X) is called an orientation of the CW complex X. Such an
orientation exists under the assumption (3.1). The initial (resp. terminal) vertex of an edge
e ∈ E(X) is denoted by o(e) (resp. t(e)). We represent an edge e by the symbol
(
o(e)
e−→ t(e)
)
.
For a 2-cell in X, the attaching map ∂∆2 → X(1) is represented by a combinatorial cycle
τ = e1 · · · en such that o(e1) = t(en) ∈ C0(T ). Here ∆2 denotes the standard 2-simplex, and we
read the combinatorial cycle from the left to the right by convention.
10 T. Ishibashi
Definition 3.1. A choice of the following data (T, P,E+, F ) is called a data of representatives
for the action of G on X:
1. T ⊆ X(1) is a tree of representatives for X mod G.
2. P : C1(X)→ E(X) is an orientation of X.
3. E+ ⊆ P (C1(X)) is a set of representatives mod G such that P (C1(T )) ⊆ E+ and o(e) ∈
C0(T ) for each edge e in E+.
4. F ⊆ C2(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 ∈ E+, there is a unique vertex we ∈ C0(T ) which is G-equivalent to the terminal vertex t(e).
Choose an element ge ∈ G such that
t(e) = ge(we). (3.2)
Here we choose ge := 1 ∈ G if e is contained in T by convention. These elements together with
the elements of the isotropy group Gv := {g ∈ G | g(v) = v} (v ∈ C0(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 ∈ E(X) such that v := o(e) ∈ C0(T ) has one of the following
forms:
(a) e =
(
v
he−→ hgewe
)
, where e ∈ E+ and h ∈ Gv.
(b) e =
(
we
hg−1
e ē−−−−→ hg−1
e v
)
, where e ∈ E+, ē is the opposite edge and h ∈ Gwe .
Then we set
g :=
{
hge (case (a)),
hg−1
e (case (b)).
For a combinatorial cycle τ = e1 · · · en such that o(e1) = t(en) ∈ C0(T ), the above rule provides
us with elements gτ1 , . . . , g
τ
n ∈ G such that t(ei) ∈ gτ1 · · · gτi (C0(T )). Let gτ := gτ1 · · · gτn. Note
that gτ ∈ Go(e1).
Theorem 3.2 (Brown [3]). Let G be a group acting on a simply-connected CW complex X
preserving the cell structure. Assume the condition (3.1). Then G is generated by the elements ge
(e ∈ E+) and the isotropy groups Gv (v ∈ C0(T )), subject to the following relations:
1. ge = 1 if e is contained in T .
2. g−1
e ie(g)ge = ce(g) for any e ∈ E+ and g ∈ Ge, where ie : Ge → Go(e) is the inclusion and
ce : Ge → Gwe is defined by g 7→ g−1
e gge.
3. gτ = gτ1 · · · gτn for any τ ∈ F .
We call the relations of type (2) the isotropy relations, and those of type (3) the face relations.
3.2 Application to the cluster modular groups
In this section, we collect basic properties of the action of the saturated cluster modular group
on the saturated cluster modular complex which are relevant to the application of the above
method. Let ε = (εij)i,j∈I be an exchange matrix. Let V := C0
(
M̂|ε|
)
be the set of vertices of
the saturated modular complex. First note that we have a natural surjective map Mat: V → |ε|
which extracts the exchange matrix assigned to each vertex.
Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists 11
Lemma 3.3. The map Mat induces a bijection Mat: V/Γ̂|ε| → |ε|.
Proof. The surjective map Mat: V → |ε| descends to a map Mat: V/Γ̂|ε| → |ε|, since the
action of the saturated cluster modular group preserves the assigned exchange matrices. If two
vertices v0 and v1 are assigned the same matrices, then there exists an element γ ∈ Aut(D) such
that γ(v0) = v1. Hence v0 is sent to v1 by an element of Γ̂|ε|. Thus the map Mat: V/Γ̂|ε| → |ε|
is bijective. �
Lemma 3.4. The action of the saturated cluster modular group on the saturated modular com-
plex M̂|ε| satisfies the condition (3.1).
