Nielsen equivalence in triangle groups

Abstract

We extend the result of \cite{Dutra} to generating pairs of triangle groups, that is, we show that any generating pair of a triangle group is represented by a special almost orbifold covering.

Full text

NIELSEN EQUIVALENCE IN TRIANGLE GROUPS
EDERSON R. F. DUTRA
Abstract. We extend the result of [1] to generating pairs of triangle groups,
that is, we show that any generating pair of a triangle group is represented by
a special almost orbifold covering.
1. Introduction
Let Gbe a group and n>1. Recall that an elementary Nielsen transformation
on a n-tuple T= (g1, . . . , gn)of elements of Gis one of the following three types
of moves:
(T1) For some i∈ {1, . . . , n}replace giby g−1
iin T.
(T2) For some i6=j,i, j ∈ {1, . . . , n}interchange giand gjin T.
(T3) For some i6=j,i, j ∈ {1, . . . , n}replace giby gigjin T.
We say that two n-tuples Tand T0are Nielsen equivalent if there exists a finite
sequence of n-tuples
T=T0, T1, . . . , Tk−1, Tk=T0
such that Tiis obtained from Ti−1by an elementary Nielsen transformation for
16i6k. Nielsen equivalence clearly defines an equivalence relation for tuples of
elements of G.
Nielsen equivalence in Fuchsian groups have been extensively studied by many
authors, see [2, 5, 6, 7, 8, 11, 12, 13, 15] for example. However the techniques
deployed so far are mainly algebraic such as normal form and K-theoretic argu-
ments. In [1] we give a new geometric insight to the problem by showing that
any generating tuple of the fundamental group of a sufficiently large 2-orbifold is
represented by a special almost orbifold covering. We still do not know if the same
holds for non sufficiently large orbifolds. The interesting case are orbifolds with
three cone points whose underlying surface is a sphere. The purpose of this note is
to show that generating pairs of the fundamental group of such orbifolds also have
this geometric description.
Theorem 1.1. Let O=S2(p1, p2, p3)and
G=πo
1(O)∼
=∆(p1, p2, p3) = hs1, s2, s3|sp1
1=sp2
2=sp3
3=s1s2s3= 1i.
Then for any generating pair Tof Gthere is a special almost orbifold covering
η0:O0→Oand a generating pair T0of πo
1(O0)such that η0
∗(T0)and Tare Nielsen
equivalent.
A natural question that arises is whether this holds for arbitrary generating
tuples, that is:
Author supported by FAPESP, São Paulo Research Foundation, grant 2018/08187-6.
1
arXiv:2101.05344v1 [math.GR] 13 Jan 2021

2 EDERSON R. F. DUTRA
Question 1.2. Are all generating tuples of a triangle group also represented by a
(special) almost orbifold coverings?
The proof of Theorem 1.1 relies heavily on the description of all generating
pairs of two-generated Fuchsian groups given by Fine-Knapp-Matelsky-Purzitsky-
Rosenberger [2, 4, 9, 11, 13].
2. Orbifolds and almost orbifold coverings
In this section we give a quick review about orbifolds, orbifold fundamental
groups and orbifold coverings. More details about orbifolds can be found in the
beautiful article of P. Scott [14].
A2-orbifold Ois a pair (F, p)where Fis a connected surface, called the under-
lying surface of O, and p:F→Nis a function such that
Σ(O) := {x∈F|p(x)>2}
is finite and contained in the interior of F. The number p(x)is called the order of
xand the points of Σ(O)are called cone points. An orbifold is said to be compact
(resp. closed) if its underlying surface is compact (resp. closed).
A compact orbifold O= (F, p)with Σ(O) = {x1, . . . , xr}will also be denoted by
F(p1, . . . , pr)where p(xi) = pifor i= 1, . . . , r.
The fundamental group πo
1(O)of O= (F, p)is defined in terms of the fundamental
group of the underlying surface Fas follows. Let x1, x2, . . . be the cone points of
Oand for each i= 1,2, . . . let Di⊆Int(F)be a disk centered at xisuch that
Di∩Dj=∅for i6=j. Set
SO=F− ∪
i=1,2,...Int(Di).
Then πo
1(O)is the group obtained from π1(SO)by adding the relations spi
i= 1 for
all i= 1,2, . . ., where siis the homotopy class represented by ∂Diand piis the order
of xi. For example the fundamental group of O=S2(p1, p2, p3)has presentation
πo
1(O) = hs1, s2, s3|sp1
1=sp2
2=sp3
3=s1s2s3= 1i.
p1
p2
p3
s1
s2
s3
Figure 1. The generator siis represented by a simple closed curve
that goes once around the cone point xi.
In order to define almost orbifold coverings we first recall the definition of an
orbifold covering. Let O= (F, p)and O0= (F0, p0)be orbifolds. An orbifold
covering η:O0→Ois a continuous surjective map η:F0→Fwith the following
properties:
(1) For each point y∈F0, the order of ydivides the order of η(y).

