# Nielsen equivalence in triangle groups

Ederson R. F. Dutra  Universidade Federal de São Carlos, São Carlos, Brazil
<EMAIL_ADDRESS>
###### 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.

Author supported by FAPESP, São Paulo Research Foundation, grant 2018/08187-6.

##  1\. Introduction

Let $G$ be a group and $n\geqslant 1$. Recall that an _elementary Nielsen
transformation_ on a $n$-tuple $T=(g_{1},\ldots,g_{n})$ of elements of $G$ is
one of the following three types of moves:

  1. (T1)

For some $i\in\\{1,\ldots,n\\}$ replace $g_{i}$ by $g_{i}^{-1}$ in $T$.

  2. (T2)

For some $i\neq j$, $i,j\in\\{1,\ldots,n\\}$ interchange $g_{i}$ and $g_{j}$
in $T$.

  3. (T3)

For some $i\neq j$, $i,j\in\\{1,\ldots,n\\}$ replace $g_{i}$ by $g_{i}g_{j}$
in $T$.

We say that two $n$-tuples $T$ and $T^{\prime}$ are _Nielsen equivalent_ if
there exists a finite sequence of $n$-tuples

$T=T_{0},T_{1},\ldots,T_{k-1},T_{k}=T^{\prime}$

such that $T_{i}$ is obtained from $T_{i-1}$ by an elementary Nielsen
transformation for $1\leqslant i\leqslant k$. 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 arguments. 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 $\mathcal{O}=S^{2}(p_{1},p_{2},p_{3})$ and

$G=\pi_{1}^{o}(\mathcal{O})\cong\Delta(p_{1},p_{2},p_{3})=\langle
s_{1},s_{2},s_{3}\ |\
s_{1}^{p_{1}}=s_{2}^{p_{2}}=s_{3}^{p_{3}}=s_{1}s_{2}s_{3}=1\rangle.$

Then for any generating pair $T$ of $G$ there is a special almost orbifold
covering $\eta^{\prime}:\mathcal{O}^{\prime}\rightarrow\mathcal{O}$ and a
generating pair $T^{\prime}$ of $\pi_{1}^{o}(\mathcal{O}^{\prime})$ such that
$\eta_{\ast}^{\prime}(T^{\prime})$ and $T$ are Nielsen equivalent.

A natural question that arises is whether this holds for arbitrary generating
tuples, that is:

######  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].

A $2$-_orbifold_ $\mathcal{O}$ is a pair $(F,p)$ where $F$ is a connected
surface, called the _underlying surface_ of $\mathcal{O}$, and
$p:F\rightarrow\mathbb{N}$ is a function such that

$\Sigma(\mathcal{O}):=\\{x\in F\ |\ p(x)\geqslant 2\\}$

is finite and contained in the interior of $F$. The number $p(x)$ is called
the _order_ of $x$ and the points of $\Sigma(\mathcal{O})$ are called _cone
points_. An orbifold is said to be _compact_ (resp. _closed_) if its
underlying surface is compact (resp. closed).

A compact orbifold $\mathcal{O}=(F,p)$ with
$\Sigma(\mathcal{O})=\\{x_{1},\ldots,x_{r}\\}$ will also be denoted by
$F(p_{1},\ldots,p_{r})$ where $p(x_{i})=p_{i}$ for $i=1,\ldots,r$.

The _fundamental group_ $\pi_{1}^{o}(\mathcal{O})$ of $\mathcal{O}=(F,p)$ is
defined in terms of the fundamental group of the underlying surface $F$ as
follows. Let $x_{1},x_{2},\ldots$ be the cone points of $\mathcal{O}$ and for
each $i=1,2,\ldots$ let $D_{i}\subseteq\text{Int}(F)$ be a disk centered at
$x_{i}$ such that ${D}_{i}\cap{D}_{j}=\emptyset$ for $i\neq j$. Set

$S_{\mathcal{O}}={F-\underset{i=1,2,\ldots}{\cup}\text{Int}(D_{i})}.$

Then $\pi_{1}^{o}(\mathcal{O})$ is the group obtained from
$\pi_{1}(S_{\mathcal{O}})$ by adding the relations $s_{i}^{p_{i}}=1$ for all
$i=1,2,\ldots$, where $s_{i}$ is the homotopy class represented by $\partial
D_{i}$ and $p_{i}$ is the order of $x_{i}$. For example the fundamental group
of $\mathcal{O}=S^{2}(p_{1},p_{2},p_{3})$ has presentation

$\pi_{1}^{o}(\mathcal{O})=\langle s_{1},s_{2},s_{3}\ |\
s_{1}^{p_{1}}=s_{2}^{p_{2}}=s_{3}^{p_{3}}=s_{1}s_{2}s_{3}=1\rangle.$

Figure 1. The generator $s_{i}$ is represented by a simple closed curve that
goes once around the cone point $x_{i}$.

In order to define almost orbifold coverings we first recall the definition of
an orbifold covering. Let $\mathcal{O}=(F,p)$ and
$\mathcal{O}^{\prime}=(F^{\prime},p^{\prime})$ be orbifolds. An _orbifold
covering_ $\eta:\mathcal{O}^{\prime}\rightarrow\mathcal{O}$ is a continuous
surjective map $\eta:F^{\prime}\rightarrow F$ with the following properties:

  1. (1)

For each point $y\in F^{\prime}$, the order of $y$ divides the order of
$\eta(y)$.

  2. (2)

For each point $x\in\text{Int}(F)$ the set $\eta^{-1}(x)\subseteq F^{\prime}$
is discrete and, over a small disk in $F$ centered at $x$, the map $\eta$ is
equivalent to the map

$(z,y)\in\mathbb{D}^{2}\times\eta^{-1}(x)\mapsto e^{(\frac{2\pi
p^{\prime}(y)}{p(x)}i)}z\in\mathbb{D}^{2}$

and $x$ corresponds to $0$ in $\mathbb{D}^{2}$.