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 φ ∈ Γ̂|ε|, φ 6= 1, fixes an edge e
of M̂|ε|. Let x and y be endpoints and i ∈ I the label of e. Then φ is of the form φ = σµi as an
element of Γ̂[x], where σ is an involutive quiver isomorphism. Now suppose we have φ(x) = y and
φ(y) = x. Then we have φ2 ∈ Aut([x]), namely, φ2 must be given by a seed isomorphism τ as
an element of Γ̂[x]. Hence we have a non-trivial relation τ = µσ−1(i)µi, 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 v0 and v1, the above lemma
provides an inclusion of isotropy groups
(
Γ̂|ε|
)
e
⊆
(
Γ̂|ε|
)
v0
∩
(
Γ̂|ε|
)
v1
. An element σ of (Γ̂|ε|)e is
called a simultaneous seed isomorphism of the pair (v0, v1). We also say that σ is a simultaneous
seed isomorphism of the pair of underlying exchange matrices ([v0], [v1]).
Lemma 3.5. The Γ̂|ε|-orbit of an oriented edge in M̂|ε| is determined by a triple (ε1, ε2; k)
of two exchange matrices ε1, ε2 ∈ |ε| and an index k ∈ I such that ε2 = µk(ε1), modulo the
equivalence relation generated by (ε1, ε2; k) ∼ (σ.ε1, σ.ε2;σ(k)) for seed permutations σ.
Proof. Fix a tree T of representative. Then each oriented edge is translated by an element
of Γ̂|ε| so that its origin belongs to C0(T ). If we have two oriented edges with the same origin,
they are translated each other by an element of Γ̂|ε| if and only if the corresponding triples are
equivalent. �
Lemma 3.6. The Γ̂|ε|-orbit of a standard (h+ 2)-gon cycle C in M̂|ε| is determined by a triple
(ε′; k, l) of an exchage matrix ε′ ∈ |ε| and indices k, l ∈ I− I0 such that ε′kl = −pε′lk = p, modulo
the equivalence relation generated by (ε′; k, l) ∼ (σ.ε′;σ(k), σ(l)). 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 Γ̂|ε| 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 ∈ V and determined by a pair
of indices k, l is denoted by Ch+2(k, l)v. A cycle may have several expressions.
Definition 3.7 (the saturated modular graph). We define the saturated modular graph Ĝ|ε|
to be the graph whose vertices are exchange matrices in |ε| and edges are Γ̂|ε|-orbits of edges
in M̂|ε|.
From Lemma 3.6, each Γ̂|ε|-orbit of a standard cycle is represented by a circuit in the saturated
modular graph Ĝ|ε|. The following lemma is clear from the definition.
Lemma 3.8. If ε is of finite mutation type, then the saturated modular graph is a finite graph.
12 T. Ishibashi
0
1
2 3
4
56 Q0
0
1
2 3
4
56 Q1
Figure 3. Two quivers in |X7|.
Q0 Q1[e0]
[e1] [e2]
Figure 4. The modular graph ĜX7 . The tree T is shown by a thick line. The set E+ is shown by dashed
lines.
4 Presentations of Γ̂X7
and Γ̂X6
In this section, based on the method established in Section 3, we give finite presentations of the
saturated cluster modular groups of type X7 and X6.
4.1 The saturated cluster modular group of type X7
The mutation class X7 consists of two quivers Q0 and Q1. They are shown in Fig. 3. Let v0 ∈ V
be a vertex of the saturated modular complex such that Mat(v0) = Q0. Let us fix a data of
representatives as follows:
1. The tree T consists of two vertices v0 and v1 := µ0(v0) together with an edge e0 :=
(
v0
µ0−→
v1
)
.
2. The set E+ of representative of oriented edges consists of three oriented edges e0, e1 :=(
v0
µ1−→ µ1(v0)
)
and e2 :=
(
v1
µ1−→ µ1(v1)
)
. There is an orientation of M̂X7 which extends
them.
3. The set F := {τ1, τ2, τ3, τ4} of representative of 2-cells consists of two square cycles τ1 :=
C4(1, 4)v0 , τ2 := C4(1, 3)v1 and two pentagon cycles τ3 := C5(0, 1)v0 , τ4 := C5(1, 4)v1 .
The tree T and the set E+ are shown in Fig. 4. Then the following proposition can be easily
verified using Lemmas 3.3–3.6.
Proposition 4.1. The data (T, P,E+, F ) determines a data of representatives for the action
of Γ̂X7 on M̂X7.
