Irreducible generating tuples of Fuchsian groups Ederson Dutraand Richard Weidmann October 10 2022

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Irreducible generating tuples of Fuchsian groups

Ederson Dutra∗and Richard Weidmann

October 10, 2022

Abstract

L. Louder showed in [Lou] that any generating tuple of a surface group is Nielsen

equivalent to a stabilized standard generating tuple i.e. (a1,...,ak,1. . . , 1) where

(a1,...,ak) is the standard generating tuple. This implies in particular that irre-

ducible generating tuples, i.e. tuples that are not Nielsen equivalent to a tuple of the

form (g1,...,gk,1), are minimal. In [Dut] the ﬁrst author generalized Louder’s ideas

and showed that all irreducible and non-standard generating tuples of suﬃciently large

Fuchsian groups can be represented by so-called almost orbifold covers endowed with

a rigid generating tuple.

In the present paper a variation of the ideas from [W2] is used to show that this

almost orbifold cover with a rigid generating tuple is unique up to the appropriate

equivalence. It is moreover shown that any such generating tuple is irreducible. This

provides a way to exhibit many Nielsen classes of non-minimal irreducible generating

tuples for Fuchsian groups.

As an application we show that generating tuples of fundamental groups of Haken

Seifert manifolds corresponding to irreducible horizontal Heegaard splittings are irre-

ducible.

1 Introduction

Studying Nielsen equivalence classes of generating tuples of surface groups and more gen-

erally Fuchsian groups has a long history, starting with the work of Zieschang [Zie] on

fundamental groups of orientable surfaces. He showed that any minimal generating tuple

of a surface group is Nielsen equivalent to the standard generating tuple. Variations of the

cancellation methods developed by Zieschang were then successfully employed by Rosen-

berger to study Nielsen classes of minimal generating tuples for many Fuchsian groups

[R1, R2, R3]. Nielsen classes of minimal generating tuples were then studied by Lustig and

Moriah using innovative algebraic ideas [L, LM1, LM2]; recently this lead to a classiﬁcation

in all but a few exceptional cases [LM3].

Non-minimal generating tuples where ﬁrst studied in the groundbreaking work of Louder

[Lou] who proved that any generating tuple of a surface group is Nielsen equivalent to a

stabilized standard tuple, i.e. a tuple of the form (a1, . . . , ak,1. . . , 1) where (a1, . . . , ak) is

the standard generating tuple. In particular any two generating tuples of the same size

are Nielsen equivalent, thus a true analogue of Nielsen’s theorem for free groups holds for

surface groups. Louder’s proof can be thought of as a folding argument in an appropriate

category of square complexes and he shows that any square complex representing a gen-

erating tuple can be folded and unfolded onto a square complex representing a stabilized

standard generating tuple.

In the case of Fuchsian groups the situation is more subtle and more interesting. In

[Dut] the ﬁrst author generalized the ideas of Louder to the context of suﬃciently large

Fuchsian groups, i.e. Fuchsian groups that are not triangle groups. Dutra proved that any

∗The ﬁrst author was supported by FAPESP, S˜ao Paulo Research Foundation, grants 2018/08187-6 and

2021/12276-7.

arXiv:2210.03611v1 [math.GT] 7 Oct 2022

non-standard irreducible generating tuple can be represented by a so called almost orbifold

cover with a rigid generating tuple, in the case that all elliptic elements are of order 2 this

implies a direct generalization of Louder’s result. Recall that a tuple is reducible if it is

equivalent to a tuple of the form (g1, . . . , gk,1) and irreducible otherwise. Clearly minimal

generating tuples are irreducible, the converse does not hold in general.

Almost orbifold covers are branched maps that are close to being orbifold covers.

Deﬁnition 1.1. Let Obe a closed cone-type 2-orbifold and O0a compact cone-type 2-

orbifold with a single boundary component. A map η:O0→ O is called an almost orbifold

cover if exists a point x∈ O and a closed disk D⊂ O containing xsuch that D\ {x}

contains no cone points such the following hold:

1. η−1(D) = D1∪. . . ∪Dm∪Swhere Diis a disk for all 1 ≤i≤mand S=∂O0.