Note that the map
$\eta|_{F^{\prime}-\eta^{-1}(\Sigma(\mathcal{O}))}:F^{\prime}-\eta^{-1}(\Sigma(\mathcal{O}))\rightarrow
F-\Sigma(\mathcal{O})$ is a genuine covering. The _degree_ of
$\eta:\mathcal{O}^{\prime}\rightarrow\mathcal{O}$ is defined as the degree of
$\eta|_{F^{\prime}-\eta^{-1}(\Sigma(\mathcal{O}))}$. It is not hard to see
that an orbifold covering $\eta:\mathcal{O}^{\prime}\rightarrow\mathcal{O}$
induces a monomorphism
$\eta_{\ast}:\pi_{1}^{o}(\mathcal{O}^{\prime})\rightarrow\pi_{1}^{o}(\mathcal{O}).$
Conversely, for any subgroup $H\leq\pi_{1}^{o}(\mathcal{O})$ there is an
orbifold covering $\eta:\mathcal{O}_{H}\rightarrow\mathcal{O}$ such that
$\eta_{\ast}(\pi_{1}^{o}(\mathcal{O}_{H}))=H$.

######  Definition (Almost orbifold covering).

Let $\mathcal{O}^{\prime}=(F^{\prime},p^{\prime})$ and $\mathcal{O}=(F,p)$ be
compact orbifolds. An _almost orbifold covering_
$\eta:\mathcal{O}^{\prime}\rightarrow\mathcal{O}$ is a (non necessarily
surjective) continuous map $\eta:F^{\prime}\rightarrow F$ having the following
properties:

  1. (C1)

For each $y\in F^{\prime}$ the order of $y$ divides the order of $\eta(y)$.

  2. (C2)

There is a point $x\in\text{Int}(F)$ of order $p\geqslant 1$, called the
_exceptional point_ , and a disk ${D}\subseteq\text{Int}(F)$ centered at $x\in
F$, called the _exceptional disk_ , with

$(D-\\{x\\})\cap\Sigma(\mathcal{O})=\emptyset$

such that $\eta$ restricted to $F^{\prime}-\eta^{-1}(\text{Int}(D))$ defines
an orbifold covering of finite degree between the compact $2$-orbifolds

$\mathcal{Q}^{\prime}:=(F^{\prime}-\eta^{-1}(\text{Int}(D)),p^{\prime}|_{F^{\prime}-\eta^{-1}(\text{Int}(D))})\subseteq\mathcal{O}^{\prime}$

and

$\mathcal{Q}:=(F-\text{Int}(D),p|_{F-\text{Int}(D)})\subseteq\mathcal{O}.$

  3. (C3)

$\eta^{-1}(D)=D_{1}\sqcup D_{2}\sqcup\ldots\sqcup D_{t}\sqcup
C_{1}\sqcup\ldots\sqcup C_{u}$, where $t\geqslant 0$, $u\geqslant 1$ and

    1. (C3.a)

$C_{1},\ldots,C_{u}\subseteq\partial F^{\prime}$ are boundary components of
$\mathcal{O}^{\prime}$, called the _exceptional boundary components_ of
$\mathcal{O}^{\prime}$.

    2. (C3.b)

Each $D_{j}\subseteq\text{Int}(F^{\prime})$ ($1\leqslant j\leqslant t$) is a
disk and $\eta|_{D_{j}}:D_{j}\rightarrow D$ is equivalent to the map

$z\in\mathbb{D}^{2}\longmapsto e^{\frac{2\pi p}{q}i}z\in\mathbb{D}^{2}$

and $x\in D$ corresponds to $0$ in $\mathbb{D}^{2}$, where $q$ is the order of
the point $\eta^{-1}(x)\cap D_{j}$.

The degree of $\eta:\mathcal{O}^{\prime}\rightarrow\mathcal{O}$ is defined as
the degree of the orbifold covering

$\eta|_{\mathcal{Q}^{\prime}}:\mathcal{Q}^{\prime}\rightarrow\mathcal{Q}.$

We further call an almost orbifold covering
$\eta:\mathcal{O}^{\prime}\rightarrow\mathcal{O}$ _special_ if $u=1$ (i.e.
$\mathcal{Q}^{\prime}$ has a single exceptional boundary component) and the
degree of the map $\eta|_{C_{1}}:C_{1}\rightarrow\partial D$ is at most the
order $p$ of the exceptional point $x$.

######  Example 2.1.

Let $\mathcal{O}=(F,p)$ be a closed orbifold and $x$ an arbitrary point of
$F$. Let further $D\subseteq F$ be a small disk centered at $x$ such that $D$
contains no cone points except possibly $x$. Denote by $\mathcal{O}_{0}$ the
orbifold $(F-\text{Int}(D),p|_{F-\text{Int}(D)})$ and by $j$ the inclusion map
$F-\text{Int}(D)\hookrightarrow F$.

Any orbifold covering
$\eta^{\prime}:\mathcal{O}^{\prime}\rightarrow\mathcal{O}_{0}$ of finite
degree induces an almost orbifold covering of $\mathcal{O}$, namely the map
$\eta:=j\circ\eta^{\prime}:F^{\prime}\rightarrow F.$ Note that the exceptional
point is $x$, the exceptional disk is $D$ and

$\eta^{-1}(D)=\partial\mathcal{O}^{\prime}=C_{1}\sqcup\ldots\sqcup C_{u}$

are the exceptional boundary components of $\eta$. When the degree of
$\eta^{\prime}$ is one, an hence the degree of $\eta$ is also one, then
$\mathcal{O}^{\prime}=\mathcal{O}_{0}$. In this case we say that $\eta$ is
trivial.

##  3\. Proof of Theorem 1.1

Throughout this section $\mathcal{O}$ will denote the orbifold
$S^{2}(p_{1},p_{2},p_{3})$ and $G$ its fundamental 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 $T$ is a generating pair of $G$ (the arguments in [9] work also for non-
hyperbolic triangle groups) then $T$ is equivalent to $T^{\prime}$ where one
of the following holds:

  1. (1)

$T^{\prime}=(s_{i_{1}}^{\nu_{1}},s_{i_{2}}^{\nu_{2}})$ with $1\leqslant
i_{1}<i_{2}\leqslant 3$ and $(p_{i_{j}},\nu_{j})=1$ for $j=1,2$.

  2. (2)

$p_{1}=2$, $p_{3}\geqslant 3$ odd and
$T^{\prime}=(s_{2}^{\nu},s_{1}s_{2}^{\nu^{\prime}}s_{1}^{-1})=(s_{2}^{\nu},s_{1}s_{2}^{\nu^{\prime}}s_{1})$
with $\nu$ and $\nu^{\prime}$ coprime with $p_{2}$.

  3. (3)

$p_{1}=2$, $p_{2}=3$ and $p_{3}\geqslant 3$ is odd, and
$T^{\prime}=(s_{1},s_{2}^{-1}\cdot s_{3}^{\nu}\cdot s_{2})$ with
$(\nu,p_{3})=1$.

