# The fundamental group of Galois covers of surfaces of degree $8$

Meirav Amram  Meirav Amram, Department of Mathematics, Shamoon College of
Engineering, Ashdod, Israel<EMAIL_ADDRESS>,  Cheng Gong  Cheng Gong,
Department of Mathematics, Soochow University, Suzhou 215006, Jiangsu, P. R.
China<EMAIL_ADDRESS>,  Praveen Kumar Roy  Praveen Kumar Roy, UM-DAE
Centre for excellence in basic sciences, Mumbai, India
<EMAIL_ADDRESS>,  Uriel Sinichkin  Uriel Sinichkin, School of
Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel
<EMAIL_ADDRESS>and  Uzi Vishne  Uzi vishne, Department of
Mathematics, Bar-Ilan University, Israel<EMAIL_ADDRESS>
###### Abstract.

We compute the fundamental group of the Galois cover of a surface of degree
$8$, with singularities of degree $4$ whose degeneration is homeomorphic to a
sphere. The group is shown to be a metabelian group of order $2^{23}$. The
computation amalgamates local groups, classified elsewhere, by an iterative
combination of computational and group theoretic methods. Three simplified
surfaces, for which the fundamental group of the Galois cover is trivial, hint
toward complications that depend on the homotopy of the degenerated surface.

###### Key words and phrases:

fundamental group, degeneration, Galois cover, classification of surfaces
MSC2010: Primary: 14D06, 14H30, 14J10.

##  1\. Introduction

The classification of algebraic surfaces is a classical problem in algebraic
geometry. The fundamental group of the complement of a branch curve and the
fundamental group of the Galois cover of the surface, are important invariants
of the surface. Degeneration is a crucial tool in classifying surfaces,
leading to a presentation of these groups by generators and relations. Some
recent studies of degenerations and their structure appear in [10, 11, 13].

Given a surface $X$ embedded in $\mathbb{C}\mathbb{P}^{N}$ for some $N$, and a
generic projection to ${\mathbb{C}}{\mathbb{P}}^{2}$ with a branch curve $S$,
the fundamental group $G=\pi_{1}(\mathbb{C}\mathbb{P}^{2}-S)$ is an invariant
of the degeneration class of $X$. The standard generators in the Van Kampen
presentation of this group are denoted by
$\gamma_{1},\gamma_{1^{\prime}},\dots,\gamma_{q},\gamma_{q^{\prime}}$, where
$q$ is the number of lines in the degenerated object. There is a natural map
from the quotient $G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}$ to the
symmetric group $S_{n}$, where $n=\deg X$ is the degree of the projection and
${{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}$ is the normal subgroup
generated by
$\gamma_{1}^{2},\gamma_{1^{\prime}}^{2},\dots,\gamma_{q}^{2},\gamma_{q^{\prime}}^{2}$.
The fundamental group of the Galois cover ${X_{\operatorname{Gal}}}$ is the
kernel of this map [14], so we have a short exact sequence

$1\longrightarrow\pi_{1}({X_{\operatorname{Gal}}})\longrightarrow\pi_{1}(\mathbb{C}\mathbb{P}^{2}-S)/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}\longrightarrow
S_{n}\longrightarrow 1.$

The presentation of group $G$ is fairly complicated and it is often a
challenging task to ascertain basic properties of this group. The computation
has been carried out in many cases, for example, for
${\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$ [15], the
Veronese surfaces [17, 18], Hirzebruch surfaces $F_{k}(a,b)$ for $a$ and $b$
relatively prime [16], ${\mathbb{C}}{\mathbb{P}}^{1}\times T$ [2, 9] where $T$
is the complex torus, toric varieties [7], and surfaces with Zappatic
singularities of type $E_{k}$ [4]. In one of the most complex cases, that of
the surface $T\times T$ [8], the Van Kampen presentation had 54 generators and
nearly 2000 relations.

To the degeneration of a surface $X$ we associate the shadow, which is a
$2$-dimensional simplicial complex. There is one $2$-cell (i.e., a triangle)
for each irreducible component of the degeneration, which are all copies of
$\mathbb{C}\mathbb{P}^{2}$. Two triangles share a $1$-cell (an edge),
respectively a $0$-cell (a vertex), if the corresponding planes intersect in a
line, respectively at a point. In particular, the number of triangles in the
shadow is the degree of $X$. Edges that belong to a single triangle are
redundant. The number of non-redundant edges meeting in a vertex is the
multiplicity (or the degree) of the singularity at this point. The shadow is
planar if it can be embedded in the Euclidean plane ${\mathbb{R}}^{2}$.
Experience shows that the computation of the fundamental group is typically
easier when the shadow is planar.

Most examples studied so far have planar shadows [4, 7, 16, 17, 18].
Relatively new works on surfaces of degrees $5$ and $6$, with planar shadows,
appear in [5, 3]. Recently, the first named author studied surfaces with non-
planar shadows that have degrees $4$ and $6$, see [1]. The shadow of the
surface $\mathbb{C}\mathbb{P}^{1}\times T$ (where $T$ is the torus), which was
studied in [2], is homeomorphic to a cylinder.

In this work we focus on surfaces of degree $8$, whose shadow is homeomorphic
to the sphere $S^{2}$, such that every vertex in the shadow has degree $4$.
Such a shadow can be described as the union of two triangulated squares glued
along all four edges, see Figure 1; for that reason we denote the shadow as
$X^{4}$. As expected, $\pi_{1}({X^{4}_{\operatorname{Gal}}})$ has rather
complicated computations. We thus implement the following technique: We
introduce intermediate shadows, obtained by “cutting open” one glued pair of
edges in each step. There are thus four shadows,

$X^{1}\rightarrow X^{2}\rightarrow X^{3}\rightarrow X^{4},$

where each arrow denotes a covering map obtained from identifying two edges,
see Figures 2, 3, and 4, respectively. Computing the four fundamental groups
in this manner becomes a sequence of problems of increasing complexity, each
of which requires additional ideas to solve. For $i=1,2,3$, we were able to
use Magma to simplify the presentation of
$\pi_{1}({X^{i}_{\operatorname{Gal}}})$, resulting in each case in a trivial
group. Singularities of high multiplicity complicate the computation, so for
$X^{4}$ we needed to incorporate the methods of [6] and [19].

For every graph $T$ there are a Coxeter group $C(T)$ and a natural quotient
$C_{Y}(T)$. This is relevant here because in all cases computed so far, the
fundamental group of the surface turns out to be a quotient of $C_{Y}(T)$,
where $T$ is the dual graph of the shadow. Naively, we would want to make all
computations in $C_{Y}(T)$, because this group is fully understood. However,
not all the defining relations of $C_{Y}(T)$ are given a priori. This is
resolved by a technical innovation of a multi-layered computation. We run
Magma on the given presentation. When we cannot show that the fundamental
group is indeed a quotient of $C_{Y}(T)$, we show that a subgroup is a
quotient of $C_{Y}(T^{\prime})$ for a subgraph $T^{\prime}$, use this to
obtain more relations, run Magma again, and repeat. This is best demonstrated
in the computation for $X^{3}$ below.

As we progress from $X^{1}$ to $X^{2}$ to $X^{3}$ and finally to $X^{4}$, the
situation becomes more complex. For $X^{1}$ the traditional techniques
suffice. For $X^{2}$ we used Magma and methods from [19]. For $X^{3}$ we used
the multi-layered computation described above. For $X^{4}$ we needed to also
combine ideas from [6], which computes the Artin covers of $C_{Y}(T)$.

The remainder of this paper is organized as follows. In Section 2 we give the
constructions of $X^{i}$, $i=1,2,3,4$, we recall the Van Kampen Theorem to get
presentations of fundamental groups, and give some necessary details on
Coxeter groups and dual graphs from the papers [6] and [19]. In Theorems 3.1,
3.2, 3.3, and 3.4 (Section 3) the groups $\pi_{1}({X_{\operatorname{Gal}}})$
related to $X^{i}$, $i=1,2,3,4$, are determined.

##  2\. Preliminaries and methods

In this section we describe the shadows $X^{1}$, $X^{2}$, $X^{3}$, and $X^{4}$
in detail, and apply standard techniques to exhibit presentations for their
fundamental groups $\pi_{1}({X^{i}_{\operatorname{Gal}}})$. We recommend [10,
11, 13] as sources for relevant information about degenerations and [3] for
detailed background about fundamental groups of complements of branch curves
and fundamental groups of Galois covers.

###  2.1. Projective degeneration

We begin by defining a degeneration. Let $\Delta$ denote the unit disc, and
let $X,Y$ be projective algebraic surfaces. Let
$p:Y\rightarrow\mathbb{CP}^{2}$ and $p^{\prime}:X\rightarrow\mathbb{CP}^{2}$
be generic projections. We say that $p^{\prime}$ is a projective degeneration
of $p$ if there is a flat family $\pi:V\rightarrow\Delta$ and an embedding
$F:V\rightarrow\Delta\times\mathbb{CP}^{2}$, such that $F$ composed with the
first projection is $\pi$, and:

  * (a)

$\pi^{-1}(0)\simeq X$;

  * (b)

there is a $t_{0}\neq 0$ in $\Delta$ such that $\pi^{-1}(t_{0})\simeq Y$;

  * (c)

the family $V-\pi^{-1}(0)\rightarrow\Delta-{0}$ is smooth;

  * (d)

restricting to $\pi^{-1}(0)$, $F={0}\times p^{\prime}$ under the
identification of $\pi^{-1}(0)$ with $X$;

  * (e)

restricting to $\pi^{-1}(t_{0})$, $F={t_{0}}\times p$ under the identification
of $\pi^{-1}(t_{0})$ with $Y$.

In the above process, we construct a flat degeneration of $X$ into a union of
planes. We consider degenerations with only two planes intersecting at a line,
with each plane homeomorphic to $\mathbb{CP}^{2}$.

###  2.2. Constructions and figures of degenerations

Here we describe the four shadows $X^{i}$, $i=1,2,3,4$, which as explained
above are all of degree $8$ and have singularities of degree at most (and in
$X^{4}$: precisely) $4$.

######  Construction 1 (The shadow $X^{4}$).

A triangulation of degree $8$ of the sphere is obtained by cutting the
Euclidean sphere by three orthogonal planes passing through the origin, see
Figure 1. This triangulation has $6$ vertices which we denote
$V_{1},\dots,V_{6}$ and $12$ edges; the repeated edges, $5,6,8,9$, are glued
in pairs. We numerate the vertices and edges for future reference.