2. η|O0\η−1(D):O0\η−1(D)→ O \ Dis an orbifold cover.

3. η|D1∪...∪Dm:D1∪. . . ∪Dm→Dis an orbifold cover.

4. η|S:S→∂D is a cover.

The degree of ηis deﬁned as the degree of the cover η|O0\η−1(D). We say that ηis a

special almost orbifold cover if deg(η|S)< m and deg(η|S) does not divide m, where mis

the order of x.

Remark 1.2. Throughout this paper we are mostly interested in the case where the point

x∈ O is a cone point. This is particular always the case if ηis a special almost orbifold

covering.

Remark 1.3. It is easily veriﬁed that ηcan be extended to an orbifold cover by gluing in

a disk with cone point of order m

deg(η|S)iﬀ deg(η|S) divides m.

Example 1.4. Let O=T2(15,14) and O0=F(15,14,7) where Fis a once punctured

orientable surface of genus two. Consider the map η:O0→ O described in Fig. 1 where

the eﬀect of ηon the component containing the cone point of order 7 is described in Fig. 2.

Let x∈ O be the cone point of order 15 and Dbe the disk depicted in Fig. 1. Then

η−1(D) = D1∪Sand ηdeﬁnes an orbifold cover O0\D1∪S→ O \ Dof degree three.

Thus ηis an almost orbifold cover of degree 3. As η|S:S→∂D is of degree two we

conclude that ηis special.

−→

x∈D

Figure 1: ηis special of degree 3.

Deﬁnition 1.5. A marking of Ois a pair (η:O0→ O,[T0]) where ηis an almost orbifold

cover and [T0] is a Nielsen equivalence class of generating tuples of πo

1(O0). We say that

the Nielsen class [η∗(T0)] is represented by the marking (η:O0→ O,[T0]).

We say that a generating tuple Tof the fundamental group of a compact cone-type

2-orbifold with q≥1 boundary components is rigid if Tis not Nielsen-equivalent to a tuple

(g1,...gl, γ1, . . . , γq) where γ1, . . . , γqcorrespond to the boundary components.

−→

Figure 2: A half turn rotation.

Deﬁnition 1.6. Let (η:O0→ O,[T0]) be a marking of the orbifold O.

1. We say that (η:O0→ O,[T0]) is standard if the following hold:

(a) ηhas degree one.

(b) if Ois not a surface, then the exceptional point of ηis of order ≥2.

2. We say that (η:O0→ O,[T0]) is special if ηis special and [T0] consists of rigid

generating tuples of πo

1(O0).

Deﬁnition 1.7. We say that a generating tuple Tof πo

1(O) is standard if there is a

standard marking (η:O0→ O,[T0]) of Osuch that [η∗(T0)] = [T].

In [Dut] it is shown that any non-standard irreducible generating tuple of a suﬃciently

large 2-orbifold is represented by a special marking. The main purpose of this paper is to

establish the uniqueness of this marking and the fact that generating tuples represented

by special markings are irreducible:

Theorem 1.8. Let Obe a suﬃciently large cone type 2-orbifold and let Tbe a non-

standard irreducible generating tuple of πo

1(O). Then there is a unique special marking

(η:O0→ O,[T0]) such that [T] = [η∗(T0)].

Theorem 1.9. Let Obe a suﬃciently large cone type 2-orbifold. If

(η:O0→ O,[T0])

is a special marking such that η0

∗is surjective, then η∗(T0)is irreducible.

Example 1.10. Let η:O0→ O be the special almost orbifold cover given in Example 1.4.

A presentation for πo

1(O) is given by

ha1, b1, s1, s2|s15

1, s14

2, s1s2= [a1, b1]i.

Consider the generating tuple

T0= (σ1, σ2, α1, α2, β1, γ1, σ3)

of πo

1(O0) as described in Fig. 3. Then

η∗(T0)=(s1, s2, a1, b−1

1a1b1, b3

1, b1s1b1, b1s2

2b−1

1).

