On the isoperimetric functions of a class of Artin groups

Arye Juhász

doi:10.4171/ggd/950

JournalsggdVol. 20, No. 3pp. 771–796

On the isoperimetric functions of a class of Artin groups

Arye Juhász
Technion – Israel Institute of Technology, Haifa, Israel

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Abstract

We find a polynomial (

n^{6}

) isoperimetric function for Artin groups, the defining graph of which contains no edges labelled by

3 .

This in particular shows that even Artin groups have solvable word problem. We use small cancellation theory of relative extended presentations.

1. Introduction

Let

m, n \in ℕ

(

ℕ

– the natural numbers),

m \geq 0,

n \geq 1,

and let

Γ

be a simple graph without loops, with vertex set

V = {v_{1}, \dots, v_{n}}

and edge set

E = {e_{1}, \dots, e_{m}} .

Label the edges by natural numbers via a labelling function

λ : E \to ℕ ∖ {1} .

Denote

λ (e) = n_{i j},

where the edge

e

connects

v_{i}

to

v_{j},

i \neq j

(

n_{i j} = n_{j i}

), and let

n_{i j} = 0

if

v_{i}

and

v_{j}

are not connected by an edge. For every such labelled graph corresponds a group presentation

(1.1)

𝒫 (Γ) = ⟨ X ∣ ℛ ⟩

such that

X = {x_{v}, v \in V},

ℛ = {R_{e}, e \in E},

denote

x_{v_{i}}

by

x_{i},

R_{e} = {(x_{i} x_{j})}^{\frac{1}{2} n_{i j}} {(x_{i}^{- 1} x_{j}^{- 1})}^{\frac{1}{2} n_{i j}},

if

n_{i j}

is even and

R_{e} = {(x_{i} x_{j})}^{\frac{n_{i j} - 1}{2}} x_{i} {(x_{i}^{- 1} x_{j}^{- 1})}^{\frac{n_{i j} - 1}{2}} x_{j}^{- 1},

if

n_{i j}

is odd.

The group presented by

𝒫 (Γ)

is denoted by

A (Γ)

and is called the Artin group defined by $Γ$ .

Examples 1.

$n_{i j} = 0$ for every $i, j,$ $1 \leq i < j \leq n .$

Then $A (Γ)$ is the free group on $X .$
$n_{i j} = 2$ for every $i, j,$ $1 \leq i < j \leq n .$

Then $A (Γ)$ is the free abelian group on $X .$
$n_{i j} \in {0, 2}$ for every $i, j,$ $1 \leq i < j \leq n .$

Then $A (Γ)$ is called right-angled Artin group.
$n_{i j} \in 2 ℕ$ for every $i, j,$ $1 \leq i < j \leq n .$

Then $A (Γ)$ is called an even Artin group.
$n_{i j} \neq 2$ for every $i,$ $j,$ $1 \leq i < j \leq n .$

Then $A (Γ)$ is said to be of large type.

To each Artin group

A (Γ),

there corresponds a Coxeter group

W_{A},

obtained by adding the relators

x_{i}^{2},

i = 1, \dots, n .

An Artin group is said to be of spherical type if

W_{A}

is a finite group. The graphs of the spherical type Artin groups are classically known. An Artin group is said to be of FC type if it belongs to the smallest class of Artin groups which are closed under amalgamation along standard parabolic subgroups and contain all the spherical type Artin groups. Standard parabolic subgroups are the subgroups which are generated by subsets of

X .

We recall isoperimetric functions. Let

G

be a group presented by

𝒫 = ⟨ X ∣ ℛ ⟩ .

Let

W \in F (X),

F (X)

be the free group freely generated by

X,

W

cyclically reduced (i.e.,

W W

contains neither

x x^{- 1}

nor

x^{- 1} x,

x \in X

). Then

W

represents

1

if and only if

(1.2)

\begin{matrix} W & = C_{1} \dots C_{k_{w}}, & k_{w} \geq 1, \\ C_{i} & = f_{i} R_{i}^{ε_{i}} f_{i}^{- 1}, & f_{i} \in F (X), R_{i} \in ℛ and ε \in {1, - 1} . \end{matrix}

A function

g : ℕ \to ℕ

is an isoperimetric function for the presentation

𝒫 = ⟨ X ∣ ℛ ⟩

if for every word

W

which represents

1

of the group

G,

k_{W}

in (1.2) satisfies

k_{W} \leq g (| W |) .

Here

| W |

denotes the length of

W

in

F (X) .

The isoperimetric functions are known for spherical type, FC type, large type, right-angled Artin groups and some other classes of Artin groups. See [1

R. Charney, Artin groups of finite type are biautomatic. Math. Ann. 292 (1992), no. 1, 671–683 Zbl 0736.57001 MR 1157320

, 4

D. F. Holt and S. Rees, Artin groups of large type are shortlex automatic with regular geodesics. Proc. Lond. Math. Soc. (3) 104 (2012), no. 3, 486–512 Zbl 1275.20034 MR 2900234

, 5

J. Huang and D. Osajda, Helly meets Garside and Artin. Invent. Math. 225 (2021), no. 2, 395–426 Zbl 1482.20023 MR 4285138

].

Main Theorem. Let $A$ be an Artin group, $A = A (Γ) .$ If $n_{i j} \neq 3$ for every $n_{i j},$ $1 \leq i < j \leq n,$ then $f (n) = n^{6}$ is an isoperimetric function for $𝒫 (Γ) .$

We believe that “

6

” can be replaced by “

2

”.

Corollary.

Let $A (Γ)$ be an Artin group. If $n_{i j} \neq 3$ for every $i, j,$ $1 \leq i < j \leq n,$ then $A$ has solvable word problem. In particular, even Artin groups have solvable word problem.

The corollary has been established recently also in [9

R. Blasco-Garcéa, M. Cumplido, and R. Morris-Wright, The word problem is solvable for 3-free Artin groups in quadratic time. [v1] 2022 [v4] 2025, arXiv:2204.03523v4

], using a completely different method.

Our basic method is small cancellation theory of extended (relative) presentations via van Kampen and Howie diagrams (see [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

, Lemma 1.2.2, Remark 1.2.3]). (Recall that a van Kampen

ℛ -

diagram over

F (X)

is a connected, simply connected labelled planar

2 -

complex, labelled by elements of

F (X)

such that the labels of the boundary of the

2 -

cells are elements of

ℛ .

We say that the corresponding

2 -

cell (region) realises the element of

ℛ .

)

The original version of this article appeared in [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

] and has been approved by the referee. The present article is a shortened version of it.

Outline of the proof of the Main Theorem. In order to find an isoperimetric function for the presentation

𝒫,

we have to find an upper bound on the number

k_{w}

in (1.2) in terms of the length

| W |

of

W

for every

W

representing

1

in the group defined by

𝒫 .

By the basic theorem of van Kampen diagrams, for every cyclically reduced

W

representing

1,

there is a van Kampen diagram

M

with boundary label

W

such that the number of regions in

M

is at most

k_{W} .

See [10

R. C. Lyndon and P. E. Schupp, Combinatorial group theory. Classics in Math., Ergeb. Math. Grenzgeb. 89, Springer, Berlin, 1977, 339 pp. Zbl 0368.20023 MR 0577064

, Chapter V]. So our problem can be considered as a counting problem in van Kampen diagrams, and this is our approach. We use the following simple basic principles, together with the well-known result in Proposition 1. Thus let

S

be a finite set. In order to count the elements of

S,

we subdivide

S

into subsets

S_{1}, \dots, S_{m}

such that we know

m

and know the maximal possible number

n

of elements in

S_{i},

i = 1, \dots, m .

Then

S

has at most

m n

elements. In our case, Proposition 1 provides the first approximation to the numbers

m

and

n,

in terms of the length of

W .

Proposition 1.

Let $M$ be a reduced van Kampen $ℛ -$ diagram over $F (X) .$ Let $W$ be a boundary label of $M,$ cyclically reduced. Assume $M$ satisfies conditions $C (4)$ and $T (4) .$ Suppose that each piece has length $1 .$ Then each of the following holds:

For each region $Δ$ of $M$ , we have $| \partial Δ | \leq | W | .$
The number $| M |$ of regions $Δ$ in $M$ is at most ${| W |}^{2} .$

Hence if our diagram

M

would be a

C (4)

and

T (4)

diagram, then by part (b) of the proposition, the number of regions in

M

would be bounded by

{| W |}^{2} .

Being this not the case, we first modify our approach. The core idea is not to try to estimate the number of all the regions via a result like Proposition 1 but to use that result for the estimation of a part of the regions only, namely,

| {Reg}_{4^{+}} (M) |,

the number of regions of

M

having boundary cycle of length at least

8,

and then relate

| {Reg}_{2} (M) |,

the number of remaining regions, to

| {Reg}_{4^{+}} (M) | .

In this direction, we have the following.

Theorem 1.

Let $M$ be a van Kampen $ℛ -$ diagram, $ℛ$ given by (1.1). Assume

(1.3)

ℛ = {R_{e} ∣ e \in E}, λ (e) \neq 3 for every e \in E .

Then there is a van Kampen diagram $M^{'}$ with the same boundary label as $M$ such that $| {Reg}_{4^{+}} (M^{'}) | \leq {| \partial M |}^{4} .$

For the proof, see [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

, pp. 5–11,105].

Now, we relate the number of regions in

{Reg}_{2} (M)

to the number of regions in

{Reg}_{4^{+}} (M) .

First, let us state a definition.

Definition 1

(

a -

Bands). Let

D_{1}, D_{2} \in {Reg}_{2} (M),

and let

Supp (D_{1}) = {a, x}, Supp (D_{2}) = {a, y}

(

x = y

is not excluded). Say that

D_{1}

and

D_{2}

are $a -$ neighbours if

θ := \partial D_{1} \cap \partial D_{2}

contains a subpath

θ_{0}

with label

a^{\pm 1} .

(We consider

D_{i}

as a neighbour of itself,

i = 1, 2 .

) The transitive closure of “

a -

neighbourhood” is an equivalence relation. For

D \in {Reg}_{2} (M)

with

Supp (D) = {a, x},

denote by

{\hat{L}}_{a} (D)

the

a -

equivalence class of

D .

Let

L_{a} (D)

be the subdiagram of

M

generated by

{\hat{L}}_{a} (D) .

Call

L_{a} (D)

the $a -$ band generated by $D$ . Denote by

Str (M)

the collection of all the bands in

M .

Clearly,

{Reg}_{2} (M) = ⋃_{L \in Str (M)} Reg (L) .

We make the observation that every band either starts on the boundary of a region in

{Reg}_{4^{+}} (M)

or on the boundary of

M .

The same is true for the ends of bands. Hence we can estimate the number of bands in terms of the number of regions from

{Reg}_{4^{+}} (M) .

Next, we show that the number of regions in a band is bounded by the number of bands. Relying on Theorem 1, this easily implies that

| {Reg}_{2} (M) |

is bounded by

{| \partial M |}^{6} .

See [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

, pp. 10–12].

The proof of the Main Theorem is reduced to the proofs of Theorems A–C and some results on the way to their proofs. In the rest of the work, we concentrate on their proofs. We assume that

ℛ

is given by (1.3).

See Definition 2.1.4 for

ℳ_{3} (W) .

Theorem A.

Let $M$ be a van Kampen $ℛ -$ diagram over $F$ with cyclically reduced boundary label $W .$ If $M \in ℳ_{3} (W),$ then ${\tilde{𝕄}}^{t}$ satisfies the small cancellation conditions $C (4)$ and $T (4)$ for every $t \in T (M) .$

Theorem B.

Let $M$ be a van Kampen $ℛ -$ diagram over $F$ with cyclically reduced boundary label $W .$ If $M \in ℳ_{3} (W),$ then for every region $D$ in $Reg (M)$ realising a relation $R (a, t),$ ${∥ W ∥}_{t} \geq n (a, t)$ and ${∥ W ∥}_{a} \geq n (a, t),$ $a$ and $t$ in $T .$

The main theorem obtained from the proof of Theorem 1 and the main result is Theorem C, which is a variant of Greendlinger’s lemma.

Theorem C

(Greendlinger’s lemma). Let $A (Γ)$ be an Artin group given by (1.3). Let $M_{0} \in ℳ_{3} (W)$ (see Definition 2.1.4) be a connected, simply connected minimal $ℛ -$ diagram over $F$ with cyclically reduced boundary label $W, W \neq 1$ in $A (Γ_{2}) .$

Let $W = W_{1} W_{2}^{- 1}$ be reduced as written. Then there is a van Kampen diagram $M, M \in ℳ_{3} (W),$ with boundary cycle $ω,$ such that $ω$ decomposes into $ω = ω_{1} ω_{2}^{- 1},$ $ω_{i}$ labelled with $W_{i},$ $i = 1, 2$ and one of the following holds:

There is a boundary region $K \in {Reg}_{4^{+}} (𝕄)$ with $\partial K \cap ω_{i}$ connected, $∥ \partial K \cap ω_{i} ∥ \geq n (K)$ , for $i = 1$ or $i = 2$ , where ${Reg}_{4^{+}} (𝕄) = {K \in Reg (𝕄) ∣ n (K) \geq 4} .$ Moreover, $| ξ | \leq | \partial K \cap ω_{i} |$ , where $ξ$ is the complement of $\partial K \cap ω_{i}$ on $\partial K .$
There are boundary regions $D_{1}, D_{2} \in {Reg}_{2} (M)$ with $\partial D_{i} \cap ω_{j}$ connected, $| \partial D_{i} \cap ω_{j} | \geq 2$ , for $j = 1$ or for $j = 2 .$
There is a region $D \in {Reg}_{2} (M)$ with $\partial D \cap ω_{j}$ connected, $| \partial D \cap ω_{j} | \geq 2$ , and there is a region $K \in {Reg}_{4^{+}} (𝕄)$ with $\partial K \cap ω_{j}$ connected, $∥ \partial D \cap ω_{j} ∥ \geq 2$ , for $j = 1$ or for $j = 2 .$

Overview of the present work. The goals of the work are

to estimate the number of regions in ${Reg}_{4^{+}} (M),$
to estimate the number of regions in ${Reg}_{2} (M) .$

The idea behind (1) is to count the number of regions in

{\tilde{𝕄}}^{t}, t \in T (M)

(for notations, see [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

, the three constructions on pp. 6, 7, 11]) where due to Theorem A, conditions

C (4)

and

T (4)

are satisfied, hence we may do this, and then to show that the number of regions in

{\tilde{𝕄}}^{t}

and

{Reg}_{4^{+}, t} 𝕄

is the same, via a natural mapping

ψ_{t}

of diagrams which sends a region

Δ

in

𝕄_{t}

to a uniquely defined region

\tilde{Δ}

in

\tilde{𝕄^{t}} .