$\textstyle{V_{5}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{9}$$\scriptstyle{2}$$\scriptstyle{5}$$\textstyle{V_{5}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{5}$$\scriptstyle{10}$$\scriptstyle{9}$$\textstyle{V_{6}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\scriptstyle{8}$$\textstyle{V_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{4}$$\scriptstyle{3}$$\textstyle{V_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{6}$$\textstyle{V_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{7}$$\scriptstyle{6}$$\textstyle{V_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{12}$$\scriptstyle{11}$$\textstyle{V_{6}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{8}$$\textstyle{V_{2}}$$\textstyle{V_{2}}$
Figure 1. Degree $4$ degenerations with four identified edges give $X^{4}$

It is worth noting that following [11, Corollary 12.2], each degree $4$
degeneration can be deformed to a cone over an elliptic curve $E_{0}$, where
$E_{0}$ is a curve of degree $4$ in $\mathbb{P}^{3}$. The Zappatic
singularities of type $E_{4}$ are obtained as the intersection of $4$ lines.
For more details about such surfaces, see [4].

Now consider the two connected components of Figure 1, without the gluing. We
will construct $X^{1}$, $X^{2}$, and $X^{3}$ by gluing one pair, then two
pairs, and eventually, three pairs of edges.

######  Construction 2 (The shadow $X^{1}$).

The shadow $X^{1}$ appears in Figure 2. It has $8$ vertices and $15$ edges, of
which only $9$ edges are non-redundant (the redundant edges, those belonging
to a single triangle, are not numerated, as they do not appear in the
computation. Edge $5$ appears twice in the figure, and the two copies are of
course glued.

Notice that the degree of $V_{1}$ and $V_{2}$ is $4$; the degree of $V_{3}$
and $V_{4}$ is $3$; and the other vertices have degree $1$.

$\textstyle{V_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{2}$$\scriptstyle{5}$$\textstyle{V_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{5}$$\scriptstyle{7}$$\textstyle{V_{5}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\textstyle{V_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{4}$$\scriptstyle{3}$$\textstyle{V_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{6}$$\textstyle{V_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{9}$$\scriptstyle{8}$$\textstyle{V_{8}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{6}}$$\textstyle{V_{7}}$
Figure 2. Degree $4$ degenerations with one identified edge give $X^{1}$

We now glue in $X^{1}$ the edge connecting the vertices $V_{4}$ and $V_{6}$
with the edge connecting the vertices $V_{4}$ and $V_{7}$, to obtain $X^{2}$
(the enumeration of the vertices and edges changes).

######  Construction 3 (Degeneration $X^{2}$).

The degeneration $X^{2}$ appears in Figure 3. This shadow has $7$ vertices and
$10$ non-redundant edges. Edges $5$ and $6$ appear twice, and the two copies
are being glued. The degree of $V_{1}$, $V_{2}$, and $V_{3}$ is $4$; the
degree of $V_{4}$ and $V_{5}$ is $3$; and the degree of $V_{6}$ and $V_{7}$ is
$1$.

$\textstyle{V_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{2}$$\scriptstyle{5}$$\textstyle{V_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{5}$$\scriptstyle{8}$$\textstyle{V_{6}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\textstyle{V_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{4}$$\scriptstyle{3}$$\textstyle{V_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{6}$$\textstyle{V_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{7}$$\scriptstyle{6}$$\textstyle{V_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{10}$$\scriptstyle{9}$$\textstyle{V_{7}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{5}}$$\textstyle{V_{5}}$
Figure 3. Degree $4$ degenerations with two identified edges give $X^{2}$

We now glue the edge connecting $V_{5}$ and $V_{6}$ with the edge connecting
$V_{5}$ and $V_{7}$ (in the numeration of $X^{2}$) to obtain $X^{3}$.

######  Construction 4 (The shadow $X^{3}$).

There are now $6$ vertices and $11$ non-redundant edges, see Figure 4. The
degrees of $V_{1}$, $V_{2}$, $V_{3}$, and $V_{4}$ are $4$; and the degrees of
$V_{5}$ and $V_{6}$ are $3$.

$\textstyle{V_{5}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{2}$$\scriptstyle{5}$$\textstyle{V_{5}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{5}$$\scriptstyle{9}$$\textstyle{V_{6}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\scriptstyle{8}$$\textstyle{V_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{4}$$\scriptstyle{3}$$\textstyle{V_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{6}$$\textstyle{V_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{7}$$\scriptstyle{6}$$\textstyle{V_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{11}$$\scriptstyle{10}$$\textstyle{V_{6}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{8}$$\textstyle{V_{2}}$$\textstyle{V_{2}}$
Figure 4. Degree $4$ degenerations with three identified edges give $X^{3}$

One can also depict $X^{1}$, $X^{2}$, and $X^{3}$ as planar shadows, without
gluing, as shown in Figure 5. We maintain the numeration of the vertices and
edges for each shadow as given above. To obtain $X^{4}$ from $X^{3}$ we must
glue the two edges connecting $V_{5}$ and $V_{6}$ (one from the triangle
$V_{5}V_{6}V_{4}$ and one from $V_{5}V_{6}V_{1}$), and the resulting shadow is
no longer planar.

$\textstyle{V_{8}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{9}$$\textstyle{V_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{7}$$\scriptstyle{2}$$\scriptstyle{5}$$\textstyle{V_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{8}$$\scriptstyle{6}$$\textstyle{V_{7}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{5}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\textstyle{V_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{3}$$\scriptstyle{4}$$\textstyle{V_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{6}}$
$\textstyle{V_{7}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{10}$$\textstyle{V_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{8}$$\scriptstyle{2}$$\scriptstyle{5}$$\textstyle{V_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{7}$$\scriptstyle{9}$$\textstyle{V_{6}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\textstyle{V_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{3}$$\scriptstyle{4}$$\textstyle{V_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{6}$$\textstyle{V_{5}}$
$\textstyle{V_{5}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{9}$$\scriptstyle{2}$$\scriptstyle{5}$$\textstyle{V_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{7}$$\scriptstyle{10}$$\textstyle{V_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{3}$$\scriptstyle{4}$$\textstyle{V_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{6}$$\textstyle{V_{2}}$$\textstyle{V_{6}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{8}$$\scriptstyle{11}$$\scriptstyle{1}$
Figure 5. Alternative presentations for the shadows $X^{1}$, $X^{2}$ and
$X^{3}$.

###  2.3. Fundamental groups

We now describe how the fundamental groups for $X^{i}$, $i=1,2,3,4$, are
computed from the diagrams. Let $X$ be an algebraic surface of degree $n$
embedded in some projective space, with a generic projection
$f{\,{:}\,}X\rightarrow{\mathbb{C}}{\mathbb{P}}^{2}$ of degree $n$. Let

${X_{\operatorname{Gal}}}=\overline{(X\times_{\mathbb{CP}^{2}}\ldots\times_{\mathbb{CP}^{2}}X)-\triangle},$

be the Galois cover of $X$, where the product is taken $n$ times, and
$\triangle$ is the extended diagonal that is defined as

$\triangle=\\{(x_{1},\dots,x_{k})\in X^{k}\bigm{|}x_{i}=x_{j}\quad\text{for
some}\quad i\neq j\\}.$

The projection has a branch curve $S$, and we turn to compute
$G=\pi_{1}(\mathbb{CP}^{2}-S)$ and $\pi_{1}({X_{\operatorname{Gal}}})$. The
degeneration of $X$ into a union of planes has a branch curve which is a line
arrangement of degree $q$, the number of edges in the degeneration.

The vertices of the line arrangement are the points where more than two lines
intersect. The multiplicity of a vertex is the number of lines $k$ that
intersect at the point. In this work, we have singularities with
multiplicities $1$, $3$, and $4$. There is a regeneration process (explained,
for example, in [3]), which “opens” each of the lines to either a double line
or a conic. We obtain a local description for the regenerated curve $S$, which
is a cuspidal curve of degree $2q$. To compute the presentation we first must
choose a global ordering of the edges in the shadow. Then, the relations
arising from each vertex depend on the local ordering at this point. Now, the
Van Kampen Theorem [20] states that group
$G=\pi_{1}(\mathbb{C}\mathbb{P}^{2}-S)$ is generated by
$\gamma_{i},\gamma_{i^{\prime}}$ for $i=1,\dots,q$, where each generator
represents an edge between two planes in the degeneration, subject to
relations of the following five types.

  1. $($1$)$

for every branch point of a conic, a relation of the form
$\gamma=\gamma^{\prime}$ where $\gamma,\gamma^{\prime}$ are certain conjugates
of the generators $\gamma_{j}$ and $\gamma_{j^{\prime}}$,

  2. $($2$)$

for every node, $[\gamma,\gamma^{\prime}]=1$ where $\gamma,\gamma^{\prime}$
are certain conjugates of $\gamma_{i}$ and $\gamma_{j}$, where $i,j$ are the
lines meeting in this node,

  3. $($3$)$

for every cusp, $\langle\gamma,\gamma^{\prime}\rangle=1$ where
$\gamma,\gamma^{\prime}$ are as in (2),

  4. $($4$)$

the “projective relation”
$\prod\limits_{i=d}^{1}\gamma_{i^{\prime}}\gamma_{i}=1$,

  5. (5*)

a commutator relation for every pair of edges which meet in the branch curve.

As a consequence, other than the projective relation, the relations can be
computed locally. In other words, $G$ is an amalgamated product of “local
groups”, one for each vertex, generated by the generators corresponding to the
edges meeting in the vertex — modulo the projective relation of (4) and the
relations of type (5*). The amalgamation is done by identifying the generators
associated to the same edge, each appearing in the two local groups of the
vertices of this edge.

The following lemmas summarize the local groups for vertices of multiplicity
$3$ or $4$ after regeneration, for a particular order of the edges. Our
numeration of the edges in each of the shadows $X^{i}$ was chosen so that
locally we only see the orderings appearing in these lemmas.

In Figure 6 we have an intersection of three lines, being globally ordered as
$i<j<k$.