  4. (4)

$p_{1}=2$, $p_{2}=3$ and $p_{3}\geqslant 4$ coprime with $3$, and

$T^{\prime}=(s_{1}s_{2}s_{1}\cdot
s_{3}^{\nu}\cdot(s_{1}s_{2}s_{1})^{-1},s_{2})$

with $\nu$ coprime with $p_{3}$.

  5. (5)

$p_{1}=2$, $p_{2}=3$ and $p_{3}\geqslant 5$ coprime with $4$, and

$T^{\prime}=((s_{2}s_{1})^{2}\cdot
s_{3}^{\nu}\cdot(s_{2}s_{1})^{-2},s_{3}^{\nu^{\prime}})$

with $\nu$ and $\nu^{\prime}$ coprime with $p_{3}$.

  6. (6)

$(p_{1},p_{2},p_{3})=(2,3,5)$, and one of the following holds:

    1. (6.a)

$T^{\prime}=(s_{1},(s_{2}^{2}s_{1})^{2}\cdot
s_{2}\cdot(s_{2}^{2}s_{1})^{-2})$.

    2. (6.b)

$T^{\prime}=(s_{2}\cdot s_{1}\cdot s_{2}^{-1},(s_{1}s_{2})^{2}s_{1}\cdot
s_{3}^{\nu}\cdot s_{1}^{-1}(s_{1}s_{2})^{-1})$ with $\nu\in\\{1,2\\}$.

    3. (6.c)

$T^{\prime}=(s_{2}s_{1}\cdot s_{2}\cdot(s_{2}s_{1})^{-1},s_{3}^{3}s_{1}\cdot
s_{3}^{\nu}\cdot(s_{3}^{3}s_{1})^{-1})$ with $\nu\in\\{1,2\\}$.

    4. (6.d)

$T^{\prime}=(s_{2}s_{1}\cdot
s_{2}\cdot(s_{2}s_{1})^{-1},(s_{1}s_{2})^{2}s_{1}\cdot s_{3}^{\nu}\cdot
s_{1}^{-1}(s_{1}s_{2})^{-1})$.

  7. (7)

$(p_{1},p_{2},p_{3})=(2,3,7)$, and one of the following holds:

    1. (7.a)

$T^{\prime}=(s_{2},s_{3}^{-3}s_{1}\cdot s_{3}^{\nu}\cdot s_{1}s_{3}^{3})$ with
$\nu$ coprime with $7$

    2. (7.b)

$T^{\prime}=((s_{1}s_{2}^{2}s_{1}s_{2})^{2}s_{2}s_{1}s_{2},(s_{2}(s_{2}s_{1})^{2})^{2}s_{2}s_{1})$.

  8. (8)

$p_{1}=p_{2}=3$ and $p_{3}\geqslant 4$ coprime with $3$, and

$T^{\prime}=(s_{1}s_{2}^{2},s_{2}^{2}s_{1}).$

  9. (9)

$p_{1}=2$, $p_{2}=4$ and $p_{3}\geqslant 7$ odd, and

$T^{\prime}=(s_{1}s_{2}^{2},s_{2}^{3}s_{1}s_{2}^{3}).$

  10. (10)

$p_{1}=2$, $p_{2}=3$ and $p_{3}\geqslant 7$ coprime with $6$, and

$T^{\prime}=(s_{1}s_{2}s_{1}s_{2}^{2},s_{2}^{2}s_{1}s_{2}s_{1}).$

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
$T^{\prime}=(s_{2}^{\nu},s_{1}s_{2}^{\nu^{\prime}}s_{1}^{-1})$ given in (2)
with $\nu=\nu^{\prime}=1$ is Nielsen equivalent to the standard pair
$(s_{1},s_{3}^{2})$. 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
orbifold covering that represents $T$. First note that standard generating
tuples, that is, those given in (1), are represented by trivial almost
orbifold coverings, see Example 2.1 and [1, Example 1.9]. Thus we are left
with the generating pairs given in items (2)-(10).

Let $D\subseteq S^{2}$ be a disk centered at the cone point $x_{3}$ that does
not contain $x_{1}$ and $x_{2}$ as shown in Fig. 2. Denote the orbifold
obtained from $\mathcal{O}$ by removing the interior of $D$ by
$\mathcal{O}_{0}$. Then $\mathcal{O}_{0}$ is a disk
$D_{0}:=S^{2}-\text{Int}(D)$ with two cone points $x_{1}$ and $x_{2}$ of order
$p_{1}$ and $p_{2}$ respectively. Consequently $\pi_{1}^{o}(\mathcal{O}_{0})$
has the following presentation

$\langle S_{1},S_{2}\ |\ S_{1}^{p_{1}}=S_{2}^{p_{2}}=1\rangle.$

The inclusion map $i:D_{0}\rightarrow S^{2}$ induces a surjective homomorphism

$i_{\ast}:\pi_{1}^{o}(\mathcal{O}_{0})\twoheadrightarrow\pi_{1}^{o}(\mathcal{O})$

that sends $S_{i}$ onto $s_{i}$ ($i=1,2$) and $S_{1}S_{2}$ onto $s_{3}^{-1}$.

Let further $\alpha$ and $\beta$ be arcs in $S^{2}$ with
$\partial\alpha=\\{x_{1},x_{3}\\}$ and $\partial\beta=\\{x_{2},x_{3}\\}$ as
shown in Fig. 2. Let further $\alpha_{0}$ and $\beta_{0}$ be the arcs
$D_{0}\cap\alpha$ and $D_{0}\cap\beta$ in $D_{0}$. We also label the cone
point $x_{1}$ (of order $p_{1}$) by a small square an the cone point $x_{2}$
(of order $p_{2}$) by a small disk. The arcs $\alpha$, $\alpha_{0}$, $\beta$
and $\beta_{0}$ and the labels of the points $x_{1}$ and $x_{2}$ will help us
to visualize the defined orbifold maps.

Figure 2.  The arcs $\alpha$ and $\beta$ in $S^{2}$ and the orbifold
$\mathcal{O}_{0}=D_{0}(p_{1},p_{2})\subseteq\mathcal{O}$.

In all cases we will describe an almost orbifold covering
$\mathcal{O}^{\prime}\rightarrow\mathcal{O}$ where the orbifold
$\mathcal{O}^{\prime}$ is either a disk with two cone points or a torus with
no cone points that has an open disk removed. Moreover, in all cases $x_{3}$
will be the exceptional point and $D$ the exceptional disk.