For

Ψ_{t},

see [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

, pp. 79–81]. One of the problems is that

{\tilde{𝕄}}^{t}

is obtained from

𝕄_{t}

by shrinking edges and identifying vertices; hence in principle,

\tilde{Δ}

may shrink to an edge or a vertex. Thus our task is to show that

ψ_{t}

(and

ψ_{t}^{- 1}

) does not cause deformation and collapse of regions. One of the classical methods to avoid deformations in diagrams is by Greendlinger’s lemma. It roughly says that a diagram with at least two regions has at least two Greendlinger regions, the boundary of which contains a big portion of the relators. See [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

, Definition 1.3.1]. In general terms, it shows that if there is a kind of deformation in certain subdiagrams, then certain events are simultaneously unavoidable and forbidden, which of course is absurd. Hence no deformation occurs. For example, consider a reduced diagram in which every region has boundary label

U^{n},

U

cyclically reduced, not proper power,

n \geq 7 .

Suppose that the boundary of a region

E

is not simple closed. (This is the deformation which we would like to avoid.) Then there is a loop in the boundary of the region which surrounds a disc

Q,

the boundary of which is labelled by a subword of the cyclic word

U^{n} .

Suppose that in

Q,

we have Greendlinger’s lemma. Then

Q

has a boundary region

D

with

\partial D \cap \partial Q

connected such that

| \partial D \cap \partial Q | \geq 3 | U | .

Hence

D

and

E

necessarily have a common (unavoidable) piece of length at least

3 | U | .

But such a word is forbidden, because due to

U

not being a proper power, this would mean that

D

cancels

E,

violating that the diagram is reduced. (Though we do not use this example, we use its underlying idea. See the proofs of Propositions 2.1.7 and 2.1.8 in [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

].) Coming back to the context of our work, notice that by Theorem C, every bigonal

ℛ -

diagram

M,

that is,

\partial M = ω_{1} ω_{2}^{- 1}

has a generalised Greendlinger region on one of the sides

ω_{i}

of the bigon. Since

\tilde{𝕄^{t}}

does satisfy

C (4)

and

T (4),

hence it has Greendlinger regions. The natural candidates for Greendlinger’s regions in

M

are the images (by

Ψ_{t}^{- 1}

) of the Greendlinger regions of

\tilde{𝕄^{t}} .

The problem is that while Greendlinger regions are boundary regions, our candidates in

M

need not be. So we need a technique to “move” inner regions to the boundary. More precisely, to show that there is a diagram with the same boundary label as

M

in which the candidates are boundary regions. We do this by observing that since the generators of the second homotopy group of the defining complex are prisms, we can replace one half of the prism by its complement by a rotation of the prism. In

M,

this has the effect of moving a region which occurs both in

M

and in the prism. See [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

, Section 3]. The involvement of the second homotopy group requires the introduction of extended presentations.

Finally, we have to count

| {Reg}_{2} (M) | .

To this end, it is enough to show that two bands cannot intersect more than once (see Section 4). Essentially, we prove this simultaneously with the other main proposition (see Section 4).

The ideas developed here lead to further results for Artin groups

A

dealt with in the present work. Recall that a parabolic subgroup of an Artin group is a conjugate of a standard parabolic subgroup. In [7

A. Juhász, Fusion in Artin groups II. In preparation

], we show that the intersection of parabolic subgroups is parabolic. Also, we describe fusion in

A .

In particular, we show that in even Artin groups, every standard parabolic subgroup

P

controls fusion in

P

(i.e., if two elements of

P

are conjugate in

A,

then they are already conjugate in

P

).

Together with further ideas, we show in [8

A. Juhász, On locally reducible Artin groups. In preparation

] that locally reducible Artin groups (i.e., no (2.3.3), (2.3.4) and (2.3.5) type standard parabolics occur) have polynomial

(n^{6})

isoperimetric functions.

The work is organised as follows:

In Section 2, we introduce the diagrams $𝕄$ and $𝕄_{t}$ and show that they are adequate. Also, $M$ is adequate.
In Section 3, we consider extended presentations, Howie diagrams, $ℐ -$ moves and banded diagrams.
In Section 4, we consider the connection between $𝕄_{t}$ and $\tilde{𝕄^{t}}$ and prove the main theorems.

2. The diagrams $𝕄,$ $𝕄_{t}$ and ${\tilde{𝕄}}^{t}, t \in T (M)$

2.1. The diagrams $𝕄$ and $𝕄_{t}, t \in T (M)$

Let

M

be a simply connected

ℛ -

diagram over

F,

ℛ

given by (1.3), and let

t \in T (M) .

Denote by

{Reg}_{H_{t}} (M)

the set of all the regions of

M

the boundaries of which are labelled only by letters from

H_{t}

(

H_{t} = ⟨ X - {t} ⟩

). Denote by

{Reg}_{t} (M)

the set of all the regions of

M

the boundaries of which have labels from

⟨ t ⟩

and also from

H_{t} .

Thus

Reg (M) = {Reg}_{t} (M) \cup {Reg}_{H_{t}} (M) .

We subdivide

{Reg}_{t} (M)

further by

{Reg}_{t} (M) = {Reg}_{4^{+}, t} (M) \cup {Reg}_{2, t} (M),

where

{Reg}_{4^{+}, t} (M)

is the set of regions in

{Reg}_{t} (M)

with boundary length at least

8

and

{Reg}_{2, t} (M)

is the set of regions in

{Reg}_{t} (M)

with boundary length

4 .

We have for

t \in T

(2.1)

Reg (M) = {Reg}_{4^{+}, t} (M) \cup {Reg}_{H_{t}} (M) \cup {Reg}_{2, t} (M) .

For the definition of the diagrams

𝕄

and

𝕄_{t},

we introduce equivalence relations “

\underset{t}{\sim}

”, “

\underset{H_{t}}{\sim}

” and “

\underset{B_{t}}{\sim}

” on

{Reg}_{4^{+}, t} (M),

{Reg}_{H_{t}} (M)

and

{Reg}_{2, t} (M),

respectively.

Definition 2.1.1

(of

\underset{t}{\sim}

). Let

D_{1}

and

D_{2}

be regions in

{Reg}_{4^{+}, t} (M) .

Say that they are $t -$ friends if the following hold:

$\partial D_{1} \cap \partial D_{2}$ contains an edge.
$Supp (\partial D_{1}) = Supp (\partial D_{2}) .$

Let

\underset{t}{\sim}

be the transitive closure of

t -

friendness. It is easy to see that

\underset{t}{\sim}

is an equivalence relation on

{Reg}_{4^{+}, t} (M) .

Denote by

{[D]}_{t}

the

\underset{t}{\sim}

equivalence class of

D .

Definition 2.1.2

(of

\underset{H_{t}}{\sim}

). Let

D_{1}

and

D_{2}

be regions in

{Reg}_{H_{t}} (M) .

Say that they are $H_{t} -$ friends if

\partial D_{1} \cap \partial D_{2}

contains an edge. Let

\underset{H_{t}}{\sim}

be the transitive closure of

H_{t} -

friendness. Then

\underset{H_{t}}{\sim}

is an equivalence relation on

{Reg}_{H_{t}} (M) .

Denote by

{[D]}_{H_{t}}

the equivalence class of

D \in {Reg}_{H_{t}} (M) .

Definition 2.1.3

(of

\underset{B_{t}}{\sim}

). Let

D_{1}

and

D_{2}

be regions in

{Reg}_{2, t} (M) .

Say that

D_{1}

and

D_{2}

are $B_{t} -$ friends if

\partial D_{1} \cap \partial D_{2}

consists of an edge with label

t^{\pm 1} .

Let

\underset{B_{t}}{\sim}

be the transitive closure of

B_{t} -

friendness. Then

\underset{B_{t}}{\sim}

is an equivalence relation. For

D \in {Reg}_{2, t} (M),

denote by

{[D]}_{B_{t}}

the equivalence class of

D

in

{Reg}_{2, t} (M) .

Definition 2.1.4

(The class

ℳ_{3} (W)

of diagrams). The following statements hold:

Let $W$ be a cyclically reduced word which represents $1$ in $A (Γ) .$ Assume that $W \neq 1$ in $A (Γ_{2}) .$ Here $Γ_{2}$ is the graph obtained from $γ$ by removing all edges not labelled by $2 .$ Denote by $ℳ (W)$ the set of all the van Kampen diagrams with boundary label $W .$ By [10
R. C. Lyndon and P. E. Schupp, Combinatorial group theory. Classics in Math., Ergeb. Math. Grenzgeb. 89, Springer, Berlin, 1977, 339 pp. Zbl 0368.20023 MR 0577064
, Chapter V], $ℳ (W) \neq \emptyset .$ Let $ℳ_{1} (W)$ denote the set of all the diagrams in $ℳ (W)$ for which $| {Reg}_{4^{+}} (M) |$ is minimal possible. Clearly, $ℳ_{1} (W) \neq \emptyset .$ Let $ℳ_{2} (W)$ be the set of the diagrams in $ℳ_{1} (W)$ which contain minimal number of equivalence classes ${[D]}_{M}$ of regions in ${Reg}_{4^{+}} (M) .$ Clearly, $ℳ_{2} (W) \neq \emptyset .$ Finally, let $ℳ_{3} (W)$ be the set of the diagrams in $ℳ_{2} (W)$ with adequate bands. Say that $M$ is minimal if $M \in ℳ_{2} (W)$ and $| {Reg}_{2} (M) |$ is minimal possible. It follows that if $M$ is minimal, then $M \in ℳ_{3} (W) .$ Hence $ℳ_{3} (W) \neq \emptyset .$
Definition 2.1.1 makes sense without the requirement that $D_{1}$ and $D_{2}$ are in ${Reg}_{4^{+}, t} (M),$ namely, $D_{1} \sim D_{2}$ if $Supp (\partial D_{1}) = Supp (\partial D_{2})$ and $\partial D_{1} \cap \partial D_{2}$ contains an edge. We shall call the corresponding equivalence relation “equivalence”. For $D \in {Reg}_{Z} (M),$ define $κ_{0} (D) = Int (\cup {\bar{E} ∣ E \in {[D]}_{Z}}) .$ Then $κ_{0} (D)$ is connected. If $Z = t^{4^{+}},$ then define $n (κ_{0} (D)) = n (D) .$ Notice that $n (κ_{0} (D))$ is well defined. Also, denote $κ_{0} (D)$ by $Δ (D)$ for this case. Denote the diagram obtained by $𝕄 .$

Definition 2.1.5

(Adequate). Let

M

be a connected, simply connected

ℛ -

diagram over

F,

M \in ℳ_{3} (W) .

Say that

M

is adequate, if for every

D \in Reg (M), D

is homeomorphic to the open unit disc,

\partial D

is simple such that the boundary label is not

1

and is cyclically reduced.

Definition 2.1.6

(

𝕄_{t}

). Let

M

be a van Kampen

ℛ -

diagram over

F,

and let

t \in T (M) .

If

κ_{0} (D)

is homeomorphic to the open unit disc, for every

D \in Reg (M),

then we may consider

{κ_{0} (D) ∣ D \in Reg (M)}

as a set of regions such that their totality generates a diagram which we denote by

𝕄_{t} .

Thus

Reg (𝕄_{t}) = {κ_{0} (D) ∣ D \in Reg (M)} .

We have

\partial 𝕄_{t} = \partial M .

Also,

(2.2)

Reg (𝕄_{t}) = {Reg}_{4^{+}, t} (𝕄_{t}) \cup {Reg}_{B_{t}} (𝕄_{t}) \cup {Reg}_{H_{t}} (𝕄_{t}) for every t \in T (M) .

The main results of this section are Propositions 2.1.7 and 2.1.8.

Proposition 2.1.7.

Let $M \in ℳ_{3} (W) .$ Then $𝕄_{t}$ is adequate for every $t \in T (M) .$ Moreover, if $K_{1}$ and $K_{2}$ are regions of $𝕄_{t},$ not both bands, then $\partial K_{1} \cap \partial K_{2} \neq \emptyset$ implies that $\partial K_{1} \cap \partial K_{2}$ is connected.

Proposition 2.1.8.

Let $M \in ℳ_{3} (W) .$ Then $M$ is adequate. Moreover, if $\partial D_{1} \cap \partial D_{2} \neq \emptyset,$ then $\partial D_{1} \cap \partial D_{2}$ is connected.

Remark 2.1.9.

Let

K \in {Reg}_{4^{+}, t} (𝕄_{t}) .

