# The fundamental group of quotients of products of some topological spaces by a finite group — A generalization of a theorem of Bauer-Catanese-Grunewald-Pignatelli

Rodolfo Aguilar Aguilar

**Abstract.** We provide a description of the fundamental group of the quotient of a product of topological spaces $X_i$, each admitting a universal cover, by a finite group $G$, provided that there is only a finite number of path-connected components in $X_i^g$ for every $g \in G$. This generalizes previous work of Bauer-Catanese-Grunewald-Pignatelli and Dedieu-Perroni.

**Keywords.** fundamental group; quotients by finite group; orbifolds

**2020 Mathematics Subject Classification.** 14F35; 54B15; 18A32; 20E34

[Français]

Le groupe fondamental de quotients de produits de certains espaces topologiques par un groupe fini — Généralisation d'un théorème de Bauer-Catanese-Grunewald-Pignatelli

**Résumé.** Nous fournissons une description du groupe fondamental du quotient d'un produit d'espaces topologiques $X_i$, chacun admettant un revêtement universel, par un groupe fini $G$, pourvu qu'il n'existe qu'un nombre fini de composantes connexes par arcs dans $X_i^g$ pour chaque $g \in G$. Cela généralise des résultats antérieurs de Bauer-Catanese-Grunewald-Pignatelli et de Dedieu-Perroni.

Received by the Editors on April 23, 2020.

Accepted on September 1, 2021.

Rodolfo Aguilar Aguilar

Université Grenoble-Alpes, Institut Fourier, 100 rue de Maths, 384610, Gières, France.

e-mail: rodolfo.aguilar-aguilar@univ-grenoble-alpes.fr

Partially supported by ERC ALKAGE, grant No. 670846 and ANR project Hodgefun ANR-16-CE40-0011-01.

© by the author(s)

This work is licensed under http://creativecommons.org/licenses/by-sa/4.0/
# Contents

<table><tr><td>1. Introduction.</td><td>2</td></tr><tr><td>2. Preliminaries</td><td>3</td></tr><tr><td>3. The fundamental group of the product of topological spaces</td><td>5</td></tr><tr><td>4. Applications.</td><td>10</td></tr><tr><td>References.</td><td>13</td></tr></table>

## 1. Introduction

The fundamental group of a quotient of a Haussdorf space $X$ by a finite group $G$ acting freely can be computed noticing that $X \to X/G$ is a covering map, and then using the long exact sequence of homotopy groups associated to a fibration. When $X = X_1 \times \cdots \times X_k$ and $G$ acts on each $X_i$ freely and diagonally on $X$, the fundamental group of $X_1 \times \cdots \times X_k$ sits as a finite-index normal subgroup of $\pi_1(X/G)$.

In the case where each $X_i$ is a projective smooth curve and the action of $G$ is only *faithful*, the following Theorem was shown in [BCGP12].

**Theorem 1.1.** [BCGP12, Theorems 0.10 and 4.1] Let $C_1, \dots, C_k$ be smooth projective curves and let $G$ be a finite group acting faithfully by automorphisms on each of them. Consider the diagonal action of $G$ on the product $C_1 \times \cdots \times C_k$, then the fundamental group of $(C_1 \times \cdots \times C_k)/G$ admits a normal finite index subgroup $N$ isomorphic to a product of fundamental groups of smooth projective curves.

It was later extended in [DP12] to the case when the action of $G$ is non-necessarily faithful. There, they quotient $G$ to obtain a group acting faithfully, follow the subsequent arguments and then extend again to $G$.

Let us explain briefly the method of proof in [BCGP12]. First, they consider the orbifold surface groups $T_i$ of $C_i/G$, which are an extension of $G$ by $\pi_1(C_i)$ and hence come with a surjective morphism to $G$ (see Subsection 2.2.1). They show that the fundamental group $\pi_1((C_1 \times \cdots \times C_k)/G)$ is isomorphic to the quotient of the fiber product $\mathbb{H} := T_1 \times_G \cdots \times_G T_k$ by the normal subgroup $\text{Tors}(\mathbb{H})$ generated by the elements of torsion.

The second part relies on the following proposition whose proof uses abstract group theoretic arguments.

**Proposition 1.2.** [DP12, Proposition 3.5] There exists a short exact sequence of groups

$$1 \to E \to \mathbb{H}/\text{Tors}\mathbb{H} \to T \to 1$$

where $E$ is finite and $T$ is a group of finite index in a product of orbifold surfaces groups.

They finally use Proposition 1.2 and properties of the orbifold surface groups such as residually finiteness and cohomological goodness to construct a subgroup of $\mathbb{H}/\text{Tors}\mathbb{H}$ intersecting $E$ trivially and satisfying the required properties.

Here, a more geometric approach is used via fundamental groups of stacks or orbispaces [Noo05, Che01]. This theory permits to see $X \to [X/G]$ as a covering map under some conditions on $X$, where $[X/G]$ denotes the quotient stack, and a long exact sequence of homotopy groups is available. We will denote the fundamental group of the stack $[X/G]$ by $\pi_1([X/G])$.

For $i = 1, \dots, k$, let $X_i$ be a connected, locally path-connected and semi-locally simply connected topological space with an action of a finite group $G$, consider the diagonal action of $G$ in $X := X_1 \times \cdots \times X_k$
and denote by $I$ the subgroup generated by the elements having a fixed point in every $X_i$ for $i = 1, \dots, k$. We
can formulate now our first main Theorem.

**Theorem 1.3.** Let $X, X_1, \dots, X_k$ and $G$ as above. Suppose that the number of path connected components in the fixed locus set $X_i^g$ of an element $g \in G$ is finite for every $g \in G$ and $i = 1, \dots, k$. Then there exists a homomorphism

$$
\pi_1(X/G) \to \prod_{i=1}^{k} \pi_1((X_i/I)/(G/I))
$$

which has finite kernel and its image is a finite-index subgroup.

This can be seen as a generalization of Proposition 1.2 (Bauer-Catanese-Grunewald-Pignatelli) by the
remarks preceding the statement of the Proposition.

The action of $G/I$ on $X_i/I$ is induced by the action of $G$ in $X_i$. Note that if $k=1$ then $G/I$ can be
seen to act freely on $X_1/I$ and $\pi_1((X_1/I)/(G/I)) = \pi_1((X_1/I)/(G/I))$ but $(X_1/I)/(G/I) \cong X_1/G$. The same
argument works if we make the product of the same topological space, which gives the following corollary.

**Corollary 1.4.** Let $X_i = X_1$ for $i = 2, \dots, k$ and $G$ satisfy the hypothesis of the above theorem. Then the homomorphism $\pi_1(X/G) \to \pi_1(X_1/G)^k$ has finite kernel and its image is a finite-index subgroup.