Case (2). Recall that $p_{1}=2$ and that $p_{3}\geqslant 3$ is odd. The pair
$T^{\prime}$ is equal to $(s_{2}^{\nu},s_{1}s_{2}^{\nu^{\prime}}s_{1}^{-1})$
with $(\nu,p_{2})=(\nu^{\prime},p_{2})=1$.

Consider the surjecitve map $\eta^{\prime}:D^{2}\twoheadrightarrow D_{0}$
described in Fig. 3, that is,
$\eta^{\prime}(\alpha_{1})=\eta^{\prime}(\alpha_{2})=\alpha_{0}$,
$\eta^{\prime}(\beta_{1})=\eta^{\prime}(\beta_{2})=\beta_{0}$ and
$\eta^{\prime}$ maps each component of

$D^{2}-\alpha_{1}\cup\alpha_{2}\cup\beta_{1}\cup\beta_{2}$

homeomorphically onto $D_{0}-\alpha_{0}\cup\beta_{0}$.

Denote by $\mathcal{O}^{\prime}$ the orbifold with underlying surface $D^{2}$
and with cone points $y_{1}$ and $y_{2}$ both of order $p_{2}$. Thus
$\mathcal{O}^{\prime}=D^{2}(p_{2},p_{2})$. Then it is not hard to see that
$\eta^{\prime}$ defines an orbifold covering of degree $2$ from
$\mathcal{O}^{\prime}$ onto $\mathcal{O}_{0}\subseteq\mathcal{O}$ that
corresponds to the subgroup $\langle S_{2},S_{1}S_{2}S_{1}^{-1}\rangle$ of
$\pi_{1}^{o}(\mathcal{O}_{0})$. As
$\eta_{\ast}^{\prime}:\pi_{1}^{o}(\mathcal{O}^{\prime})\rightarrow\pi_{1}^{o}(\mathcal{O}_{0})$
is 1-1 we may assume that

$\pi_{1}^{o}(\mathcal{O}^{\prime})=\langle S_{2},S_{1}S_{2}S_{1}^{-1}\rangle.$

It follows from Example 2.1 that the map
$\eta:=i\circ\eta^{\prime}:D^{2}\rightarrow S^{2}$ defines an almost orbifold
covering from $\mathcal{O}^{\prime}$ to $\mathcal{O}$. As the degree of
$\eta|_{C}:C\rightarrow\partial D$ is equal to two, which is strictly smaller
than $p_{3}$, we conclude that $\eta$ is special. Moreover,

$\eta_{\ast}(S_{2}^{\nu})=i_{\ast}(S_{2}^{\nu})=s_{2}^{\nu}\ \ \text{ and }\ \
\eta_{\ast}(S_{1}S_{2}^{\nu^{\prime}}S_{1}^{-1})=i_{\ast}(S_{1}S_{2}^{\nu^{\prime}}S_{1}^{-1})=s_{1}s_{2}^{\nu^{\prime}}s_{1}^{-1}$

which means that $\eta$ represents the generating pair $T^{\prime}$.

Figure 3.  $\eta=\eta^{\prime}\circ i:D^{2}\rightarrow S^{2}$ defines a
special almost orbifold covering of degree $2$.

Case (3). Recall that in this case we have $p_{1}=2$, $p_{2}=3$ and
$p_{3}\geqslant 3$ odd, and $T^{\prime}=(s_{1},s_{2}^{-1}s_{3}^{\nu}s_{2})$
with $(\nu,p_{3})=1$. Let $\eta:D^{2}\rightarrow S^{2}$ be the map described
in Fig. 4, that is, $\eta(\alpha_{1})=\eta(\alpha_{2})=\alpha_{0}$,
$\eta(\alpha_{3})=\alpha$, $\eta(\beta_{1})=\eta(\beta_{2})=\beta_{0}$,
$\eta(\beta_{3})=\beta$ and $\eta$ maps each component of

$D^{2}-\text{Int}(D_{1})\cup\alpha_{1}\cup\alpha_{2}\alpha_{3}\cup\beta_{1}\cup\beta_{2}\cup\beta_{3}$

homeomorphically onto $D_{0}-\alpha_{0}\cup\beta_{0}$ and the disk $D_{1}$
homeomorphically onto $D$.

Denote by $\mathcal{O}^{\prime}$ the orifold with underlying surface $D^{2}$
and with two cone pints $y_{1}$ and $y_{2}$ of order $p_{1}=2$ and $p_{3}$
respectively, i.e. $\mathcal{O}^{\prime}=D^{2}(p_{1},p_{3})$. Then $\eta$
defines an almost orbifold covering of degree $3$ from $\mathcal{O}^{\prime}$
to $\mathcal{O}$.

By the description of $\eta$ we see that $\eta^{-1}(D)=D_{1}\sqcup C$. The
restriction of $\eta$ to the disk $D_{1}$ defines an orbifold covering of
degree $1$ from the orbifold $\mathcal{Q}^{\prime}=D_{1}(p_{2})$ (a disk with
a single cone point, $y_{2}$ , order $p_{3}$) onto $\mathcal{Q}=D(p_{3})$ (a
disk with a single cone point, $x_{3}$ , of order $p_{3}$). Moreover, the
exceptional boundary component $C$ is mapped by $\eta$ onto $\partial D$ with
degree $2$ which implies that $\eta$ is a special almost orbifold covering.

Figure 4.  Case (3). $\eta:D^{2}\rightarrow S^{2}$ defines a special almost
orbifold covering of degree $3$.