Then we choose the Brown generators as follows:
1. The isotropy group of the vertex v0 is the image of the group homomorphism
ι : S({1, 3, 5})→ S({0, 1, 2, 3, 4, 5, 6})
given by ι(σ) : 0 7→ 0, k 7→ σ(k), k + 1 7→ σ(k) + 1 for k ∈ {1, 3, 5}. Here S(X) denotes
the symmetric group of a finite set X. We write a cyclic permutation k1 7→ · · · 7→ kn 7→ k1
of {1, 3, 5} as σk1...kn . It turns out that the isotropy group of v1 is the same, hence
S3 := ι(S({1, 3, 5})) consists of simultaneous seed isomorphisms.
Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists 13
2. Set ge0 := 1, ge1 := (1 2)µ1 ∈ Γ̂v0 and ge2 := (0 1 2)(3 4 5 6)µ1 ∈ Γ̂v1 . Then they satisfy
the condition (3.2). Changing their basepoints to the vertex v0 using the edge e0 (see
Remark 2.8), we get φ1 := (1 2)µ1 and ψ1 := (0 1 2)(3 4 5 6)µ2µ1µ0, which we regard as
elements in Γ̂Q0 .
From Theorem 3.2, the elements in S3 together with φ1 and ψ1 generate the saturated cluster
modular group. Let us investigate the relations among them. To simplify the computations,
let us introduce auxiliary elements φ3 := σ135φ1σ
−1
135 and φ5 := σ153φ1σ
−1
153. Similarly define ψ3
and ψ5. Then the relations are determined as follows.
Isotropy relations. The isotropy group of the edge e1 is generated by σ35. Then the isotropy
relation implies φ−1
1 σ35φ1 = σ35. Similarly, the isotropy group of the edge e2 is generated by σ35.
Then the isotropy relation implies ψ−1
1 σ35ψ1 = σ35.
Face relations. From the cycle τ1 we get
φ3φ
−1
1 φ−1
3 φ1 = (3 4)µ3(1 2)µ2(3 4)µ4(1 2)µ1 = µ4µ1µ4µ1 = 1.
From the cycle τ3 we get
σ35φ
−1
1 ψ2
1 = (3 5)(4 6)(1 2)µ2(0 1 2)(3 4 5 6)µ2µ1µ0(0 1 2)(3 4 5 6)µ2µ1µ0
= (0 1)µ0µ1µ0µ1µ0 = 1.
In order to investigate τ2 and τ4, we represent each element in Γ̂v1 : ψ1 = (0 1 2)(3 4 5 6)µ1,
ψ3 = (0 3 4)(5 6 1 2)µ3 and ψ5 = (0 5 6)(1 2 3 4)µ5. Then from the cycle τ2 we get
ψ−1
3 ψ1ψ
−1
3 ψ1 = (0 4 3)(6 5 2 1)µ4(0 1 2)(3 4 5 6)µ1(0 4 3)(6 5 2 1)µ4(0 1 2)(3 4 5 6)µ1
= µ3µ1µ3µ1 = 1.
From the cycle τ4 we get
σ153ψ5ψ1ψ3ψ5ψ1 = (1 5 3)(2 6 4)(0 5 6)(1 2 3 4)µ5(0 1 2)(3 4 5 6)µ1
(0 3 4)(5 6 1 2)µ3(0 5 6)(1 2 3 4)µ5(0 1 2)(3 4 5 6)µ1
= (1 4)µ1µ4µ1µ4µ1 = 1.
Summarizing these relations, we obtain Theorem 1.1.
4.2 The saturated cluster modular group of type X6
Next we investigate the mutation class X6. Although the computation is a little more compli-
cated in this case, the strategy is the same as before. The mutation class X6 consists of five
quivers Q0,. . . , Q4. They are shown in Fig. 5. Let v0 ∈ V be a vertex of the saturated modular
complex such that Mat(v0) = Q0. Let us fix a data of representatives as follows, see Fig. 6.
1. The tree T consists of five vertices v0, v1 := (2 3)µ1(v0), v2 := (3 5)µ6(v1), v3 :=
(2 4)(1 6)µ6(v2) and v4 := (4 5)µ1(v3) together with four edges
e0 :=
(
v0
µ1−→ v1
)
, e1 :=
(
v1
µ6−→ v2
)
,
e2 :=
(
v2
µ6−→ v3
)
, e3 :=
(
v3
µ1−→ v4
)
.
Here the permutations are inserted in order to make the labelings of quiver vertices at
v0, . . . , v4 consistent with those shown in Fig. 5.