NIELSEN EQUIVALENCE IN TRIANGLE GROUPS 3
(2) For each point x∈Int(F)the set η−1(x)⊆F0is discrete and, over a small
disk in Fcentered at x, the map ηis equivalent to the map
(z, y)∈D2×η−1(x)7→ e(2πp0(y)
p(x)i)z∈D2
and xcorresponds to 0in D2.
Note that the map η|F0−η−1(Σ(O)) :F0−η−1(Σ(O)) →F−Σ(O)is a genuine
covering. The degree of η:O0→Ois defined as the degree of η|F0−η−1(Σ(O)) . It
is not hard to see that an orbifold covering η:O0→Oinduces a monomorphism
η∗:πo
1(O0)→πo
1(O).Conversely, for any subgroup H≤πo
1(O)there is an orbifold
covering η:OH→Osuch that η∗(πo
1(OH)) = H.
Definition (Almost orbifold covering).Let O0= (F0, p0)and O= (F, p)be compact
orbifolds. An almost orbifold covering η:O0→Ois a (non necessarily surjective)
continuous map η:F0→Fhaving the following properties:
(C1) For each y∈F0the order of ydivides the order of η(y).
(C2) There is a point x∈Int(F)of order p>1, called the exceptional point, and
a disk D⊆Int(F)centered at x∈F, called the exceptional disk, with
(D− {x})∩Σ(O) = ∅
such that ηrestricted to F0−η−1(Int(D)) defines an orbifold covering of
finite degree between the compact 2-orbifolds
Q0:= (F0−η−1(Int(D)), p0|F0−η−1(Int(D)))⊆O0
and
Q:= (F−Int(D), p|F−Int(D))⊆O.
(C3) η−1(D) = D1tD2t. . . tDttC1t. . . tCu, where t>0,u>1and
(C3.a) C1, . . . , Cu⊆∂F 0are boundary components of O0, called the excep-
tional boundary components of O0.
(C3.b) Each Dj⊆Int(F0)(16j6t) is a disk and η|Dj:Dj→Dis
equivalent to the map
z∈D27−→ e2πp
qiz∈D2
and x∈Dcorresponds to 0in D2, where qis the order of the point
η−1(x)∩Dj.
The degree of η:O0→Ois defined as the degree of the orbifold covering
η|Q0:Q0→Q.
We further call an almost orbifold covering η:O0→Ospecial if u= 1 (i.e. Q0has a
single exceptional boundary component) and the degree of the map η|C1:C1→∂D
is at most the order pof the exceptional point x.
Example 2.1. Let O= (F, p)be a closed orbifold and xan arbitrary point of F.
Let further D⊆Fbe a small disk centered at xsuch that Dcontains no cone
points except possibly x. Denote by O0the orbifold (F−Int(D), p|F−Int(D))and
by jthe inclusion map F−Int(D)→F.
Any orbifold covering η0:O0→O0of finite degree induces an almost orbifold
covering of O, namely the map η:= j◦η0:F0→F. Note that the exceptional point
is x, the exceptional disk is Dand
η−1(D) = ∂O0=C1t. . . tCu