An important case of Theorem 1.3 and Corollary 1.4 is when $X_i$ is a smooth complex algebraic variety for $i = 1, \dots, k$. Indeed, the fundamental group of a variety with quotient singularities is the fundamental group of a smooth variety.

Our second main Theorem generalizes Theorem 1.1 (Bauer-Catanese-Grunewald-Pignatelli). It can be
stated without using the language of stacks or orbispaces.

**Theorem 1.5.** Let $X, X_1, \dots, X_k$ and $G$ satisfy the hypothesis of Theorem 1.3. Suppose that $\pi_1(X/G)$ is residually finite. Then there exists a normal finite-index subgroup $N \triangleleft \pi_1(X/G)$ isomorphic to a product:

$$
\mathcal{N} \cong \prod_{i=1}^{k} H_{i}.
$$

with $H_i \triangleleft \pi_1(X_i/I)$ normal finite-index subgroups.

As a corollary, following closely the arguments used in [BCGP12], we show that for smooth curves $C_1, \dots, C_k$ the group $\pi_1((C_1 \times \dots \times C_k)/G)$ is residually finite. Hence, we have that Theorem 1.1 (Bauer-Catanese-Grunewald-Pignatelli, Dedieu-Perroni) is valid in the case when the curve is non-necessarily compact.

**Corollary 1.6.** Let $C_1, \dots, C_k$ be smooth algebraic curves and let $G$ be a finite group acting on each of them. Then there exists a normal subgroup $N \triangleleft \pi_1((C_1 \times \dots \times C_k)/G)$ of finite index, isomorphic to a product $\Pi_1 \times \dots \times \Pi_k$, where $\Pi_j$ is either the fundamental group of a smooth projective curve or a free group of finite rank.

The paper is organized as follows: in Section 2 preliminary results are given. Then the first main Theorem is proved in Section 3 and the proof of the second main Theorem together with some applications are given in Section 4.

2. Preliminaries

2.1. Properties of fundamental group of topological stacks

Let $X$ be a connected, semi-locally simply connected and locally path-connected topological space and $G$
a finite group acting continuously on it.
**2.1.1. Fiber homotopy exact sequence.**— There exists a homotopy theory for stacks and the existence of the long exact sequence of homotopy, see [Noo14], is more general than what follows, however we only need the following case: consider the topological stack $\mathcal{X} = [X/G]$, a point $x \in X$ and denote by $\bar{x} \in \mathcal{X}$ the image of $x$. We have an associated fibration $G \to X \to \mathcal{X}$ and a long exact sequence of homotopy groups,

$$ \dots \to \pi_{n+1}(\mathcal{X}, \bar{x}) \to \pi_n(G, \mathrm{Id}) \to \pi_n(X, x) \to \pi_n(\mathcal{X}, \bar{x}) \to \pi_{n-1}(G, \mathrm{Id}) \to \dots $$

the map $\pi_n(G, \mathrm{Id}) \to \pi_n(X, x)$ is induced by the orbit $G \cdot x \hookrightarrow X$.

**2.1.2. Action on the universal cover.**— The hypothesis made on $X$ ensures that there exists an universal cover $\tilde{X}$ and moreover, if we let $\mathcal{X} = [X/G]$ as in §2.1.1, we have an action of $\pi_1(\mathcal{X}, \bar{x})$ on $\tilde{X}$ (see §3.2.1). We will use several times the following lemma in what follows.

**Lemma 2.1.** Consider the action of $\pi_1(\mathcal{X}, \bar{x})$ in $\tilde{X}$, let $y \in \tilde{X}$ and denote by $I_y$ the isotropy group of the action. Then there exists a monomorphism $I_y \to G$.

*Proof.* By §2.1.1 we obtain a short exact sequence

$$ 1 \to \pi_1(X, x) \to \pi_1(\mathcal{X}, \bar{x}) \to G \to 1, $$

as the action of $\pi_1(X, x)$ on $\tilde{X}$ is free, we obtain that the restriction of $\pi_1(\mathcal{X}, \bar{x}) \to G$ to $I_y$ is injective. $\square$

**2.2. Product of topological spaces**

**2.2.1. Fundamental group of the quotient of a product.**— For $i = 1, \dots, k$ let $X_i$ as in §2.1 be a connected, semi-locally simply connected and locally path-connected topological space and $G$ a finite group acting on each of them.

By §2.1.1 we have $k$ exact sequences

$$ (2.1) \qquad 1 \to \pi_1(X_i, x_i) \to \pi_1(\mathcal{X}_i, \bar{x}_i) \xrightarrow{\varphi_i} (G, \mathrm{Id}) \to 1 $$

where $\mathcal{X}_i = [X_i/G]$, $x_i \in X_i$ and its image in $\mathcal{X}_i$ is denoted by $\bar{x}_i$.

Denote by $\mathbb{H} := \pi_1(\mathcal{X}_1, x_1) \times_G \dots \times_G \pi_k(\mathcal{X}_k, x_k)$. The exact sequences in (2.1) can be assembled as follows

$$ (2.2) \qquad 1 \to \pi_1(X_1 \times \dots \times X_k, x) \to \mathbb{H} \to G \to 1 $$

with $x = (x_1, \dots, x_k)$. The geometric nature of $\mathbb{H}$ is shown in the following lemma.

**Lemma 2.2.** Let $G$ act diagonally on $X = X_1 \times \dots \times X_k$. Consider the stack $\mathcal{X} = [X/G]$ then $\pi_1(\mathcal{X}, \bar{x}) \cong \mathbb{H}$.

*Proof.* We have natural projection maps $\mathcal{X} \to \mathcal{X}_i$ for $i = 1, \dots, k$, which together with the morphisms $\varphi_i : \pi_1(\mathcal{X}_i, \bar{x}_i) \to G$ and the universal property of the fiber product give us a morphism $\pi_1(\mathcal{X}, \bar{x}) \to \mathbb{H}$. By the exact sequence of a fibration §2.1.1 applied to the action of $G$ to $X_1 \times \dots \times X_k$ and by (2.2) we obtain

$$ \begin{tikzcd} 1 \arrow[r] & \pi_1(X_1 \times \dots \times X_k, x) \arrow[r] \arrow[d, "\mathrm{id}"] & \pi_1(\mathcal{X}, \bar{x}) \arrow[r] \arrow[d, "\mathrm{id}"] & G \arrow[r] \arrow[d, "\mathrm{id}"] & 1 \\ 1 \arrow[r] & \pi_1(X_1 \times \dots \times X_k, x) \arrow[r] & \mathbb{H} \arrow[r] & G \arrow[r] & 1 \end{tikzcd} $$

which implies the result. $\square$

**Lemma 2.3.** Let $X, X_i$ and $G$ be as above. Then

$$ \pi_1(X/G, [x]) \cong \pi_1(\mathcal{X}, \bar{x})/N \cong \pi_1(\mathcal{X}, \bar{x})/\mathbb{I} $$