14 T. Ishibashi
1
2
3 4
5
6 Q0
1
2
3 4
5
6 Q1
1
62
4
3 5 Q2
1
2
3 4
5
6 Q3
1
2
3 4
5
6 Q4
Figure 5. Five quivers in |X6|.
2. The set E+ :=
{
e0, . . . , e3, e0, . . . , e6
}
of representatives of oriented edges consists of eleven
oriented edges, where
e0 :=
(
v0
µ6−→ (v4)
)
, e1 :=
(
v0
µ2−→ µ2(v0)
)
, e2 :=
(
v4
µ2−→ µ2(v4)
)
,
e3 :=
(
v1
µ2−→ µ2(v1)
)
, e4 :=
(
v1
µ3−→ µ3(v1)
)
, e5 :=
(
v2
µ4−→ µ4(v2)
)
,
e6 :=
(
v2
µ3−→ µ3(v2)
)
.
There is an orientation of M̂X6 which extends them.
3. The set F := {τ1, . . . , τ11} of representatives of 2-cells consists of six square cycles
τ1 := C4(2, 6)v0 , τ2 := C4(2, 5)v0 , τ3 := C4(2, 5)v4 ,
τ4 := C4(3, 4)v1 , τ5 := C4(2, 5)v0 , τ6 := C4(4, 6)v1
and five pentagon cycles
τ7 := C5(1, 6)v0 , τ8 := C5(1, 2)v0 , τ9 := C5(1, 5)v4 ,
τ10 := C5(2, 4)v1 , τ11 := C5(2, 4)v3 .
Then the following proposition can be easily verified using Lemmas 3.3–3.6.
Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists 15
Q0
Q1
Q2
Q3
Q4
e0
e1 e2
e3
e0
e1
e2
e3
e4
e5 e6
Figure 6. The modular graph ĜX6
. The tree T is shown by thick lines. The set E+ is shown by dashed
lines.
Proposition 4.2. The data (T,E+, F ) determines a data of representatives for the action of Γ̂X6
on M̂X6.
Then we get the Brown generators as follows:
1. The isotropy group of the vertices v0, v1, v3 and v4 are the same, and generated by an
involution σ. It is written as σ = (2 4)(3 5) in Γ̂Q0 and Γ̂Q4 , σ = (2 5)(3 4) in Γ̂Q1 and Γ̂Q3 .
The isotropy group of the vertex v2 is generated by a permutation ρ := (1 2 5)(3 6 4) ∈ Γ̂Q2
of order 3.
2. Set gei := 1 for i = 0, . . . , 3, and
ge0 := (1 6)µ1µ6µ1µ6µ1 ∈ Γ̂v0 ,
ge1 := (2 3)µ2 ∈ Γ̂v0 ,
ge2 := (2 3)µ2 ∈ Γ̂v4 ,
ge3 := (4 6)(1 3)µ2(3 5)µ6(2 4)(1 6)µ6 ∈ Γ̂v3 ,
ge4 := (4 6)(1 2)µ3(3 5)µ6(2 4)(1 6)µ6 ∈ Γ̂v3 ,
ge5 := (3 5)(2 1 5 4 6 3)µ4 ∈ Γ̂v2 ,
ge6 := (2 4)(1 6)µ6(1 3 6 5)(2 4)µ3 ∈ Γ̂v2 .
Then they satisfy the condition (3.2). Changing their basepoints to the vertex v0 along
the tree T , we get the corresponding elements in Γ̂Q0 :
φ0 := (1 6)µ1µ6µ1µ6µ1, φ1 := (2 3)µ2,
φ2 := (2 3)µ6µ2µ6, φ3 := (1 5 2 6 3 4)µ4µ2µ4µ3µ1,
φ4 := (1 4)(2 5)(3 6)µ4µ3µ4µ2µ1, φ5 := (1 5 3)(2 4 6)µ3µ4µ6µ1,
φ6 := (1 5)(2 6)µ5µ2µ6µ1.
For example, φ3 = ((2 4)(1 6)µ6(3 5)µ6(2 3)µ1)−1ge3((2 4)(1 6)µ6(3 5)µ6(2 3)µ1). From Theo-
rem 3.2, these elements generate the saturated cluster modular group of type X6. Let us
investigate the relations among them.
Isotropy relations. We have the relations σ2 = 1, ρ3 = 1 and ρ = σφ6.
Face relations. The verification of the following face relations is tedious but straightforward.