Put $F:=D^{2}-\text{Int}(D_{1})$. It is easy to see that the map

$\eta^{\prime}:=\eta|_{F}:F:=D^{2}-\text{Int}(D_{1})\twoheadrightarrow D_{0}$

defines an orbifold covering of degree $3$ from the orbifold
$\mathcal{O}_{0}^{\prime}=F(p_{1})$ (an annulus with a single cone point
$y_{1}$ of order $p_{1}=2$) onto the orbifold
$\mathcal{O}_{0}\subseteq\mathcal{O}$ and that $\eta^{\prime}$ corresponds to
the subgroup $\langle S_{1},S_{2}^{-1}\cdot S_{1}S_{2}\cdot S_{2}\rangle$ of
$\pi_{1}^{o}(\mathcal{O}_{0})$. Thus we may assume that

$\pi_{1}^{o}(\mathcal{O}_{0}^{\prime})=\langle S_{1},S_{2}^{-1}\cdot
S_{1}S_{2}\cdot S_{2}\rangle.$

The inclusion map $j:F\hookrightarrow D^{2}$ induces a surjective homomorphism

$j_{\ast}:\pi_{1}^{o}(\mathcal{O}_{0}^{\prime})\rightarrow\pi_{1}^{o}(\mathcal{O}^{\prime}).$

Consequently $\pi_{1}^{o}(\mathcal{O}^{\prime})$ is generated by

$t_{1}:=j_{\ast}(S_{1})\ \ \text{ and }\ \ t_{2}:=j_{\ast}(S_{2}^{-1}\cdot
S_{1}S_{2}\cdot S_{2}).$

As $\eta_{\ast}\circ j_{\ast}=i_{\ast}\circ\eta_{\ast}^{\prime}$ and
$i_{\ast}(S_{1}S_{2})=s_{3}$, we see that $\eta_{\ast}$ sends the generating
pair $(t_{1},t_{2}^{-\nu})$ of $\pi_{1}^{o}(\mathcal{O}^{\prime})$ onto the
pair $T^{\prime}$, and hence the special almost orbifold covering
$\eta:\mathcal{O}^{\prime}\rightarrow\mathcal{O}$ represents the generating
pair $T^{\prime}$.

Case (4). In this case we have $p_{1}=2$, $p_{2}=3$ and $p_{3}\geqslant 4$
with $(p_{3},3)=1$, and

$T^{\prime}=(s_{1}s_{2}s_{1}\cdot
s_{3}^{\nu}\cdot(s_{1}s_{2}s_{1})^{-1},s_{2})$

with $(\nu,p_{3})=1$. Consider the map $\eta:D^{2}\rightarrow S^{2}$ described
in Fig. 5 (its precise description is given as in case (3)) and let
$\mathcal{O}^{\prime}$ be the orifold with underlying surface $D^{2}$ and with
two cone points $y_{1}$ and $y_{2}$ of order $p_{2}=3$ and $p_{3}$
respectively, i.e. $\mathcal{O}^{\prime}=D^{2}(p_{2},p_{3})$. Then $\eta$
defines an almost orbifold covering of degree $4$ from $\mathcal{O}^{\prime}$
to $\mathcal{O}$.

Figure 5.  Case (4). $\eta:D^{2}\rightarrow S^{2}$ defines a special almost
orbifold covering of degree $4$.

The description of $\eta$ implies that $\eta^{-1}(D)=D_{1}\sqcup C$. The
restriction of $\eta$ to the disk $D_{1}$ defines an orbifold covering of
degree $1$ from the orbifold $\mathcal{Q}^{\prime}=D_{1}(p_{2})$ (a disk with
a single cone point $y_{2}$ order $p_{3}$) onto $\mathcal{Q}=D(p_{3})$ (a disk
with a single cone point $x_{3}$ of order $p_{3}$). Moreover, the exceptional
boundary component $C$ is mapped by $\eta$ onto $\partial D$ with degree $3$,
and so $\eta$ is a special almost orbifold covering.

The restriction $\eta^{\prime}$ of $\eta$ to the surface
$F:=D^{2}-\text{Int}(D_{1})\subseteq D^{2}$ defines an orbifold covering of
degree $4$ from the orbifold $\mathcal{O}_{0}^{\prime}=F(p_{2})$ (an annulus
with a single cone point $y_{1}$ of order $p_{2}=3$) onto the orbifold
$\mathcal{O}_{0}\subseteq\mathcal{O}$ such that

$\eta_{\ast}^{\prime}(\pi_{1}^{o}(\mathcal{O}_{0}^{\prime}))=\langle
S_{1}S_{2}S_{1}\cdot S_{1}S_{2}\cdot S_{1}S_{2}^{2}S_{1},S_{2}\rangle.$

Thus we may assume that $\pi_{1}^{o}(\mathcal{O}_{0}^{\prime})=\langle
S_{1}S_{2}S_{1}\cdot S_{1}S_{2}\cdot S_{1}S_{2}^{2}S_{1},S_{2}\rangle.$

Now the inclusion map $j:F\hookrightarrow D^{2}$ induces a surjective
homomorphism $j_{\ast}$ from $\pi_{1}^{o}(\mathcal{O}_{0}^{\prime})$ onto
$\pi_{1}^{o}(\mathcal{O}^{\prime}).$ Consequently
$\pi_{1}^{o}(\mathcal{O}^{\prime})$ is generated by

$t_{1}:=j_{\ast}(S_{1}S_{2}S_{1}\cdot S_{1}S_{2}\cdot S_{1}S_{2}^{2}S_{1})\ \
\text{ and }\ \ t_{2}:=j_{\ast}(S_{2}).$

Since $\eta_{\ast}\circ j_{\ast}=i_{\ast}\circ\eta_{\ast}^{\prime}$ and since
$i_{\ast}(S_{1}S_{2})=s_{3}^{-1}$, we see that $\eta_{\ast}$ sends the
generating pair $(t_{1}^{-\nu},t_{2})$ of $\pi_{1}^{o}(\mathcal{O}^{\prime})$
onto the pair
$T^{\prime}=(s_{1}s_{2}s_{1}s_{3}^{\nu}(s_{1}s_{2}s_{1})^{-1},s_{2})$.

Case (5). Recall that $p_{1}=2$, $p_{2}=3$ and $p_{3}\geqslant 5$ with
$(p_{3},4)=1$, and that $T^{\prime}$ is equal to $((s_{2}s_{1})^{2}\cdot
s_{3}^{\nu}\cdot(s_{2}s_{1})^{-2},s_{3}^{\nu^{\prime}})$ with $\nu$ and
$\nu^{\prime}$ coprime with $p_{3}$.

Consider the map $\eta:D^{2}\rightarrow S^{2}$ described in Fig. 6 (again, its
precise definition is given as in case (3)) and let $\mathcal{O}^{\prime}$ be
the orifold with underlying surface $D^{2}$ and with two cone pints $y_{1}$
and $y_{2}$ both of order $p_{3}$, i.e.
$\mathcal{O}^{\prime}=D^{2}(p_{3},p_{3})$. Then it is easy to see that $\eta$
defines an almost orbifold covering of degree $6$ from $\mathcal{O}^{\prime}$
to $\mathcal{O}$.

Figure 6.  Case (5) $\eta:D^{2}\rightarrow S^{2}$ defines a special almost
orbifold covering of degree $6$.