When passing from

𝕄_{t}

to

\tilde{𝕄^{t}}

by shrinking edges labelled by elements of

H_{t}

to a point and shrinking

t -

bands to edges, it may happen that the interior of the obtained region

\tilde{K^{t}}

in

\tilde{𝕄^{t}}

becomes disconnected. Since

M

is adequate, we can replace

\tilde{K^{t}}

by a disc (and we shall do so), the boundary of which is labelled by

Φ (\partial (\tilde{K^{t}})) .

We prove Propositions 2.1.7 and 2.1.8 by simultaneous induction on the size of

M .

See [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

, Sections 2.2–2.4].

3. Extended presentations, $ℐ ‑$ moves and banded diagrams

3.1. Identities among relations

We recall identities among relations from [2

D. J. Collins and J. Huebschmann, Spherical diagrams and identities among relations. Math. Ann. 261 (1982), no. 2, 155–183 Zbl 0477.20019 MR 0675732

, p. 157]. Let

C

be a

2 -

complex. Let

c_{0}

be a basepoint for

C

(some designated vertex of

C

). A sequence $σ$ over $C$ at $v_{0}$ is

σ = (q_{1}, \dots, q_{n}),

where

q_{i}

is a path in

C

consisting of a concatenation

γ_{i} \partial Δ_{i} γ_{i}^{- 1},

Δ_{i}

being a region (face) of

C

and

γ_{i}

a path from

v_{0}

to

v (Δ_{i}),

1 \leq i \leq n,

where

v (Δ_{i})

is the basepoint of

Δ_{i} .

We call

σ

an identity sequence if the concatenation

q_{1} \dots q_{n}

is freely equivalent to the trivial path at

v_{0} .

Given a presentation

𝒫 = ⟨ X ∣ ℛ ⟩,

following [10

R. C. Lyndon and P. E. Schupp, Combinatorial group theory. Classics in Math., Ergeb. Math. Grenzgeb. 89, Springer, Berlin, 1977, 339 pp. Zbl 0368.20023 MR 0577064

], we construct a bouquet of tailed circles

C

with basepoint

v_{0} .

Let

σ = (q_{1}, \dots, q_{n})

be a sequence over

C

at

v_{0},

q_{i} = γ_{i} Δ_{i} γ_{i}^{- 1},

γ_{i}

tails. Label

\partial Δ_{i}

by

R_{i} \in ℛ

and label

γ_{i}

by a word in

F (X),

via a labelling function

Φ_{0} : C \to F .

After carrying out all free cancellations in

Φ_{0} (σ) = Φ_{0} (q_{1}) \dots Φ_{0} (q_{n}),

we get an

ℛ -

diagram

M

over

F

the boundary label of which is the cyclically reduced word for

Φ_{0} (q_{1} \dots q_{n})

in

F .

If

Φ_{0} (q_{1} \dots q_{n}) \neq 1_{F},

then

M

is a connected and simply connected planar diagram (van Kampen diagram), and if

Φ_{0} (q_{1} \dots q_{n}) = 1

in

F,

then

M

is the union of a finite number of spines and tessellated spheres (

≅ S^{2} \subseteq ℝ^{3}

) by the

Δ_{i} .

In this case, we call

M

a spherical diagram. We call

M

a singular sphere or just a sphere if it consists of a single sphere. When the

q_{i}

are labelled with

R_{i}

and

ϕ (q_{1} \dots q_{n}) = R_{1} \dots R_{n} = 1

in

F,

then we call the sequence

S = (R_{1}, \dots, R_{n})

an identity sequence over $ℛ$ and call it a simple identity sequence, if the corresponding spherical diagram is a simple sphere.

Definition 3.1.1

(Extended presentations). Let

𝒫 = ⟨ X ∣ ℛ ⟩

be a finite presentation of a group

G,

and let

ℐ

be a set of simple identity sequences over

ℛ .

The corresponding extended presentation is the triple

⟨ X ∣ ℛ ∣ ℐ ⟩ .

This notion was introduced in [3

R. A. Fenn, Techniques of geometric topology. London Math. Soc. Lecture Note Ser. 57, Cambridge University Press, Cambridge, 1983, 280 pp. Zbl 0517.57001 MR 0787801

] by Roger Fenn. Now let

A = A (Γ)

be an Artin group given by

⟨ X ∣ ℛ ⟩

in (1.3). The identities among relations

ℐ

we choose are obtained from the

3 -

generated standard parabolic subgroups

A (i, j, k)

with defining subgraphs

Γ_{i, j, k}

with the vertex set

V (Γ_{i, j, k}) = {x_{i}, x_{j}, x_{k}}

and the edge set

E (Γ_{i, j, k}) = {e_{i j}, e_{i, k}, e_{j, k}},

1 \leq i < j < k \leq n,

with labels

λ (e_{i j}) = λ (e_{j, k}) = 2

and

λ (e_{i, k}) = m,

m \geq 4 .

The corresponding tessellated simple spheres

Σ_{i, j, k}

are prisms with two regions of degree

2 m

(upper and lower) labelled with

R_{i k},

which we call the big regions, and a band of length

2 n

in which the regions are labelled by

R_{i j}

and

R_{j k}

in an alternating manner. See Figure 1 for

n = 4 .

Figure 1.

A big region realising

R_{i k}

in a half sphere for

n = 4 .

We introduce one of our main tools, $ℐ -$ moves.

Let

Σ = Σ_{i, j, k},

and let

ω

be a simple closed curve on

Σ .

Then

ω

subdivides

Σ

into two simply connected subdiagrams,

ω_{R}

and

ω_{L},

with connected interiors, where

ω_{R}

is the submap of all regions to the right of

ω

and

ω_{L}

is the submap of all the regions to the left of

ω .

Clearly,

\bar{ω_{R}} \cap \bar{ω_{L}} = ω

and

ω_{R} \cup ω \cup ω_{L} = Σ .

Let

S (Σ)

be the set of all the connected, simply connected submaps of

Σ

with connected interior, and let

β_{ω} : S (Σ) \to S (Σ)

be the function which for every closed curve

ω

on

Σ

sends

ω_{R}

to

ω_{L}

and

ω_{L}

to

ω_{R}

and leaves

ω

unaltered.

Let

M

be a van Kampen

ℛ -

diagram over

F,

and let

M_{0}

be a simply connected subdiagram of

M

with connected interior and simple boundary cycle

μ .

Suppose that

\bar{M_{0}}

can be embedded into

Σ = Σ_{i, j, k}

for some (uniquely defined)

i,

j

and

k

as an

ℛ -

diagram with image

N

in

Σ

with

\partial N

simple closed such that

μ

is mapped onto

\partial N .

Then

N \in S (Σ) .

Definition 3.1.2

(

ℐ -

moves). Let

ℐ

be a simple identity among relations realised by a uniquely defined sphere

Σ = Σ_{i, j, k} .

Let

M,

M_{0},

N,

S (Σ)

and

β_{ω}

be as defined above for some

ω .

An $ℐ -$ move at $M_{0}$ is a surgery in

M

by which we cut out

M_{0}

and fill in instead by

β_{ω} (N) .

Since

\partial N = \partial (β_{ω} N),

this is well defined.

Below we consider some examples which illustrate the way

ℐ -

moves are used.

Example 3.1.3.

Let

X = {a, b, c},

ℛ = {R_{1}, R_{2}, R_{3}},

where

R_{1} = a b a^{- 1} b^{- 1},

R_{2} = a c a^{- 1} c^{- 1},

R_{3} = b c b^{- 1} c^{- 1} .

Here

G ≅ ℤ \oplus ℤ \oplus ℤ .

Let

W = a b c a^{- 1} b^{- 1} c^{- 1} .

Then

W

represents

1

in

G .

Figure 2 shows the corresponding van Kampen diagrams.

Figure 2.

Application of an

ℐ -

move on

⟨ D_{1}, D_{2}, D_{3} ⟩ .

Example 3.1.4.

Now let

ℰ = ⟨ a, b, c ∣ ℛ ∣ I ⟩,

ℛ = {R_{1}, R_{2}, R_{3}},

I = R_{1},

R_{2}^{b},

R_{3},

{(R_{1}^{- 1})}^{c},

R_{2}^{- 1},

{(R_{3}^{- 1})}^{a}

be an extended presentation, where

R^{x} = x R x^{- 1}

and

R_{1},

R_{2},

R_{3}

are as in Example 3.1.3. The effect of

I

on the diagrams of Figure 2 is the replacement of the diagram in Figure 2 (a) by the diagram in Figure 2 (b).

This changes the positions of the regions realising

R_{1},

R_{2}

and

R_{3}

and may reduce the number of regions. See Figure 3.

Figure 3.

An

ℐ -

move followed by a free reduction which reduces the number of regions.

3.2. Banded subdiagrams and transfer of regions

In this subsection, we assume

M \in ℳ_{3} (W)

and assumption

ℋ .

Lemma 3.2.1.

Let $M \in ℳ_{2} (W),$ and let $I$ be an $I -$ move or extended $I -$ move. Then $M^{I} \in ℳ_{2} (W) .$

Proof.

We have to show that

I

increases neither

| {Reg}_{4^{+}} (M) |

nor

| {Reg}_{4^{+}} (𝕄) | .

Suppose that

I

replaces the simply connected subdiagram

N

with connected interior by a diagram

N^{'} .

We use notation of Definition 3.1.1. If

n = 2,

then clearly

I

cannot alter

| {Reg}_{4^{+}} (M) |

and

| {Reg}_{4^{+}} (𝕄) | .

If

n \geq 4,

then

N

contains exactly one big region

Δ

and

N^{'}

also contains exactly one big region

Δ^{'},

which is a copy of

Δ .

Since

M \in ℳ_{2} (W),

the result follows.

Definition 3.2.2

(Banded subdiagrams). Let

B = ⟨ E_{1}, \dots, E_{k} ⟩

be a band in

M

with sides

σ_{1}

and

σ_{2},

and let

θ

be a boundary path of a simply connected subdiagram

M_{0}

with connected interior.

Say that $M_{0}$ is banded at $θ$ by $B$ if $∥ θ ∥ \geq 1$ and $θ \subseteq σ_{1} \cup σ_{2} .$ See Figure 4 (a).
Let notation be as in part (a), and let $θ = v_{0} θ_{1} v_{1} θ_{2} v_{2} \dots θ_{r} v_{r}$ be a decomposition of $θ,$ $v_{i}$ vertices, $r \geq 1 .$ Say that $M_{0}$ is banded at $(θ_{1}, \dots, θ_{r})$ by bands $B_{i}$ , respectively, if $M_{0}$ is banded at $θ_{i}$ by $B_{i}$ for every $i,$ $i = 1, \dots, r .$

Definition 3.2.3

(Transfer of regions from

M_{0}

). Let

M

be a connected, simply connected

ℛ -

diagram over

F

with connected interior, and let

M_{0}

be a connected, simply connected

ℛ -

subdiagram over

F

with connected interior. Let

D

be a boundary region of

M_{0}

with

μ := \partial D \cap \partial M_{0}

connected, and let

B = B_{1} \cup B_{2} \cup B_{3}

be such that

\partial D \cap \partial B = \partial D \cap \partial B_{2} .

Observe that

D \cup B_{2}

embeds into the prism

P_{m},

m = 2 n (D),

P_{m} = D \cup B_{2} \cup D^{'} \cup B_{2}^{'},

where

B_{2} \cup B_{2}^{'}

is an annulus with

m

regions from

{Reg}_{2} (M)

(a closed band) and

D

is a region with

n (D^{'}) = n (D)

and

Supp (D) = Supp (D^{'}) .

(We identify

D

and

B_{2}

with their images in

P_{m} .

) Notice that

B_{2}

and

B_{2}^{'}

have the same poles

α

and

β,

hence

B^{'} = B_{1} \cup B_{2}^{'} \cup B_{3}

is a band. Replace

B \cup D

with

B^{'} \cup D^{'}

(which have the same boundaries) and denote the resulting diagram by

M_{0}^{J},

where

J

is the

ℐ -

move which replaced

B \cup D

with

B^{'} \cup D^{'} .

Thus

M_{0} \cup B

is replaced by

((M_{0} \cup B) ∖ (D \cup B)) \cup (D^{'} \cup B^{'}) .

Hence

(3.1)

M_{0}^{J} = M_{0} ∖ D, B^{J} = B^{'} and M_{0}^{J} is banded by B^{'} .

In this situation, we say that $J$ transferred $D$ from $M_{0}$ . See Figure 4 (b) and 4 (c). If every region can be transferred from

M_{0},

we say that

M_{0}

is transferable. The removal of

D \in Reg (M_{0})

naturally extends to transfer of regions

Δ

from

𝕄 .

(See Figure 4 (b^′) and 4 (c^′).)

Figure 4.

(a)–(c): Transfer of the region

D

via an adjacent band

B

by the application of the

ℐ -

move. (b^′) and (c^′): Transfer of the modified region

Δ

via an adjacent band

B

by the application of the

ℐ -

move.

Notice that if instead of

B

we have a band-bundle, then

D

can be transferred by a sequence of

ℐ -

moves beyond the band-bundle, without destroying the bundle structure. See Figure 5 (a) and 5 (b) for

D

and Figure 5 (a^′) and 5 (b^′) for

Δ .

Figure 5.

(a) and (b): Transfer of the region

D

via a band-bundle

(β_{1}, β_{2})

by the application of the

ℐ -

move

J .

(a^′) and (b^′): Transfer of the modified region

Δ

via a band-bundle

(β_{1}, β_{2})

by the application of the

ℐ -

move

J .

We are interested in a special type of banded subdiagrams (2-banded subdiagrams) which we consider in the next subsection.

Definition 3.2.4

(2-Banded subdiagram). Let

W \neq 1

be a cyclically reduced word in

F

which represents

1

in

A .