4 EDERSON R. F. DUTRA
are the exceptional boundary components of η. When the degree of η0is one, an
hence the degree of ηis also one, then O0=O0. In this case we say that ηis trivial.
3. Proof of Theorem 1.1
Throughout this section Owill denote the orbifold S2(p1, p2, p3)and Gits fun-
damental group. In [9] (see also Knapp [4] for the particular case of generating
pairs consisting of elliptic elements and [2] for a more general discussion) Matelski
implicitly shows that if Tis a generating pair of G(the arguments in [9] work also
for non-hyperbolic triangle groups) then Tis equivalent to T0where one of the
following holds:
(1) T0= (sν1
i1, sν2
i2)with 16i1< i263and (pij, νj)=1for j= 1,2.
(2) p1= 2,p3>3odd and T0= (sν
2, s1sν0
2s−1
1) = (sν
2, s1sν0
2s1)with νand ν0
coprime with p2.
(3) p1= 2,p2= 3 and p3>3is odd, and T0= (s1, s−1
2·sν
3·s2)with (ν, p3) = 1.
(4) p1= 2,p2= 3 and p3>4coprime with 3, and
T0= (s1s2s1·sν
3·(s1s2s1)−1, s2)
with νcoprime with p3.
(5) p1= 2,p2= 3 and p3>5coprime with 4, and
T0= ((s2s1)2·sν
3·(s2s1)−2, sν0
3)
with νand ν0coprime with p3.
(6) (p1, p2, p3) = (2,3,5), and one of the following holds:
(6.a) T0= (s1,(s2
2s1)2·s2·(s2
2s1)−2).
(6.b) T0= (s2·s1·s−1
2,(s1s2)2s1·sν
3·s−1
1(s1s2)−1)with ν∈ {1,2}.
(6.c) T0= (s2s1·s2·(s2s1)−1, s3
3s1·sν
3·(s3
3s1)−1)with ν∈ {1,2}.
(6.d) T0= (s2s1·s2·(s2s1)−1,(s1s2)2s1·sν
3·s−1
1(s1s2)−1).
(7) (p1, p2, p3) = (2,3,7), and one of the following holds:
(7.a) T0= (s2, s−3
3s1·sν
3·s1s3
3)with νcoprime with 7
(7.b) T0= ((s1s2
2s1s2)2s2s1s2,(s2(s2s1)2)2s2s1).
(8) p1=p2= 3 and p3>4coprime with 3, and
T0= (s1s2
2, s2
2s1).
(9) p1= 2,p2= 4 and p3>7odd, and
T0= (s1s2
2, s3
2s1s3
2).
(10) p1= 2,p2= 3 and p3>7coprime with 6, and
T0= (s1s2s1s2
2, s2
2s1s2s1).
Note that the list (1)-(10) above does not give a classification of generating pairs,
that is, pairs from distinct items might be Nielsen equivalent. For example the
generating pair T0= (sν
2, s1sν0
2s−1
1)given in (2) with ν=ν0= 1 is Nielsen equivalent
to the standard pair (s1, s2
3). For a complete classification of generating pairs of
triangle groups see [2].
For each generating pair given in (1)-(10) we will describe a special almost orb-
ifold covering that represents T. First note that standard generating tuples, that
is, those given in (1), are represented by trivial almost orbifold coverings, see Ex-
ample 2.1 and [1, Example 1.9]. Thus we are left with the generating pairs given
in items (2)-(10).