where $N$ is the normal subgroup generated by the image of the inertia groups $I_x$ and $\mathbb{I}$ is the subgroup generated by the elements of $\pi_1(\mathcal{X})$ having fixed points in the universal cover of $X_1 \times \dots \times X_k$.
*Proof.* By [Noo08, Theorem 8.3(i)] we have that $\pi_1(X/G, [x]) \cong \pi_1(\mathcal{X}, \bar{x})/N$. The group $\pi_1(\mathcal{X}, \bar{x})$ acts on $\tilde{X} \cong \tilde{X}_1 \times \cdots \times \tilde{X}_k$ the universal cover of $X_1 \times \cdots \times X_k$ in such a way that $[(\tilde{X}_1 \times \cdots \times \tilde{X}_k)/\pi_1(\mathcal{X}, \bar{x})] \cong \mathcal{X}$. As $G$ is finite, by Lemma 2.1 any stabilizer $I_x$ for the action of $\pi_1(\mathcal{X})$ on $\tilde{X}$ is finite, therefore it has the slice property and by [Noo08, Theorem 9.1] we obtain that $\pi_1(X/G, [x]) \cong \pi_1(\mathcal{X}, \bar{x})/I$.

<figure><img src="image_1.png" /></figure>

# 3. The fundamental group of the product of topological spaces

## 3.1. Constructing the homomorphism

**3.1.1. Finite index of the group in the product.**— Let $I_y$ denote the isotropy at the point $y$ in $\tilde{X}$ for the action of $\pi_1(\mathcal{X}, \bar{x})$. By Lemma 2.1 the map $\pi_1(\mathcal{X}, \bar{x}) \to G$ restricted to $I_y$ is injective, therefore we can identify $I_y$ with a subgroup of $G$. When we do such identification we will denote it by $I'_y < G$.

Now as $\pi_1(\mathcal{X}, \bar{x}) \cong \pi_1(\mathcal{X}_1, \bar{x}_1) \times_G \cdots \times_G \pi_1(\mathcal{X}_k, \bar{x}_k)$, if $y = (y_1, \dots, y_k)$ we define $I_i < \pi_1(\mathcal{X}_i, \bar{x}_i)$ as the image of $I_y$ via the morphism $\pi_1(\mathcal{X}, \bar{x}) \to \pi_1(\mathcal{X}_i, \bar{x}_i)$.

**Lemma 3.1.** We have that $I_y \cong I_i$ for all $i = 1, \dots, k$ and $I_y = I_1 \times_{I'_y} \cdots \times_{I'_y} I_k$.

*Proof.* For $\gamma = (\gamma_1, \dots, \gamma_k) \in I_y$ note that $\gamma_i \in \pi_1(\mathcal{X}_i, \bar{x}_i)$ fixes $y_i \in \tilde{X}_i$, otherwise $\gamma$ can not fix a point in $\tilde{X}$. As above, the restriction of $\pi_1(\mathcal{X}_i, \bar{x}_i) \to G$ to $I_{y_i}$ is injective and as $I_i \subset I_{y_i}$ we have that $\gamma_i \neq \beta_i$ for $\gamma, \beta \in I_y \subset \pi_1(\mathcal{X}_1, \bar{x}_1) \times_G \cdots \times_G \pi_1(\mathcal{X}_k, \bar{x}_k)$ with $\gamma \neq \beta$. Therefore we can construct an inverse to the projection. The result follows.

<figure><img src="image_2.png" /></figure>

Note that we obtain that $I_i < I_{y_i}$, but in general $I_{y_i}$ can be bigger. Let us define the homomorphism $I_y \to \prod I_{y_i}$ given by decomposing an element in its components. By Lemma 3.1 it is injective. Denote by $N$ the subgroup in $\pi_1(\mathcal{X}, \bar{x})$ generated by all the $I_y$ and by $N'_i$ the subgroup in $\pi_1(\mathcal{X}_i, \bar{x}_i)$ generated by $I_i$.

**Lemma 3.2.** The subgroup $N'_i$ is normal in $\pi_1(\mathcal{X}_i, \bar{x}_i)$.

*Proof.* Let $\gamma_i \in N'_i$ and $t_i \in \pi_1(\mathcal{X}_i, \bar{x}_i)$. We can write $\gamma_i = \gamma_{i_1} \cdots \gamma_{i_j}$ with each $\gamma_{i_l} \in I_{i_l}$ coming from $\gamma_l = (\gamma_{l_1}, \dots, \gamma_{l_i}, \dots, \gamma_{k_l}) \in I_{y_l} \subset \pi_1(\mathcal{X}, \bar{x})$ and the point $y_l = (y_{l_1}, \dots, y_{l_i}, \dots, y_{k_l}) \in \tilde{X}$ for $l = 1, \dots, j$. As every $\pi_1(\mathcal{X}_j, \bar{x}_j) \to G$ is surjective, for $j = 1, \dots, i-1, i+1, \dots, k$, there exists $t_j \in \pi_1(\mathcal{X}_j, \bar{x}_j)$ such that $t = (t_1, \dots, t_k) \in \pi_1(\mathcal{X}, \bar{x})$.

As $t \cdot \gamma_l \cdot t^{-1} \in I_{ty_l}$ it follows that $t_i \gamma_{i_l} t_i^{-1} \in N'_i$ and therefore

$$t_i \gamma_i t_i^{-1} = (t_i \gamma_{i_1} t_i^{-1}) \cdot t_i \cdots t_i^{-1} \cdot (t_i \gamma_{i_j} t_i^{-1}) \in N'_i.$$

<figure><img src="image_3.png" /></figure>

**Proposition 3.3.** There is an homomorphism

$$\pi_1(X/G, [x]) \to \prod_{i=1}^{k} \pi_1(\mathcal{X}_i, \bar{x}_i)/N'_i$$

such that the image has finite index.

*Proof.* By Lemma 2.2 we have that $\pi_1(\mathcal{X}, \bar{x}) \cong \pi_1(\mathcal{X}_1, \bar{x}_1) \times_G \cdots \times_G \pi_1(\mathcal{X}_k, \bar{x}_k)$. Therefore there is an injective homomorphism $\pi_1(\mathcal{X}, \bar{x}) \to \prod \pi_1(\mathcal{X}_i, \bar{x}_i)$. By Lemma 3.2 we obtain the exact sequence

$$(3.1) \qquad 1 \to \prod_{i=1}^{k} N'_{i} \to \prod_{i=1}^{k} \pi_1(\mathcal{X}_i, \bar{x}_i) \to \prod_{i=1}^{k} \pi_1(\mathcal{X}_i, \bar{x}_i)/N'_{i} \to 1,$$
**Lemma 3.4.** [Che01, Lemma 3.4.2] There exists a natural homeomorphism between the pointed topological spaces $(\Omega(\mathcal{X}, \bar{x}), x)$ and $(P(X, G, x), (x, 1_G))$.

