Finite Quotients of Surface Braid Groups and Double Kodaira Fibrations

Finite quotients of surface braid groups and double Kodaira ﬁbrations

Francesco Polizzi and Pietro Sabatino

To Professor Ciro Ciliberto on the occasion of his 70th birthday

Abstract Let Σ푏 be a closed Riemann surface of genus 푏. We give an account of some results obtained in the recent papers [CaPol19, Pol20, PolSab21] and concerning what we call here pure braid quotients, namely non-abelian ﬁnite groups appearing as quotients of the pure braid group on two strands P2 (Σ푏). We also explain how these groups can be used in order to provide new constructions of double Kodaira ﬁbrations.

0 Introduction

A Kodaira fibration is a smooth, connected holomorphic fibration 푓1 : 푆 −→ 퐵1, where 푆 is a compact complex surface and 퐵1 is a compact complex curve, which is not isotrivial (this means that not all its fibres are biholomorphic to each others). The genus 푏1 := 푔(퐵1) is called the base genus of the fibration, whereas the genus 푔 := 푔(퐹), where 퐹 is any fibre, is called the fibre genus. If a surface 푆 is the total space of a Kodaira fibration, we will call it a Kodaira fibred surface. For every Kodaira fibration we have 푏1 ≥ 2 and 푔 ≥ 3, see [Kas68, Theorem 1.1]. Since the fibration is smooth, the condition on the base genus implies that 푆 contains no rational or elliptic curves; hence it is minimal and, by the sub-additivity of the Kodaira dimension, it is of general type, hence algebraic. arXiv:2106.01743v1 [math.GT] 3 Jun 2021

Francesco Polizzi Dipartimento di Matematica e Informatica, Università della Calabria, Ponte Bucci Cubo 30B, 87036 Arcavacata di Rende (Cosenza), Italy, e-mail: [email protected] Pietro Sabatino Via Val Sillaro 5, 00141 Roma e-mail: [email protected]

1 2 Francesco Polizzi and Pietro Sabatino

Examples of Kodaira fibrations were originally constructed in [Kod67, At69] in order to show that, unlike the topological Euler characteristic, the signature 휎 of a real manifold is not multiplicative for fibre bundles. In fact, every Kodaira fibred surface 푆 satisfies 휎(푆) > 0, see for example the introduction of [LLR20], whereas 휎(퐵1) = 휎(퐹) = 0, and so 휎(푆) ≠ 휎(퐵1)휎(퐹). A double Kodaira surface is a compact complex surface 푆, endowed with a double Kodaira fibration, namely a surjective, holomorphic map 푓 : 푆 −→ 퐵1 ×퐵2 yielding, by composition with the natural projections, two Kodaira fibrations 푓푖 : 푆 −→ 퐵푖, 푖 = 1, 2. The purpose of this article is to give an account of recent results, obtained in the series of papers [CaPol19, Pol20, PolSab21], concerning the construction of some double Kodaira fibrations (that we call diagonal) by means of group-theoretical methods. Let us start by introducing the needed terminology. Let 푏 ≥ 2 and 푛 ≥ 2 be two positive integers, and let P2 (Σ푏) be the pure braid group on two strands on a closed Riemann surface of genus 푏. We say that a finite group 퐺 is a pure braid quotient of type (푏, 푛) if there exists a group epimorphism