Let

M \in ℳ_{3} (W),

and let

Q

be a connected, simply connected subdiagram of

M

with connected interior. Say that

Q

is $2 -$ banded if

there is an $a -$ band $B_{l}$ and a $b -$ band $B_{r}$ in $Q,$ $a, b \in X$ ( $a = b$ is not excluded),
there are connected and simply connected subdiagrams $Q_{r},$ $Q_{l}$ and $P$ in $Q$ and paths $θ_{r}$ and $θ_{l}$ such that
1. $Q = Q_{l} \cup B_{l} \cup_{θ_{l}} P \cup_{θ_{r}} B_{r} \cup Q_{r}$ and $\partial Q = ω_{1} α_{l}^{- 1} γ α_{r}^{- 1} ω_{3} β_{r} β_{l},$ where $θ_{l} = \partial B_{l} \cap \partial P, θ_{r} = \partial B_{r} \cap \partial P, θ_{l} and θ_{r}$ are sides of $B_{l} and B_{r},$ respectively, $ω_{1} = \partial Q_{l} \cap \partial Q, γ = \partial P \cap \partial Q,$ $ω_{3} = \partial Q_{r} \cap \partial Q$ are connected, $α_{l}$ and $β_{l}$ are poles of $B_{l}$ and $α_{r}$ and $β_{r}$ are poles of $B_{r},$
2. $o (θ_{l}) = o (θ_{r}),$
$Int (P) \neq \emptyset,$ then $Φ (\partial P)$ is cyclically reduced.

Call

Q

closed $2 -$ banded if

t (θ_{l}) = t (θ_{r}) .

Let

U

be a cyclically reduced word in

F

which represents

1

in

A .

Denote by

𝒬_{M} (U)

the collection of all the

2 -

banded subdiagrams of

M

with boundary label

U .

We shall show that under certain conditions, there is a boundary region of

M_{0}

which we can transfer from

M_{0} .

We wish to repeat the process of transferring until either no regions remain in

M_{0}

or

M_{0}

has a special structure. The next proposition shows that

ℐ -

moves preserve these “certain conditions”, hence we can repeat transferring regions as long as there are regions in

M_{0}^{J},

or

M_{0}^{J}

does not have the special structure. Here

M_{0}^{J}

is the result of applying

J

on

M_{0},

J

a sequence of

ℐ -

moves.

Definition 3.2.5

(Standard

ℐ -

moves). Say that an

ℐ -

move

J

is standard if

θ

is a maximal piece in the sense that

θ

in Figure 5 (b) cannot be extended as a common boundary of

D

and

B .

Proposition 3.2.6.

Let $W$ be a cyclically reduced word in $F,$ $W \neq 1,$ which represents $1$ in $A,$ and let $M \in ℳ_{3} (W) .$ Let $Q$ be a subdiagram of $M$ with cyclically reduced boundary label $U$ and suppose that $Q \in 𝒬_{M} (U) .$ (Thus $Q$ is $2 -$ banded.) Let notation be as in [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

, Figure 33 (c)].

Assume that $P$ contains a boundary region $Δ \in {Reg}_{4^{+}, t} (P)$ or $D \in {Reg}_{2} (P)$ such that $\partial Δ \cap \partial P$ or $\partial D \cap \partial P,$ respectively, are connected and $\partial Δ \cap \partial P = \partial Δ \cap θ,$ or $\partial D \cap \partial P = \partial D \cap θ,$ $θ = θ_{r}$ or $θ = θ_{l}$ such that $∥ \partial Δ \cap θ ∥ \geq 2,$ or $∥ \partial θ \cap \partial D ∥ \geq 2 .$ Let $I$ be the standard $ℐ -$ move on $B \cup Δ,$ or $B \cup D,$ respectively, where $B = B_{r}$ or $B = B_{l}$ according as $θ = θ_{r}$ or $θ = θ_{l},$ respectively, which transfers $D$ or $Δ,$ respectively, from $Q .$ Then $Q^{I} \in 𝒬_{M^{I}} (U) .$ See the proof of Proposition 3.2.6 in [6
A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1
].

4. Lifting $𝒟 ({\tilde{𝕄}}^{t})$ to $𝕄_{t}, t \in T (M),$ and the proofs of the main results

The idea behind showing that

{\tilde{𝕄}}^{t}

satisfies conditions

C (4)

and

T (4)

is to use it in

M .

Obviously, we cannot lift conditions

C (4)

and

T (4)

from

{\tilde{𝕄}}^{t}

to

M;

however, we can lift a version of Greendlinger’s lemma. This is the content of Theorem C. The first step towards such a lifting is to establish the basic connection between

{\tilde{𝕄}}^{t}

and

𝕄_{t} .

We consider

𝕄_{t}

as a map and consider

{(𝕄_{t})}^{1},

its

1 -

skeleton. Following the definition of

{\tilde{𝕄}}^{t},

we consider

{({\tilde{𝕄}}^{t})}^{1}

as a quotient of

{(𝕄_{t})}^{1} .

Then we extend this from the

1 -

skeleton to the whole map and show that

| {Reg}_{4^{+}, t} (𝕄_{t}) | = | Reg ({\tilde{𝕄}}^{t}) | .

Since

{\tilde{𝕄}}^{t}

satisfies conditions

C (4)

and

T (4),

this allows us to estimate

| {Reg}_{4^{+}, t} (𝕄) | .

4.1. Preliminary results

4.1.1. The mappings $Ψ_{t}$ and $Ψ_{t}^{- 1}, t \in T (M)$

Let

M

be a connected, simply connected

ℛ -

diagram,

M \in ℳ_{3} (W)

and

t \in T (M) .

Consider the diagram

𝕄_{t}

(defined in Section 2.1). Assume first that it has connected interior. Denote by

{(𝕄_{t})}^{1}

the

1 -

skeleton of

𝕄_{t} .

Define an equivalence relation “

\underset{1}{\approx}

” on the vertices of

{(𝕄_{t})}^{1}

to be the transitive closure of the relation “

\underset{1}{\sim}

” defined as follows: vertices

v_{1}

and

v_{2}

satisfy

v_{1} \underset{1}{\sim} v_{2}

if either

v_{1} = v_{2}

or there is an edge

e

labelled by a letter from

H_{t}

(

H_{t} -

edge) with

ι (e) = v_{1}

and

τ (e) = v_{2} .

Here

ι (e)

denotes the initial vertex of

e

and

τ (e)

denotes the terminal vertex of

e .

Edges

e_{1}

and

e_{2}

satisfy

e_{1} \underset{1}{\sim} e_{2}

if

e_{1} = e_{2} .

Then by the definition of

𝕄^{t},

{(𝕄_{t})}^{1} / \underset{1}{\approx}

is isomorphic to

{(𝕄^{t})}^{1} .

Let

E_{t}

be the collection of all the edges of

𝕄_{t}

which are labelled by

t^{\pm 1}

and define

E_{H_{t}}

accordingly.

Denote the projection mapping of

{(𝕄_{t})}^{1}

which realises

\underset{1}{\approx}

by

ϕ_{t} : {(𝕄_{t})}^{1} \to {(𝕄^{t})}^{1} .

Then

ϕ_{t}

sends a vertex

v

in

{(𝕄_{t})}^{1}

to the vertex

ϕ_{t} (v)

which we denote by

v^{t} .

If

e \in E_{t} ({(𝕄_{t})}^{1}),

then

ϕ_{t} (e)

is an edge which we denote by

e^{t} .

If

e \in E_{H_{t}} ({(𝕄_{t})}^{1}),

then

e^{t} := ϕ_{t} (e)

is a loop with

ι (e) = τ (e),

which we remove. We can do this because there is no interrelation between a loop and an edge (or loop). So we define

ϕ_{t} (e) = \emptyset

and

ϕ_{t} (\bar{e}) = v_{1}^{t} = v_{2}^{t},

where

ι (e) = v_{1}

and

τ (e) = v_{2} .

We define

ϕ_{t}^{- 1} : {(𝕄^{t})}^{1} \to {(𝕄_{t})}^{1}

by its set theoretical definition as follows.

Let

v^{t}

be a vertex in

𝕄^{t}, v^{t} = ϕ_{t} (v), v

a vertex of

𝕄_{t} .

Define $ϕ_{t}^{- 1} (v^{t}) = {w ∣ w is H_{t} -connected to v} .$
If $e^{t}$ is an edge in ${(𝕄^{t})}^{1},$ then $ϕ_{t}^{- 1} (e^{t})$ is the unique edge $e \in E_{t} ({(𝕄^{t})}^{1})$ such that $ϕ_{t} (e) = e^{t} .$ (Notice that $e$ is necessarily in $E_{t} ({(𝕄_{t})}^{1}) .$ )

If

Int (𝕄_{t})

is not connected, then consider each connected component individually.

Consider now

{\tilde{𝕄}}^{t} .

Assume first that it has connected interior. Define an equivalence relation “

\underset{2}{\approx}

” on

{(𝕄^{t})}^{1}

to be the transitive closure of the relation “

\underset{2}{\sim}

” defined as follows:

Vertices $v_{1}^{t}$ and $v_{2}^{t}$ of ${(𝕄^{t})}^{1}$ satisfy $v_{1} \underset{2}{\sim} v_{2}$ if and only if $v_{1} = v_{2} .$
Edges $e_{1}^{t}$ and $e_{2}^{t}$ satisfy $e_{1}^{t} \underset{2}{\sim} e_{2}^{t}$ if and only if $τ (e_{1}^{t}) = τ (e_{2}^{t})$ and $ι (e_{1}^{t}) = ι (e_{2}^{t}) .$

It follows from the definition of

{\tilde{𝕄}}^{t}

that

{(𝕄_{t})}^{1} / “ \underset{2}{\approx} ”

is isomorphic to

{({\tilde{𝕄}}^{t})}^{1} .

Denote by

ξ_{t}

the projection from

{(𝕄^{t})}^{1}

onto

{({\tilde{𝕄}}^{t})}^{1}

which realises

\underset{2}{\approx} .

Then we have

if $v^{t} \in V ({(𝕄^{t})}^{1}),$ then $ξ_{t} (v^{t}) = \tilde{v}, \tilde{v}$ a uniquely defined vertex in ${\tilde{𝕄^{t}}}^{1},$
if $e^{t} \in E ({(𝕄^{t})}^{1}),$ then $ξ_{t} (e^{t}) = \tilde{e}, \tilde{e}$ a uniquely defined edge in ${\tilde{𝕄^{t}}}^{1} .$

We define

ξ_{t}^{- 1} : {\tilde{𝕄}}^{t} \to 𝕄^{t}

as follows:

If $\tilde{v} \in V ({(\tilde{𝕄^{t}})}^{1}),$ then $ξ_{t}^{- 1} (\tilde{v}) = v^{t},$ where $v^{t}$ is the unique vertex in ${(𝕄^{t})}^{1}$ with $ξ_{t} (v^{t}) = \tilde{v} .$
If $\tilde{e} \in E ({(\tilde{𝕄^{t}})}^{1}),$ then $ξ_{t}^{- 1} (\tilde{e}) = {e^{t} \in 𝕄^{t} ∣ ι (e^{t}) = ι (e_{0}^{t}) and τ (e^{t}) = τ (e_{0}^{t})},$ where $ξ_{t} (e_{0}^{t}) = \tilde{e} .$

Finally, define

Ψ_{t} : {(𝕄_{t})}^{1} \to {({\tilde{𝕄}}^{t})}^{1}

by

Ψ_{t} = ξ_{t} \circ ϕ_{t} .

Then

Ψ_{t}^{- 1} = ϕ_{t}^{- 1} \circ ξ_{t}^{- 1} .

It follows by checking Euler characteristics that

{\tilde{𝕄}}^{t}

is planar. We have the following.

Lemma 4.1.2.

The following statements hold:

Let $e \in E_{t} ({(𝕄_{t})}^{1})$ , and let $B_{e}^{1}$ be the $1$ -skeleton of the $t$ -band which contains $e .$ Then $Ψ_{t}^{- 1} \circ Ψ_{t} (\bar{e}) = B_{e}^{1} .$ In particular, $\bar{e} \subseteq Ψ_{t}^{- 1} \circ Ψ_{t} (\bar{e}) .$
Let $e \in E_{t} ({(𝕄_{t})}^{1}) .$ Then $Ψ_{t}^{- 1} \circ Ψ_{t} (e) = e .$
Let $v \in V ({(𝕄_{t})}^{1}) .$ Then $Ψ_{t}^{- 1} \circ Ψ_{t} (v) = {[v]}_{H_{t}} .$ In particular, $v \in Ψ_{t}^{- 1} \circ Ψ_{t} (v) .$
Let $\tilde{v} \in V ({({\tilde{𝕄}}^{t})}^{1}) .$ Then $Ψ_{t} \circ Ψ_{t}^{- 1} (\tilde{v}) = \tilde{v} .$
$Ψ_{t} (ι (e)) = ι (Ψ_{t} (e))$ and $Ψ_{t} (τ (e)) = τ (Ψ_{t} (e))$ for every edge $e \in V_{t} ({(𝕄_{t})}^{1}) .$

Proof.

Immediate by the definitions of

Ψ_{t}

and

Ψ_{t}^{- 1} .

Proposition 4.1.3.

Let notation and assumptions be as above. Then $| {Reg}_{4^{+}, t} (𝕄_{t}) | = | Reg ({\tilde{𝕄}}^{t}) | .$

Proof.

See [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

, pp. 82–83].