**Remark 3.5.** When $(\mathcal{X}, \bar{x})$ is a pointed topological stack there exists $(B[R \rightrightarrows X], x')$ a pointed topological space, where $B[R \rightrightarrows X]$ is the classifying space of the topological groupoid $[R \rightrightarrows X]$, such that we can take $\pi_1(\mathcal{X}, \bar{x}) := \pi_1(B[R \rightrightarrows X], x')$. In the case of a global quotient $\mathcal{X} = [X/G]$ it happens that $B[R \rightrightarrows X]$ equals the Borel construction $X \times_G EG$, see [Noo12].

Now, the construction of Chen also gives a natural isomorphism between $\pi_1(\mathcal{X}, \bar{x})$ and $\pi_1(X \times_G EG, x')$ [Che01, Theorem 3.4.1] linking both definitions.

There exists a canonical projection $(P(X, G, x), (x, 1_G)) \to (G, 1_G)$ given by sending $(\gamma, g)$ to $g$. This map can be seen to be a fibration [Che01, Lemma 3.4.3] having as fiber at $1_G$ the space loop $\Omega(X, x)$ via the embedding $\Omega(X, x) \hookrightarrow P(X, G, x)$ where $\gamma$ maps to $(\gamma, 1_G)$.

With this description at hand, suppose there is $y \in X$ such that it is fixed by an element $g$, that is, $y \in X^g$. Denote by $\gamma_y$ a path starting at $x$ and finishing at $y$, then $\gamma_y(g\gamma_y^{-1}) \in P(X, G, x)$, where $g\gamma_y^{-1}$ denotes the action of $g$ applied to each point of the path.

**Lemma 3.6.** Let $I_y < G$ denote the inertia (stabilizer) of the action of $G$ at $y \in X$. Every homotopy class of a path $[\gamma_y] \in \pi_1(X, x, y)$ induces an injective morphism $I_y \to \pi_1(\mathcal{X}, x)$.

*Proof.* As $G$ is discrete $g \mapsto \gamma_g(g\gamma_g^{-1})$ is continuous, with $g \in I_y$. Then by taking the functor $\pi_0$ we got a morphism of groups $\pi_0(I_y) \to \pi_0(P(X, G, x)) = \pi_1(\mathcal{X}, \bar{x})$. Finally, by composing with the projection $(\pi_0(P(X, G, x), x)) \to \pi_0((G, 1_G))$ we obtain that different points under $\pi_0(I_y) \to \pi_1(\mathcal{X}, \bar{x}) \to \pi_0(G)$ have different images, thus the morphism in injective. $\square$

**Lemma 3.7.** Let $Y \in \pi_0(X^g)$, $y_1, y_2 \in Y$ and let $\gamma_{y_1}, \gamma_{y_2}$ be paths starting at $x \in X$ and finishing at $y_1$ and $y_2$ respectively, then $\gamma_{y_1}(g\gamma_{y_1}^{-1})$ is a conjugate of $\gamma_{y_2}(g\gamma_{y_2}^{-1})$ in $\pi_1(\mathcal{X}, \bar{x})$ by elements of $\pi_1(X, x)$.

*Proof.* There exists a path $\beta \subset Y$ connecting $y_1$ and $y_2$, therefore $\gamma_{y_1}\beta(g\beta^{-1}\gamma_{y_1}^{-1}) \in P(X, G, x)$ but as $g\beta = \beta$ passing to $\pi_0(P(X, G, x), x)$ it equals $[\gamma_{y_1}(g\gamma_{y_1}^{-1})]$.

Now consider the path $\gamma_{y_2}$. Note that $\theta := \gamma_{y_1}\beta\gamma_{y_2}^{-1} \in \Omega(X, x)$. There exists a continuous map

$$ \# : P(X, G, x) \times P(X, G, x) \to P(X, G, x) $$

which induces the multiplication in the fundamental group (see [Che01, Section 3.1]). The element $\theta\#(\gamma_{y_2}(g\gamma_{y_2}^{-1})\theta^{-1})$ can be seen to be $\theta(\gamma_{y_2} \cdot (g(\theta \cdot \gamma_{y_2})^{-1})) \in P(X, G, x)$. By passing to the group $\pi_1(\mathcal{X}, \bar{x}) = \pi_0(P(X, G, x), x)$ we have that $[\theta][\gamma_{y_2}(g\gamma_{y_2}^{-1})][\theta^{-1}] = [\gamma_{y_1}(g\gamma_{y_1}^{-1})]$. $\square$

Recall that given $(X, x)$ as above, we have a pointed universal cover map $p : (\tilde{X}, \bar{x}) \to (X, x)$ where $\bar{x}$ represents the constant loop of value $x$. Every element in $\gamma \in \pi_1(X)$ corresponds to a point in $p^{-1}(x)$. So given a pointed map $p_\gamma : (\tilde{X}, \gamma) \to (X, x)$ it induces a deck transformation of $\tilde{X}$ in the following way: given $y \in \tilde{X}$ take a path $\alpha_y \subset \tilde{X}$ starting at $\gamma$ and finishing at $y$. Consider the unique lift $p_\gamma(\tilde{\alpha}_y) \subset \tilde{X}$ starting at $x$ and assign to $y$ the point $p_\gamma(\tilde{\alpha}_y)(1)$. It can be seen to be a well-defined map (see [Hat00]).

Now, by the description given above of $\pi_1(\mathcal{X}, \bar{x})$, any $\gamma \in \pi_1(\mathcal{X}, \bar{x})$ such that $\varphi(\gamma) = g$ (recall that $\varphi : \pi_1(\mathcal{X}, \bar{x}) \to G$) have as a representative an element in $P(X, G, x)$ which we still denote by $\gamma$. So $\gamma$ starts at $x$ and finishes at $gx$. Denote by $\tilde{\pi} : (\tilde{X}, \bar{x}) \to (\mathcal{X}, \bar{x})$ the universal cover morphism, note that $\tilde{\pi}_\gamma : (\tilde{X}, \gamma) \to (\mathcal{X}, \bar{x})$ is also a cover morphism. By [Che01, Theorem 4.1.6] we obtain a deck transformation in the following way: given $y \in \tilde{X}$ take a path $\alpha_y \subset \tilde{X}$ starting at $\gamma$ and ending at $y$. Using the notation of the precedent paragraph, the path $p_\gamma(\tilde{\alpha}_y)$ starts at $gx$. Then the path $g^{-1}p_\gamma(\tilde{\alpha}_y)$ starts at $x$ so we can lift it to $g^{-1}\tilde{p}_\gamma(\tilde{\alpha}_y)$ in $(\tilde{X}, \bar{x})$, the end point of this lift is then defined as the image of $y$. It is shown that it is a well defined map and does not depend on the path chosen.
**Lemma 3.8.** Let $y \in X$ be fixed by $g \in G$, consider a path $\gamma_y$ connecting $x$ and $y$. Consider the action of $\pi_1(\mathcal{X}, \bar{x})$ on $\tilde{X}$ given by deck transformations Deck($\tilde{X}, \mathcal{X}$), then the element $\gamma_y(g\gamma_y^{-1}) \in \pi_1(\mathcal{X}, \bar{x})$ fixes a point in $\tilde{X}$. Moreover, any element of $\pi_1(\mathcal{X}, \bar{x})$ fixing a point in $\tilde{X}$ is of this form.