휑 : P2 (Σ푏) −→ 퐺 (1) such that 휑(퐴12) has order 푛, where 퐴12 is the braid corresponding, via the isomorphism P2 (Σ푏)' 휋1 (Σ푏 × Σ푏 − Δ), to the homotopy class in Σ푏 × Σ푏 − Δ of a loop in Σ푏 × Σ푏 “winding once" around the diagonal Δ. Since 퐴12 is a commutator in P2 (Σ푏) and 푛 ≥ 2, it follows that every pure braid quotient is a non-abelian group, see Remark2. By Grauert-Remmert’s extension theorem together with Serre’s GAGA, the existence of a pure braid quotient as in (1) is equivalent to the existence of a Galois cover f : 푆 −→ Σ푏 ×Σ푏, branched over Δ with branching order 푛. After Stein factorization, this yields in turn a diagonal double Kodaira fibration 푓 : 푆 −→ Σ푏1 × Σ푏2 . We have f = 푓 , i.e. no Stein factorization is needed, if and only if 퐺 is a strong pure braid quotient, an additional condition explained in Definition3. We are now in a position to state our first results, see Theorems1,2,3: • If 푏 ≥ 2 is an integer and 푝 ≥ 5 is a prime number, then both extra-special 푝-groups of order 푝4푏+1 are non-strong pure braid quotients of type (푏, 푝). • If 푏 ≥ 2 is an integer and 푝 is a prime number dividing 푏 + 1, then both extra- special 푝-groups of order 푝2푏+1 are pure braid quotients of type (푏, 푝). • If a finite group 퐺 is a pure braid quotient, then |퐺| ≥ 32, with equality holding if and only if 퐺 is extra-special. Moreover, in the last case, we can explicitly compute the number of distinct quotients maps of type (1), up to the natural action of Aut(퐺).

We believe that such results are significant because, although we know that P2 (Σ푏) is residually 푝-finite for all 푝 ≥ 2 (see [BarBel09, pp. 1481-1490]), it is usually tricky to explicitly describe its non-abelian finite quotients. The geometrical counterparts of the above group-theoretical statements allow us to construct infinite families of double Kodaira fibrations with interesting numerical Finite quotients of surface braid groups and double Kodaira fibrations 3 properties, for instance having slope greater than 2 + 1/3 or signature equal to 16, see Theorems4,5,6:

• Let 푓 : 푆 푝 −→ Σ푏0 ×Σ푏0 be the diagonal double Kodaira ﬁbration associated with a non-strong pure braid quotient 휑 : P2 (Σ2) −→ 퐺 of type (2, 푝), where 퐺 is an extra-special 푝-group 퐺 of order 푝9 and 푏0 = 푝4 + 1. Then the maximum slope 휈(푆 푝) is attained for precisely two values of 푝, namely 12 휈(푆 ) = 휈(푆 ) = 2 + . 5 7 35

Furthermore, 휈(푆 푝) > 2 + 1/3 for all 푝 ≥ 5. More precisely, if 푝 ≥ 7 the function 휈(푆 푝) is strictly decreasing and 1 lim 휈(푆 푝) = 2 + . 푝→+∞ 3

• Let Σ푏 be any closed Riemann surface of genus 푏. Then there exists a double Kodaira ﬁbration 푓 : 푆 −→ Σ푏 × Σ푏. Moreover, denoting by 휅(푏) the number of such ﬁbrations, we have 휅(푏) ≥ 훚(푏 + 1), where 훚: N −→ N stands for the arithmetic function counting the number of distinct prime factors of a positive integer. In particular,

lim sup 휅(푏) = +∞. 푏→+∞

• Let 퐺 be a finite group and f : 푆 −→ Σ푏 ×Σ푏 be a Galois cover, with Galois group 퐺, branched over the diagonal Δ with branching order 푛. Then |퐺| ≥ 32, and equality holds if and only if 퐺 is extra-special. If 퐺 is extra-special of order 32 and (푏, 푛) = (2, 2), then f : 푆 −→ Σ2 × Σ2 is a diagonal double Kodaira fibration such that 푏1 = 푏2 = 2, 푔1 = 푔2 = 41, 휎(푆) = 16. As a consequence of the last result, we obtain a sharp lower bound for the signature of a diagonal double Kodaira fibration, see Theorem7:

• Let 푓 : 푆 −→ Σ푏1 × Σ푏2 be a diagonal double Kodaira fibration, associated with a pure braid quotient 휑 : P2 (Σ푏) −→ 퐺 of type (푏, 푛). Then 휎(푆) ≥ 16, and equality holds precisely when (푏, 푛) = (2, 2) and 퐺 is an extra-special group of order 32. Note that our methods show that every curve of genus 푏 (and not only some special curve with extra automorphisms) is the basis of a (double) Kodaira fibration and that, in addition, the number of distinct Kodaira fibrations over a fixed base can be arbitrarily large. Furthermore, every curve of genus 2 is the base of a (double) Kodaira fibration with signature 16 and this provides, to our knowledge, the first example of positive-dimensional family of (double) Kodaira fibrations with small signature. 4 Francesco Polizzi and Pietro Sabatino

The aforementioned examples with signature 16 also provide new “double solutions” to a problem, posed by G. Mess and included in Kirby’s problem list in low-dimensional topology, see [Kir97, Problem 2.18 A], asking what is the smallest number 푏 for which there exists a real surface bundle over a real surface with base genus 푏 and non-zero signature. We actually have 푏 = 2, also for double Kodaira ﬁbrations, see Theorem8:

• Let 푆 be double Kodaira surface, associated with a pure braid quotient 휑 : P2 (Σ푏) −→ 퐺 of type (2, 2), where 퐺 is an extra-special group of order 32. Then the real manifold 푋 underlying 푆 is a closed, orientable 4-manifold of signature 16 that can be realized as a real surface bundle over a real surface of genus 2, with fibre genus 41, in two different ways. In fact, it is an interesting question whether 16 and 41 are the minimum possible values for the signature and the fibre genus of a (not necessarily diagonal) double Kodaira surface 푓 : 푆 −→ Σ2 × Σ2, but we will not develop this point here. The above results paint a rather clear picture regarding pure braid quotients and the relative diagonal double Kodaira fibrations when |퐺| ≤ 32. It is natural then to investigate further this topic for |퐺| > 32, and indeed this paper also contains the following new result, see Theorem9: • If 퐺 is a finite group with 32 < |퐺| < 64, then 퐺 is not a pure braid quotient. We provide only a sketch of the proof, which is based on calculations performed by means of the computer algebra system GAP4, see [GAP4]; the details will appear in a forthcoming paper. Acknowledgments. F. Polizzi was partially supported by GNSAGA-INdAM. Both authors thank A. Causin for drawing the figures. Notation and conventions. The order of a finite group 퐺 is denoted by |퐺|. If 푥 ∈ 퐺, the order of 푥 is denoted by 표(푥). The subgroup generated by 푥1, . . . , 푥푛 ∈ 퐺 is denoted by h푥1, . . . , 푥푛i. The center of 퐺 is denoted by 푍 (퐺) and the centralizer of an element 푥 ∈ 퐺 by 퐶퐺 (푥). If 푥, 푦 ∈ 퐺, their commutator is defined as [푥, 푦] = 푥푦푥−1푦−1. We denote both the cyclic group of order 푝 and the field with 푝 elements by Z푝. We use sometimes the IdSmallGroup(G) label from GAP4 list of small groups. For instance, S4 = 퐺(24, 12) means that S4 is the twelfth group of order 24 in this list.

1 Pure surface braid groups and ﬁnite braid quotients

Let Σ푏 be a closed Riemann surface of genus 푏 ≥ 2, and let 풫 = (푝1, 푝2) be an ordered pair of distinct points on Σ푏.A pure geometric braid on Σ푏 based at 풫 is a pair (훼1, 훼2) of paths 훼푖 : [0, 1] −→ Σ푏 such that

• 훼푖 (0) = 훼푖 (1) = 푝푖 for all 푖 ∈ {1, 2} • the points 훼1 (푡), 훼2 (푡) ∈ Σ푏 are pairwise distinct for all 푡 ∈ [0, 1], Finite quotients of surface braid groups and double Kodaira ﬁbrations 5 see Figure1.

푝1 푝2

Fig. 1 A pure braid on two strands

Deﬁnition 1 The pure braid group on two strands on Σ푏 is the group P2 (Σ푏) whose elements are the pure braids based at 풫 and whose operation is the usual concate- nation of paths, up to homotopies among braids.