4.2. The subdiagrams $Opt (Δ), {Opt}_{r} (Δ)$ and ${Opt}_{l} (Δ),$ $Δ = Ψ_{t}^{- 1} (\tilde{Δ}),$ $\tilde{Δ} \in 𝒟 ({\tilde{ω}}_{i}), i = 1, 2$

Let

M \in ℳ_{3} (W)

and assume

Int (M)

is connected. In general, this does not imply that

{\tilde{𝕄}}^{t}

has connected interior. In the rest of the work, we aim to prove Theorems A–C. It is enough to prove these theorems for every connected component of

Int ({\tilde{𝕄}}^{t}) .

Hence without loss of generality, we may work under the assumption that

{\tilde{𝕄}}^{t}

has connected interior, and we shall do so. Thus for the rest of the work, we shall assume that ${\tilde{𝕄}}^{t}$ has connected interior, unless said otherwise.

Definition 4.2.1

(

Opt (Δ), {Opt}_{r} (Δ), {Opt}_{l} (Δ), \hat{Opt} (Δ)

). Suppose

\tilde{Δ} \in 𝒟 ({\tilde{𝕄}}^{t}),

and let

Δ = Ψ_{t}^{- 1} (\tilde{Δ}) .

Assume

\partial \tilde{Δ} \cap \partial {\tilde{𝕄}}^{t} = {\tilde{v}}_{1} {\tilde{e}}_{1} {\tilde{v}}_{2} {\tilde{e}}_{2} {\tilde{v}}_{3} {\tilde{e}}_{3} \dots {\tilde{v}}_{k}, k \geq 2,

{\tilde{v}}_{i}

vertices,

{\tilde{e}}_{i}

edges. Define

Opt (Δ) = Ψ_{t}^{- 1} ({\tilde{e}}_{1} {\tilde{v}}_{2} \dots {\tilde{e}}_{k - 1}),

define

{Opt}_{r} (Δ) = Ψ_{t}^{- 1} ({\tilde{e}}_{1} {\tilde{v}}_{2} \dots {\tilde{e}}_{k - 1} {\tilde{v}}_{k})

and define

{Opt}_{l} (Δ) = Ψ_{t}^{- 1} ({\tilde{v}}_{1} {\tilde{e}}_{1} {\tilde{v}}_{2} \dots {\tilde{e}}_{k - 1}) .

Thus

{Opt}_{r} (Δ) = Opt (Δ) \cup Ψ_{t}^{- 1} ({\tilde{v}}_{k})

and

{Opt}_{l} (Δ) = Opt (Δ) \cup Ψ_{t}^{- 1} ({\tilde{v}}_{1}) .

Define

\hat{Opt} (Δ) = Ψ_{t}^{- 1} ({\tilde{v}}_{1} {\tilde{e}}_{1} \dots {\tilde{v}}_{k}) .

It follows from the definition of

ψ_{t}^{- 1}

that

Opt (Δ) = Ψ_{t}^{- 1} ({\tilde{e}}_{1}) \cup Ψ_{t}^{- 1} ({\tilde{v}}_{2}) \cup Ψ_{t}^{- 1} ({\tilde{e}}_{2}) \cup \dots \cup Ψ_{t}^{- 1} ({\tilde{e}}_{k - 1}) = B_{1} \cup Z_{2} \cup B_{2} \cup Z_{3} \cup \dots \cup B_{k - 1},

where

B_{i}

are

t -

bands emanating from

\partial Δ \cap \partial Opt (Δ)

and

Z_{i}

are connected unions of regions from

{Reg}_{H_{t}} (𝕄_{t}) .

Thus

Δ \cup Opt (Δ),

Δ \cup {Opt}_{r} (Δ), Δ \cup {Opt}_{l} (Δ)

and

\hat{Opt} (Δ)

are connected subdiagrams of

𝕄_{t} .

Due to Proposition 2.1.7,

\partial M \cap \partial B_{i}, \partial Z_{i} \cap \partial Δ

and

\partial B_{i} \cap \partial Δ

are connected, and it is easy to see that

\partial Δ \cap \partial Opt (Δ), \partial Δ \cap \partial {Opt}_{r} (Δ), \partial Δ \cap \partial {Opt}_{l} (Δ)

and

\partial Δ \cap \partial \hat{Opt} (Δ)

are connected.

We are interested in

{Opt}_{r} (Δ) ({Opt}_{l} (Δ))

for the special case when

{\tilde{v}}_{k} ({\tilde{v}}_{1})

has valency

3

in

{\tilde{𝕄}}^{t} .

Lemma 4.2.2.

Let $M \in ℳ_{3} (W) .$ Assume that $\partial M$ decomposes by $\partial M = u ω_{1} w ω_{2}^{- 1},$ $u, w$ vertices and $t \in T (M) .$ Let ${\tilde{Δ}}_{1}^{t}$ and ${\tilde{Δ}}_{2}^{t}$ be boundary regions of ${\tilde{𝕄}}^{t},$ ${\tilde{Δ}}_{1}^{t}$ and ${\tilde{Δ}}_{2}^{t}$ in $N ({\tilde{ω}}_{1}), Δ_{i} = Ψ_{t}^{- 1} ({\tilde{Δ}}_{i}^{t}), Supp (Δ_{i}) = {a, t}, {\tilde{ω}}_{1} = Ψ_{t} (ω_{1}) .$ Assume that $\partial {\tilde{Δ}}_{1} \cap \partial {\tilde{Δ}}_{2} \cap \partial {\tilde{ω}}_{1}$ is a vertex ${\tilde{v}}^{t}$ with valency $3$ in ${\tilde{𝕄}}^{t} .$ Let $Q = Ψ_{t}^{- 1} (\tilde{v}),$ and let $P = Ψ_{t}^{- 1} (N (\tilde{v})) .$ Thus $P = Ψ_{t}^{- 1} ({\tilde{Δ^{t}}}_{1} \cup \tilde{Δ_{2}^{t}} \cup (\partial {\tilde{Δ^{t}}}_{1} \cap {\tilde{\partial Δ^{t}}}_{2}) \cup {\tilde{e}}_{k} \cup {\tilde{f}}_{1} \cup \tilde{v}),$ where ${\tilde{e}}_{k}$ and ${\tilde{f}}_{1}$ are the boundary paths of ${\tilde{Δ}}_{1}^{t}$ and ${\tilde{Δ}}_{2}^{t}$ with common endpoint $\tilde{v} .$ Then each of the following holds:

$Q$ is connected and simply connected with cyclically reduced boundary label such that each connected component $Q_{i}, i = 1, \dots, k$ of $Q$ has cyclically reduced boundary labels $U_{i} .$ Moreover, $Q_{i} \in ℳ_{3} (U_{i}), i = 1, \dots, k .$
$P$ is connected and simply connected with connected interior and cyclically reduced boundary label $V,$ $P \in ℳ_{3} (V)$ .
$\partial Q = θ_{1} p_{1} α_{1} p_{2} θ_{2} p_{3} α_{2} p_{4} θ_{3} p_{5} γ^{- 1}$ , where $p_{i}$ are vertices, $θ_{i}$ are sides of $t$ -bands $i = 1, 2, 3$ and $α_{j} = \partial Δ_{j} \cap \partial Q, j = 1, 2, Φ (α_{j}) = a^{c_{j}}, c_{j} \neq 0$ and $γ = \partial Q \cap ω_{1} .$

Proof of Lemma 4.2.2.

See the proof of Lemma 5.2.2 in [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

].

4.3. Definitions and assumptions

Generalised Greendlinger regions. In our earlier work [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

, Section 1.3], we introduced (classical) Greendlinger regions. In the rest of the work, we shall need more general notions of Greendlinger regions. Recall from [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

, Section 1.3]

𝒟_{i} (M), i = 1, 2,

and

𝒟 (M)

that

M

is a connected, simply connected map with connected interior which contains at least two regions.

If

\partial M = u μ v ν^{- 1}, μ

and

ν

boundary paths, we shall call

M

bigonal with sides $μ$ and $ν$ . We define

𝒟_{i} (μ), 𝒟_{i} (ν), i = 1, 2

and

𝒟 (μ)

and

𝒟 (ν)

as follows.

Definitions 4.3.1.

Let

M

be a bigonal map with sides

μ

and

ν .

Define

$𝒟_{j} (μ) = {D \in Reg (M), \partial D \cap μ is connected and i (D) = j} j = 1, 2 .$ $𝒟_{j} (ν)$ is defined accordingly. Furthermore, define $𝒟 (μ) = 𝒟_{1} (μ) \cup 𝒟_{2} (μ) \cup 𝒟_{0} (μ),$ where $𝒟_{0} (μ)$ is defined only if $M = {D} .$ In this case, $𝒟_{0} (μ) = {D}$ if $| μ | \geq | ν |$ and $𝒟_{0} (ν) = {D}$ if $| ν | \geq | μ | .$ In case $| μ | = | ν |,$ we choose $𝒟_{0} (μ) = {D} .$
Let $M$ be an $ℛ -$ diagram, $ℛ$ given by (1.1). Assume $M \in ℳ_{3} (W)$ and $\partial M = μ ν^{- 1} .$

Define

$\begin{matrix} 𝔻 (μ) = {Δ \in {Reg}_{4^{+}} (𝕄) ∣ & \partial Δ \cap μ is connected, ∥ \partial Δ \cap μ ∥ \geq n (Δ) \\ and | Φ (ξ) | \leq | Φ (\partial Δ \cap μ) |} . \end{matrix}$

Here, $ξ$ is the complement of $\partial Δ \cap μ$ on $\partial Δ .$ Define $𝔻 (ν)$ accordingly.
Define

$\begin{matrix} 𝒟^{v} (μ) = {Δ \in {Reg}_{4^{+}} (𝕄) ∣ & Δ contains a subdiagram S with S \in O . L . (θ_{1}, θ_{2}), \\ S \subseteq N (μ), \partial S \cap μ = θ_{1}, ∥ θ_{1} ∥ \geq n (Δ) \geq 4 \\ and | Φ (θ_{2}) | \leq | Φ (θ_{1}) |} . \end{matrix}$

Define $𝒟^{v} (ν)$ accordingly.
Define

$𝒟_{2}^{1} (μ) = {D \in 𝒟_{1} (μ), n (D) = 2},$ $𝒟_{2}^{2} (μ) = {D \in 𝒟_{2} (μ), n (D) = 2} .$
Define $σ : boundary paths \to {0, 1}$ as follows:

$σ (μ) = 1$ if one of the following holds:
1. $𝒟_{2}^{1} (μ) \neq \emptyset .$
2. $| 𝒟_{2}^{2} (μ) | \geq 2 .$
3. $| 𝒟_{2}^{2} (μ) | = 1,$ and there is a region $Δ$ in $𝕄_{4^{+}}$ and a sequence $J$ of $ℐ -$ moves such that $\partial Δ^{J} \cap μ$ is connected and $| \partial Δ^{J} \cap μ | \geq 2 .$
4. $𝒟_{2}^{2} (μ) = \emptyset,$ and $𝔻 (μ) \cup 𝒟^{v} (μ) \neq \emptyset .$
In all other cases, $σ (μ) = 0 .$
Let $\tilde{μ} = Ψ_{t} (μ) .$ Define

$𝒟^{'} (\tilde{μ}) = {\tilde{Δ} \in Reg ({\tilde{𝕄}}^{t}) ∣ Δ \in 𝒟^{v} (μ), Δ = Φ_{t}^{- 1} (\tilde{Δ})} .$

Assumptions 4.3.2.

The proofs of Theorems A–C rely on other results as well (below) which are proved by simultaneous induction. They are

Theorems A–C,
Theorems S, L and T.

Theorem 4.3.3

(Theorem S). Let $M \in ℳ_{3} (W)$ and assume that $\partial M$ decomposes by $ω_{1} v_{1} ω_{2}^{- 1} v_{2},$ where $ω_{1}$ and $ω_{2}$ are boundary paths and $v_{1}$ and $v_{2}$ are boundary vertices. Let $t \in T (M),$ and let $\tilde{Δ} \in 𝒟 ({\tilde{ω}}_{1}),$ ${\tilde{ω}}_{1} = Ψ_{t} (ω_{1}),$ $Δ = Ψ_{t}^{- 1} (\tilde{Δ}), Supp (Δ) = {a, t} .$ Assume $σ (ω_{1}) = 0$ in parts (a), (b) and (c). Then there is a sequence $J$ of $ℐ -$ moves such that $Δ^{J} = Δ, M^{J} \in ℳ_{3} (W),$ and each of the following holds:

If $β$ is an edge of ${(\partial Δ \cap \partial \hat{Opt} (Δ))}^{J},$ then there is a band which has a pole on $β$ and the other pole on $ω_{1} .$ In particular, ${(\partial Δ \cap_{B} ω_{1})}^{J}$ is connected.
If $B_{1}$ and $B_{2}$ are bands emanating from ${(\partial Δ \cap_{B} ω_{1})}^{J}$ with adjacent poles on $\partial Δ,$ then $\partial B_{1} \cap \partial B_{2}$ is connected. Moreover, let $L_{1}, \dots, L_{k}$ be the bands that emanate from ${(\partial Δ \cap_{B} ω_{1})}^{J} .$ Then $\partial L_{i} \cap \partial L_{i + 1}$ is connected, $i = 1, \dots, k - 1,$ where $L_{i}$ and $L_{i + 1}$ have adjacent poles on $\partial Δ .$
Let $Z := \hat{Opt} {(Δ)}^{J} \cup ω_{1} ∖ {L_{1}, \dots, L_{k}} = Q_{1} \dot{\cup} \dots \dot{\cup} Q_{l},$ where $Q_{i}$ are the connected components.
1. $\partial Q_{i}$ are simple closed curves and $\partial Q_{i}$ decompose by $\partial Q_{i} = ρ_{i} ξ_{i} γ_{i}^{- 1},$ where $ρ_{i} = \partial Q_{i} \cap \partial L_{j},$ $ξ_{i} = \partial Q_{i} \cap \partial L_{j + 1}$ and $γ_{i} = \partial Q_{i} \cap ω_{1},$ $j \in {1, \dots, k}, i = 1, \dots, l$ .
2. Up to diamond moves, $U_{i} := Φ (\partial Q_{i})$ is cyclically reduced and $Q_{i} \in ℳ_{3} (U_{i}), i = 1, \dots, l,$ and either $Q_{i} = {D_{i}}, n (D_{i}) = 2$ or $Q_{i}$ is transferable via $L_{j} \cup L_{j + 1} .$ See Figure 6.
3. For every $D \in Reg (Q_{i}), i = 1, \dots, l, Supp (Φ (\partial D)) \subseteq X ∖ {a, t}$ .
Let $η = {(\partial Δ \cap_{B} ω_{1})}^{J} .$ Then the following hold:
1. If $Δ \in 𝔻 (η),$ then $σ (ω_{1}) = 1 .$
2. If ${\tilde{𝕄}}^{t} \in O . L ., {\tilde{𝕄}}^{t} = ⟨ {\tilde{Δ}}_{1}, \dots, {\tilde{Δ}}_{m} ⟩,$ $m \geq 2$ and $Δ_{j} \in 𝒟^{v} (η),$ for $j = 1$ and $j = m,$ then $σ (ω_{1}) = 1 .$

Figure 6.