16 T. Ishibashi
• τ1: φ2 = φ1. This relation implies that τ2 and τ3 induces the same relation.
• τ2(= τ3): [σ, φ3]2 = 1.
• τ4: σ−1φ−1
4 φ6φ4 = 1.
• τ5: (φ1σ)2
(
φ−1
1 σ
)2
= 1.
• τ6:
(
σφ−1
4 σ−1
)
φ4φ5 = 1.
• τ7: φ0 = 1.
• τ8: φ−1
1 φ−1
3 φ4 = 1.
• τ9:
(
σφ1σ
−1
)
φ3φ
−1
4 = 1.
• τ10: φ−1
4
(
σφ3σ
−1
)
φ5φ3 = σ.
• τ11:
(
σφ4σ
−1
)
φ−1
3
(
σφ6σ
−1
)
φ−1
3 = 1.
For example, the relation induced by τ4 is verified as follows. First we change the basepoint
to v1 using the path (2 3)µ1 from v0 to v1. Then we have φ4 = (1 4)(2 6)(3 5)µ2µ4µ3 and
φ6 = (1 5)(3 6)µ3µ6, represented as elements of Γ̂v1 . Then we compute
φ−1
4 φ6φ4 = (1 4)(2 6)(3 5)µ5µ1µ6(1 5)(3 6)µ3µ6(1 4)(2 6)(3 5)µ2µ4µ3
= (2 5)(3 4)µ4µ3µ4µ3 = σ.
Now we prove Theorem 1.2.
Proof of Theorem 1.2. From the face relations τ1, τ4, τ6, τ7 and τ8, we can reduce the number
of generators as φ2 = φ1, φ6 = φ4σφ
−1
4 , φ5 = φ−1
4 σφ4σ = [φ−1
4 , σ], φ0 = 1 and φ4 = φ3φ1. Then
the remaining relations are summarized as
σ2 = 1, (4.1)
[σ, φ3φ1]3 = 1 (from ρ3 = 1), (4.2)
[σ, φ3]2 = 1 (from τ2), (4.3)
[σ, φ1] =
[
φ−1
1 , σ
]
(from τ5), (4.4)
(σφ1σ
−1)φ3(φ3φ1)−1 = 1 (from τ9), (4.5)
(φ3φ1)−1
(
σφ3σ
−1
)[
(φ3φ1)−1, σ
]
φ3 = σ (from τ10), (4.6)(
σφ3φ1σ
−1
)
φ−1
3 σ[φ3φ1, σ]φ−1
3 = 1 (from τ11). (4.7)
Using the relations (4.1) and (4.5) in the form φ3φ1 = σφ1σ
−1φ3, the last relation is slightly
simplified as
φ3 = φ1[σ, φ3]σ[φ3φ1, σ]. (4.7′)
Set α1 := φ1, β1 := φ3. Let us introduce the auxiliary elements α2 := σα1σ
−1 and β2 := σβ1σ
−1
to simplify the presentation. Then the relations (4.1)–(4.5) can be rewritten as follows:
σ2 = 1, (4.8)(
α2β2β
−1
1 α−1
1
)3
= 1, (4.9)(
β2β
−1
1
)2
= 1, (4.10)
α1α2 = α2α1, (4.11)
β−1
1 α2β1 = α1. (4.12)
Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists 17
The relation (4.12) means that β1 conjugates α1 to α2. Equivalently, β2 conjugates α2 to α1.
The relation (4.6) is equivalently transformed as
α−1
1 β−1
1 β2
[
(β1α1)−1, σ
]
β1 = σ,
α−1
1 β−1
1 β2
(
α−1
1 β−1
1 β2α2
)
β1 = σ,(
β1α
−1
1 β−1
1
)
β2α
−1
1 β−1
1 β2α2 = β1σβ
−1
1 ,
α−1
2 β2α
−1
1 β−1
1 β2α2 = β1σβ
−1
1 ,
where in the last line we used (4.12). Finally we get β2(β1α1)−1β2 = Adα2β1 σ. The relation (4.7′)
is equivalent to
β1 = α1[σ, β1]σ[β1α1, σ] = α1
(
β2β
−1
1 β2
)
α2α
−1
1 β−1
1 σ−1
= α1β1α2
(
α−1
1 β−1
1
)
σ−1 (use (4.10))
= α1β1α2
(
β−1
1 α−1
2
)
σ−1. (use (4.12))
Thus we get β1 = α1
(
β1α2β
−1
1
)
α−1
2 σ−1. Summarizing the relations, we get the desired asser-
tion. �
5 Generation by cluster Dehn twists
In this section, we verify that several (saturated) cluster modular groups of finite mutation type
are generated by cluster Dehn twists. We will use the notation of [6] and [5] for quivers of finite
mutation type (recall Theorems 2.13 and 2.14).