Put $F:=D^{2}-\text{Int}(D_{1})\cup\text{Int}(D_{2})$. It is not hard to see
that the map

$\eta^{\prime}:=\eta|_{F}:F\rightarrow D_{0}$

defines an orbifold covering of degree $6$ from the orbifold
$\mathcal{O}_{0}^{\prime}=F$ (a pair of pants without cone points) onto the
orbifold $\mathcal{O}_{0}\subseteq\mathcal{O}$. Moreover,

$\eta_{\ast}^{\prime}(\pi_{1}^{o}(\mathcal{O}_{0}^{\prime}))=\langle(S_{2}S_{1})^{2}\cdot
S_{2}^{2}S_{1}\cdot(S_{2}S_{1})^{-2},S_{2}^{2}S_{1})\rangle.$

Thus we can assume that
$\pi_{1}^{o}(\mathcal{O}_{0}^{\prime})=\langle(S_{2}S_{1})^{2}\cdot
S_{2}^{2}S_{1}\cdot(S_{2}S_{1})^{-2},S_{2}^{2}S_{1})\rangle$.

The inclusion map $j:F\hookrightarrow D^{2}$ induces a surjective homomorphism
$j_{\ast}$ from $\pi_{1}^{o}(\mathcal{O}_{0}^{\prime})$ onto
$\pi_{1}^{o}(\mathcal{O}^{\prime})$. Consequently
$\pi_{1}^{o}(\mathcal{O}^{\prime})$ is generated by

$t_{1}:=j_{\ast}((S_{2}S_{1})^{2}\cdot S_{2}^{2}S_{1}\cdot(S_{2}S_{1})^{-2})\
\ \text{ and }\ \ t_{2}:=j_{\ast}(S_{2}^{2}S_{1}).$

As $\eta_{\ast}\circ j_{\ast}=i_{\ast}\circ\eta_{\ast}^{\prime}$ and since
$i_{\ast}$ maps the element $(S_{2}^{2}S_{1})$ onto $s_{3}$, we see that the
pair $(t_{1}^{-\nu},t_{2})$ is mapped by $\eta_{\ast}$ onto the pair
$T^{\prime}$.

Case (6). $(p_{1},p_{2},p_{3})=(2,3,5)$. 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,
$T^{\prime}=(s_{2}s_{1}s_{2}^{-1},(s_{1}s_{2})^{2}s_{1}\cdot s_{3}^{\nu}\cdot
s_{1}(s_{1}s_{2})^{-2})$ with $\nu\in\\{1,2\\}$. If $\nu=1$ then we have

$T^{\prime}\sim_{NE}(s_{1},s_{3}^{2})$

which is a standard generating pair, and if $\nu=2$ then we have

$T^{\prime}\sim_{NE}(s_{3}^{2},s_{1}s_{3}^{2}s_{1}^{-1})$

which is case (2.b) with $p_{2}=3$ and $p_{3}=5$. We point out that for all
cases (6.a)-(6.d) it is also possible to construct a special almost orbifold
covering $\eta:\mathcal{O}^{\prime}\rightarrow\mathcal{O}$ and find a
generating pair $T_{\mathcal{O^{\prime}}}$ of
$\pi_{1}^{o}(\mathcal{O}^{\prime})$ that is mapped by $\eta_{\ast}$ onto
$T^{\prime}$. If we take case (6.b) for example, then the almost orbifold
covering $\eta$ described in Figure 7 represents $\tau^{\prime}$.

Figure 7.  Case (6.b). $\eta$ has degree $6$ and $\mathcal{O}^{\prime}$ is
equal to $D^{2}(p_{1},p_{3})$.

Case (7). $(p_{1},p_{2},p_{3})=(2,3,7)$. We start with the generating par
given in (7.a), that is, we define a special almost orbifold covering that
represents

$T^{\prime}=(s_{2},s_{3}^{-3}s_{1}\cdot s_{3}^{\nu}\cdot s_{1}s_{3}^{3})$

where $(\nu,7)=1$.

Let $\eta:D^{2}\rightarrow S^{2}$ be the map described in Fig. 8 and let
$\mathcal{O}^{\prime}$ be the orbifold with underlying surface $D^{2}$ and
with two cone pints $y_{1}$ and $y_{2}$ of order $p_{2}=3$ and $p_{3}=7$
respectively, that is, $\mathcal{O}^{\prime}=D^{2}(3,7)$. Then $\eta$ defines
an almost orbifold covering of degree $10$ from $\mathcal{O}^{\prime}$ to
$\mathcal{O}$.

The description of $\eta$ implies that $\eta^{-1}(D)=D_{1}\sqcup D_{2}\sqcup
C$ and that the restriction of $\eta$ to the disk $D_{1}$ defines an orbifold
covering of degree $7$ from the orbifold $\mathcal{Q}^{\prime}=D_{1}$ (a disk
without cone points) onto $\mathcal{Q}=D(p_{3})$ (a disk with a single cone
point $x_{3}$ of order $p_{3}=7$). Moreover, $\eta|_{D_{2}}:D_{2}\rightarrow
D$ defines a genuine covering (of spaces) of degree one. The exceptional
boundary component $C$ is mapped by $\eta$ onto $\partial D$ with degree $3$
which is smaller than $p_{e}=7$. Thus $\eta$ is a special almost orbifold
covering.

Figure 8.  Case (7.a). $\eta:D^{2}\rightarrow S^{2}$ defines a special almost
orbifold covering of degree $10$.

It is not hard to see that the restriction $\eta^{\prime}$ of $\eta$ to the
surface

$P:=D^{2}-\text{Int}(D_{1})\cup\text{Int}(D_{2})\subseteq D^{2}$

defines an orbifold covering of degree $10$ from the orbifold
$\mathcal{O}_{0}^{\prime}=P(p_{2})$ (a pair of pants with a single cone point
$y_{1}$ of order $p_{2}=3$) onto the orbifold
$\mathcal{O}_{0}\subseteq\mathcal{O}$. Moreover, we have
$\eta_{\ast}^{\prime}(\pi_{1}^{o}(\mathcal{O}_{0}^{\prime}))=\langle
T_{1},T_{2},T_{1}T_{2}\rangle$ where $T_{1}=S_{2}$,