The possible configurations of the adjacent bands in

Opt (Δ)

emanating from

Δ .

Remarks

(Pushing up

Δ

).

It follows from part (c) that if $𝒟_{2}^{2} (ω_{1}) = \emptyset,$ then $Opt (Δ)$ is a homogeneous band-bundle, hence we can push up $Δ$ by a sequence $J_{1}$ of $ℐ -$ moves to $ω_{1},$ such that $Δ^{j} \in 𝔻 (μ),$ obtaining the classical result on Greendlinger regions.
The same results hold if we use $ν$ in place of $μ .$

Theorem 4.3.4

(Theorem L). Let $M \in ℳ_{3} (W),$ $\partial M = u ω_{1} v ω_{2}^{- 1}, u$ and $v$ vertices, $ω_{1}$ and $ω_{2}$ boundary paths of $M .$ Assume $T (M) \neq \emptyset,$ and let $t \in T (𝕄) .$ Let ${\tilde{ω}}_{1} = Ψ_{t} (ω_{1}),$ and let $\tilde{Δ} \in Reg ({\tilde{𝕄}}^{t}) .$ Let $Δ = Ψ_{t}^{- 1} (\tilde{Δ}),$ and let $η = \partial Δ \cap_{B} ω_{1} .$ Let $Supp (Δ) = {a, t} .$

If $\tilde{Δ} \in 𝒟 ({\tilde{ω}}_{1})$ , then $σ (ω_{1}) = 1, {∥ η ∥}_{a} \geq 2$ and ${∥ η ∥}_{t} \geq 2 .$
If $\tilde{Δ} \in 𝒟^{'} ({\tilde{ω}}_{1})$ , then $σ (ω_{1}) = 1, {∥ η ∥}_{a} \geq 2$ and ${∥ η ∥}_{t} \geq 2 .$

Theorem 4.3.5

(Theorem T). Let $M \in ℳ_{3} (W),$ and let $P$ be a simply connected, connected subdiagram of $M$ with at least one region. Assume that $P$ is closed $2 -$ banded in $M$ by bands $B_{r}$ and $B_{l} .$ If $\partial P$ has cyclically reduced label $U$ and $P \in ℳ_{3} (U),$ then $P$ is transferable. In particular, there is a sequence $J$ of $ℐ -$ moves such that $B_{r}^{J} \cap B_{l}^{J}$ is connected and $B_{r}^{J} \cup B_{l}^{J}$ is a band-bundle. Moreover, $| B_{r}^{J} | = | B_{r} |$ and $| B_{l}^{J} | = | B_{l} |$ .

In each of these theorems, $M \in ℳ_{3} (W)$ and $\partial M = u μ v ν^{- 1},$ $u$ and $v$ vertices. The induction hypothesis claims that if $N$ is a proper subdiagram of $M$ with cyclically reduced boundary label $U_{N}$ such that $N \in ℳ_{3} (U_{N}),$ then the corresponding result holds for $N .$ We shall denote these hypotheses by $ℋ (A), ℋ (B), ℋ (C),$ $ℋ (S), ℋ (L)$ and $ℋ (T),$ respectively. So we assume in the sequel that

(ℋ₁)

ℋ (A), ℋ (B), ℋ (C), ℋ (S), ℋ (L) and ℋ (T) hold .

4.4. The proof of Theorem A

Lemma 4.4.1.

Let $W$ be a cyclically reduced word in $F,$ $W = 1$ in $A$ and $W \neq 1$ in $A (Γ_{2}) .$ Let $M$ be a van Kampen $ℛ -$ diagram over $F,$ $ℛ$ given by (1.1). Assume that $M \in ℳ_{1} (W)$ and all bands are adequate. Let $Δ_{1}$ and $Δ_{2}$ be regions in ${Reg}_{4 +} (𝕄),$ with $Supp (Δ_{1}) = Supp (Δ_{2}) = {a, t}, Δ_{1} \neq Δ_{2} .$ Assume that there are bands $B_{1}$ and $B_{2}$ connecting adjacent edges $e_{1} v_{1} e_{2}$ of $\partial Δ_{1}$ with adjacent edges $f_{1} w_{1} f_{2}$ of $\partial Δ_{2},$ $v_{1}$ and $w_{1}$ are vertices such that $Φ (e_{1}) = a$ and $Φ (e_{2}) = t .$ Then $M \notin ℳ_{2} (W) .$ In particular, if $M \in ℳ_{3} (W),$ then the poles of $B_{1}$ and $B_{2}$ cannot be adjacent both on $\partial Δ_{1}$ and on $\partial Δ_{2} .$

Proof.

See the proof of Lemma 5.4.1 in [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

].

Lemma 4.4.2.

Let $M \in ℳ_{3} (W)$ and assume $T (M) \neq \emptyset .$ Let $t \in T (M) .$ Then

${\tilde{𝕄}}^{t}$ has no inner vertices with valency $2,$
${\tilde{𝕄}}^{t}$ satisfies condition $C (4) .$

Proof.

See the proof of Lemma 5.4.2 in [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

].

4.4.3. Proof of Theorem A

If

{\tilde{𝕄}}^{t}

does not satisfy condition

T (4),

then due to Lemma 4.4.2,

{\tilde{𝕄}}^{t}

contains an inner vertex

\tilde{v}

with valency

3 .

Let

{\tilde{Δ}}_{1}, {\tilde{Δ}}_{2}

and

{\tilde{Δ}}_{3}

be the regions of

{\tilde{𝕄}}^{t}

which contain

\tilde{v}

on their boundary. Let

Δ_{i} = Ψ_{t}^{- 1} ({\tilde{Δ}}_{i}), i = 1, 2, 3,

and let

Q = Ψ_{t}^{- 1} (\tilde{v}) .

It is not difficult to show by arguments similar to those used in the proof of Lemma 4.2.2 that

ψ_{t}^{- 1} (⋃_{i = 1}^{i = 3} {\tilde{Δ}}_{i})

is connected and simply connected, and

\partial Q = α_{1} θ_{1} α_{2} θ_{2} α_{3} θ_{3}, α_{i} = \partial Δ_{i} \cap \partial Q

and

θ_{i}

are sides of bands

B_{i}, θ_{i} = \partial B_{i} \cap \partial Q, i = 1, 2, 3,

and

α_{i}

is either a point or a path with label

a_{i}^{k_{i}}, k_{i} \neq 0, Supp (Δ_{i}) = {t, a_{i}}, i = 1, 2, 3 .

Consider

U := Φ (\partial Q) .

Since

Supp (Φ (θ_{i})) \in C_{Γ} (t)

while if

α_{i}

is not a point, then

α_{i}

is labelled with

a_{i}^{k_{i}} \notin C_{Γ} (t),

the only occurrences of letters in

U

which do not commute with

t

are the non-trivial labels of

α_{i} .

In particular,

(*)

\begin{matrix} there are at most three disjoint occurrences of letters in U \\ which do not commute with t \end{matrix}} .

We claim that if

α_{i}

and

α_{j}

are not points, then

a_{i} = a_{j} .

Suppose not. By [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

, Lemma 1.1.c], there are at least two disjoint occurrences of

a_{i}

and two disjoint occurrences of

a_{j}

in

U,

hence

U

has at least four such occurrences violating (*). Hence

a_{i} = a_{j} .

Denote

a = a_{i} = a_{j} .

It is easy to see that

U

is cyclically reduced and

Q \in ℳ_{3} (U) .

In particular, Theorem B applies to

Q

(

| Q | < | M |

). Hence if

D

is a region of

Q

with

a \in Supp (D),

then

n (D) \leq 3,

due to (*). Since

n (D) \neq 3,

hence

n (D) = 2 .

Hence

Q

is

a -

abelian. Consequently, if

α_{i}

is not a point, then from each edge of

α_{i},

there emanates a band

L

in

Q

which has one of its poles on

α_{i}

and the other on

α_{j} .

Observe that

L

and

B_{i}

have adjacent poles on

Δ_{i}

and on

Δ_{j} .

But this violates Lemma 4.4.1. Hence all

α_{i}

are points. But then since three is odd, the three changes of orientations of the poles of

β_{i}, i = 1, 2, 3,

force

t = t^{- 1} .

This violates [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

, Lemma 1.1 (a)]. Hence

d (\tilde{v}) \neq 3 .

It follows from Lemma 4.4.2 (b) that

{\tilde{𝕄}}^{t}

satisfies conditions

C (4)

and

T (4) .

∎

4.5. Preparatory results for the proofs of Theorems S, L, T and B

Lemma 4.5.1.

Let $M \in ℳ_{3} (W)$ be a connected, simply connected $ℛ -$ diagram with connected interior. Suppose $\partial M = u_{1} ω_{1} u_{2} ω_{2}^{- 1}$ and $T (M) \neq \emptyset,$ where $u_{1}$ and $u_{2}$ are vertices.

Let $t \in T (M)$ and consider ${\tilde{𝕄}}^{t} .$ Assume $Reg ({\tilde{M}}^{t}) = {\tilde{Δ}} .$ Let $Δ = Ψ_{t}^{- 1} (\tilde{Δ}),$ $Supp (Δ) = {t, a}$ and define ${Opt}_{ω_{i}} (Δ) := Ψ_{t}^{- 1} ({\tilde{ω}}_{i}), i = 1, 2 .$ Assume Theorem S holds for $M .$ Then $σ (ω_{i}) = 1$ for $i = 1$ or $i = 2,$ and ${∥ ω_{i} ∥}_{t} \geq 2 .$

Proof.

See the proof of Lemma 5.5.1 in [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

].

Lemma 4.5.2.

Let $M \in ℳ_{3} (W)$ with boundary cycle $μ u ν^{- 1} v,$ where $u$ and $v$ are vertices. Assume that $T (M) \neq \emptyset,$ and let $t \in T (M) .$ Let $\tilde{Δ} \in Reg ({\tilde{𝕄}}^{t}),$ and let $Δ = Ψ_{t}^{- 1} (\tilde{Δ}) .$ Let $η = (\partial Δ \cap \partial \hat{Opt} (Δ)) \cap_{B} μ,$ and let $η_{0} = (\partial Δ \cap \partial Opt (Δ)) \cap_{B} μ .$ Assume $η$ is connected. Then in each of the following cases, $Δ \in 𝔻 (η)$ and ${∥ Φ (μ) ∥}_{t} \geq 2$ :

$\tilde{Δ} \in 𝒟_{1} (\tilde{μ}) .$
$\tilde{Δ} \in 𝒟_{2} (\tilde{μ})$ , and there is an $H_{t}$ -band $L$ emanating from $η ∖ η_{0}$ and ending on $μ .$

Moreover, in case (a), ${∥ Φ (μ) ∥}_{t} \geq 3,$

Proof.

See the proof of Lemma 5.5.2 in [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

].

Remark.

Lemma 4.5.2 remains true if we replace

μ

by

ν .

Lemma 4.5.3.

Let $M \in ℳ_{3} (W)$ be a connected, simply connected $ℛ -$ diagram over F with connected interior. Let $\partial M = v_{1} ω_{1} v_{2} ω_{2}^{- 1},$ $v_{1}$ and $v_{2}$ vertices and $ω_{1}$ and $ω_{2}$ boundary paths. Let $t \in T (M)$ and assume that $| {\tilde{𝕄}}^{t} | \geq 2$ and ${\tilde{𝕄}}^{t} \in O . L . (\tilde{ω_{1}}, \tilde{ω_{2}}), \tilde{ω_{i}} = Ψ_{t} (ω_{i}), {\tilde{𝕄}}^{t} = ⟨ {\tilde{Δ}}_{1}, \dots, {\tilde{Δ}}_{k} ⟩, k \geq 2 .$ If $σ (ω_{1}) = 0$ and Theorem S holds for $M,$ then each of the following holds:

$σ (\partial Opt (Δ_{i}) \cap_{B} ω_{2}) = 1, i = 1, k .$
${∥ ω_{2} ∥}_{t} \geq 3 .$

Proof.

See [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

, pp. 98–99].

Lemma 4.5.4.