*Proof.* As the endpoint of $\gamma_y(g\gamma_y^{-1})$ is $gx$ we have a pointed covering morphism

$$ \tilde{\pi}_{\gamma_y(g\gamma_y^{-1})} : (\tilde{X}, \gamma_y(g\gamma_y^{-1})) \to (\mathcal{X}, \bar{x}) $$

and we can consider $g\gamma_y$ as a path in $\tilde{X}$ connecting $\gamma_y(g\gamma_y^{-1})$ and $\gamma_y$ as follows: let us define $f(t) = \gamma(g\gamma_y^{-1}) \cdot (g\gamma_y|_t)$ where $g\gamma_y|_t(t') := g\gamma_y(t'/t)$ denote the path starting at $gx$ and finishing at $g\gamma_y(t)$ in time $t$ for $t \neq 0$ and being the constant path with value $gx$ if $t = 0$. We project then $f(t)$ to $X$ and obtain $g\gamma_y$ which starts at $gx$ and finishes at $\bar{y}$. By the discussion before the lemma, we obtain that it lifts to $\gamma_y$ in $(\tilde{X}, \bar{x})$, as $g$ fixes $y$ we obtain that the point $\gamma_y \in \tilde{X}$ is fixed by the induced deck transformation.

Consider the exact sequence

$$ 1 \to \pi_1(X, x) \to \pi_1(\mathcal{X}, \bar{x}) \xrightarrow{\varphi} G \to 1, $$

let $\gamma \in \pi_1(\mathcal{X}, \bar{x})$ and $z \in \tilde{X}$ such that $\gamma$ fixes $z$. Let $p : (\tilde{X}, \bar{x}) \to (X, x)$ be the projection, as it is $\varphi$-invariant we have that $\varphi(\gamma)p(z) = p(z)$. Then by considering the path in $X$ corresponding to $z$, we can construct an element $z\varphi(\gamma)z^{-1}$, which fixes $z \in \tilde{X}$. As in the isotropy $\varphi$ is injective by Lemma 2.1, we have that $z\varphi(\gamma)z^{-1} = \gamma$. $\square$

**Proposition 3.9.** Suppose that there are only a finite number of elements in $\pi_0(X^g)$ for each $g \in G$, then there exists a finite set $L \subset \pi_1(\mathcal{X}, \bar{x})$ consisting of elements having fixed points in $\tilde{X}$ such that if $\gamma \in \pi_1(\mathcal{X}, \bar{x})$ fixes a point in $\tilde{X}$ then it is conjugate to an element of $L$ by elements in $\pi_1(X, x)$.

*Proof.* By Lemma 3.7 for every element in $Y \in \pi_0(X^g)$ it suffices to fix an element $\gamma_y(g\gamma_y^{-1})$ with $y \in Y$. For every $g \in G$ and every element in $\pi_0(X^g)$ we pick such an element. We define $L$ as the set consisting of such elements. By Lemma 3.8 every such element fixes a point in $\tilde{X}$ and any other fixing a point will be conjugate of the element in $L$ corresponding to its connected component. $\square$

**3.2.2. Proof that the homomorphism has finite kernel.**— Let us return to the case of $k$-topological spaces $X_1, \dots, X_k$ and let $G$ be a finite group acting on each one of them on the left as in 2.2.1. Proposition 3.9 gives us $k$ subsets $L(\mathcal{X}_i) \subset \pi_1(\mathcal{X}_i, \bar{x}_i)$ whose elements correspond to the element of $\pi_0(X_i^g)$ with $g \in G$. Now consider the subsets $L_i \subset L(\mathcal{X}_i)$ consisting of elements corresponding to $\pi_0(X_i^g)$ where $g$ fixes a point in $X_i$ for $i = 1, \dots, k$.

Recall that $N < \pi_1(\mathcal{X}, \bar{x})$ (with $\mathcal{X} = [(X_1 \times \dots \times X_k)/G]$) is the subgroup generated by the inertia subgroups $I_y$ given by the action of $\pi_1(\mathcal{X}, \bar{x})$ in $\tilde{X}$ and $N'_i < \pi_1(\mathcal{X}_i, \bar{x}_i)$ is the image of the $i$-projection of $N$. The following lemma is immediate from Proposition 3.9

**Lemma 3.10.** We have that $N'_i = \langle \gamma_i l_i \gamma_i^{-1} | l_i \in L_i, \gamma_i \in \pi_1(X_i, x_i) \rangle$ in $\pi_1(\mathcal{X}_i, \bar{x}_i)$ for $i = 1, \dots, k$.

**Definition 3.11.** Let us define

$$ C_i = C_i(\pi_1(X_i), L_i) := \langle \langle \gamma_i l_i \gamma_i^{-1} l_i^{-1} | \gamma_i \in \pi_1(X_i, x_i), l_i \in L_i \rangle \rangle_{\pi_1(\mathcal{X}_i, \bar{x}_i)} $$

to be the normal subgroup generated by the commutators of elements in $\pi_1(X_i, x_i)$ and in $L_i$. Denote by $\mathbb{T}_i := \pi_1(\mathcal{X}_i, \bar{x}_i)/C_i$ and by $\hat{L}_i$ the image of $L_i$ in $\mathbb{T}_i$.

**Lemma 3.12.** It happens that $C_i < N'_i$ and moreover we can consider $C_i$ as a subgroup of $N$ via $\{e\} \times \dots \times C_i \times \dots \times \{e\}$ and $C_1 \times \dots \times C_k < N$.
**3.2.3. Geometric interpretation of the groups $\pi_1(\mathcal{X}_i, \bar{x}_i)/N_i'$.**— Let us denote by $I$, the subgroup of $G$ generated by the elements having a fixed point in every $X_i$ for $i = 1, \dots, k$. Note that $I$ is a normal subgroup. Let $x_i'$ denote the class of $x_i$ in $X/I$ and $\bar{x}_i'$ the image of $x_i'$ in $[(X_i/I)/(G/I)]$.