NIELSEN EQUIVALENCE IN TRIANGLE GROUPS 5
Let D⊆S2be a disk centered at the cone point x3that does not contain x1
and x2as shown in Fig. 2. Denote the orbifold obtained from Oby removing the
interior of Dby O0. Then O0is a disk D0:= S2−Int(D)with two cone points
x1and x2of order p1and p2respectively. Consequently πo
1(O0)has the following
presentation
hS1, S2|Sp1
1=Sp2
2= 1i.
The inclusion map i:D0→S2induces a surjective homomorphism
i∗:πo
1(O0)πo
1(O)
that sends Sionto si(i= 1,2) and S1S2onto s−1
3.
Let further αand βbe arcs in S2with ∂α ={x1, x3}and ∂β ={x2, x3}as
shown in Fig. 2. Let further α0and β0be the arcs D0∩αand D0∩βin D0. We
also label the cone point x1(of order p1) by a small square an the cone point x2
(of order p2) by a small disk. The arcs α,α0,βand β0and the labels of the points
x1and x2will help us to visualize the defined orbifold maps.
x1
x2
x3
D
α
β
p1
p2
Figure 2. The arcs αand βin S2and the orbifold O0=
D0(p1, p2)⊆O.
In all cases we will describe an almost orbifold covering O0→Owhere the
orbifold O0is either a disk with two cone points or a torus with no cone points that
has an open disk removed. Moreover, in all cases x3will be the exceptional point
and Dthe exceptional disk.
Case (2). Recall that p1= 2 and that p3>3is odd. The pair T0is equal to
(sν
2, s1sν0
2s−1
1)with (ν, p2) = (ν0, p2)=1.
Consider the surjecitve map η0:D2D0described in Fig. 3, that is, η0(α1) =
η0(α2) = α0,η0(β1) = η0(β2) = β0and η0maps each component of
D2−α1∪α2∪β1∪β2
homeomorphically onto D0−α0∪β0.
Denote by O0the orbifold with underlying surface D2and with cone points y1
and y2both of order p2. Thus O0=D2(p2, p2). Then it is not hard to see that
η0defines an orbifold covering of degree 2from O0onto O0⊆Othat corresponds
to the subgroup hS2, S1S2S−1
1iof πo
1(O0). As η0
∗:πo
1(O0)→πo
1(O0)is 1-1 we may
assume that
πo
1(O0) = hS2, S1S2S−1
1i.
It follows from Example 2.1 that the map η:= i◦η0:D2→S2defines an almost
orbifold covering from O0to O. As the degree of η|C:C→∂D is equal to two,
which is strictly smaller than p3, we conclude that ηis special. Moreover,
η∗(Sν
2) = i∗(Sν
2) = sν
2and η∗(S1Sν0
2S−1
1) = i∗(S1Sν0
2S−1
1) = s1sν0
2s−1
1
which means that ηrepresents the generating pair T0.

6 EDERSON R. F. DUTRA
y2
y1
β1
β2
α1
α2
η0
→
x3
D
,→
α0
β0
α
β
i
x1
x2
x1
x2
Figure 3. η=η0◦i:D2→S2defines a special almost orbifold
covering of degree 2.
Case (3). Recall that in this case we have p1= 2,p2= 3 and p3>3odd, and
T0= (s1, s−1
2sν
3s2)with (ν, p3)=1. Let η:D2→S2be the map described in
Fig. 4, that is, η(α1) = η(α2) = α0,η(α3) = α,η(β1) = η(β2) = β0,η(β3) = βand
ηmaps each component of
D2−Int(D1)∪α1∪α2α3∪β1∪β2∪β3
homeomorphically onto D0−α0∪β0and the disk D1homeomorphically onto D.
Denote by O0the orifold with underlying surface D2and with two cone pints y1
and y2of order p1= 2 and p3respectively, i.e. O0=D2(p1, p3). Then ηdefines an
almost orbifold covering of degree 3from O0to O.
By the description of ηwe see that η−1(D) = D1tC. The restriction of ηto
the disk D1defines an orbifold covering of degree 1from the orbifold Q0=D1(p2)
(a disk with a single cone point, y2, order p3) onto Q=D(p3)(a disk with a single
cone point, x3, of order p3). Moreover, the exceptional boundary component C
is mapped by ηonto ∂D with degree 2which implies that ηis a special almost
orbifold covering.
η
C
D1
y1
y2
x1
x2
x3
D
→
α
β
α1
α3
α2
β1
β2
β3
Figure 4. Case (3).η:D2→S2defines a special almost orbifold
covering of degree 3.
Put F:= D2−Int(D1). It is easy to see that the map
η0:= η|F:F:= D2−Int(D1)D0
defines an orbifold covering of degree 3from the orbifold O0
0=F(p1)(an annulus
with a single cone point y1of order p1= 2) onto the orbifold O0⊆Oand that η0
corresponds to the subgroup hS1, S−1
2·S1S2·S2iof πo
1(O0). Thus we may assume
that
πo
1(O0
0) = hS1, S−1
2·S1S2·S2i.
The inclusion map j:F →D2induces a surjective homomorphism
j∗:πo
1(O0
0)→πo
1(O0).