$\textstyle{\bullet}$$\scriptstyle{i}$$\scriptstyle{k}$$\scriptstyle{j}$
Figure 6. Intersection of three lines

######  Lemma 2.1.

For each vertex whose numeration is as in Figure 6, the relations induced on
$G$ are:

(1)
$\langle\gamma_{i^{\prime}},\gamma_{j}\rangle=\langle\gamma_{i^{\prime}},\gamma_{j^{\prime}}\rangle=\langle\gamma_{i^{\prime}},\gamma_{j}^{-1}\gamma_{j^{\prime}}\gamma_{j}\rangle=e$
(2)
$\gamma_{i}=\gamma_{j^{\prime}}\gamma_{j}\gamma_{i^{\prime}}\gamma_{j}^{-1}{\gamma_{j^{\prime}}}^{-1}$
(3)
$\begin{split}\langle\gamma_{k},\gamma_{j^{\prime}}\gamma_{j}\gamma_{i^{\prime}}\gamma_{j}{\gamma_{i^{\prime}}}^{-1}\gamma_{j}^{-1}{\gamma_{j^{\prime}}}^{-1}\rangle=&\langle\gamma_{k},\gamma_{j^{\prime}}\gamma_{j}\gamma_{i^{\prime}}\gamma_{j^{\prime}}{\gamma_{i^{\prime}}}^{-1}\gamma_{j}^{-1}{\gamma_{j^{\prime}}}^{-1}\rangle=\\\
=&\langle\gamma_{k},\gamma_{j^{\prime}}\gamma_{j}\gamma_{i^{\prime}}\gamma_{j}^{-1}\gamma_{j^{\prime}}\gamma_{j}{\gamma_{i^{\prime}}}^{-1}\gamma_{j}^{-1}{\gamma_{j^{\prime}}}^{-1}\rangle=e\end{split}$
(4)
$\gamma_{k^{\prime}}=\gamma_{k}\gamma_{j^{\prime}}\gamma_{j}\gamma_{i^{\prime}}\gamma_{j^{\prime}}\gamma_{j}{\gamma_{i^{\prime}}}^{-1}\gamma_{j}^{-1}{\gamma_{j^{\prime}}}^{-1}\gamma_{k}\gamma_{j^{\prime}}\gamma_{j}\gamma_{i^{\prime}}\gamma_{j}^{-1}{\gamma_{j^{\prime}}}^{-1}{\gamma_{i^{\prime}}}^{-1}\gamma_{j}^{-1}{\gamma_{j^{\prime}}}^{-1}\gamma_{k}^{-1}$
(5)
$[\gamma_{i},\gamma_{k}]=[\gamma_{i},\gamma_{k^{\prime}}]=[\gamma_{i^{\prime}},\gamma_{k}]=[\gamma_{i^{\prime}},\gamma_{k^{\prime}}]=e.$

Figure 7 depicts an intersection of four lines, being globally ordered as
$i<j<k<l$.

$\textstyle{\bullet}$$\scriptstyle{k}$$\scriptstyle{j}$$\scriptstyle{i}$$\scriptstyle{\,l\,}$
Figure 7. Intersection of four lines

######  Lemma 2.2.

A point that is an intersection of four lines contributes to $G$ the following
list of relations:

(6)
$\langle\gamma_{i^{\prime}},\gamma_{j}\rangle=\langle\gamma_{i^{\prime}},\gamma_{j^{\prime}}\rangle=\langle\gamma_{i^{\prime}},\gamma_{j}^{-1}\gamma_{j^{\prime}}\gamma_{j}\rangle=e$
(7)
$\langle\gamma_{k},\gamma_{l}\rangle=\langle\gamma_{k^{\prime}},\gamma_{l}\rangle=\langle\gamma_{k}^{-1}\gamma_{k^{\prime}}\gamma_{k},\gamma_{l}\rangle=e$
(8)
$[\gamma_{j^{\prime}}\gamma_{j}\gamma_{i^{\prime}}\gamma_{j}^{-1}{\gamma_{j^{\prime}}}^{-1},\gamma_{l}]=e$
(9)
$[\gamma_{j^{\prime}}\gamma_{j}\gamma_{i^{\prime}}\gamma_{j}^{-1}{\gamma_{j^{\prime}}}^{-1},\gamma_{k}^{-1}{\gamma_{k^{\prime}}}^{-1}\gamma_{l}^{-1}\gamma_{l^{\prime}}\gamma_{l}\gamma_{k^{\prime}}\gamma_{k}]=e$
(10)
$\langle\gamma_{i},\gamma_{j}\rangle=\langle\gamma_{i},\gamma_{j^{\prime}}\rangle=\langle\gamma_{i},\gamma_{j}^{-1}\gamma_{j^{\prime}}\gamma_{j}\rangle=e$
(11)
$\langle\gamma_{k},\gamma_{l}^{-1}\gamma_{l^{\prime}}\gamma_{l}\rangle=\langle\gamma_{k^{\prime}},\gamma_{l}^{-1}\gamma_{l^{\prime}}\gamma_{l}\rangle=\langle\gamma_{k}^{-1}\gamma_{k^{\prime}}\gamma_{k},\gamma_{l}^{-1}\gamma_{l^{\prime}}\gamma_{l}\rangle=e$
(12)
$[\gamma_{j^{\prime}}\gamma_{j}\gamma_{i}\gamma_{j}^{-1}{\gamma_{j^{\prime}}}^{-1},\gamma_{l}^{-1}\gamma_{l^{\prime}}\gamma_{l}]=e$
(13)
$[\gamma_{j^{\prime}}\gamma_{j}\gamma_{i}\gamma_{j}^{-1}{\gamma_{j^{\prime}}}^{-1},\gamma_{k}^{-1}{\gamma_{k^{\prime}}}^{-1}\gamma_{l}^{-1}{\gamma_{l^{\prime}}}^{-1}\gamma_{l}\gamma_{l^{\prime}}\gamma_{l}\gamma_{k^{\prime}}\gamma_{k}]=e$
(14)
$\gamma_{j^{\prime}}\gamma_{j}\gamma_{i^{\prime}}\gamma_{j}\gamma_{i^{\prime}}^{-1}\gamma_{j}^{-1}{\gamma_{j^{\prime}}}^{-1}=\gamma_{l}\gamma_{k^{\prime}}\gamma_{l}^{-1}$
(15)
$\gamma_{j^{\prime}}\gamma_{j}\gamma_{i^{\prime}}\gamma_{j^{\prime}}\gamma_{i^{\prime}}^{-1}\gamma_{j}^{-1}{\gamma_{j^{\prime}}}^{-1}=\gamma_{l}\gamma_{k^{\prime}}\gamma_{k}{\gamma_{k^{\prime}}}^{-1}\gamma_{l}^{-1}$
(16)
$\gamma_{j^{\prime}}\gamma_{j}\gamma_{i}\gamma_{j}\gamma_{i}^{-1}\gamma_{j}^{-1}{\gamma_{j^{\prime}}}^{-1}=\gamma_{l}^{-1}\gamma_{l^{\prime}}\gamma_{l}\gamma_{k^{\prime}}\gamma_{l}^{-1}{\gamma_{l^{\prime}}}^{-1}\gamma_{l}$
(17)
$\gamma_{j^{\prime}}\gamma_{j}\gamma_{i}\gamma_{j^{\prime}}\gamma_{i}^{-1}\gamma_{j}^{-1}{\gamma_{j^{\prime}}}^{-1}=\gamma_{l}^{-1}\gamma_{l^{\prime}}\gamma_{l}\gamma_{k^{\prime}}\gamma_{k}\gamma_{k^{\prime}}^{-1}\gamma_{l}^{-1}{\gamma_{l^{\prime}}}^{-1}\gamma_{l}.$

###  2.4. The dual graph

Group $G$ comes with a natural projection to the symmetric group $S_{n}$,
where each generator $\gamma_{i}$ or $\gamma_{i^{\prime}}$ is mapped to the
transposition switching the planes meeting in edge number $i$.

From a presentation of $G$ we can find a presentation of
$\pi_{1}({X_{\operatorname{Gal}}})$, using the exact sequence

(18) $0\rightarrow\pi_{1}({X_{\operatorname{Gal}}})\rightarrow
G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}\rightarrow S_{n}\rightarrow
0;$

this can be done using the Reidmeister-Schreier method; however, the number of
generators gets multiplied by $n!$, so one must find ways to make the
computation manageable.

Consider a shadow, such as $X^{i}$ for $i=1,2,3,4$. We now define a map from
the fundamental group $G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}$ to
the symmetric group $S_{n}$, where $n$ is the number of $2$-cells in the
shadow. Recall that $G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}$ is
generated by $\gamma_{i}$ and $\gamma_{i^{\prime}}$, one pair for each
nonredundant edge of the shadow. The map to $S_{n}$ is then defined by mapping
both $\gamma_{i}$ and $\gamma_{i^{\prime}}$ to the transposition $(ab)$, where
$a,b$ are the triangles meeting in the common edge $i$.

A set of transpositions in $S_{n}$ can be encoded as the edges of a graph $T$
on $n$ vertices. The transpositions generate the whole group if and only if
$T$ is connected. Furthermore, $S_{n}$ can be presented on the given
transpositions with respect to a certain set of relations. Group $C_{Y}(T)$,
which we define below, is defined by imposing the “local” relations, and so
there is a projection $C_{Y}(T)\rightarrow S_{n}$. In all cases studied so
far, it was possible to prove that the generators of
$G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}$ satisfy the defining
relations of $C_{Y}(T)$.

The dual graph of the shadow is graph $T$, whose vertices are the $2$-cells,
with an edge connecting every intersecting pair of $2$-cells. For example, the
dual graph of $X^{1}$ (from Figure 2), is given in Figure 10.

Because the product of two transpositions is of order $2$ or $3$, depending on
whether they intersect, we are led to associate to the dual graph $T$ an
abstract group $C_{Y}(T)$ defined as follows.

######  Definition 2.3.

Let $T$ be a connected graph, not necessarily simple, but without loops. Let
$C(T)$ be the Coxeter group whose generators are the edges of $T$, subject to
three types of relations: $u^{2}=1$ for every generator; $(uv)^{2}=1$ if the
edges corresponding to $u,v$ are disjoint; and $(uv)^{3}=1$ if $u,v$ intersect
in one vertex. (No relation is assumed if $u,v$ connect the same two
vertices).

Furthermore, we define the group $C_{Y}(T)$ as the quotient of $C(T)$ with
respect to four types of relations:

(19) $\displaystyle{}[wuw,v]$ $\displaystyle=$ $\displaystyle e\qquad\mbox{if
$u,v,w$ are as in Figure \ref{0knifeX} (left)},$ (20) $\displaystyle{}\langle
wuw,{v}\rangle$ $\displaystyle=$ $\displaystyle e\qquad\mbox{if $u,v,w$ are as
in Figure \ref{0knifeX} (right)},$ (21) $\displaystyle{}[wuw,vxv]$
$\displaystyle=$ $\displaystyle e\qquad\mbox{if $x,u,v,w$ are as in Figure
\ref{0plateX} (left)},$ (22) $\displaystyle{}\langle wuw,vxv\rangle$
$\displaystyle=$ $\displaystyle e\qquad\mbox{if $x,u,v,w$ are as in Figure
\ref{0plateX} (right)}$

$\textstyle{u}$$\textstyle{v}$$\textstyle{w}$
$\textstyle{u}$$\textstyle{v}$$\textstyle{w}$ Figure 8.
$\textstyle{u}$$\textstyle{v}$$\textstyle{w}$$\textstyle{x}$
$\textstyle{x}$$\textstyle{u}$$\textstyle{w}$$\textstyle{v}$ Figure 9.