Let notation and assumptions be as in Lemma 4.2.2. Let $η_{i} = \partial Δ_{i} \cap_{B} ω_{i},$ $i = 1, i = 2 .$ Let $ω_{1} = γ$ and $ω_{2} = θ_{1} α_{1} θ_{3} α_{2} θ_{2} .$ (Thus $| Q | < | M |,$ and $ω_{1} u ω_{2}^{- 1}$ is a boundary cycle of $Q .$ ) If $σ (ω_{1}) = 0,$ then $Q$ is $a -$ abelian.

Proof.

See the proof of Lemma 5.5.4 in [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

].

Corollary.

Let notation and assumptions be as in the lemma. If $\tilde{Δ} \in 𝒟_{2} (\tilde{μ}),$ then $Δ \in 𝔻 (μ) .$

Proof.

Due to the lemma, condition (b) of Lemma 4.5.2 is satisfied. Hence the result follows from part (b) of Lemma 4.5.2.

4.6. Proof of Theorem S

Let $τ_{1} w_{1} α w_{2} τ_{2}$ be a subpath of $\partial Δ \cap Opt (Δ),$ $w_{1}$ and $w_{2}$ vertices, $Φ (τ_{1}) = t^{\pm 1},$ $Φ (τ_{2}) = t^{\pm 1},$ $Φ (α) = a^{f}, f \neq 0 .$ It follows from the definition of $Opt (Δ)$ that there are $t -$ bands $L_{1}$ and $L_{2}$ emanating from $τ_{1}$ and $τ_{2},$ respectively, and ending on $\partial M .$ Let $θ_{1}$ be the side of $L_{1}$ which has $w_{1}$ as an endpoint, and let $θ_{2}$ be the side of $L_{2}$ which has $w_{2}$ as an endpoint. Then $Supp (Φ (θ_{i})) \subseteq C_{Γ} (t),$ $i = 1$ and $i = 2 .$ Let $z_{1}$ and $z_{2}$ be the endpoints of $θ_{1}$ and $θ_{2},$ respectively, on $\partial M,$ and let $γ$ be the subpath of $\partial M \cap \partial Opt (Δ)$ which starts at $z_{1}$ and ends at $z_{2} .$ Then $σ (γ) = 0$ as $γ \subseteq ω_{1}$ and $σ (ω_{1}) = 0 .$ Also, $U := Φ (θ_{1} α θ_{2} γ^{- 1})$ is cyclically reduced, since $a \notin C_{Γ} (t) .$ Let $Q$ be the bounded component of $𝔼^{2} ∖ (θ_{1} α θ_{2} γ^{- 1}) .$ Assume first that $\partial Q (= θ_{1} α θ_{2} γ^{- 1})$ is a simple closed path. Hence $Int (Q)$ is connected. It follows from [6
A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1
, Lemma 2.2.1] that $Q \in ℳ_{3} (U) .$ Hence Theorem A applies to $Q,$ and it follows from the proof of Lemma 4.5.4 that if $Q$ is not $a -$ abelian, then one of the four cases mentioned in the proof of Lemma 5.5.4 in [6
A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1
] occurs. From this point on, we follow the proof of Lemma 4.5.4, with ${∥ ω_{2} ∥}_{a} = 1$ (here) instead of ${∥ ω_{2} ∥}_{a} = 2,$ showing that each case leads to a contradiction, hence $Q$ is $a -$ abelian, more precisely, in the first case, by Lemma 4.5.1, in the second case, by Lemma 4.5.3, in the third case, by Lemma 4.5.2 and in the fourth case, by Theorem L. Therefore, if $β = β_{1} \dots β_{k},$ $Φ (β_{i}) = a^{\pm 1},$ then from every $β_{i},$ there emanates an $a -$ band which ends on $γ,$ since $a \notin C_{Γ} (t) .$ (Notice that it cannot end on $β .$ ) If $Int (Q)$ is not connected, then a similar argument applies to each component.
Let $B_{1}$ and $B_{2}$ be bands emanating from $\partial Δ \cap \partial Opt (Δ)$ and ending on $ω_{1} .$ Assume that their poles on $\partial Δ$ are adjacent. Let $ξ_{1}$ and $ξ_{2}$ be the sides of $B_{1}$ and $B_{2},$ respectively, with $w := ξ_{1} \cap ξ_{2} \cap \partial Δ \neq \emptyset$ a point. We claim that $ξ_{1} \cap ξ_{2}$ is connected. Suppose not and consider $𝔼^{2} ∖ (ξ_{1} \cup ξ_{2}) .$ Let $P$ be a bounded connected component of it. Then $\partial P = (\partial P \cap ξ_{1}) \cup (\partial P \cap ξ_{2}) .$ Let $V := Φ (\partial P) .$ Then applying diamond moves at the endpoints of $\partial P \cap ξ_{1}$ if needed, we may assume that $V$ is cyclically reduced. It follows from [6
A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1
, Lemma 2.2.1] that $P \in ℳ_{3} (V) .$ Hence Theorem T applies to $P$ ( $| P | < | M |$ ). Hence by a sequence $J_{0}$ of $ℐ -$ moves, $P$ is transferable beyond $B_{1}$ and $B_{2}$ such that $P^{J}$ contains no regions. Doing so independently for every connected component $P_{i},$ by $J_{i}, i = 1, \dots, r,$ we get that ${(B_{1} \cap B_{2})}^{J}$ is connected, where $J = J_{1} \circ J_{2} \circ \dots \circ J_{r} .$

Consider now $𝔼^{2} ∖ {L_{1}, \dots, L_{k}} .$ Let $Q_{1}, \dots, Q_{m}$ be its bounded connected components.

Assume $k = 3$ and $\partial L_{1} \cap \partial L_{2}$ is connected. Assume that for some $i, 1 \leq i \leq m,$ $Q_{i}$ occurs between $L_{2}$ and $L_{3}$ such that $\partial Q_{i} = (\partial Q_{i} \cap \partial L_{2}) \cup (\partial Q_{i} \cap \partial L_{3}) .$ Suppose that we would like to transfer a region $D$ from $Q_{i}$ via $L_{2} .$ In principle, it is possible that a previous step of transferring regions $D$ occurred between $L_{1}$ and $L_{2}$ and was transferred to $Q_{1} .$ Hence now we transfer it back, and by this we create a loop in the process. The point is that because $\partial L_{1} \cap \partial L_{2}$ is connected, we may regard $L_{1} \cup L_{2}$ as a single band, and this enables us to transfer $D$ further beyond $L_{1},$ out of $\hat{Opt} (Δ),$ and, by this, to “empty” (by $ℐ -$ moves $J$ ) each $Q_{i},$ so to get $L_{1}^{J} \cap L_{2}^{J}$ and $L_{2}^{J} \cap L_{3}^{J}$ simultaneously connected. From this, it is clear how to prove for general $k .$
(i) This is clear from the construction of $Z$ and the definition of $Q_{i} .$

(ii) and (iii) It follows from [6
A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1
, Lemma 2.2.1] that $Q_{i} \in ℳ_{3} (U_{i}) .$ It remains to show that either $Q_{i} = {D}$ with $n (D) = 2$ or $Q_{i}$ is transferable. Thus Theorem C is applicable to $Q_{i},$ implying that $σ (ρ_{i} ξ_{i}) = 1 .$ (Recall that $σ (γ) = 0 .$ ) Consequently, if $Q_{i}$ is not abelian, then $N (ρ_{i} ξ_{i})$ contains a transferable region. Hence by applying a finite number of $ℐ -$ moves, we can transfer all the regions $D$ with $n (D) \geq 4$ beyond $L_{i}$ and/or $L_{i + 1}$ and remaining either with an abelian diagram $Q_{i}^{'}$ or with a diagram with $𝒟_{2}^{2} (ω_{1}^{'}) = \emptyset,$ where $ω_{1}^{'}$ is obtained from $ω_{1}$ by the appropriate $I -$ move. Applying [6
A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1
, Lemma 4.1.2], the result follows. Finally, since either every region $D$ in $Q$ is transferable via $L_{j} \cup L_{j + 1}$ or $Q = {D},$ $Supp (\partial D) \subseteq C_{Γ} (a) \cup C_{Γ} (t) .$ But ${a, t} \cap (C_{Γ} (a) \cup C_{Γ} (t)) = \emptyset .$ Hence ${a, t} \cap Supp (D) = \emptyset .$
We follow notation of part (c).

(i) By assumption, $Δ \in 𝔻 (η) .$ Hence $∥ \partial Δ \cap η ∥ \geq ∥ n (Δ) ∥ \geq 4 .$ Consider $\hat{Opt} (Δ) .$ Let ${L_{1}, \dots, L_{k}}$ be the bands which emanate from $η$ and end on $ω_{1} .$ Consider $L_{1} \cup Q_{1} \cup L_{2} .$ If $Q_{1} \neq {D}$ with $n (D) = 2,$ then we can transfer $Q_{1}$ beyond $L_{1}$ and $L_{2}$ via a suitable sequence $J$ of $ℐ -$ moves such that in $M^{J},$ configurations (1), (2), (3) and (4) as in [6
A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1
, Figure 41] occur. In configuration (1), $𝒟_{2}^{2} (ω_{1}) \geq 2,$ in configurations (2) and (3), $𝒟_{2}^{2} (ω_{1}) \geq 1,$ and in configuration (4), $𝒟_{2}^{2} (ω_{1}) \geq 0 .$ Similar arguments apply to $L_{k - 1} \cup Q_{k - 1} \cup L_{k} .$ Notice that if $L_{1} \cup Q_{1} \cup L_{2}$ and $L_{k - 1} \cup Q_{k - 1} \cup L_{k}$ are in configurations (1), (2) or (3), then $| 𝒟_{2}^{2} (ω_{1}) | \geq 2 .$ This is part (2) in the definition of $σ (ω_{1}) = 1 .$ Hence we have to show that if $𝒟_{2}^{2} (ω_{1}) \leq 1,$ then $σ (ω_{1}) = 1,$ which contradicts $σ (ω_{1}) = 0 .$

Let $L_{1} \cup L_{2}$ be in configuration $i$ and $L_{k - 1} \cup L_{k}$ in configuration $j,$ $1 \leq i \leq 4$ and $1 \leq j \leq 4 .$ If $| 𝒟_{2}^{2} (ω_{1}) | \leq 1$ then either $i = 4$ or $j = 4$ or both. Assume $L_{1} \cup L_{2}$ is in configuration (4) and $L_{k - 1} \cup L_{k}$ in configuration $j \leq 3 .$ Then $L_{1} \cup L_{2}$ is a homogeneous band-bundle. See [6
A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1
] for the definition. Let $𝔹_{0}$ be the maximal band-bundle that contains $L_{1} \cup L_{2} .$ Then we claim that $𝔹_{0} = L_{1} \cup \dots \cup L_{k - 1} .$ Thus suppose $𝔹_{0} = L_{1} \cup \dots \cup L_{l},$ $l < k - 1$ and consider $L_{l + 1} .$ If $Reg (Q_{l}) \neq \emptyset,$ then we transfer $Q_{l}$ via $L_{l} \cup L_{l + 2}$ by a sequence $J$ of $ℐ -$ moves such that $Reg (Q_{l}^{J}) = \emptyset .$ Hence $L_{l} \cup L_{L + 1}$ is in configurations (1), (2) or (3), since $| 𝒟_{2}^{2} (ω_{1}) | = 1$ implies that $l = k - 1 .$ Observe that we can push up $Δ$ by a sequence $J_{1}$ of $ℐ -$ moves along $𝔹_{0}$ and $∥ \partial Δ^{J} \cap ω_{1} ∥ \geq n (Δ) - 2 \geq 2 .$ This is part (3) in the definition of $σ (ω_{1}) = 1 .$ So it remains to deal with the case when both $L_{1} \cup L_{2}$ and $L_{k - 1} \cup L_{k}$ are in configuration (4). Let $𝔹_{1}$ be the maximal band-bundle which contains $L_{k - 1} \cup L_{k} .$ Then $𝔹_{0} \cup 𝔹_{1} = Opt (Δ) .$ If $𝔹_{0} = 𝔹_{1},$ then $Opt (Δ)$ is a band-bundle, and by pushing up $Δ$ via $Opt (Δ),$ we get $Δ^{J_{2}} \in 𝒟 (ω_{1}) .$ But then by part (4) of the definition of $σ (ω_{1}),$ $σ (ω_{1}) = 1 .$ Finally, if $𝔹_{0} = 𝔹_{1},$ then either $∥ η \cap \partial 𝔹_{0} ∥ \geq 2$ or $∥ η \cap \partial 𝔹_{1} ∥ \geq 2,$ and we fulfil part (3) of the definition of $σ (ω_{1}),$ showing that $σ (ω_{1}) = 1 .$ If one of the $Q_{i}$ consists of a single region $D$ with $n (D) = 2,$ then $| 𝒟_{2}^{2} (ω_{1}) | \geq 1,$ and the above arguments apply to this case.