NIELSEN EQUIVALENCE IN TRIANGLE GROUPS 7
Consequently πo
1(O0)is generated by
t1:= j∗(S1)and t2:= j∗(S−1
2·S1S2·S2).
As η∗◦j∗=i∗◦η0
∗and i∗(S1S2) = s3, we see that η∗sends the generating pair
(t1, t−ν
2)of πo
1(O0)onto the pair T0, and hence the special almost orbifold covering
η:O0→Orepresents the generating pair T0.
Case (4). In this case we have p1= 2,p2= 3 and p3>4with (p3,3) = 1, and
T0= (s1s2s1·sν
3·(s1s2s1)−1, s2)
with (ν, p3)=1. Consider the map η:D2→S2described in Fig. 5 (its precise
description is given as in case (3)) and let O0be the orifold with underlying surface
D2and with two cone points y1and y2of order p2= 3 and p3respectively, i.e.
O0=D2(p2, p3). Then ηdefines an almost orbifold covering of degree 4from O0to
O.
C
y1
D1
η
x1
x2
x3
D
→
α
β
y2
Figure 5. Case (4).η:D2→S2defines a special almost orbifold
covering of degree 4.
The description of ηimplies that η−1(D) = D1tC. The restriction of ηto the
disk D1defines an orbifold covering of degree 1from the orbifold Q0=D1(p2)(a
disk with a single cone point y2order p3) onto Q=D(p3)(a disk with a single
cone point x3of order p3). Moreover, the exceptional boundary component Cis
mapped by ηonto ∂D with degree 3, and so ηis a special almost orbifold covering.
The restriction η0of ηto the surface F:= D2−Int(D1)⊆D2defines an orbifold
covering of degree 4from the orbifold O0
0=F(p2)(an annulus with a single cone
point y1of order p2= 3) onto the orbifold O0⊆Osuch that
η0
∗(πo
1(O0
0)) = hS1S2S1·S1S2·S1S2
2S1, S2i.
Thus we may assume that πo
1(O0
0) = hS1S2S1·S1S2·S1S2
2S1, S2i.
Now the inclusion map j:F →D2induces a surjective homomorphism j∗from
πo
1(O0
0)onto πo
1(O0).Consequently πo
1(O0)is generated by
t1:= j∗(S1S2S1·S1S2·S1S2
2S1)and t2:= j∗(S2).
Since η∗◦j∗=i∗◦η0
∗and since i∗(S1S2) = s−1
3, we see that η∗sends the generating
pair (t−ν
1, t2)of πo
1(O0)onto the pair T0= (s1s2s1sν
3(s1s2s1)−1, s2).
Case (5). Recall that p1= 2,p2= 3 and p3>5with (p3,4) = 1, and that T0is
equal to ((s2s1)2·sν
3·(s2s1)−2, sν0
3)with νand ν0coprime with p3.
Consider the map η:D2→S2described in Fig. 6 (again, its precise definition is
given as in case (3)) and let O0be the orifold with underlying surface D2and with
two cone pints y1and y2both of order p3, i.e. O0=D2(p3, p3). Then it is easy to
see that ηdefines an almost orbifold covering of degree 6from O0to O.