It is easy to see that if we map each generator to the transposition,
switching its two vertices, all the relations defining $C_{Y}(T)$ hold in
$S_{n}$. We thus have a well-defined projection $C_{Y}(T)\rightarrow S_{n}$,
where $n$ is the number of vertices.

It is not hard to see that if $T$ is a tree, then $C_{Y}(T)$ is isomorphic to
$S_{n}$. In general, the transpositions arranged in a cycle satisfy another
relation, the “cyclic relation”, which does not hold in $C_{Y}(T)$. This was
used to compute the kernel of the map $C_{Y}(T)\rightarrow S_{n}$, as we now
explain.

###  2.5. The group $S_{n}\ltimes A_{t,n}$

The kernel of the map $C_{Y}(T)\rightarrow S_{n}$ was computed in [19], when
$T$ is a simple graph. For a planar graph $T$, this was extended in [6] to a
similarly defined group $A_{Y}(T)$, which naturally covers Artin’s braid group
$B_{n}$; the kernel of the map $A_{Y}(T)\rightarrow B_{n}$ was computed there.
From this result one can deduce the structure of the kernel of
$C_{Y}(T)\rightarrow S_{n}$ for any planar graph $T$, simple or not. We
believe that the same description holds for any graph (see [12] for details),
but this is not needed for the current paper.

Let $t\geq 0$ and $n$ be natural numbers. Recall the definition of group
$A_{t,n}$ from [19]. Let $U:=\\{u,u^{\prime},\dots\\}$ be a set consisting of
$t$ elements.

######  Definition 2.4.

Group $A_{t,n}$ is generated by the $n^{2}{\left|U\right|}$ elements
$u_{x,y}$, for $u\in U$ and $x,y\in{\\{{1,\dots,n}\\}}$, satisfying the
following relations for any $u,u^{\prime}\in U$ and any $x,y,z$:

  1. $($1$)$

$u_{x,x}=e$

  2. $($2$)$

$u_{x,y}u_{y,z}=u_{x,z}=u_{y,z}u_{x,y}$

  3. $($3$)$

$[u_{x,y},u_{w,z}^{\prime}]=e$ (for distinct $x,y,w$ and $z$)

Another description of the same group is provided by the following.

######  Definition 2.5.

For $x\in{\\{{1,\dots,n}\\}}$, let $F^{(x)}$ denote the free group on $t$
letters $u_{x}$ $(u\in U)$. Put
$F_{t,n}^{*}:=\prod\limits_{x\in{\\{{1,\dots,n}\\}}}F^{(x)}$. Map
$\textit{ab}:F^{(x)}\rightarrow\mathbb{Z}^{t}$, defined by
$\textit{ab}(u_{x})=u$, (where $\mathbb{Z}^{t}$ is thought of as a free
Abelian group generated by $U$), can be extended naturally to give map
$F_{t,n}^{*}\rightarrow\mathbb{Z}^{t}$. We define the kernel of this map to be
$F_{t,n}$ [19, Definition 5.6].

The main result of [19] is that $C_{Y}(T)\cong S_{n}\ltimes A_{t,n}$ where $n$
is the number of vertices in $T$ and $t$ is the rank of $\pi_{1}(T)$. The
symmetric group acts on $A_{t,n}$, by its action on the indices. It is also
verified in [19, Theorem 5.7] for $n\geq 5$, that $A_{t,n}\cong F_{t,n}$. In
particular, there is a short exact sequence

$1\longrightarrow F_{t,n}\longrightarrow C_{Y}(T)\longrightarrow
S_{n}\longrightarrow 1.$

##  3\. Results

In this section we describe the groups $\pi_{1}({X^{i}_{\operatorname{Gal}}})$
for each $i=1,2,3,4$. We apply the algorithm and methods described in Section
2.

We have the following notation that will be used in commutation relations.

######  Notation 1.

A formula involving $\gamma_{i,i^{\prime}}$ is a shorthand for all the
formulas that are obtained by replacing this term with $\gamma_{i}$ or
$\gamma_{i^{\prime}}$.

###  3.1. The surface $X^{1}$

######  Theorem 3.1.

The fundamental group of the Galois cover of $X^{1}$ is trivial.

###### Proof.

Let us describe group $G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}$ for
the shadow $X^{1}$ depicted in Figure 2. The group is generated by
${\\{{\gamma_{i}}\\}}$ for $i=1,1^{\prime},2,2^{\prime},\dots,9,9^{\prime}$.
Vertices $V_{1}$ and $V_{2}$ give relations as in Lemma 2.2, where $(i,j,k,l)$
are equal to $(1,2,3,4)$ and $(6,7,8,9)$, respectively. For the vertices
$V_{3}$ and $V_{4}$ we have the relations from Lemma 2.1, with $(i,j,k)$ equal
to $(2,5,7)$ and $(4,5,6)$, respectively. The vertices $V_{5}$, $V_{6}$,
$V_{7}$, and $V_{8}$ give the relations $\gamma_{i}=\gamma_{i^{\prime}}$ for
$i=1,3,8$, and $9$ respectively. Then we have the relations of type (5*):

$[\gamma_{i,i^{\prime}},\gamma_{j,j^{\prime}}]=e\ \ \mbox{for}\ \
(i,j)\in{({\\{{1,3}\\}}\times{\\{{5,6,7,8,9}\\}})}\cup{({\\{{2}\\}}\times{\\{{6,8,9}\\}})}\cup{({\\{{4}\\}}\times{\\{{7,8,9}\\}})}\cup{({\\{{5}\\}}\times{\\{{8,9}\\}})}.$

The last relation is the projective one
$\prod\limits_{i=9}^{1}\gamma_{i^{\prime}}\gamma_{i}=e$.

Next we compare (14) and (15) with $(i,j,k,l)$ equal to $(1,2,3,4)$, and
together with $\gamma_{1}=\gamma_{1^{\prime}}$ and
$\gamma_{3}=\gamma_{3^{\prime}}$, we get $\gamma_{2}=\gamma_{2^{\prime}}$.
Then we can get from (14) and (16) that $\gamma_{4}=\gamma_{4^{\prime}}$.
Similarly, we use $(i,j,k,l)$ equal to $(6,7,8,9)$, along with
$\gamma_{8}=\gamma_{8^{\prime}}$ and $\gamma_{9}=\gamma_{9^{\prime}}$ to get
$\gamma_{7}=\gamma_{7^{\prime}}$ and $\gamma_{6}=\gamma_{6^{\prime}}$ as well.

Now we turn to relation (2) and substitute $i=2$ and $j=5$ along with
$\gamma_{2}=\gamma_{2^{\prime}}$, which was obtained before, to get
$\gamma_{5}=\gamma_{5^{\prime}}$. In a similar way, we substitute $i=4$ and
$j=5$ and $\gamma_{5}=\gamma_{5^{\prime}}$, and get
$\gamma_{4}=\gamma_{4^{\prime}}$.

It follows that group $G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}$ is
generated by ${\\{{\gamma_{i}}\\}}$, $1\leq i\leq 9$. We now present the dual
graph of the edges in $X^{1}$, see Figure 10, from which one can easily read
the map $G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}\rightarrow S_{8}$
(where $8$ is the number of vertices in the dual graph).

$\textstyle{\bullet}$$\scriptstyle{2}$$\textstyle{\bullet}$$\scriptstyle{9}$$\textstyle{\bullet}$$\scriptstyle{1}$$\scriptstyle{3}$$\textstyle{\bullet}$$\scriptstyle{5}$$\textstyle{\bullet}$$\scriptstyle{7}$$\scriptstyle{6}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\scriptstyle{4}$$\textstyle{\bullet}$$\scriptstyle{8}$
Figure 10. The dual graph $T_{1}$ related to $X^{1}$

The relations that we have in
$G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}$, from the above
simplification, are of type

(23) $\displaystyle[\gamma_{i},\gamma_{j}]=e,\ \ \mbox{if i,j are disjoint
lines;}$ $\displaystyle\langle\gamma_{i},\gamma_{j}\rangle=e,\ \
\mbox{otherwise.}$

We also have two relations
$\gamma_{1}\gamma_{2}\gamma_{1}=\gamma_{3}\gamma_{4}\gamma_{3}$ and
$\gamma_{6}\gamma_{7}\gamma_{6}=\gamma_{8}\gamma_{9}\gamma_{8}$ that relate to
the two cycles in the dual graph. Moreover, relations
$[\gamma_{4},\gamma_{2}\gamma_{5}\gamma_{2}]=e$ and
$[\gamma_{5},\gamma_{6}\gamma_{7}\gamma_{6}]=e$ appear in our presentation,
and they are associated to the two sets of three lines each meeting at a point
in the graph. This means that
$G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}\rightarrow S_{8}$ is an
isomorphism, and since $\pi_{1}({X^{1}_{\operatorname{Gal}}})$ is the kernel
of this map, the fundamental group of the Galois cover of $X^{1}$ is trivial.
∎

###  3.2. The surface $X^{2}$

######  Theorem 3.2.

The fundamental group of the Galois cover related to $X^{2}$ is trivial.

###### Proof.

The group $G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}$ is generated by
${\\{{\gamma_{i},\gamma_{i^{\prime}}}\\}}$ for $i=1,\dots,10$. Following
Figure 3, vertices $V_{1}$, $V_{2}$ and $V_{3}$ give the relations from Lemma
2.2, where $(i,j,k,l)$ equal to $(1,2,3,4)$ for $V_{1}$, to $(4,5,6,7)$ for
$V_{2}$, and to $(7,8,9,10)$ for $V_{3}$. Vertices $V_{4}$ and $V_{5}$ give
the relations from Lemma 2.1, with $(i,j,k)$ equal to $(2,5,8)$ and $(3,6,9)$,
respectively. The vertices $V_{6}$ and $V_{7}$ give the relations
$\gamma_{i}=\gamma_{i^{\prime}}$ for $i=1$ and $i=10$, respectively. The
relations of type (5*) are:

$[\gamma_{i,i^{\prime}},\gamma_{j,j^{\prime}}]=e\ \ \mbox{for}\ \
(i,j)\in\begin{array}[]{l}{({\\{{1}\\}}\times{\\{{5,6,7,8,9,10}\\}})}\cup{({\\{{2}\\}}\times{\\{{6,7,9,10}\\}})}\cup{({\\{{3}\\}}\times{\\{{5,7,8,10}\\}})}\\\
\cup{({\\{{4}\\}}\times{\\{{8,9,10}\\}})}\cup{({\\{{5}\\}}\times{\\{{9,10}\\}})}\cup{({\\{{6}\\}}\times{\\{{8,10}\\}})}.\end{array}$

And as always, the projective relation is
$\prod\limits_{i=10}^{1}\gamma_{i^{\prime}}\gamma_{i}=e$.