**Proposition 3.15.** *There is an isomorphism*

$$ \pi_1(\mathcal{X}_i, \bar{x}_i)/N_i' \xrightarrow{\sim} \pi_1([[(X_i/I)/(G/I)], \bar{x}_i)]. $$

*Proof.* Observe that the action of $G$ on $X_i$ descends to an action of $G/I$ on $X_i/I$ and therefore we can define $[(X_i/I)/(G/I)]$. Recall by the previous subsection 3.2.1 that $\pi_1(\mathcal{X}_i, \bar{x}_i)$ can be identified with the set of path-components of $P(X_i, G, x)$. Therefore an element $[\gamma] \in \pi_1(\mathcal{X}_i, \bar{x}_i)$ can be represented by a path $\gamma$ in $X_i$ starting at $x_i$ and finishing at $gx_i$ for some $g \in G$. Denote by $p_i : X_i \to X_i/I$ the quotient map. By considering $p_i(\gamma)$, we obtain a morphism between $\pi_1(\mathcal{X}_i, \bar{x}_i)$ and $\pi_1([[(X_i/I/G/I)], \bar{x}_i')]$.

It is immediate to see that the paths coming from the inertia of $I$ in $X_i$, that is, the elements of the form $\gamma_y(g\gamma_y^{-1})$ with $g \in I$ and $y \in X_i^g$, are sent to the trivial element in $\pi_1(X_i/I, x_i')$.

Now consider $\gamma \in \ker(\pi(\mathcal{X}_i, \bar{x}_i) \to \pi_1([[(X_i/I)/(G/I)], \bar{x}_i]))$. Then $\gamma$ is represented by a path in $X_i$, which we still denote by $\gamma$, starting at $x_i$ and finishing at $gx_i$ with $g \in G$. Note that moreover $g \in I$, otherwise by the projection $\pi_1([X_i/I/G/I], \bar{x}_i') \to G$ the element would be sent to a non-zero element. Hence the image of $\gamma$ lies in $\pi_1(X_i/I, x_i')$ and it is trivial. By the exact sequence

$$ 1 \to N_{[X_i/I]} \to \pi_1([X_i/I], \bar{x}_i) \to \pi_1(X/I, x_i') \to 1 $$

and noticing that $N_{[X_i/I]} = N_i'$ we have that $\gamma \in N_i'$ which proves the result. $\square$

# 4. Applications

## 4.1. Product of the same topological space

Now let us describe a case where $N_i'$ equals the whole subgroup $N_i$ generated by the elements having a fixed point in the universal cover.

**Corollary 4.1.** Let $X_i = X_1$ for $i = 2, \dots, k$ and $G$ finite acting on $X_1$. Then the morphism

$$ \pi_1((X_1 \times \dots \times X_1)/G, [x]) \to \prod_{i=1}^k \pi_1(X_1/G, [x_i]) $$

has finite kernel.

*Proof.* We only have to show that $N_1' = N_1$ and then we obtain the result by applying Theorem 3.14. By construction we have that $N_1' \subset N_1$. Let us show the inverse inclusion. Take $\gamma_1 \in N_1$, then we can write $\gamma_1 = \gamma_{1_1} \cdots \gamma_{1_l}$ such that there exists $y_{1_j} \in \tilde{X}_1$ satisfying $\gamma_{1_j} \in I_{y_{1_j}}$ for $j = 1, \dots, l$. As $\tilde{X} = \tilde{X}_1 \times \dots \times \tilde{X}_k$ by taking $y_j = (y_{1_j}, \dots, y_{1_j}) \in \tilde{X}$ we have that $\gamma^j = (\gamma_{1_j}, \dots, \gamma_{1_j}) \in I_{y_j}$ and therefore $\gamma = \gamma^1 \cdots \gamma^l \in N$ and the image of $\gamma$ in $N_1$ equals $\gamma_1$. $\square$

Another proof using Proposition 3.15 can be obtained as follows: the action of $G/I$ is free in $X_1/I$ and since $X_1/G \cong (X_1/I)/(G/I)$ we have $\pi_1([[(X_1/I)/(G/I)]]) \cong \pi_1(X_1/G)$.

## 4.2. Second Main Theorem

**Theorem 4.2.** Let $X_1, \dots, X_k$ admit a universal cover and let $G$ be a finite group acting on each of them such that $|\pi_0(X_i^g)| < +\infty$ for every $g \in G$ and $i = 1, \dots, k$. Denote $X = X_1 \times \dots \times X_k$ and consider the diagonal action of $G$ on it. Suppose $\pi_1(X/G, [x])$ is residually finite, then $\pi_1(X/G, [x])$ has a normal finite-index subgroup $N \cong H_1 \times \dots \times H_k$ isomorphic to a product of normal finite index subgroups $H_i \subset \pi_1(X_i/I, [x_i])$.
**Proof.** By Theorem 1.3 we get a morphism $\Theta : \pi_1(X/G, [x]) \to \prod_{i=1}^k \pi_1([X_i/I/G/I])$ having finite kernel $E$. As $\pi_1(X/G, [x])$ is residually finite we can construct a finite-index normal subgroup $\Gamma \subset \pi_1(X/G, [x])$ such that $\Gamma \cap E = \{e\}$.

The morphism $\Theta|_{\Gamma} : \Gamma \to \prod_{i=1}^k \pi_1([X_i/I]/(G/I)), \bar{x}'_i)$ is therefore injective and moreover as the subgroup $\Theta(\pi_1(X/G)) < \prod_{i=1}^k \pi_1([X_i/I]/(G/I)), \bar{x}'_i)$ has finite index it follows that $\Theta(\Gamma) < \prod_{i=1}^k \pi_1([X_i/I]/(G/I)), \bar{x}'_i)$ has finite index.

For every $i = 1, \dots, k$, we have $\pi_1(X_i/I, [x_i]) < \pi_1([X_i/I]/(G/I)), \bar{x}'_i)$ as a normal finite-index subgroup. Define the subgroup

$$ \Theta(\Gamma)_i := \Theta(\Gamma) \cap (\{e_1\} \times \dots \times \pi_1(X_i/I, [x_i]) \times \dots \times \{e_k\})) $$

where $e_k \in \pi_1(X_j/I, [x_j])$ is the identity element. As $\Theta(\Gamma)_i$ has finite index in $\pi_1(X_i/I, [x_i])$, there exists a normal subgroup of finite index $H_i$ of $\pi_1([X_i/I]/(G/I))$ contained in $\Theta(\Gamma)_i$. Set $H := H_1 \times \dots \times H_k$, then $H \subset \Theta(\Gamma)$ and it is a finite-index normal subgroup of $\prod_{i=1}^k \pi_1([X_i/I]/(G/I)), \bar{x}'_i)$. The subgroup $\mathcal{N} := \Theta^{-1}(H) \cap \Gamma$ satisfies the stated properties. $\square$

### 4.2.1. Case of smooth curves.—

**Corollary 4.3.** Let $C_1, \dots, C_k$ be smooth algebraic curves and let $G$ be a finite group acting on each $C_i$. Denote $C = C_1 \times \dots \times C_k$. Consider $\mathcal{C} = [C/G]$ with $G$ acting diagonally on $C$. Then $\pi_1(C/G)$ has a normal subgroup $\mathcal{N}$ of finite index isomorphic to $\prod_1 \times \dots \times \prod_k$ where $\prod_i$ is either a surface group or a finitely generated free group for $i = 1, \dots, k$.