$T_{2}=S_{1}S_{2}S_{1}S_{2}S_{1}S_{2}^{2}\cdot S_{1}S_{2}\cdot
S_{2}S_{1}S_{2}^{2}S_{1}S_{2}^{2}S_{1}$

and $T_{3}=(S_{1}S_{2})^{7}$. By the injectivity of $\eta_{\ast}^{\prime}$ we
can assume that $\pi_{1}^{o}(\mathcal{O}_{0}^{\prime})=\langle
T_{1},T_{2},T_{3}\rangle.$ Using the relations in $G$ we see that
$i_{\ast}\circ\eta_{\ast}^{\prime}(T_{1})=s_{2}$, that

$\displaystyle i_{\ast}\circ\eta_{\ast}^{\prime}(T_{2})$ $\displaystyle=$
$\displaystyle s_{1}s_{2}s_{1}s_{2}s_{1}s_{2}^{2}\cdot s_{1}s_{2}\cdot
s_{2}s_{1}s_{2}^{2}s_{1}s_{2}^{2}s_{1}$ $\displaystyle=$ $\displaystyle
s_{3}^{-3}s_{1}\cdot s_{3}^{-1}\cdot s_{1}s_{3}^{3}$

and that
$i_{\ast}\circ\eta_{\ast}^{\prime}(T_{3})=(s_{1}s_{2})^{7}=s_{3}^{-7}=1.$ The
inclusion map $j:P\hookrightarrow D^{2}$ induces a surjective homomorphism
$j_{\ast}:\pi_{1}^{o}(\mathcal{O}_{0}^{\prime})\rightarrow\pi_{1}^{o}(\mathcal{O}^{\prime})$
whose kernel is normally generated by $T_{3}$. Consequently
$\pi_{1}^{o}(\mathcal{O}^{\prime})$ is generated by

$t_{1}:=j_{\ast}(T_{1})\ \ \text{ and }\ \ t_{2}:=j_{\ast}(T_{2}).$

As we clearly have $\eta_{\ast}\circ
j_{\ast}=i_{\ast}\circ\eta_{\ast}^{\prime}$ we see that $\eta_{\ast}$ sends
the generating pair $(t_{1},t_{2}^{\nu})$ of
$\pi_{1}^{o}(\mathcal{O}^{\prime})$ onto the pair $T^{\prime}$.

For the reaming of this paper $F$ will denote the surface obtained from a
torus by removing an open open disk and $\mathcal{O}^{\prime}$ will denote the
orbifold with underlying surface $F$ and without cone points, that is,
$\mathcal{O}^{\prime}$ is 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

$T^{\prime}=((s_{1}s_{2}^{2}s_{1}s_{2})^{2}s_{2}s_{1}s_{2},(s_{2}(s_{2}s_{1})^{2})^{2}s_{2}s_{1}).$

The map $\eta:F\rightarrow S^{2}$ described in Fig. 9 defines an almost
orbifold covering of degree $17$ from the orbifold $\mathcal{O}^{\prime}$ to
$\mathcal{O}$. The description of $\eta$ implies that

$\eta^{-1}(D)=D_{1}\sqcup D_{2}\sqcup C$

and that

$\eta|_{D_{i}}:D_{i}\rightarrow D$

defines an orbifold covering of degree $7$ from the orbifold
$\mathcal{Q}_{i}^{\prime}=D_{i}$ (a disk without cone points) onto
$\mathcal{Q}=D(p_{3})$ (a disk with a single point of order $p_{3}=7$) for
$i=1,2$. As $\eta|_{C}:C\rightarrow\partial D$ has degree $4$ we conclude that
$\eta$ is a special almost orbifold covering.

Figure 9.  Case (7.b). $\eta:F\rightarrow S^{2}$ defines a special almost
orbifold covering of degree $17$.

Put $F^{\prime}:=F-\text{Int}(D_{1})\cup\text{Int}(D_{2})$. It is not hard to
see that the map

$\eta^{\prime}:=\eta|_{F^{\prime}}:F^{\prime}\rightarrow D_{0}$

defines an orbifold covering of degree $17$ from the orbifold
$\mathcal{O}_{0}^{\prime}=F^{\prime}$ onto the orbifold
$\mathcal{O}_{0}\subseteq\mathcal{O}$. Moreover, we have

$\eta_{\ast}^{\prime}(\pi_{1}^{o}(\mathcal{O}_{0}^{\prime}))=\langle
T_{1},T_{2},T_{3},T_{4},T_{5}\rangle\leq\pi_{1}^{o}(\mathcal{O}_{0})$

where

$\displaystyle T_{1}$ $\displaystyle=$
$\displaystyle(S_{1}S_{2}^{2}S_{1}S_{2})^{2}S_{2}S_{1}S_{2}$ $\displaystyle
T_{2}$ $\displaystyle=$ $\displaystyle(S_{2}(S_{2}S_{1})^{2})^{2}S_{2}S_{1})$
$\displaystyle T_{3}$ $\displaystyle=$ $\displaystyle
S_{1}S_{2}^{2}\cdot(S_{1}S_{2})^{7}\cdot S_{1}S_{2}^{2}$ $\displaystyle T_{4}$
$\displaystyle=$
$\displaystyle(S_{1}S_{2})^{3}S_{2}S_{1}\cdot(S_{1}S_{2})^{7}S_{1}S_{2}(S_{1}S_{2})^{3}$
$\displaystyle T_{5}$ $\displaystyle=$ $\displaystyle(S_{1}S_{2})^{7}$

By the injectivity of $\eta_{\ast}^{\prime}$ we can assume that
$\pi_{1}^{o}(\mathcal{O}_{0}^{\prime})=\langle
T_{1},T_{2},T_{3},T_{4},T_{5}\rangle.$ Using the relations in $G$ we see that
$i_{\ast}\circ\eta_{\ast}^{\prime}(T_{1})=(s_{1}s_{2}^{2}s_{1}s_{2})^{2}s_{2}s_{1}s_{2}$,
that
$i_{\ast}\circ\eta_{\ast}^{\prime}(T_{2})=(s_{2}(s_{2}s_{1})^{2})^{2}s_{2}s_{1})$
and that

$i_{\ast}\circ\eta_{\ast}^{\prime}(T_{3})=i_{\ast}\circ\eta_{\ast}^{\prime}(T_{4})=i_{\ast}\circ\eta_{\ast}^{\prime}(T_{5})=1.$

Now the inclusion map $j:F^{\prime}\hookrightarrow F$ induces a surjective
homomorphism

$j_{\ast}:\pi_{1}^{o}(\mathcal{O}_{0}^{\prime})\rightarrow\pi_{1}^{o}(\mathcal{O}^{\prime})$

whose kernel is normally generated by $\\{T_{3},T_{4},T_{5}\\}$. Consequently
$\pi_{1}^{o}(\mathcal{O}^{\prime})$ is generated by

$t_{1}:=j_{\ast}(T_{1})\ \ \text{ and }\ \ t_{2}:=j_{\ast}(T_{2}).$

Since $\eta_{\ast}\circ j_{\ast}=i_{\ast}\circ\eta_{\ast}^{\prime}$ we see
that $\eta_{\ast}$ sends the pair $(t_{1},t_{2})$ onto the pair $T$, and hence
the special almost orbifold covering
$\eta:\mathcal{O}^{\prime}\rightarrow\mathcal{O}$ represents $T^{\prime}$.