Now we explain the simplification of the presentation of
$G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}$. As written above, we
have 20 generators and a quite a few relations. We try a run of the
presentation in Magma and get as an output a simplified presentation of
$G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}$ with eight generators
$\gamma_{1},\gamma_{3^{\prime}},\gamma_{5},\gamma_{5^{\prime}},\gamma_{6^{\prime}},\gamma_{7},\gamma_{9},\gamma_{10}$,
still with a long list of relations.

We construct the dual graph according to the list of generators, see Figure
11.

|
---|---
$\textstyle{\bullet}$$\scriptstyle{5}$$\textstyle{\bullet}$$\scriptstyle{5^{\prime}}$$\textstyle{\bullet}$$\scriptstyle{1}$$\textstyle{\bullet}$$\scriptstyle{3^{\prime}}$$\textstyle{\bullet}$$\scriptstyle{6^{\prime}}$$\textstyle{\bullet}$$\scriptstyle{7}$$\scriptstyle{9}$$\textstyle{\bullet}$$\scriptstyle{10}$$\textstyle{\bullet}$
Figure 11. The dual graph after the first run in Magma

We see in the output that there is a missing relation
$[\gamma_{7},\gamma_{10}]=e$, which can be derived from another relation
$[\gamma_{1}\gamma_{7}\gamma_{6^{\prime}}\gamma_{3^{\prime}}\gamma_{6^{\prime}}\gamma_{1}\gamma_{7},\gamma_{10}]=e$
in the output. We then add it to the presentation, and run Magma for the
second time.

The output of the second run on Magma consists of
$\gamma_{1},\gamma_{2},\gamma_{3^{\prime}},\gamma_{5^{\prime}},\gamma_{6^{\prime}},\gamma_{7},\gamma_{9},\gamma_{10}$,
as generators, again with a long list of relations. We pick up all relations
of type (23) from the simplified presentation, and construct the dual graph
related to these generators, see Figure 12.

$\textstyle{{}e\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\scriptstyle{2}$$\textstyle{f\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{5^{\prime}}$$\textstyle{a\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{7}$$\textstyle{{}d\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{3^{\prime}}$$\textstyle{c\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{6^{\prime}}$$\textstyle{b\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{9}$$\textstyle{g\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{10}$$\textstyle{h}$
Figure 12. The dual graph after the second run in Magma

In the output we could not find the relation
$[\gamma_{5^{\prime}},\gamma_{6^{\prime}}]=e$, but the relation
$[\gamma_{6^{\prime}}\gamma_{3^{\prime}}\gamma_{6^{\prime}},\gamma_{5^{\prime}}]=e$
appearing in the output simplifies to
$[\gamma_{5^{\prime}},\gamma_{6^{\prime}}]=e$. At this point, a cyclic
relation for the cycle appearing in the graph, is still missing.

For $T$, the graph depicted in Figure 12, we get a well-defined surjection
$\varphi:C_{Y}(T)\to G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}$, see
Definition 2.3. This map sends the generator corresponding to an edge $E$ to
the generator $\gamma_{i}$, where $i$ is the label on edge $E$ in Figure 12.
It is well-defined by the definition of $C_{Y}(T)$. By [19], the map
$\psi:C_{Y}(T)\to S_{8}\ltimes A_{1,8}$, defined by

$\gamma_{1}\mapsto(e\ d),\ \ \gamma_{2}\mapsto(e\ f),\ \
\gamma_{3^{\prime}}\mapsto(c\ d),\ \ \gamma_{5^{\prime}}\mapsto(a\ f)u_{a,f},\
\ \gamma_{6^{\prime}}\mapsto(b\ c),$

and

$\gamma_{7}\mapsto(a\ b),\ \ \gamma_{9}\mapsto(b\ g),\ \ \gamma_{10}\mapsto(g\
h)$

is an isomorphism. Here, $S_{8}$ acts on the label set
$L={\\{{a,b,c,d,e,f,g,h}\\}}$, and $A_{1,8}$ is generated by the elements of
the form $u_{x,y}$ (see Subsection 2.5), for $x,y\in L$. So, we are left to
compute $\ker\varphi\circ\psi^{-1}$. This is done by pulling back the defining
relations of $G$ to $S_{8}\ltimes A_{1,8}$, which gives us a unique non-
vanishing relation: $u_{h,f}u_{g,d}=e$. However, we can act with $S_{8}$ on
this relation, and obtain $u_{x,y}=u_{x^{\prime},y^{\prime}}^{-1}$ for any
distinct $x,y,x^{\prime},y^{\prime}\in L$. In particular,
$u_{a,b}^{2}=u_{c,d}u_{d,e}=u_{c,e}=u_{a,b}$, so $u_{a,b}=1$ and all the
generators of $A_{1,8}$ are contained in $\ker\varphi\circ\psi^{-1}$.

We thus proved that $G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}\cong
S_{8}$. In particular, the fundamental group of ${X^{2}_{\operatorname{Gal}}}$
is trivial. ∎

###  3.3. The surface $X^{3}$

######  Theorem 3.3.

The fundamental group of the Galois cover related to $X^{3}$ is trivial.

###### Proof.

Group $G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}$ related to $X^{3}$
is generated by ${\\{{\gamma_{i}}\\}}$ for
$i=1,1^{\prime},2,2^{\prime},\dots,11,11^{\prime}$. Vertices $V_{1}$, $V_{2}$,
$V_{3}$, and $V_{4}$ give the relations from Lemma 2.2, where $(i,j,k,l)$ are
equal to $(1,2,3,4)$ for $V_{1}$, $(3,6,8,10)$ for $V_{2}$, $(4,5,6,7)$ for
$V_{3}$, and $(7,9,10,11)$ for $V_{4}$. Vertices $V_{5}$ and $V_{6}$ give the
relations from Lemma 2.1, with $(i,j,k)$ equals $(2,5,9)$ and $(1,8,11)$,
respectively. The relations of type (5*) are:

$[\gamma_{i,i^{\prime}},\gamma_{j,j^{\prime}}]=e$

for

$(i,j)\in\begin{array}[]{l}{({\\{{1}\\}}\times{\\{{5,6,7,9,10}\\}})}\cup{({\\{{2}\\}}\times{\\{{6,7,8,10,11}\\}})}\cup{({\\{{3}\\}}\times{\\{{5,7,9,11}\\}})}\cup{({\\{{4}\\}}\times{\\{{9}\\}})}\\\
\cup{({\\{{4,5}\\}}\times{\\{{8,10,11}\\}})}\cup{({\\{{6}\\}}\times{\\{{9,11}\\}})}\cup{({\\{{8}\\}}\times{\\{{7,9}\\}})},\end{array}$

and the projective relation is
$\prod\limits_{i=11}^{1}\gamma_{i^{\prime}}\gamma_{i}=e$.

Group $G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}$ now has $22$
generators and a long list of relations. This input was fed into Magma, and
the output is now the group with the generators
$\gamma_{1^{\prime}},\gamma_{2^{\prime}},\gamma_{5},\gamma_{5^{\prime}},\gamma_{6^{\prime}},\gamma_{7},\gamma_{8^{\prime}},\gamma_{9}$,
and $\gamma_{10}$, and a long list of relations. Let us label the triangles of
$X^{3}$, as in Figure 13.

$\textstyle{V_{5}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{2}$$\scriptstyle{5}$$\textstyle{V_{5}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{5}$$\scriptstyle{9}$$\textstyle{V_{6}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\scriptstyle{8}$$\textstyle{V_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ⓐ}$$\scriptstyle{ⓑ}$$\scriptstyle{ⓒ}$$\scriptstyle{ⓗ}$$\scriptstyle{4}$$\scriptstyle{3}$$\textstyle{V_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{6}$$\textstyle{V_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{7}$$\scriptstyle{6}$$\textstyle{V_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ⓖ}$$\scriptstyle{ⓕ}$$\scriptstyle{ⓔ}$$\scriptstyle{ⓓ}$$\scriptstyle{11}$$\scriptstyle{10}$$\textstyle{V_{6}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{8}$$\textstyle{V_{2}}$$\textstyle{V_{2}}$
Figure 13. The shadow $X^{3}$ with triangles labelled

The dual graph $T$ with these generators is now given in Figure 14.

---
$\textstyle{b\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{2^{\prime}}$$\textstyle{a\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{5}$$\scriptstyle{5^{\prime}}$$\textstyle{f\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{7}$$\scriptstyle{9}$$\textstyle{g}$$\textstyle{c\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{8^{\prime}}$$\scriptstyle{1^{\prime}}$$\textstyle{d\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{10}$$\textstyle{e\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{6^{\prime}}$$\textstyle{h}$
Figure 14. The dual graph for the remaining generators

The list of simplified relations we got from Magma contains all the defining
relations of $C_{Y}(T)$ of a type (23) corresponding to $T$ except
$\langle\gamma_{5},\gamma_{7}\rangle=e,\langle\gamma_{5},\gamma_{9}\rangle=e,\langle\gamma_{5^{\prime}},\gamma_{7}\rangle=e,\langle\gamma_{6^{\prime}},\gamma_{10}\rangle=e$
and $\langle\gamma_{7},\gamma_{10}\rangle=e$. It also contains the relations
$\langle\gamma_{6^{\prime}}\gamma_{7}\gamma_{6^{\prime}},\gamma_{5^{\prime}}\rangle=e$,
$\langle\gamma_{6^{\prime}}\gamma_{5^{\prime}}\gamma_{8^{\prime}}\gamma_{6^{\prime}},\gamma_{7}\gamma_{10}\gamma_{8^{\prime}}\gamma_{7}\rangle=e$,
and
$\langle\gamma_{6^{\prime}}\gamma_{8^{\prime}}\gamma_{7}\gamma_{6^{\prime}}\gamma_{8^{\prime}},\gamma_{1^{\prime}}\gamma_{5}\gamma_{2^{\prime}}\gamma_{5}\gamma_{1^{\prime}}\rangle=e$,
which can be simplified to $\langle\gamma_{5^{\prime}},\gamma_{7}\rangle=e$,
$\langle\gamma_{7},\gamma_{10}\rangle=e$, and
$\langle\gamma_{5},\gamma_{7}\rangle=e$ respectively. We thus obtained the
relations of type (23) corresponding to $T$, except
$\langle\gamma_{5},\gamma_{9}\rangle=e$ and
$\langle\gamma_{6^{\prime}},\gamma_{10}\rangle=e$.