8 EDERSON R. F. DUTRA
D2
C
y2
η
x1
x2
x3
D
→
α
β
y1
D1
Figure 6. Case (5) η:D2→S2defines a special almost orbifold
covering of degree 6.
Put F:= D2−Int(D1)∪Int(D2). It is not hard to see that the map
η0:= η|F:F→D0
defines an orbifold covering of degree 6from the orbifold O0
0=F(a pair of pants
without cone points) onto the orbifold O0⊆O. Moreover,
η0
∗(πo
1(O0
0)) = h(S2S1)2·S2
2S1·(S2S1)−2, S2
2S1)i.
Thus we can assume that πo
1(O0
0) = h(S2S1)2·S2
2S1·(S2S1)−2, S2
2S1)i.
The inclusion map j:F →D2induces a surjective homomorphism j∗from
πo
1(O0
0)onto πo
1(O0). Consequently πo
1(O0)is generated by
t1:= j∗((S2S1)2·S2
2S1·(S2S1)−2)and t2:= j∗(S2
2S1).
As η∗◦j∗=i∗◦η0
∗and since i∗maps the element (S2
2S1)onto s3, we see that the
pair (t−ν
1, t2)is mapped by η∗onto the pair T0.
Case (6). (p1, p2, p3) = (2,3,5). It is not hard to see that all generating pairs
given in (6.a)-(6.d) are equivalent to those given in (1)-(2). For example, suppose
that (6.b) holds, that is, T0= (s2s1s−1
2,(s1s2)2s1·sν
3·s1(s1s2)−2)with ν∈ {1,2}.
If ν= 1 then we have
T0∼NE (s1, s2
3)
which is a standard generating pair, and if ν= 2 then we have
T0∼NE (s2
3, s1s2
3s−1
1)
which is case (2.b) with p2= 3 and p3= 5. We point out that for all cases (6.a)-
(6.d) it is also possible to construct a special almost orbifold covering η:O0→O
and find a generating pair TO0of πo
1(O0)that is mapped by η∗onto T0. If we take
case (6.b) for example, then the almost orbifold covering ηdescribed in Figure 7
represents τ0.
Case (7). (p1, p2, p3) = (2,3,7). We start with the generating par given in (7.a),
that is, we define a special almost orbifold covering that represents
T0= (s2, s−3
3s1·sν
3·s1s3
3)
where (ν, 7) = 1.
Let η:D2→S2be the map described in Fig. 8 and let O0be the orbifold with
underlying surface D2and with two cone pints y1and y2of order p2= 3 and p3= 7
respectively, that is, O0=D2(3,7). Then ηdefines an almost orbifold covering of
degree 10 from O0to O.
The description of ηimplies that η−1(D) = D1tD2tCand that the restriction
of ηto the disk D1defines an orbifold covering of degree 7from the orbifold Q0=D1