Case (8). Recall that $p_{1}=p_{2}=3$ and $p_{3}\geqslant 4$ with
$(p_{3},3)=1$. We must find a special almost orbifold covering that represents
the generating pair $T^{\prime}=(s_{1}s_{2}^{2},s_{2}^{2}s_{1}).$

Let $\eta^{\prime}:F\twoheadrightarrow D_{0}\subseteq S^{2}$ be the surjective
map described in Fig. 10, that is, $\eta^{\prime}(\alpha_{i})=\alpha_{0}$,
$\eta^{\prime}(\beta_{i})=\beta$ for $1\leqslant i\leqslant 3$ and
$\eta^{\prime}$ maps each component of

$F-\alpha_{1}\cup\alpha_{2}\cup\alpha_{3}\cup\beta_{1}\cup\beta_{2}\cup\beta_{3}$

homeomorphically onto $D_{0}-\alpha_{0}\cup\beta_{0}$.

Figure 10.  Case (8). $\eta=i\circ\eta^{\prime}:F\rightarrow S^{2}$ defines an
almost orbifold covering of degree $3$.

It is not hard to see that $\eta^{\prime}$ defines an orbifold covering of
degree $3$ from $\mathcal{O}^{\prime}$ onto $\mathcal{O}_{0}$. Note further
that the orbifold covering $\eta^{\prime}$ corresponds to the subgroup
$\langle t_{1},t_{2}\rangle$ of $\pi_{1}^{o}(\mathcal{O}_{0})$ where
$t_{1}=S_{1}S_{2}^{2}$ and $t_{2}=S_{2}^{2}S_{1}$. Thus we can assume that
$\pi_{1}^{o}(\mathcal{O}^{\prime})=\langle t_{1},t_{2}\rangle$. Moreover,
$\eta^{\prime}$ maps the boundary $C$ of $\mathcal{O}^{\prime}$ onto $\partial
D$ with degree $3$.

It follows from Example 2.1 that $\eta:=i\circ\eta^{\prime}:F\rightarrow
S^{2}$ defines an almost orbifold covering from $\mathcal{O}^{\prime}$ to
$\mathcal{O}$. Since $\eta|_{C}$ has degree $3$, which is smaller than
$p_{3}$, we conclude that $\eta$ is a special almost orbifold covering. Now
the generating pair $(t_{1},t_{2})$ is clearly mapped by
$\eta_{\ast}=i_{\ast}\circ\eta_{\ast}^{\prime}$ onto the pair $T^{\prime}$.

Case (9). In this case $p_{1}=2$, $p_{2}=4$ and $p_{3}\geqslant 7$ odd, and
$T^{\prime}=(s_{1}s_{2}^{2},s_{2}^{3}s_{1}s_{2}^{3})$.

Let $\eta^{\prime}:F\twoheadrightarrow D_{0}\subseteq S^{2}$ be the surjective
map described in Fig. 11. Thus $\eta^{\prime}$ defines an orbifold covering of
degree $4$ from $\mathcal{O}^{\prime}$ onto $\mathcal{O}_{0}$. Moreover,

$(\eta_{n}^{\prime})_{\ast}\pi_{1}^{o}(\mathcal{O}^{\prime})=\langle
T_{1},T_{2}\rangle$

where $t_{1}=S_{1}S_{2}^{2}$ and $t_{2}=S_{2}^{3}S_{1}S_{2}^{3}$, consequently
we may assume that $\pi_{1}^{o}(\mathcal{O}^{\prime})$ is generated by the
pair $(t_{1},t_{2})$.

Figure 11.  Case (9). $\eta=i\circ\eta^{\prime}:F\rightarrow S^{2}$ defines a
special almost orbifold covering of degree $4$.

It follows from Example 2.1 that $\eta:=i\circ\eta^{\prime}:F\rightarrow
S^{2}$ defines an almost orbifold covering from $\mathcal{O}^{\prime}$ to
$\mathcal{O}$ which is special as the degree of the map $\eta|_{C}$ is equal
to $4$ which is smaller than $p_{3}$. The generating pair $(t_{1},t_{2})$ of
$\pi_{1}^{o}(\mathcal{O}^{\prime}$) clearly gets mapped by
$\eta_{\ast}=i_{\ast}\circ\eta_{\ast}^{\prime}$ onto the pair $\tau^{\prime}$.

Case (10). In this case we have $p_{1}=2$, $p_{2}=3$ and $p_{3}\geqslant 7$
with $(p_{3},6)=1$, and $T^{\prime}$ is equal to
$(s_{1}s_{2}s_{1}s_{2}^{2},s_{2}^{2}s_{1}s_{2}s_{1}).$

The map $\eta^{\prime}:F\twoheadrightarrow D_{0}$ described in Fig. 12 defines
an orbifold covering of degree $6$ from $\mathcal{O}^{\prime}$ onto
$\mathcal{O}_{0}$ such that

$\eta_{\ast}^{\prime}(\pi_{1}^{o}(\mathcal{O}^{\prime}))=\langle
S_{1}S_{2}S_{1}S_{2}^{2},S_{2}^{2}S_{1}S_{2}S_{1}\rangle.$

By applying Example 2.1 we conclude that $\eta:=i\circ\eta^{\prime}$ defines a
almost orbifold covering from $\mathcal{O}^{\prime}$ to $\mathcal{O}$.
Moreover it is easy to see that

$\eta_{\ast}(S_{1}S_{2}S_{1}S_{2}^{2},S_{2}^{2}S_{1}S_{2}S_{1})=T^{\prime}$

which concludes the proof.

Figure 12.  Case (10). $\eta=i\circ\eta^{\prime}:F\rightarrow S^{2}$ defines a
special almost orbifold covering of degree $6$.

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