Denote by $T_{0}^{\prime}$ the subgraph of $T$ resulting by removing the edges
labeled $6^{\prime}$, $9$, and $5^{\prime}$, and by $T_{0}$ the subgraph of
$T$ resulting by removing the edges labeled $6^{\prime}$, $9$, and $5$. Since
we have derived the relations corresponding to $T_{0}$ and $T_{0}^{\prime}$,
we conclude that there are natural maps $C_{Y}(T_{0})\to
G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}$ and
$C_{Y}(T_{0}^{\prime})\to G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}$.
Those maps agree on the generators that appear in both groups, i.e.,
$\gamma_{1^{\prime}},\gamma_{2^{\prime}},\gamma_{7},\gamma_{8^{\prime}},\gamma_{10}$,
and so we get a map from their amalgamated product. Each of the graphs $T_{0}$
and $T_{0}^{\prime}$ has one cycle, so by [19], $C_{Y}(T_{0})$ and
$C_{Y}(T_{0}^{\prime})$ are both isomorphic to $S_{6}\ltimes A_{1,6}$ by

$\gamma_{1^{\prime}}\mapsto(b\ c),\gamma_{2^{\prime}}\mapsto(a\
b),\gamma_{5}\mapsto(a\ f)u_{a,f},\gamma_{5^{\prime}}\mapsto(a\
f)u_{a,f}^{\prime},\gamma_{7}\mapsto(e\ f),\gamma_{8^{\prime}}\mapsto(c\
d),\gamma_{10}\mapsto(d\ e).$

Here, $S_{6}$ acts on the label set $L={\\{{a,b,c,d,e,f}\\}}$, the copy of
$A_{1,6}$ corresponding to $C_{Y}(T_{0})$ is generated by the elements of the
form $u_{x,y}$ for $x,y\in L$, and the copy of $A_{1,6}$ corresponding to
$C_{Y}(T_{0}^{\prime})$ is generated by elements of the form
$u^{\prime}_{x,y}$ for $x,y\in L$ (see Subsection 2.5). Thus, their
amalgamated product is isomorphic to $S_{6}\ltimes(A_{1,6}*A_{1,6})$, which we
denote by $H$. The resulting map to
$G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}$ is of course not
surjective, so we take the free product with the missing generators to get the
map

$\varphi_{1}:H*\langle\gamma_{6^{\prime}},\gamma_{9}\rangle\twoheadrightarrow
G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}},$

of which we need to find the kernel after adding the remaining relations.

One of the relations in the output pulls back by $\varphi_{1}$ to

$\displaystyle(a\ b\ c)\gamma_{6^{\prime}}(a\ b)(c\ d\
e)\gamma_{6^{\prime}}(a\ b\ c)(d\ e)\gamma_{6^{\prime}}(a\ b)(c\ d\
e)\gamma_{6^{\prime}}(a\ b\ d\ c)\gamma_{6^{\prime}}(a\ b)(c\ e\ d)=e.$

Using the fact that $\gamma_{6^{\prime}}$ commutes with all the permutations
that fix $e$, this relation simplifies to
$\left\langle\gamma_{6^{\prime}},\left(d\;e\right)\right\rangle=e$ which is
the pullback (by $\varphi_{1}$) of
$\langle\gamma_{6^{\prime}},\gamma_{10}\rangle=e$, one of the missing
relations of type (23).

Adding edge $6^{\prime}$ to $T_{0}$ and $T_{0}^{\prime}$ and performing the
procedure described above by counting the newly derived relation
$\langle\gamma_{6^{\prime}},\gamma_{10}\rangle=e$, we get a map

$\varphi_{2}:(S_{7}\ltimes(A_{1,7}*A_{1,7}))*\langle\gamma_{9}\rangle\twoheadrightarrow
G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}.$

Here, $S_{7}$ acts on the label set $L={\\{{a,b,c,d,e,f,h}\\}}$; one of the
copies of $A_{1,7}$ is generated by the elements of the form $u_{x,y}$ and the
other by elements of the form $u_{x,y}^{\prime}$ for $x,y\in L$.

Another relation in the output pulls back by $\varphi_{2}$ to
$u_{f,a}u_{c,d}^{\prime}u_{a,f}u_{d,c}^{\prime}=e$ which, after conjugating
with transpositions, implies that $u_{x,y}$ commutes with $u_{w,z}^{\prime}$
for all distinct $x,y$ and $w,z$ (recall that $u_{y,x}=u_{x,y}^{-1}$).

Next we consider the relation from the output that pulls back by $\varphi_{2}$
to

$\displaystyle
u_{e,a}^{\prime}u_{h,e}u_{a,h}^{\prime}u_{c,a}u_{h,c}^{\prime}u_{a,h}u_{c,e}^{\prime}=e,$

and simplify it. We use $\sim$ to denote conjugacy in the group.

$\displaystyle u_{c,e}$ $\displaystyle\sim$ $\displaystyle
u_{c,h}^{\prime}u_{c,e}u_{h,c}^{\prime}$ $\displaystyle=$ $\displaystyle
u_{c,e}^{\prime}u_{e,h}^{\prime}u_{c,a}u_{a,e}u_{h,c}^{\prime}$
$\displaystyle\sim$ $\displaystyle
u_{e,h}^{\prime}u_{c,a}u_{h,c}^{\prime}u_{a,e}u_{c,e}^{\prime}$
$\displaystyle=$ $\displaystyle
u_{e,h}^{\prime}u_{c,a}u_{h,c}^{\prime}u_{a,h}u_{d,e}u_{h,d}u_{c,e}^{\prime}$
$\displaystyle\sim$ $\displaystyle
u_{h,d}u_{e,h}^{\prime}u_{c,a}u_{h,c}^{\prime}u_{a,h}u_{d,e}u_{c,e}^{\prime}$
$\displaystyle=$ $\displaystyle
u_{h,d}u_{e,a}^{\prime}u_{a,h}^{\prime}u_{c,a}u_{h,c}^{\prime}u_{a,h}u_{d,e}u_{c,e}^{\prime}$
$\displaystyle=$ $\displaystyle
u_{e,a}^{\prime}u_{h,d}u_{d,e}u_{a,h}^{\prime}u_{c,a}u_{h,c}^{\prime}u_{a,h}u_{c,e}^{\prime}=e.$

This implies that $u_{x,y}=e$ for all distinct $x$ and $y$. In particular,
$\gamma_{5}$ pulls back by $\varphi_{2}$ to $(a\;f)$, i.e., $\gamma_{5}$ is a
conjugation on $\gamma_{7}$ by
$\gamma_{2^{\prime}}\gamma_{1^{\prime}}\gamma_{8^{\prime}}\gamma_{10}$. Since
$\gamma_{1^{\prime}},\gamma_{2^{\prime}},\gamma_{8^{\prime}}$, and
$\gamma_{10}$ commute with $\gamma_{9}$, and since
$\langle\gamma_{7},\gamma_{9}\rangle=e$ holds, we conclude that
$\langle\gamma_{5},\gamma_{9}\rangle=e$ holds as well.

Moreover, $\gamma_{5}$ can be expressed as product of other generators. This
means that we have a surjection $C_{Y}(\hat{T})\twoheadrightarrow
G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}$ where $\hat{T}$ is the
subgraph of $T$ that we get by removing edge $5$. We want to find the kernel
of this surjection.

As before, $C_{Y}(\hat{T})$ is isomorphic to $S_{8}\ltimes A_{1,8}$ by

$\gamma_{1^{\prime}}\mapsto(b\ c),\ \ \gamma_{2^{\prime}}\mapsto(a\ b),\ \
\gamma_{5^{\prime}}\mapsto(a\ f)u_{a,f}^{\prime},\ \
\gamma_{6^{\prime}}\mapsto(e\ h),\ \ \gamma_{7}\mapsto(e\ f),$

and

$\gamma_{8^{\prime}}\mapsto(c\ d),\ \ \gamma_{9}\mapsto(f\ g),\ \
\gamma_{10}\mapsto(d\ e).$

Pulling back some of the defining relations of
$G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}$ to $C_{Y}(\hat{T})$ and
using the above isomorphism with $S_{8}\ltimes A_{1,8}$, we get the relations
$u_{f,d}^{\prime 2}=e$ and
$u_{c,g}^{\prime}u_{c,d}^{\prime}u_{f,d}^{\prime}u_{e,a}^{\prime}u_{h,b}^{\prime}=e$.
We can use them to find that $u_{w,z}^{\prime}=e$ for all $w,z$. To this end,
we first conjugate $u_{f,d}^{\prime 2}=e$ by a transposition $(f\ c)$ to get
$u_{c,d}^{\prime 2}=e$, and then simplify the other relation:

(24)
$e=u_{c,g}^{\prime}u_{c,d}^{\prime}u_{f,d}^{\prime}u_{e,a}^{\prime}u_{h,b}^{\prime}=u_{d,g}^{\prime}u_{c,d}^{\prime
2}u_{f,d}^{\prime}u_{e,a}^{\prime}u_{h,b}^{\prime}=u_{f,g}^{\prime}u_{e,a}^{\prime}u_{h,b}^{\prime}.$

Conjugating by the transposition $(h\ c)$ we get

(25) $u_{f,g}^{\prime}u_{e,a}^{\prime}u_{c,b}^{\prime}=e.$

The difference of (24) and (25) is $u_{h,c}^{\prime}=e$, and after conjugating
by elements of $S_{8}$ we get $u_{w,z}^{\prime}=e$ for all $w,z$. This implies
that $A_{1,8}$ is contained in the kernel we are computing, and therefore
$G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}\cong S_{8}$. This means
that the fundamental group of the Galois cover related to $X^{3}$ is trivial.
∎

###  3.4. The surface $X^{4}$

Finally, we consider a surface $X$ whose degeneration is the shadow $X^{4}$.
Recall that $G=\pi_{1}(\mathbb{C}\mathbb{P}^{2}-S)$ is the fundamental group
of the complement of the branch curve. We let $\bar{G}$ denote
$\pi_{1}({X_{\operatorname{Gal}}})=G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}$.