By Theorem 1.3 we have a morphism $\pi_1(C/G) \to \prod_{i=1}^k \pi_1([C_i/I/G/I])$ with finite kernel, however if the action of $G/I$ is not faithful on $C_i/I$ then $\pi_1([C_i/I]/(G/I))$ is not necessarily an orbifold surface group. This can be overcome as follows: let $K_i := \ker(G/I \to \operatorname{Aut} C_i/I)$ and $H_i := (G/I)/K_i$. Denote by $C_i := [(C_i/I)/G/I]$ and by $C'_i := [(C_i/I)/H_i]$, we have a canonical morphism $C_i \to C'_i$.

**Lemma 4.4.** The induced homomorphism $q_i : \pi_1(C_i) \to \pi_1(C'_i)$ is surjective and has finite kernel.

**Proof.** By choosing a point $x_i \in C_i$ and denoting by $\bar{x}_i$ its image in both $C_i$ and $C'_i$ we obtain a fibration $[pt/K, pt] \hookrightarrow (C_i, \bar{x}_i) \to (C'_i, \bar{x}_i)$. By taking the long homotopy exact sequence

$$ \dots \to \pi_2(C'_i, \bar{x}_i) \to \pi_1(pt/K, pt) \to \pi_1(C_i, \bar{x}_i) \to \pi_1(C'_i, \bar{x}_i) \to 1, $$

as there is an isomorphism between $\pi_1(pt/K, pt)$ and $\pi_0(K, 1_K)$, the result follows. $\square$

So by composing, we obtain a morphism $\Theta : \pi_1(C/G) \to \prod_{i=1}^k \pi_1(C_i) \to \prod_{i=1}^k \pi_1(C'_i)$, this allows us to prove the following lemma, which together with Theorem 4.2 will imply Corollary 4.3.

**Lemma 4.5.** The group $\pi_1(C/G)$ is residually finite.

**Proof.** First note that, as $\pi_1(C'_i)$ is an orbifold surface group, it is in particular residually finite. Now, it follows that $\Theta(\pi_1(C/G))$ is residually finite as it is a finite-index subgroup of a direct product of residually finite groups.

We need another property of these groups to continue. Let $H$ be a group and let $\hat{H}$ be its profinite completion. A group $H$ is called *good* if for each $k \ge 0$ and for each finite $H$-module $M$ the natural homomorphism

$$ H^k(\hat{H}, M) \to H^k(H, M) $$

is an isomorphism. In [GJZZ08, Lemmas 3.2 and 3.4, Proposition 3.6] it is shown that a finite-index subgroup of a good group is good, the product of good groups is good and that $\pi_1(C)$ for $C$ an algebraic orbifold curve is good. We obtain therefore that $\Theta(\pi_1(C/G))$ is good.

Finally, [GJZZ08, Proposition 6.1] asserts that if $T$ is a residually finite good group and $\varphi : H \to T$ is a surjective homomorphism with finite kernel then $H$ is residually finite. Applying this to $\Theta' : \pi_1(C/G) \to \Theta(\pi_1(C/G))$ we obtain the result. $\square$
### 4.3. Partial compactifications of arrangement of lines

The original motivation of this work was to study the partial compactifications of the complement of an arrangement of lines in $\mathbb{P}_C^2$ which is the topic of my Ph.D. thesis. In [Agu19] a general method for computing a presentation of the fundamental group was given and some examples studied. A family of arrangements related to the studied in op. cit. is available for any $n \in \mathbb{N}$, however this will require a treatment one by one. The results obtained here can be used to study some of these partial compactifications in family.

**4.3.1. Partial compactification of the complement of an arrangement of lines.**— Consider the projective plane $\mathbb{P}_C^2$ with homogeneous coordinates $(z_1 : z_2 : z_3)$.

Let $\mathcal{A} = \sum_{i=1}^k L_i$ be a divisor in $\mathbb{P}^2$ such that the irreducible components $L_i$ are copies of $\mathbb{P}^1$ (lines). Then the singular set $\text{Sing} \mathcal{A}$ of $\mathcal{A}$ consists only of points. Consider $\pi : \text{Bl}_{\text{Sing} \mathcal{A}} \mathbb{P}^2 \to \mathbb{P}^2$ the blow up of the projective plane at the points $\text{Sing} \mathcal{A}$. The divisor $\pi^*\mathcal{A} = \sum_{i=1}^{k+|\text{Sing} \mathcal{A}|} D_i$ has as irreducible components copies of $\mathbb{P}^1$, with $D_1, \dots, D_k$ being the strict transform of $L_1, \dots, L_k$ respectively and $D_{k+1}, \dots, D_{k+|\text{Sing} \mathcal{A}|}$ being the exceptional divisors. Take a subset $J \subset \{1, 2, \dots, k + |\text{Sing} \mathcal{A}|\}$. The surface $\text{Bl}_{\text{Sing} \mathcal{A}} \mathbb{P}^2 \setminus (\cup_{j \in J} D_j)$ is called a *partial compactification of* $\mathbb{P}^2 \setminus (\cup L_i)$. We are interested in how the fundamental group changes when we partially compactify the complement of such an arrangement $\mathcal{A} \subset \mathbb{P}^2$.

**4.3.2. Examples.**— The subvariety of $\mathbb{P}^2$

$$ \text{Ceva}(n) := \{(z_1 : z_2 : z_3) \mid (z_1^n - z_2^n)(z_1^n - z_3^n)(z_2^n - z_3^n) = 0\} $$

can be seen as the union of the closure of the three singular fibers of the rational map $f : \mathbb{P}^2 \to \mathbb{P}^1$ given by $(z_1 : z_2 : z_3) \mapsto ((z_1^n - z_2^n) : (z_2^n - z_3^n))$. The map $f$ is not defined in a subset $S = \{p_1, \dots, p_{n^2}\} \subset \text{Sing}\text{Ceva}(n)$ consisting of $n^2$ points where $\mathcal{A}_1 := \{z_1^n - z_2^n = 0\} = \sum_{i=1}^n L_i$ intersects $\mathcal{A}_2 := \{z_1^n - z_3^n = 0\} = \sum_{i=n+1}^{2n} L_i$. It actually happens that $S \subset \mathcal{A}_3 := \{z_2^n - z_3^n = 0\} = \sum_{i=2n+1}^{3n} L_i$ and $S$ consists of points where 3 lines of Ceva($n$) meet. We have another 3 points $p_{n^2+i}$ in $\text{Sing}(\text{Ceva}(n))$ which correspond to each singular point of $\mathcal{A}_i$ for $i = 1, 2, 3$ and hence of multiplicity $n$.