5.1 Type X7 and X6
Recall the presentations given in Theorems 1.1 and 1.2. Let us prove the following:
Proposition 5.1. The cluster modular group of type X7 is generated by the six cluster Dehn
twists φk and ψk (k = 1, 3, 5). The cluster modular group of type X6 has a subgroup of index at
most two generated by the four cluster Dehn twists αk for k = 1, 2, 3, 4, where α3 := β1α1β
−1
1
and α4 := σα3σ
−1.
Proof. It suffices to show that the corresponding saturated cluster modular groups have desired
generators, since the cluster modular group is its quotient.
Type X7. We can delete the permutations using the relations σ35 = ψ−2
1 φ1, σ51 = ψ−2
3 φ3
and σ13 = ψ−2
5 φ5. Hence the saturated cluster modular group is generated by ψk’s and φk’s.
The elements φk are clearly cluster Dehn twists by definition. The elements ψk are also cluster
Dehn twists, since ψ4
k = φ2
k.
Type X6. The element α1 is clearly a cluster Dehn twist by definition. Note that any
conjugate of a cluster Dehn twist is also a cluster Dehn twist. In particular, the elements α2,
α3 and α4 are cluster Dehn twists.
From the relations
β1 = α1
(
β1α2β
−1
1
)
α−1
2 σ−1 = α1α3α
−1
2 σ, β2 = σβ1σ
−1 = α2α4α
−1
1 σ,
any element g ∈ Γ̂X6 can be written as a product of the cluster Dehn twists αk for k = 1, 2, 3, 4
and σ. Since σ2 = 1 and we know the commutation relations of σ and αk, we can move all the
σ’s to the right and g can be written as a product of the cluster Dehn twists αk for k = 1, 2, 3, 4
and only one σ on the extreme right. Thus the subgroup of Γ̂X6 generated by the cluster Dehn
twists αk for k = 1, 2, 3, 4 has index 2. �
Note that for the X6 case, it can possibly occur that the whole cluster modular group is
generated by αk for k = 1, 2, 3, 4 with a help of an additional “non-standard” relation involving σ.
18 T. Ishibashi
3 3
1 2
3 4
Figure 7. The quiver of type G
(∗,∗)
2 .
3 3
1 2
3 4
Qa
3 3
1 2
3 4
Qb
Figure 8. Two quivers equivalent to G
(∗,∗)
2 .
5.2 Type G
(∗,∗)
2
The weighted quiver representing the mutation class G
(∗,∗)
2 is shown in Fig. 7.
Theorem 5.2 (Fock–Goncharov [7, Corollary 4.2]). The saturated cluster modular group of
type G
(∗,∗)
2 is isomorphic to the braid group of type G2:
Γ̂
G
(∗,∗)
2
∼= 〈a, b | (ab)3 = (ba)3〉.
Although this theorem is obtained by a more topological observation, it can be also verified
by a computation based on Section refapply cluster using the data of representative (which is
called the “spanning tree”) chosen loc. cit.
Proposition 5.3. The generators a and b are given by a := pa((3 4)µ4)p−1
a and b := φb′φ−1,
which are cluster Dehn twists. Here b′ := pb((1 2)µ2)p−1
b and φ := (3 4)µ4µ3µ4µ2µ1 are elements
of Γ
G
(∗,∗)
2
, pa := (1 2)(3 4)µ1 and pb := (3 4)µ4µ2µ1 are mutation sequences. In particular the
cluster modular group of type G
(∗,∗)
2 is generated by the two cluster Dehn twists a and b.
Proof. The elements a and b′ are two of the five Brown generators associated with the data
of representatives fixed in [7]. The mutation sequences p−1
a and p−1
b are the ones which con-
nect G
(∗,∗)
2 with the quivers sa and sb shown in Fig. 8, respectively. Then observe that the
vertices 3 and 4 are connected by two arrows in the quiver sa. Hence the element (3 4)µ4 ∈ Γsa
is a cluster Dehn twist, so is a ∈ Γ
G
(∗,∗)
2
. Similarly b′ is a cluster Dehn twist, so is b = φb′φ−1. �
5.3 Type Ẽ6, Ẽ7 and Ẽ8
In [1], 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
Aut+(A) (here A stands for the cluster algebra), which is a subgroup of index ≤ 2 of the cluster
automorphism group. See [14]. Hence we can rephrase the results shown in [1, Table 1] as
follows, restricting our attention to the quivers of affine and finite mutation type.
Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists 19
1 2
3
4
5 6 7
Figure 9. A quiver equivalent to Ẽ6.
Theorem 5.4 (Assem–Schiffler–Shramchenko [1]). The cluster modular groups of affine type
Ẽ6, Ẽ7 and Ẽ8 are given as follows:
• ΓẼ6
∼= Z×S3.
• ΓẼ7
∼= Z× Z/2.
• ΓẼ8
∼= Z.
Proposition 5.5. The cluster modular groups of type Ẽ6, Ẽ7 and Ẽ8 are generated by cluster
Dehn twists.
Proof. Since each group is virtually cyclic, it suffices to find one cluster Dehn twist φ. Indeed, a
sufficiently large power of another generator ψ of infinite order lies in the cyclic group generated
by φ and hence ψ is also a cluster Dehn twist from the definition.
One can directly verify that the initial quiver of type Ẽ6 is mutation-equivalent to the quiver
shown in Fig. 9. Then we have two arrows from the vertex 4 to 3 in s, and other vertices x are
either disjoint from these vertices or connected in the form 3 → x → 4. Hence by the proof of
[15, Lemma 2.32], φ := (3 4)µ3 ∈ Γs is a cluster Dehn twist. In each of the mutation classes Ẽ7
and Ẽ8, one can similarly find a quiver with a pair of vertices u, v and two arrows from v to u,
such that the other vertices x are either disjoint from them or connected in the form u→ x→ v.
Hence we get a cluster Dehn twist. �
A Relations with the mapping class group of an annulus
In this section, we investigate several relations between the cluster modular group of type X7
with the mapping class group of an annulus. Recall from Example 2.3 and Theorem 2.13 that
an ideal triangulation ∆ of a marked surface Σ gives rise to the seed s∆ of finite mutation type.
The following lemma is repeatedly used in this section.
Proposition A.1. For a seed s = (ε, (Ai)i∈I , (Xi)i∈I), define its unfrozen part to be
suf :=
(
(εij)i,j∈I\I0 , (Ai)i∈I\I0 , (Xi)i∈I\I0
)
.
Then we have a group homomorphism e : Γs → Γsuf
which fits into the following exact sequence:
1→ Aut0(s)→ Γs
e−→ Γsuf
,
where Aut0(s) := {σ ∈ Aut(s) |σ(i) = i for all i ∈ I − I0}.
We call the homomorphism e : Γs → Γsuf
the elimination homomorphism.
20 T. Ishibashi
0
1 2
Figure 10. The quiver t.
0
2
1
Figure 11. The ideal triangulation ∆ of the annulus with (2+1) marked points.
Proof. Each element φ ∈ Γs can be written as φ = σµik · · ·µi1 , where i1, . . . , ik ∈ I − I0 and σ
is a seed isomorphism. Here we can write σ = σ0σuf , where σ0 is a permutation of I0 and σuf
is a permutation of I − I0. Then we define e(φ) := σufµil · · ·µi1 ∈ Γsuf
. The map e is a group
homomorphism, since the permutation σ0 commutes with mutations. Moreover e(φ) = 1 if and
only if φ = σ0 ∈ Aut0(s). Thus we get the desired exact sequence. �
Similarly we can define the elimination homomorphism between the corresponding saturated
cluster modular groups, and we have a similar exact sequence. The elimination homomorphism
is not surjective in general, and the image e(Γs) ⊂ Γsuf
can have infinite index.
Let t be the quiver shown in Fig. 10. Note that it gives a subquiver of both X6 and X7.
Lemma A.2. The quiver t coincides with the unfrozen part of the quiver associated with the
ideal triangulation ∆ of the annulus S with (1 + 2) marked points shown in Fig. 11. Moreover,
the elimination homomorphism induces an isomorphism e : Γs∆ = MC(S)
∼−→ Γt. Under this
isomorphism, the generator s ∈ MC(S) ∼= Z which rotates the two marked points on one boundary
component (called a “half twist”) corresponds to the element (0 1 2)µ2µ1µ0 ∈ Γt. The Dehn
twist t := s2 ∈ MC(S) corresponds to the element (1 2)µ1 ∈ Γt.