NIELSEN EQUIVALENCE IN TRIANGLE GROUPS 9
C
D1
D2
p1
p3
x1
x2
x3
D
α
β
η
→
Figure 7. Case (6.b).ηhas degree 6and O0is equal to
D2(p1, p3).
(a disk without cone points) onto Q=D(p3)(a disk with a single cone point x3of
order p3= 7). Moreover, η|D2:D2→Ddefines a genuine covering (of spaces) of
degree one. The exceptional boundary component Cis mapped by ηonto ∂D with
degree 3which is smaller than pe= 7. Thus ηis a special almost orbifold covering.
x1
x2
x3
D
α
β
C
y2
D1
D2
y1
η
→
Figure 8. Case (7.a).η:D2→S2defines a special almost
orbifold covering of degree 10.
It is not hard to see that the restriction η0of ηto the surface
P:= D2−Int(D1)∪Int(D2)⊆D2
defines an orbifold covering of degree 10 from the orbifold O0
0=P(p2)(a pair
of pants with a single cone point y1of order p2= 3) onto the orbifold O0⊆O.
Moreover, we have η0
∗(πo
1(O0
0)) = hT1, T2, T1T2iwhere T1=S2,
T2=S1S2S1S2S1S2
2·S1S2·S2S1S2
2S1S2
2S1
and T3= (S1S2)7. By the injectivity of η0
∗we can assume that πo
1(O0
0) = hT1, T2, T3i.
Using the relations in Gwe see that i∗◦η0
∗(T1) = s2, that
i∗◦η0
∗(T2) = s1s2s1s2s1s2
2·s1s2·s2s1s2
2s1s2
2s1
=s−3
3s1·s−1
3·s1s3
3
and that i∗◦η0
∗(T3)=(s1s2)7=s−7
3= 1.The inclusion map j:P →D2induces a
surjective homomorphism j∗:πo
1(O0
0)→πo
1(O0)whose kernel is normally generated
by T3. Consequently πo
1(O0)is generated by
t1:= j∗(T1)and t2:= j∗(T2).

10 EDERSON R. F. DUTRA
As we clearly have η∗◦j∗=i∗◦η0
∗we see that η∗sends the generating pair (t1, tν
2)
of πo
1(O0)onto the pair T0.
For the reaming of this paper Fwill denote the surface obtained from a torus by
removing an open open disk and O0will denote the orbifold with underlying surface
Fand without cone points, that is, O0is isomorphic to the surface F.
We now turn our attention to case (7.b). Thus we must find a special almost
orbifold covering that represents the pair
T0= ((s1s2
2s1s2)2s2s1s2,(s2(s2s1)2)2s2s1).
The map η:F→S2described in Fig. 9 defines an almost orbifold covering of
degree 17 from the orbifold O0to O. The description of ηimplies that
η−1(D) = D1tD2tC
and that
η|Di:Di→D
defines an orbifold covering of degree 7from the orbifold Q0
i=Di(a disk without
cone points) onto Q=D(p3)(a disk with a single point of order p3= 7) for
i= 1,2. As η|C:C→∂ D has degree 4we conclude that ηis a special almost
orbifold covering.
η
D1
D2
C
→
x1
x2
x3
D
α
β
Figure 9. Case (7.b).η:F→S2defines a special almost
orbifold covering of degree 17.
Put F0:= F−Int(D1)∪Int(D2). It is not hard to see that the map
η0:= η|F0:F0→D0
defines an orbifold covering of degree 17 from the orbifold O0
0=F0onto the orbifold
O0⊆O. Moreover, we have
η0
∗(πo
1(O0
0)) = hT1, T2, T3, T4, T5i ≤ πo
1(O0)
where
T1= (S1S2
2S1S2)2S2S1S2
T2= (S2(S2S1)2)2S2S1)
T3=S1S2
2·(S1S2)7·S1S2
2
T4= (S1S2)3S2S1·(S1S2)7S1S2(S1S2)3
T5= (S1S2)7