Using the relation s1= [a1, b1]s−1

2in πo

1(O) we can rewrite η∗(T0) as

(a1b1a−1

1b−1

1s−1

2, s2, a1, b−1

1a1b1, b3

1, b1a1b1a−1

1b−1

1s−1

2b1, b1s2

2b−1

1).

Thus η∗(T0) is Nielsen equivalent to

(b1a−1

1b−1

1, s2, a1, b−1

1a1b1, b3

1, b1, b1s2

2b−1

σ1

α1

α2

β1

σ3

γ1

σ2

Figure 3: The tuple T0.

which can be shown to be Nielsen equivalent to (a1, b1, s2,1,1,1,1). Therefore, η∗(T0) is

reducible. This shows that the restriction to rigid generating tuples of πo

1(O0) is necessary

to guarantee that the corresponding tuple in πo

1(O) is irreducible. On the other hand,

according to Lemma 3.2,

T0:= (σ2

1, σ3

2, α1, α2, β1, γ1, σ2

is a rigid generating tuple of πo

1(O0). The previous Theorem therefore implies that

η∗(T0)=(s2

1, s3

2, a1, b−1

1a1b1, b3

1, b1s1b1, b1s4

2b−1

1).

is irreducible. This is an example of a non-minimal irreducible generating tuple of πo

1(O).

We expect that our approach can also be used to classify standard generating tuples

up to Nielsen equivalence, completing the work of Moriah and Lustig. We plan to address

this question in a future paper. While the underlying ideas we employ should also be able

to study generating tuples of triangle groups, it is clear that new language needs to be

developed to carry out this approach in these remaining cases.

There is an interesting relationship between almost orbifold covers and horizontal Hee-

gaard splittings of Seifert 3-manifolds. If Mis a Seifert 3-manifold and Tis generating

tuple of π1(M) corresponding to a horizontal Heegaard splitting then the image T0of Tin

the fundamental group of the base orbifold is naturally represented by an almost orbifold

cover which is induced by the Heegaard splitting. Theorem 1.9 gives us therefore a way to

establish the irreducibility of T0and therefore of T. We obtain the following:

Theorem 1.11. Let Mbe a Haken orientable Seifert 3-manifold with orientable base space

and Tbe a tuple corresponding to a horizontal Heegaard splitting. Then Tis irreducible if

and only if the Heegaard splitting is irreducible.

Note that we need to exclude the small Seifert manifolds as their base orbifolds are not

suﬃciently large. We actually give a new proof of Theorem 8.1 of [Se] in this setting.

2 Strategy of proof

In this section we brieﬂy sketch the strategy of the proof of the main theorems. At this

point we cannot introduce the subtle notions needed to be precise. Thus this section

remains vague, maybe even pointless.

Nielsen’s theorem states that any two generating m-tuples of the free groups Fnare

Nielsen equivalent. The folding proof of Nielsen’s theorem (in the case m=n) goes as

follows): Identify the free group with π1(Rn) where Rnis the rose with npetals. Any

generating n-tuple Tcan be represented by a tuple (Γ, u0, Y, E, f) where Γ is a graph with

base point u0∈VΓ, Y⊂Γ is a maximal tree, E= (e1, . . . , en) is an orientation of EΓ\EY

and f: Γ →Rnis a π1-surjective morphism. The represented tuple is (f∗(s1), . . . , f∗(sn))

where si∈π1(Γ, u0) is represented by the closed path whose only edge not contained in Y

is ei. Now the graph can be folded onto Rnusing Stallings folds [St] yielding a sequence

Γ=Γ0,Γ1,...,Γk−1,Γk=Rn

such that Γi+1 is obtained from Γiby a single Stalling fold pi. One observes that we can

deﬁne tuples (Γi, ui

0, Yi, Ei, fi) with fi+1 ◦pi=fifor all isuch that all tuples represent

the same Nielsen class. As the initial tuple deﬁnes Tand the terminal tuples deﬁnes the

standard basis, the theorem follows. Note that it is crucial that any folding sequence yields

essentially the same (folded) object. Thus there is only one Nielsen-class and therefore

there is no need to distinguish distinct classes.