Proof. The injectivity is easily verified using the exact sequence in Proposition A.1. It can
be verified that Γt is generated by the element (0 1 2)µ2µ1µ0, by observing that the mutation
class |t| consists of two seeds and the element acts on the set Mat−1(t) ⊂ V
(
M̂|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 Γ̂s∆
∼= Γs∆ in this case, see Theorem 9.17 in [11]. Hence we also have
Γ̂t
∼= Γt.
Let Γ
(12)
X7
be the subgroup of ΓX7 which consists of elements of the form φ = σµik · · ·µi1 ,
where i1, . . . , ik ∈ {0, 1, 2} and σ is a permutation such that σ({0, 1, 2}) = {0, 1, 2}. Note that
the group Γ
(12)
X7
is naturally identified with the cluster modular group of the seed obtained from
the seed X7 by freezing the vertices {3, 4, 5, 6}.
Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists 21
Lemma A.4. We have a split exact sequence
1→ Z/2 i(12)
−−→ Γ
(12)
X7
e(12)
−−−→ Γt = MC(S)→ 1.
Here i(12) is given by 1 7→ σ35 = (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)(ψ1). The
commutativity relations ψ1σ35 = σ35ψ1 and φ1σ35 = σ35φ1 in Theorem 1.1 show that the image
of i(12) is central. Since MC(S) ∼= Z, there exists a section of e(12). �
Having this lemma in mind, we write MC(S)(12) := Γ
(12)
X7
∼= MC(S)×Z/2. The relation t = s2
in MC(S) corresponds to the relation φ1 = ψ2
1σ35 in MC(S)(12).
Similarly we define the subgroups MC(S)(34), MC(S)(56) of the cluster modular group ΓX7 .
They are isomorphic to MC(S) × Z/2. Summarizing, we have proved the following geometric
interpretation of Proposition 5.1 in terms of these extended mapping class groups:
Theorem A.5. The extended mapping class groups MC(S)(12), MC(S)(34) and MC(S)(56) ge-
nerate the cluster modular group ΓX7. The six cluster Dehn twists given in Proposition 5.1
corresponds to the half twist and Dehn twist in MC(S) via the elimination homomorphisms.
Acknowledgements
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 Γ̂X6 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.
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1 Introduction and main results
2 Basic definitions
2.1 Seed mutations
2.2 Saturated cluster modular groups and saturated modular complexes
2.3 Seeds of finite mutation type
3 Brown's algorithm for a group acting on a CW complex
3.1 General setting
3.2 Application to the cluster modular groups
4 Presentations of "0362X7 and "0362X6
4.1 The saturated cluster modular group of type X7
4.2 The saturated cluster modular group of type X6
5 Generation by cluster Dehn twists
5.1 Type X7 and X6
5.2 Type G2(*,*)
5.3 Type 6, 7 and 8
A Relations with the mapping class group of an annulus
References
|
| id | nasplib_isofts_kiev_ua-123456789-210585 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-17T12:04:17Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Ishibashi, Tsukasa 2025-12-12T10:30:46Z 2020 Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists. Tsukasa Ishibashi. SIGMA 16 (2020), 025, 22 pages 1815-0659 2020 Mathematics Subject Classification: 13F60; 05E15; 30F60 arXiv:1711.07785 https://nasplib.isofts.kiev.ua/handle/123456789/210585 https://doi.org/10.3842/SIGMA.2020.025 We give a method to compute presentations of saturated cluster modular groups. Using this, we obtain finite presentations of the saturated cluster modular groups of finite mutation type X₆ and X₇. We verify that the cluster modular groups of finite mutation type Ẽ₆, Ẽ₇, Ẽ₈, G⁽*'*⁾₂, X₆ and X₇ are virtually generated by cluster Dehn twists. 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 Гₓ₆ in the first version of this paper. Most of the 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. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists Article published earlier |
| spellingShingle | Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists Ishibashi, Tsukasa |
| title | Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists |
| title_full | Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists |
| title_fullStr | Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists |
| title_full_unstemmed | Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists |
| title_short | Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists |
| title_sort | presentations of cluster modular groups and generation by cluster dehn twists |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210585 |
| work_keys_str_mv | AT ishibashitsukasa presentationsofclustermodulargroupsandgenerationbyclusterdehntwists |