######  Theorem 3.4.

The fundamental group $\bar{G}$ of the Galois cover related to $X^{4}$ is an
abelian-by-abelian group of order $2^{23}$. (In fact, a central extension
$1\rightarrow{\mathbb{Z}}_{2}^{3}\rightarrow\bar{G}\rightarrow{\mathbb{Z}}_{2}^{20}\rightarrow
1$.)

###### Proof.

Group $G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}$, in this case, is
generated by ${\\{{\gamma_{i}}\\}}$ for
$i=1,1^{\prime},2,2^{\prime},\dots,12,12^{\prime}$. This time the degree of
each of the six vertices $V_{1},\dots,V_{6}$ is $4$, and the relations they
induce are as in Lemma 2.2. For $V_{1}$ we have $(i,j,k,l)=(1,2,3,4)$, for
$V_{2}$ we have $(i,j,k,l)=(3,6,8,11)$, for $V_{3}$ we have
$(i,j,k,l)=(4,5,6,7)$, for $V_{4}$ we have $(i,j,k,l)=(7,10,11,12)$, for
$V_{5}$ the indices are $(i,j,k,l)=(2,5,9,10)$, and finally, for $V_{6}$, we
have $(i,j,k,l)=(1,8,9,12)$. The relations of type (5*) are:

$[\gamma_{i,i^{\prime}},\gamma_{j,j^{\prime}}]=e$

for

$(i,j)\in\begin{array}[]{l}{({\\{{1}\\}}\times{\\{{5,6,7,10,11}\\}})}\cup{({\\{{2}\\}}\times{\\{{6,7,8,11,12}\\}})}\cup{({\\{{3}\\}}\times{\\{{5,7,9,10,12}\\}})}\\\
\cup{({\\{{4}\\}}\times{\\{{8,9,10,11,12}\\}})}\cup{({\\{{5}\\}}\times{\\{{8,11,12}\\}})}\cup{({\\{{6}\\}}\times{\\{{9,10,12}\\}})}\\\
\cup{({\\{{7}\\}}\times{\\{{8,9}\\}})}\cup{({\\{{8}\\}}\times{\\{{10}\\}})}\cup{({\\{{9}\\}}\times{\\{{11}\\}})}.\end{array}$

The projective relation is, as before,
$\prod\limits_{i=12}^{1}\gamma_{i^{\prime}}\gamma_{i}=e$.

Group $G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}$ now has 24
generators and a long list of relations. We used Magma to simplify the
relations. Again, some generators can be expressed in terms of others, and we
are left with the generators
$\gamma_{2^{\prime}},\gamma_{3^{\prime}},\gamma_{4},\gamma_{5},\gamma_{5^{\prime}},\gamma_{6^{\prime}},\gamma_{7},\gamma_{8^{\prime}},\gamma_{10},\gamma_{10^{\prime}}$.
We label the triangles as in Figure 15, and consider the dual graph which is
generated by these generators, see Figure 16.

$\textstyle{V_{5}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{9}$$\scriptstyle{2}$$\scriptstyle{5}$$\textstyle{V_{5}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{5}$$\scriptstyle{10}$$\scriptstyle{9}$$\textstyle{V_{6}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\scriptstyle{8}$$\textstyle{V_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ⓒ}$$\scriptstyle{ⓑ}$$\scriptstyle{ⓖ}$$\scriptstyle{ⓗ}$$\scriptstyle{4}$$\scriptstyle{3}$$\textstyle{V_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{6}$$\textstyle{V_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{7}$$\scriptstyle{6}$$\textstyle{V_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ⓐ}$$\scriptstyle{ⓓ}$$\scriptstyle{ⓔ}$$\scriptstyle{ⓕ}$$\scriptstyle{12}$$\scriptstyle{11}$$\textstyle{V_{6}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{8}$$\textstyle{V_{2}}$$\textstyle{V_{2}}$ Figure 15. $X^{4}$ with labelled triangles |
---|---
$\textstyle{b\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{2^{\prime}}$$\textstyle{c\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{4}$$\scriptstyle{5}$$\scriptstyle{5^{\prime}}$$\textstyle{d\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{7}$$\scriptstyle{10}$$\scriptstyle{10^{\prime}}$$\textstyle{a}$$\textstyle{f\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{8^{\prime}}$$\textstyle{g\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{3^{\prime}}$$\textstyle{h\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{6^{\prime}}$$\textstyle{e}$
Figure 16. The dual graph after the first run in Magma

We collected all relations that are of form (23), noticing that relations
$\langle\gamma_{5},\gamma_{10^{\prime}}\rangle=e,\langle\gamma_{5},\gamma_{10}\rangle=e$,
and $\langle\gamma_{5},\gamma_{7}\rangle=e$ are missing. Relation
$\langle\gamma_{10^{\prime}},\gamma_{5^{\prime}}\gamma_{5}\gamma_{2^{\prime}}\gamma_{5}\gamma_{5^{\prime}}\gamma_{5}\gamma_{2^{\prime}}\gamma_{5}\gamma_{5^{\prime}}\rangle=e$
from the output can be simplified to
$\langle\gamma_{5},\gamma_{10^{\prime}}\rangle=e$, and this is enough for us
to run Magma on the original presentation, with this additional relation,
which is now proven. We now get another simplified presentation for
$G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}$, this time with the
generators
$\gamma_{1^{\prime}},\gamma_{2^{\prime}},\gamma_{3^{\prime}},\gamma_{5},\gamma_{5^{\prime}},\gamma_{7},\gamma_{8^{\prime}},\gamma_{9},\gamma_{11},\gamma_{11^{\prime}}$,
whose graph, denoted by $T$, is depicted in Figure 17.

---
$\textstyle{{}a\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{9}$$\textstyle{b\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1^{\prime}}$$\scriptstyle{2^{\prime}}$$\textstyle{c\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{5}$$\scriptstyle{5^{\prime}}$$\textstyle{d\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{7}$$\textstyle{{}h\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{3^{\prime}}$$\textstyle{g\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{8^{\prime}}$$\textstyle{f\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{11}$$\scriptstyle{11^{\prime}}$$\textstyle{e}$
Figure 17. The dual graph after the second run in Magma

We want to verify that our group is a quotient of $C_{Y}(T)$, where $T$ is
given in Figure 17. First we notice that the relations
$\langle\gamma_{2^{\prime}},\gamma_{9}\rangle=e,\langle\gamma_{5},\gamma_{7}\rangle=e$,
$[\gamma_{5},\gamma_{9}]=e$ and $[\gamma_{5^{\prime}},\gamma_{9}]=e$ are
missing; but these can be deduced using other relations from the output; for
example, we simplified
$\langle\gamma_{2^{\prime}}\gamma_{5^{\prime}}\gamma_{2^{\prime}},\gamma_{9}\rangle=e$
to get $\langle\gamma_{2^{\prime}},\gamma_{9}\rangle=e$. Thereafter, we proved
the following relations:

$\displaystyle{}[\gamma_{1^{\prime}}\gamma_{2^{\prime}}\gamma_{1^{\prime}},\gamma_{9}]$
$\displaystyle=$ $\displaystyle e$
$\displaystyle{}[\gamma_{1^{\prime}}\gamma_{3^{\prime}}\gamma_{1^{\prime}},\gamma_{8^{\prime}}]$
$\displaystyle=$ $\displaystyle e$
$\displaystyle{}\langle\gamma_{5^{\prime}},\gamma_{2^{\prime}}\gamma_{5}\gamma_{2^{\prime}}\rangle$
$\displaystyle=$ $\displaystyle e$
$\displaystyle{}\langle\gamma_{5^{\prime}},\gamma_{7}\gamma_{5}\gamma_{7}\rangle$
$\displaystyle=$ $\displaystyle e$
$\displaystyle{}\langle\gamma_{11^{\prime}},\gamma_{7}\gamma_{11}\gamma_{7}\rangle$
$\displaystyle=$ $\displaystyle e$
$\displaystyle{}\langle\gamma_{11^{\prime}},\gamma_{8^{\prime}}\gamma_{11}\gamma_{8^{\prime}}\rangle$
$\displaystyle=$ $\displaystyle e$
$\displaystyle{}[\gamma_{2^{\prime}}\gamma_{5^{\prime}}\gamma_{2^{\prime}},\gamma_{7}\gamma_{5}\gamma_{7}]$
$\displaystyle=$ $\displaystyle e$
$\displaystyle{}[\gamma_{8^{\prime}}\gamma_{11^{\prime}}\gamma_{8^{\prime}},\gamma_{7}\gamma_{11}\gamma_{7}]$
$\displaystyle=$ $\displaystyle e.$

By that, we verified all the defining relations (19)–(22), so our group is
indeed a quotient of $C_{Y}(T)$.

Let $F_{t,n}$ be the group defined in Subsection 2.5. As explained in
Subsection 2.4, $C_{Y}(T)\,\cong\,S_{8}\ltimes F_{3,8}$. To compute
$G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}$, it remains to cast all
the relations in the language of the generators of $F_{3,8}$. Recall that
$F_{3,8}$ is a subgroup of the direct product $({\mathbb{F}}_{3})^{8}$; let
$u_{i},v_{i},w_{i}$ be the generators of the $i$th component. So the free
groups ${\langle{u_{i},v_{i},w_{i}}\rangle}$ commute elementwise. In this
language, $F_{3,8}$ is generated by all the elements $u_{j}u_{i}^{-1}$,
$v_{j}v_{i}^{-1}$ and $w_{j}w_{i}^{-1}$.