NIELSEN EQUIVALENCE IN TRIANGLE GROUPS 11
By the injectivity of η0
∗we can assume that πo
1(O0
0) = hT1, T2, T3, T4, T5i.Using
the relations in Gwe see that i∗◦η0
∗(T1) = (s1s2
2s1s2)2s2s1s2, that i∗◦η0
∗(T2) =
(s2(s2s1)2)2s2s1)and that
i∗◦η0
∗(T3) = i∗◦η0
∗(T4) = i∗◦η0
∗(T5)=1.
Now the inclusion map j:F0→Finduces a surjective homomorphism
j∗:πo
1(O0
0)→πo
1(O0)
whose kernel is normally generated by {T3, T4, T5}. Consequently πo
1(O0)is gener-
ated by
t1:= j∗(T1)and t2:= j∗(T2).
Since η∗◦j∗=i∗◦η0
∗we see that η∗sends the pair (t1, t2)onto the pair T, and
hence the special almost orbifold covering η:O0→Orepresents T0.
Case (8). Recall that p1=p2= 3 and p3>4with (p3,3) = 1. We must
find a special almost orbifold covering that represents the generating pair T0=
(s1s2
2, s2
2s1).
Let η0:FD0⊆S2be the surjective map described in Fig. 10, that is,
η0(αi) = α0,η0(βi) = βfor 16i63and η0maps each component of
F−α1∪α2∪α3∪β1∪β2∪β3
homeomorphically onto D0−α0∪β0.
η0
→
x1
x2
x3
D
,→
x1
x2
α0
β0
α
β
α1
α2
α3
β1
C
β2
β3
i
Figure 10. Case (8).η=i◦η0:F→S2defines an almost
orbifold covering of degree 3.
It is not hard to see that η0defines an orbifold covering of degree 3from O0onto
O0. Note further that the orbifold covering η0corresponds to the subgroup ht1, t2iof
πo
1(O0)where t1=S1S2
2and t2=S2
2S1. Thus we can assume that πo
1(O0) = ht1, t2i.
Moreover, η0maps the boundary Cof O0onto ∂D with degree 3.
It follows from Example 2.1 that η:= i◦η0:F→S2defines an almost orbifold
covering from O0to O. Since η|Chas degree 3, which is smaller than p3, we conclude
that ηis a special almost orbifold covering. Now the generating pair (t1, t2)is clearly
mapped by η∗=i∗◦η0
∗onto the pair T0.
Case (9). In this case p1= 2,p2= 4 and p3>7odd, and T0= (s1s2
2, s3
2s1s3
2).
Let η0:FD0⊆S2be the surjective map described in Fig. 11. Thus η0defines
an orbifold covering of degree 4from O0onto O0. Moreover,
(η0
n)∗πo
1(O0) = hT1, T2i
where t1=S1S2
2and t2=S3
2S1S3
2, consequently we may assume that πo
1(O0)is
generated by the pair (t1, t2).

12 EDERSON R. F. DUTRA
η0
C
→
x1
x2
x3
D
,→
x1
x2
α0
β0
α
β
i
Figure 11. Case (9).η=i◦η0:F→S2defines a special almost
orbifold covering of degree 4.
It follows from Example 2.1 that η:= i◦η0:F→S2defines an almost orbifold
covering from O0to Owhich is special as the degree of the map η|Cis equal to 4
which is smaller than p3. The generating pair (t1, t2)of πo
1(O0) clearly gets mapped
by η∗=i∗◦η0
∗onto the pair τ0.
Case (10). In this case we have p1= 2,p2= 3 and p3>7with (p3,6) = 1, and
T0is equal to (s1s2s1s2
2, s2
2s1s2s1).
The map η0:FD0described in Fig. 12 defines an orbifold covering of degree
6from O0onto O0such that
η0
∗(πo
1(O0)) = hS1S2S1S2
2, S2
2S1S2S1i.
By applying Example 2.1 we conclude that η:= i◦η0defines a almost orbifold
covering from O0to O. Moreover it is easy to see that
η∗(S1S2S1S2
2, S2
2S1S2S1) = T0
which concludes the proof.
η0
→
C
x1
x2
x3
D
x1
x2
α0
β0
α
β
,→
i
Figure 12. Case (10).η=i◦η0:F→S2defines a special
almost orbifold covering of degree 6.
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Universidade Federal de São Carlos, São Carlos, Brazil
Email address:edersondutra@dm.ufscar.br