A similar argument can be used to prove Grushko’s theorem [Gr] which states that

any generating tuple of a free product A∗Bis Nielsen equivalent to a tuple of the

form (a1, . . . , ak, b1, . . . , bl). This argument relies on folding sequences in the category

of (marked) graphs of groups with trivial edge groups, see [Du] and [BF] where folding

sequences for graphs of groups were studied. In this case one obtains a sequence of graphs

of groups with trivial edge groups where each vertex is marked with a generating tuple of

its vertex group and where the terminal object is the 1-edge graph of groups corresponding

to the free product A∗Band the vertices are marked with generating tuples of (a1, . . . , ak)

of Aand (b1, . . . , bl) of B, respectively. Unlike in the case of free groups, the terminal

object is not unique, however in the case of irreducible generating tuples it was shown in

[W2] that any terminal object that occurs is marked by generating tuples (a0

1, . . . , a0

k) and

(b0

1, . . . , b0

l) that are Nielsen-equivalent to (a1, . . . , ak) and (b1, . . . , bl), respectively. This

implies the strongest possible uniqueness result for the output of Grushko’s theorem. The

simple but somewhat subtle idea underlying the proof in [W2] relies on considering appro-

priate equivalence classes of marked graphs or groups (or more precisely A-graphs) to be

vertices where vertices are connected by edges if one can be obtained by the other by a

fold. The result then follows by observing connectivity of the graph and establishing that

there is unique vertex of minimal complexity.

The proofs of Louder [Lou] and Dutra [Dut] can be thought of as a variation of the

proof of Grushko, but in a diﬀerent category. It is shown that any Nielsen equivalence

class of irreducible generating tuples is represented by a folded object which represents a

standard generating tuple in the case of surfaces (Louder) or is a special marking in the

case of suﬃciently large cone type 2-orbifolds. In the ﬁrst case the uniqueness is then

immediate and in the second case it follows if all cone points are of order 2. The main

objective of this paper is to show that in the second case the folded object is unique up

to the appropriate equivalence. To do so we broadly follow the strategy of [W2], however

both the language developed and the arguments applied are signiﬁcantly more involved.

We will use the language of A-graph as developed in [KMW] and rely on a number

of results of [Dut]. Moreover a certain degree of familiarity with both of these papers is

assumed. Following Louder and Dutra we decompose the fundamental group of the orbifold

Ounder consideration as the fundamental group of a graph of groups Acorresponding to a

decomposition of Oalong essential simple closed curves. The paper has two main chapters,

Chapter 3 and Chapter 4:

1. In Chapter 3 we study generating tuples of fundamental groups of orbifolds with

boundary, those are the groups that occur as vertex groups of A. In Section 3.1 we

collect basic facts about Nielsen classes generating tuples, in particular we charac-

terize rigid generating tuples. In the subsequent sections we study ﬁnitely generated

subgroups of the vertex groups of Awhere some of the generators are assumed to

be peripheral, i.e. conjugate to elements of the adjacent edge groups. We study

these so-called partitioned tuples up to the natural equivalence. As the groups under

consideration are fundamental groups of graphs of groups with trivial edge groups

we can rely on a variation of the arguments from [W2] to prove Proposition 3.11,

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IrreduciblegeneratingtuplesofFuchsiangroupsEdersonDutra*andRichardWeidmannOctober10,2022AbstractL.Loudershowedin[Lou]thatanygeneratingtupleofasurfacegroupisNielsenequivalenttoastabilizedstandardgeneratingtuplei.e.(a1;:::;ak;1:::;1)where(a1;:::;ak)isthestandardgeneratingtuple.Thisimpliesinparticulart...

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Irreducible generating tuples of Fuchsian groups Ederson Dutraand Richard Weidmann October 10 2022

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