The rational map $f$ extends to a morphism $\tilde{f} : \text{Bl}_{\text{Sing}\text{Ceva}(n)} \mathbb{P}^2 \to \mathbb{P}^1$ having as generic fiber the *Fermat curve of degree $n$* defined as $F(n) := \{z_1^n + z_2^n + z_3^n = 0\} \subset \mathbb{P}^2$. Therefore $\tilde{f}$ is an isotrivial fibration.

Denote by $\mu(n)$ the group of $n^{\text{th}}$ roots of unity. By taking 3 copies of it we define $H(n) := \mu_1(n) \oplus \mu_2(n) \oplus \mu_3(n) / \langle \mu_1 \mu_2 \mu_3 = 1 \rangle$ where $\mu_i \in \mu_i(n)$. It acts on $F(n)$ via $(z_1 : z_2 : z_3) \mapsto (\mu_1 z_1 : \mu_2 z_2 : \mu_3 z_3)$. The proof of the following theorem will appear elsewhere.

**Theorem 4.6.** Consider the diagonal action of $H(n)$ in $F(n) \times F(n)$. Denote by $S$ the minimal resolution of $F(n) \times F(n)/H(n)$.

(1) The fibration $S \to (F(n) \times F(n))/H(n) \to F(n)/H(n) \cong \mathbb{P}^1$ is isomorphic to $\tilde{f}$.

(2) Every singular point in $F(n) \times F(n)/H(n)$ corresponds to the contraction of the strict transform $D_i$ of some line $L_i \in \text{Ceva}(n)$.

(3) The contraction of the $n$ lines corresponding to $\mathcal{A}_i$ lie in the line $E_i$ which is the exceptional divisor corresponding to the unique singular point in $\mathcal{A}_i$.

(4) $E_i$ is mapped to a point via $F(n) \times F(n)/H(n) \to \mathbb{P}^1$.

The Fermat curve $F(n)$ of degree $n$ can be seen as a branched covering of $\mathbb{P}^1$ of degree $n^2$ via the morphism in $\mathbb{P}^2$ given by $F(n) \ni (z_1 : z_2 : z_3) \to (z_1^n : z_2^n : z_3^n) \in \{w_0 + w_1 + w_2 = 0\}$ which branches at the points $(1 : -1 : 0), (1 : 0 : -1), (0 : 1 : -1)$. Over each branching point there are $n$ points, we denote by $X_1, \dots, X_n$ for those over $(1 : -1 : 0)$, by $Y_1, \dots, Y_n$ over $(1 : 0 : -1)$ and $Z_1, \dots, Z_n$ over $(1 : -1 : 0)$.

Recall that for $S \to S'$ be a resolution of singularities of $S'$, if $S'$ has only quotient singularities, by [Kol93, Theorem 7.8.1] we have that $\pi_1(S) \to \pi_1(S')$ is an isomorphism.
**Example 4.7.** Consider the surface $S_1 := (F(n) \times (F(n) \setminus \{X_1, \dots, X_n\})) / H(n)$. The subgroup $I$ generated by the elements of $H(n)$ having fixed points both in $F(n)$ and in $F(n) \setminus \{X_1, \dots, X_n\}$ equals $H(n)$. As $F(n)/H(n) \cong \mathbb{P}^1$, $F(n) \setminus \{X_1, \dots, X_n\}/H(n) \cong \mathbb{C}$ and by Theorem 1.3 the morphism

$$ \pi_1(S_1) \to \pi_1(\mathbb{P}^1) \times \pi_1(\mathbb{C}) $$

has finite kernel, it follows that $\pi_1(S_1)$ is finite.

The minimal resolution of singularities $S'_1 \to S_1$ can be identified with the following partial compactification of Ceva(n). Consider

$$ J := \{1, \dots, n, 3n + n^3 + 1\} \subset \{1, \dots, 3n + n^2 + 3\} $$

then following the construction given in §4.3.1 we have that

$$ \mathrm{Bl}_{\mathrm{Sing}\,\mathrm{Ceva}(n)} \mathbb{P}^2 \setminus \{\cup_{j \in J} D_j\} \cong S'_1. $$

That is, from the surface $\mathrm{Bl}_{\mathrm{Sing}\,\mathrm{Ceva}(n)} \mathbb{P}^2$ we remove only the strict transform of $\mathcal{A}_1$ and the exceptional divisor coming from the singular point of $\mathcal{A}_1$. This can be identified with a singular fiber or $\tilde{f}$.

**Example 4.8.** Consider now $S_2 := (F(n) \times F(n) \setminus \{X_i, Y_i\}) / H(n)$. In this case the subgroup $I$, defined as in the previous paragraph, is isomorphic to $\mu(n)$. As $F(n)/\mu(n) \cong \mathbb{P}^1$, $F(n) \setminus \{X_i, Y_i\}/\mu(n) \cong \mathbb{C}^*$ and by Theorem 1.3 the morphism

$$ \pi_1(S_2) \to \pi_1([\mathbb{P}^1/\mu(n)]) \times \pi_1([\mathbb{C}^*/\mu(n)]) $$

has finite kernel and the image is a finite-index subgroup.

By Theorem 4.2 and Corollary 4.3, we have that $\mathbb{Z} \triangleleft \pi_1(S_2)$ has finite index. As in Example 4.7 the minimal resolution of singularities $S'_2 \to S_2$ can be identified with $\mathrm{Bl}_{\mathrm{Sing}\,\mathrm{Ceva}(n)} \mathbb{P}^2$ minus two singular fibers of $\tilde{f}$.

**Example 4.9.** If we consider $S_3 := (F(n) \times F(n) \setminus \{X_i, Y_i, Z_i\}) / H(n)$ it can be identified with $\mathrm{Bl}_{\mathrm{Sing}\,\mathcal{A}} \mathbb{P}^2$ minus the three singular fibers of $\tilde{f}$. As $H(n)$ acts freely in $F(n) \times F(n) \setminus \{X_i, Y_i, Z_i\}$, the long exact sequence of homotopy associated to the covering map $F(n) \times F(n) \setminus \{X_i, Y_i, Z_i\} \to S_3$ yields

$$ 1 \to \pi_1(F(n)) \times \pi_1(F(n) \setminus \{X_i, Y_i, Z_i\}) \to \pi_1(S_3) \to H(n) \to 1. $$

*Remark 4.10.* We can remove points also in the first component $F(n)$ of the product. However, we can not get more partial compactifications of Ceva(n) in this way. This can be shown by drawing the dual graph of the divisor $\pi^*\mathrm{Ceva}(n)$ and noticing that the lines obtained by removing points does not satisfy the intersection pattern of the graph.

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