(ii) The case when ${\tilde{𝕄}}^{t} \in O . L . (ω_{1}, ω_{2})$ is dealt with similarly.∎

4.7. Proof of Theorem L

If $| {\tilde{𝕄}}^{t} | = 1,$ then by Lemma 4.5.1, $Δ \in 𝔻 (η) .$ Assume $| {\tilde{𝕄}}^{t} | \geq 2 .$ By Theorem S part (d), it is enough to show that $Δ \in 𝔻 (η) .$ By [6
A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1
, Lemma 1.3.2], $i (\tilde{Δ}) \leq 2 .$ If $i (\tilde{Δ}) = 1,$ then by Lemma 4.5.2 (a), $Δ \in 𝔻 (η) .$ If $i (\tilde{Δ}) = 2,$ then by [6
A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1
, Lemma 1.3.4], $\partial \tilde{Δ} \cap {\tilde{ω}}_{1}$ has an endpoint $\tilde{v}$ with valency $3 .$ Let $Q = Ψ_{t}^{- 1} (\tilde{v}) .$ If $σ (ω_{1}) = 0,$ then $Q$ is an $a -$ abelian, by Lemma 4.5.4, hence there is an $a -$ band-bundle emanating from $α_{1}$ and ending at $ω_{1} .$ (We follow the notation of Lemma 4.5.4.) Consequently, $Δ \in 𝔻 (η),$ by Lemma 4.5.2 (b), as required. Since $n (Δ) \geq 4,$ the rest follow.
Let ${\tilde{𝕄}}^{t} = ⟨ {\tilde{Δ}}_{1}, \dots, {\tilde{Δ}}_{k} ⟩, k \geq 2$ and consider $Δ_{1} .$ By [6
A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1
, Proposition 1.5.2], $\partial S \cap \partial Δ_{1}$ is connected, $∥ \partial S \cap \partial Δ_{1} ∥ \geq n (Δ_{1})$ and $| ξ | \leq | \partial S \cap \partial Δ_{1} |,$ where $ξ$ is the complement of $\partial S \cap \partial Δ_{1}$ on $\partial Δ_{1} .$ Consequently, $S \in 𝔻^{v} (η) .$ See Definition 4.3.1 (iii). Therefore, the result follows from Theorem S part (d). The rest follows from the fact that $∥ \partial S \cap \partial Δ_{1} ∥ \geq n (Δ_{1}) \geq 4,$ hence ${∥ \partial S \cap \partial Δ_{1} ∥}_{a} \geq 2$ and ${∥ \partial S \cap \partial Δ_{1} ∥}_{t} \geq 2,$ due to the alternating nature (in $a$ and $t$ ) of the boundary label of $Δ_{1} .$ ∎

4.8. Proof of Theorem T

By induction on

| P | .

Let

ω_{r} = \partial P \cap \partial B_{r},

and let

ω_{l} = \partial P \cap \partial B_{l} .

Then

u ω_{r} v ω_{l}^{- 1} = \partial P,

u

and

v

boundary vertices. We start with all the transferable regions from

{Reg}_{2} (P)

by applying successively a sequence

J_{1}

of

ℐ -

moves. Let

P_{1} = P^{J_{1}}

be the resulting diagram. Assume

Reg (P_{1}) \neq \emptyset .

If

P_{1} \notin ℳ_{3} (U_{1}),

then we can replace

P_{1}

by

P_{1}^{'} \in ℳ_{3} (U_{1})

and make the same argument. Hence we may assume that

P_{1} \in ℳ_{3} (U_{1}),

where

U_{1}

is a boundary label of

P_{1},

which we may assume to be cyclically reduced. Since we removed only regions of

{Reg}_{2} (M)

and this cannot increase

| {Reg}_{4^{+}} (M) |

and

| {Reg}_{4^{+}} (𝕄) |,

Theorem C applies to

P_{1}

and implies that

σ (ω_{1}^{*}) = 1

or

σ (ω_{2}^{*}) = 1,

where

ω_{i}^{*} = ω_{i}^{J_{1}}, i = 1, 2 .

Consequently, there is a region

Δ

with

n (Δ) \geq 4

or a submap

S \in 𝔻^{v} (ω_{i}^{*})

which can be transferred from

P_{1} .

See Remarks following Theorem S. Let the resulting diagram obtained from

P_{1}

be denoted by

P_{2} .

Notice that

| P_{2} | < | P | .

Then we may continue transferring from

P

all the regions

Δ

with

n (Δ) \geq 4,

ending up either with a diagram without regions proving the theorem or with an abelian diagram. In this case, [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

, Lemma 4.1.2] completes the proof.∎

4.9. Proofs of Theorems B and C

Proof of Theorem B.

We start with the proof of Theorem B.

Case $1 .$

T (M) = \emptyset .

Then

{Reg}_{4^{+}} (M) = \emptyset,

hence

Reg (M) = {Reg}_{2} (M) .

Hence if

Supp (D) = {a, b},

D \in Reg (M),

then

{∥ \partial D ∥}_{a} = {∥ \partial D ∥}_{b} = 2 .

Now, by [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

, Lemma 1.1.1 (c)],

{∥ \partial M ∥}_{a} \geq 2 .

Hence

{∥ \partial M ∥}_{a} \geq {∥ \partial D ∥}_{a}

and

{∥ \partial M ∥}_{b} \geq {∥ \partial D ∥}_{b},

as required.

Case $2 .$

T (M) \neq \emptyset .

Let

t \in T

and consider

{\tilde{𝕄}}^{t} .

Recall that for every region

Δ

in

𝕄_{t},

{∥ \partial Δ ∥}_{t} = ∥ \partial \tilde{Δ} ∥

and

n (Δ) = n (\tilde{Δ}),

where

\tilde{Δ} = Ψ_{t} (Δ) .

Without loss of generality, we may assume that

{\tilde{𝕄}}^{t}

has connected interior. Hence due to Theorem A and [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

, Lemma 1.3.2], either

{\tilde{𝕄}}^{t} = {\tilde{Δ}}

or

| 𝒟 ({\tilde{𝕄}}^{t}) | \geq 2 .

In the first case, the result is clear. Assume, therefore, that

𝒟 ({\tilde{𝕄}}^{t}) = {{\tilde{Δ}}_{1}, \dots, {\tilde{Δ}}_{k}},

k \geq 2 .

We show that

(4.1)

∥ \partial {\tilde{𝕄}}^{t} ∥ \geq n (Δ_{j}), j = 1, \dots, k .

From this, the result follows. We show this for

j = 1 .

The proof is the same for

j = 2, \dots, k .

Recall the notation of Lemma 4.5.2. By (3) in the proof of Lemma 4.5.2,

{∥ η_{0} ∥}_{t} \geq n (Δ_{j}) - i ({\tilde{Δ}}_{j}),

j = 1, \dots, k .

Hence by Theorem A and due to [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

, Lemma 1.3.2], the individual contribution of

Δ_{j}

to

{∥ \partial M ∥}_{t}

is at least

n (Δ_{j}) - 2 .

However, due to possible consolidation of labels of adjacent

Δ_{j},

the actual contribution of

Δ_{j}

to

{∥ \partial M ∥}_{t}

is at least

n (Δ_{j}) - 3 .

Observe that

(4.2)

{∥ \partial 𝕄 ∥}_{t} \geq {∥ η_{0} ∥}_{t} .

Consequently, due to (4.2),

(4.3)

{∥ \partial 𝕄 ∥}_{t} \geq \sum_{j = 1}^{k} [n (Δ_{j}) - 3] .

Now,

\sum_{j = 1}^{k} [n (Δ_{j}) - 3] = n (Δ_{1}) + (\sum_{j = 2}^{k} [n (Δ_{j}) - 3]) - 3 \geq n (Δ_{1}) + (\sum_{j = 2}^{k} [4 - 3]) - 3 = n (Δ_{1}) + (k - 1) - 3 = n (Δ_{1}) + k - 4,

that is,

(4.4)

{∥ \partial 𝕄 ∥}_{t} \geq n (Δ_{1}) + k - 4 (recall that n (Δ_{j}) \geq 4) .

Hence if

k \geq 4,

then

{∥ \partial 𝕄 ∥}_{t} \geq n (Δ_{1}),

as required. If

k = 3,

then due to [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

, Lemma 1.3.2],

i ({\tilde{Δ}}_{j}) = 1, j = 1, 2, 3,

hence

{∥ \partial 𝕄 ∥}_{t} \geq (n (Δ_{1}) - 2) + (n (Δ_{2}) - 2) + (n (Δ_{3}) - 2) \geq n (Δ_{1}) + (4 - 2) + (4 - 2) - 2 > n (Δ_{1}) .

Finally, if

k = 2,

then by [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

, Lemma 1.3.2],

i (Δ_{j}) = 1, j = 1, 2,

hence

(n (Δ_{1}) - 2) + (n (Δ_{2}) - 2) = n (Δ_{1}) + [(n (Δ_{2}) - 2) - 2] \geq n (Δ_{1}),

that is, due to (4.3),

{∥ \partial 𝕄 ∥}_{t} \geq n (Δ_{1}),

as required.

Proof of Theorem C.

We have to show that either

σ (ω_{1}) = 1

or

σ (ω_{2}) = 1 .

Let

t \in T (M)

and consider

{\tilde{𝕄}}^{t} .

By [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

, Lemma 1.3.2], either

{\tilde{𝕄}}^{t} = {\tilde{Δ}}

or

| 𝒟 ({\tilde{𝕄}}^{t}) | \geq 2 .

If

| 𝒟 ({\tilde{𝕄}}^{t}) | \geq 3,

then by the Pigeon Hole Principle (P.H.P), either

| 𝒟 ({\tilde{ω}}_{1}) | \neq \emptyset

or

| 𝒟 ({\tilde{ω}}_{2}) | \neq \emptyset .

Assume

| 𝒟 ({\tilde{ω}}_{2}) | = \emptyset .

Then

| 𝒟 ({\tilde{ω}}_{1}) | \neq \emptyset,

hence

σ (ω_{1}) = 1,

by Theorem L, as required. If

| 𝒟 ({\tilde{𝕄}}^{t}) | = 2,

then by [6

A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1

, Lemma 1.3.2],

{\tilde{𝕄}}^{t} \in O . L .

Hence, again, by Theorem L,

σ (ω_{1}) = 1 .

Finally, if

{\tilde{𝕄}}^{t} = {\tilde{Δ}},

then by Lemma 4.5.3,

σ (ω_{1}) = 1 .

Acknowledgements

I am very grateful to the referee for his important remarks on a previous version of the work, in particular for pointing out several omissions.

References

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[3] R. A. Fenn, Techniques of geometric topology. London Math. Soc. Lecture Note Ser. 57, Cambridge University Press, Cambridge, 1983, 280 pp. Zbl 0517.57001 MR 0787801
[4] D. F. Holt and S. Rees, Artin groups of large type are shortlex automatic with regular geodesics. Proc. Lond. Math. Soc. (3) 104 (2012), no. 3, 486–512 Zbl 1275.20034 MR 2900234
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[6] A. Juhasz, On the isoperimetric functions of a class of Artin groups. 2025 arXiv:2507.16770v1
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Cite this article

Arye Juhász, On the isoperimetric functions of a class of Artin groups. Groups Geom. Dyn. 20 (2026), no. 3, pp. 771–796

DOI 10.4171/GGD/950

Abstract

1. Introduction

Examples 1.

Corollary.

Proposition 1.

Theorem 1.

Definition 1

Theorem A.

Theorem B.

Theorem C

2. The diagrams 𝕄, 𝕄t and 𝕄~t,t∈T(M)

2.1. The diagrams 𝕄 and 𝕄t,t∈T(M)

Definition 2.1.1

Definition 2.1.2

Definition 2.1.3

Definition 2.1.4

Definition 2.1.5

Definition 2.1.6

Proposition 2.1.7.

Proposition 2.1.8.

Remark 2.1.9.

3. Extended presentations, ℐ‑moves and banded diagrams

3.1. Identities among relations

Definition 3.1.1

Definition 3.1.2

Example 3.1.3.

Example 3.1.4.

3.2. Banded subdiagrams and transfer of regions

Lemma 3.2.1.

Proof.

Definition 3.2.2

Definition 3.2.3

Definition 3.2.4

Definition 3.2.5

Proposition 3.2.6.

4. Lifting 𝒟(𝕄~t) to 𝕄t,t∈T(M), and the proofs of the main results

4.1. Preliminary results

4.1.1. The mappings Ψt and Ψt−1,t∈T(M)

Lemma 4.1.2.

Proof.

Proposition 4.1.3.

Proof.

4.2. The subdiagrams Opt(Δ),Optr(Δ) and Optl(Δ), Δ=Ψt−1(Δ~), Δ~∈𝒟(ω~i),i=1,2

Definition 4.2.1

Lemma 4.2.2.

Proof of Lemma 4.2.2.

4.3. Definitions and assumptions

Definitions 4.3.1.

Assumptions 4.3.2.

Theorem 4.3.3

Remarks

Theorem 4.3.4

Theorem 4.3.5

4.4. The proof of Theorem A

Lemma 4.4.1.

Proof.

Lemma 4.4.2.

Proof.

4.4.3. Proof of Theorem A

4.5. Preparatory results for the proofs of Theorems S, L, T and B

Lemma 4.5.1.

Proof.

Lemma 4.5.2.

Proof.

Remark.

Lemma 4.5.3.

Proof.

Lemma 4.5.4.

Proof.

Corollary.

Proof.

4.6. Proof of Theorem S

4.7. Proof of Theorem L

4.8. Proof of Theorem T

4.9. Proofs of Theorems B and C

Proof of Theorem B.

Case 1.

Case 2.

Proof of Theorem C.

Acknowledgements

2. The diagrams $𝕄,$ $𝕄_{t}$ and ${\tilde{𝕄}}^{t}, t \in T (M)$

2.1. The diagrams $𝕄$ and $𝕄_{t}, t \in T (M)$

3. Extended presentations, $ℐ ‑$ moves and banded diagrams

4. Lifting $𝒟 ({\tilde{𝕄}}^{t})$ to $𝕄_{t}, t \in T (M),$ and the proofs of the main results

4.1.1. The mappings $Ψ_{t}$ and $Ψ_{t}^{- 1}, t \in T (M)$

4.2. The subdiagrams $Opt (Δ), {Opt}_{r} (Δ)$ and ${Opt}_{l} (Δ),$ $Δ = Ψ_{t}^{- 1} (\tilde{Δ}),$ $\tilde{Δ} \in 𝒟 ({\tilde{ω}}_{i}), i = 1, 2$

Case $1 .$

Case $2 .$