Based on the spanning subgraph $T_{0}$ that was obtained from $T$ by removing
edges $5$, $5^{\prime}$ and $11^{\prime}$, the final stage is to reinterpret
the defining relations of
$\bar{G}=G/{{\langle{{\langle{\gamma^{2}}\rangle}}\rangle}}$, which we do by
mapping $\bar{G}$ to a quotient of $S_{8}\ltimes F_{3,8}$ by

$\gamma_{1^{\prime}}\mapsto(bg),\ \ \gamma_{2^{\prime}}\mapsto(bc),\ \
\gamma_{3^{\prime}}\mapsto(gh),\ \ \gamma_{7}\mapsto(de),\ \
\gamma_{8^{\prime}}\mapsto(fg),\ \ \gamma_{9}\mapsto(ab)$

and

$\gamma_{5}\mapsto(cd)u_{c}u_{d}^{-1},\ \
\gamma_{5^{\prime}}\mapsto(cd)v_{c}v_{d}^{-1},\ \
\gamma_{11^{\prime}}\mapsto(ef)w_{e}w_{f}^{-1}.$

As may be expected, most of the defining relations of $\bar{G}$ now become
trivial. One exception is the relation $\omega=e$, where

(26)
$\omega=[u_{d},v_{d}][v_{c},w_{c}]u_{a}^{-1}w_{a}v_{b}^{-1}u_{b}w_{b}^{-1}v_{b}u_{c}w_{c}w_{d}^{-1}u_{d}^{-1}w_{e}^{-1}u_{e}^{-1}w_{f}u_{f}^{-1}u_{g}w_{g}u_{h}w_{h}^{-1}.$

The other nontrivial relations are much nicer, and by acting with $S_{8}$, we
obtain the following rules:

(27) $\displaystyle{}[u_{i},v_{i}]$ $\displaystyle=$
$\displaystyle[u_{j},v_{j}],$ (28) $\displaystyle{}[u_{i},w_{i}]$
$\displaystyle=$ $\displaystyle[u_{j},w_{j}],$ (29)
$\displaystyle{}[v_{i},w_{i}]$ $\displaystyle=$ $\displaystyle[v_{j},w_{j}],$
(30) $\displaystyle{}(u_{i})^{2}$ $\displaystyle=$ $\displaystyle(u_{j})^{2},$
(31) $\displaystyle{}(v_{i})^{2}$ $\displaystyle=$ $\displaystyle(v_{j})^{2},$
(32) $\displaystyle{}(w_{i})^{2}$ $\displaystyle=$ $\displaystyle(w_{j})^{2}.$

Let $C^{*}$ and $C$, respectively, denote the quotient of $F^{*}_{3,8}$ and
$F_{3,8}$ by the relations (27)–(32). It is easier to compute in $C^{*}$:
Write $[u,v]$ for all $[u_{i},v_{i}]$, and likewise for $[u,w]$, $[v,w]$, and
the squares $u^{2},v^{2},w^{2}$. We have that
$v_{i}^{2}=v_{j}^{2}=u_{i}v_{j}^{2}u_{i}^{-1}=u_{i}v_{i}^{2}u_{i}^{-1}=[u,v]^{2}v_{i}^{2}$,
so $[u,v]^{2}=1$, and similarly, $[u,w]^{2}=[v,w]^{2}=1$. Modulo the
commutators $[u,v]$, $[v,w]$, and $[w,u]$, which are central, the group is
abelian.

######  Corollary 3.5.

Groups $C$ and $C^{*}$ are central extensions, as described in the following
commutative diagram:

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{Z}}_{2}^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{ab}}$$\textstyle{({\mathbb{Z}}_{2}^{3})^{7}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{Z}}_{2}^{3}\times{\mathbb{Z}}^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{ab}}$$\textstyle{({\mathbb{Z}}_{2}^{3})^{8}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{Z}}^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cdot
2}$$\textstyle{{\mathbb{Z}}^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{Z}}_{2}^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{1}$$\textstyle{1}$$\textstyle{1}$

Here the group at the upper-left corner is generated by the commutator; the
arrows so-marked are the abelianization maps, and the middle row is induced by
the short exact series

$1\rightarrow F_{3,8}\rightarrow
F_{3,8}^{*}\rightarrow\mathbb{Z}^{3}\rightarrow 1$

described in Definition 2.5.

Notice that (for distinct $i,j,k$) $u_{i}u_{j}^{-1}$ commutes with
$v_{i}v_{j}^{-1}$ in $C$, but $[u_{i}u_{j}^{-1},v_{i}v_{k}^{-1}]=[u,v]$. In
light of Corollary 3.5 we can simplify the element $\omega$ defined in (26).
The $b$-entry of this element relation cancels the commutators, and we get
$\omega=u_{a}^{-1}w_{a}u_{b}w_{b}^{-1}u_{c}w_{c}w_{d}^{-1}u_{d}^{-1}w_{e}^{-1}u_{e}^{-1}w_{f}u_{f}^{-1}u_{g}w_{g}u_{h}w_{h}^{-1}$.
Using $u_{i}^{2}=u_{j}^{2}$ and $w_{i}^{2}=w_{j}^{2}$, as well as
$[u,w]^{2}=1$. We then get a further simplification to

(33)
$\omega=[u,w](u_{a}u_{b}u_{c}u_{d}u_{e}^{-1}u_{f}^{-1}u_{g}^{-1}u_{h}^{-1})(w_{a}w_{b}w_{c}w_{d}w_{e}^{-1}w_{f}^{-1}w_{g}^{-1}w_{h}^{-1}).$

Again, by (30) and (32), this element is symmetric under the action of $S_{8}$
on the indices, and has order $2$, so $C/{\langle{\omega}\rangle}$ has order
$2^{23}$. Recall that this is the group
$\pi_{1}({X^{4}_{\operatorname{Gal}}})$, whose computation is the main goal of
this paper.

Recalling that the fundamental group of the complement of the branch curve is
the semidirect product of this group with $S_{8}$, it is desirable to describe
the $S_{8}$-module structure of the abelianization.

######  Remark 3.6.

Considering all groups as $S_{8}$-modules, the abelianization
$C^{*}/[C^{*},C^{*}]$ is a direct sum of three copies of the natural module
$V={\mathbb{Z}}_{2}^{8}$, namely ${\langle{u_{i}}\rangle}$,
${\langle{v_{i}}\rangle}$, and ${\langle{w_{i}}\rangle}$.

For the sake of comparison, note that for $p>2$, ${\mathbb{Z}}_{p}^{8}$ is a
direct sum of two irreducible submodules,
${\mathbb{Z}}_{p}^{8}=({\mathbb{Z}}_{p}^{8})_{0}\oplus D$, where the left
component is the subspace of zero-sum vectors and $D$ is the “diagonal”
subspace, spanned by $(1,\dots,1)$. On the other hand, for $p=2$ the
irreducible modules are $0\subset D\subset V_{0}\subset V$, of dimensions
$0<1<7<8$ over ${\mathbb{Z}}_{2}$, respectively. Now, while
$C^{*}/[C^{*},C^{*}]=V\oplus V\oplus V$, it follows from Corollary 3.5 that
the abelianization of $C$ is $C/[C,C]\cong V_{0}\oplus V_{0}\oplus V_{0}$.

Finally, $\omega$ glues the $u$ and $w$ copies of $D$ in this sum, so that the
abelianization of $\pi_{1}({X^{4}_{\operatorname{Gal}}})$, is

$(C/{\langle{\omega}\rangle})/[C/{\langle{\omega}\rangle},C/{\langle{\omega}\rangle}]\cong
V_{0}\oplus(V_{0}\oplus V_{0})/D(1,1),$

where $D(1,1)$ is the diagonal copy of $D$ in the direct sum $D\oplus D\subset
V_{0}\oplus V_{0}$.

∎

## References

  * [1] M. Amram, Non-planar degenerations and related fundamental groups, submitted.
  * [2] M. Amram, D. Goldberg, Higher degree Galois covers of $\mathbb{CP}^{1}\times T$, Algebr. Geom. Topol. 4 (2004), 841–859. Zbl 1069.14065
  * [3] M. Amram, C. Gong, U. Sinichkin, S.-L. Tan, W.-Y. Xu, M. Yoshpe, Fundamental group of Galois covers of degree $6$ surfaces, (2021), https://doi.org/10.1142/S1793525321500412.
  * [4] M. Amram, C. Gong, S. L. Tan, M. Teicher, W.-Y. Xu, On the fundamental groups of Galois covers of planar Zappatic deformations of type $E_{k}$, Int. J. Algebr. Comput. 29 (2019), 905–925.
  * [5] M. Amram, C. Gong, M. Teicher , W.-Y. Xu, Fundamental group of Galois covers of degree $5$ surfaces, Turkish Journal of Mathematics 45(4) (2021), 1517–1542.
  * [6] M. Amram, R. Lawrence, U. Vishne, Artin covers of the braid groups, Journal of Knot Theory and Its Ramifications Vol. 21 (7) (2012), 44 pages, https://doi.org/10.1142/S0218216512500617.
  * [7] M. Amram, S. Ogata, Toric varieties–degenerations and fundamental groups, Michigan Math. J. 54 (2006), 587–610.
  * [8] M. Amram, M. Teicher, U. Vishne, The fundamental group of the Galois cover of the surface $T\times T$, International J. of Algebra and Computation, 18(8) (2008), 1259–1282.
  * [9] M. Amram, S.-L. Tan, W.-Y. Xu, M. Yoshpe, Calculating the Fundamental Group of Galois Cover of the (2,3)-embedding of ${\mathbb{C}}{\mathbb{P}}^{1}\times T$, Acta Math. Sin. 36 (2020), 273–291.
  * [10] A. Calabri, C. Ciliberto, F. Flamini, et al., On the $K^{2}$ of degenerations of surfaces and the multiple point formula, Ann. of Math. 165 (2007), 335–395.
  * [11] A. Calabri, C. Ciliberto, F. Flamini, et al., On degenerations of surfaces, (2008), arXiv:math/0310009v2.
  * [12] N. Cohen, “Local covers of the symmetric groups”, MSc thesis, Bar Ilan University, in preparation.
  * [13] C. Ciliberto, A. Lopez, R. Miranda, Projective degenerations of K3 surfaces, Gaussian maps, and Fano threefolds, Invent. Math. 114 (1993), 641–667.
  * [14] C. Liedtke, Fundamental groups of Galois closures of generic projections, Trans. Amer. Math. Soc. 362 (2010), 2167–2188. Zbl 1198.14017
  * [15] B. Moishezon, M. Teicher, Simply connected algebraic surfaces of positive index, Invent. Math. 89 (1987), 601–643.
  * [16] Moishezon, B., Robb, A., Teicher, M., On Galois covers of Hirzebruch surfaces, Math. Ann. 305 (1996), 493-539.
  * [17] B. Moishezon, M. Teicher, Braid group technique in complex geometry III: Projective degeneration of $V_{3}$, Contemp. Math. 162 (1993), 313–332.
  * [18] B. Moishezon, M. Teicher, Braid group technique in complex geometry IV: Braid monodromy of the branch curve $S_{3}$ of $V_{3}\rightarrow{\mathbb{C}}{\mathbb{P}}^{2}$ and application to $\pi_{1}({\mathbb{C}}{\mathbb{P}}^{2}-S_{3},\ast)$, Contemp. Math. 162 (1993), 332–358.
  * [19] L. Rowen, M. Teicher, U. Vishne, Coxeter covers of the symmetric groups, _J. Group Theory_ 8 (2005), 139–169. Zbl 1120.20040
  * [20] E. R. Van Kampen, On the fundamental group of an algebraic curve, Amer. J. Math. 55 (1933), 255–260.