It can be shown that P2 (Σ푏) does not depend on the choice of the set 풫 = (푝1, 푝2), and that there is an isomorphism

P2 (Σ푏)' 휋1 (Σ푏 × Σ푏 − Δ, 풫) (2) where Δ ⊂ Σ푏 × Σ푏 is the diagonal. The group P2 (Σ푏) is ﬁnitely presented for all 푏, and explicit presentations can be found in [Bel04, Bir69, GG04, S70]. Here we follow the approach in [GG04, Sections 1-3], referring the reader to that paper for further details.

Proposition 1([GG04, Theorem 1]) Let 푝1, 푝2 ∈ Σ푏, with 푏 ≥ 2. Then the map of pointed topological spaces given by the projection onto the ﬁrst component

(Σ푏 × Σ푏 − Δ, 풫) −→ (Σ푏, 푝1) induces a split short exact sequence of groups

1 −→ 휋1 (Σ푏 − {푝1}, 푝2) −→ P2 (Σ푏) −→ 휋1 (Σ푏, 푝1) −→ 1. (3)

For all 푗 ∈ {1, . . . , 푏}, let us consider now the 2푏 elements

휌1 푗 , 휏1 푗 , 휌2 푗 , 휏2 푗 (4) of P2 (Σ푏) represented by the pure braids shown in Figure2. 6 Francesco Polizzi and Pietro Sabatino

Fig. 2 The pure braids 휌1 푗 , 휏1 푗 , 휌2 푗 , 휏2 푗 on Σ푏

If ℓ ≠ 푖, the path corresponding to 휌푖 푗 and 휏푖 푗 based at 푝ℓ is the constant path. Moreover, let 퐴12 be the pure braid shown in Figure3. In terms of the isomorphism (2), the generators 휌푖 푗 , 휏푖 푗 correspond to the generators of 휋1 (Σ푏 × Σ푏 − Δ, 풫) coming from the usual description of Σ푏 as the identiﬁcation space of a regular 2푏- gon, whereas 퐴12 corresponds to the homotopy class in Σ푏 × Σ푏 − Δ of a topological loop in Σ푏 × Σ푏 that “winds once” around Δ.

Fig. 3 The pure braid 퐴12 on Σ푏

The elements 휌21, . . . , 휌2푏, 휏21, . . . , 휏2푏, 퐴12 (5) can be seen as generators of the kernel 휋1 (Σ푏 −{푝1}, 푝2) in (3), whereas the elements

휌11, . . . , 휌1푏, 휏11, . . . , 휏1푏 (6) are lifts of a set of generators of 휋1 (Σ푏, 푝1) via the quotient map P2 (Σ푏) −→ 휋1 (Σ푏, 푝1), namely, they form a complete system of coset representatives for 휋1 (Σ푏, 푝1). By Proposition1, the braid group P2 (Σ푏) is a semi-direct product of the two groups 휋1 (Σ푏 −{푝1}, 푝2) and 휋1 (Σ푏, 푝1), whose presentations are both well-known; then, in order to write down a presentation for P2 (Σ푏), it only remains to specify how the generators in (6) act by conjugation on those in (5). This is provided by the following result, cf. [CaPol19, Theorem 1.6], where the conjugacy relations are expressed in the commutator form (i.e., instead of 푥푦푥−1 = 푧 we write [푥, 푦] = 푧푦−1).

Proposition 2([GG04, Theorem 7]) The group P2 (Σ푏) admits the following presentation.

Generators 휌1 푗 , 휏1 푗 , 휌2 푗 , 휏2 푗 , 퐴12 푗 = 1, . . . , 푏.

Relations Finite quotients of surface braid groups and double Kodaira ﬁbrations 7

• Surface relations:

−1 −1 −1 −1 −1 −1 −1 −1 −1 [휌1푏 , 휏1푏 ] 휏1푏 [휌1 푏−1, 휏1 푏−1] 휏1 푏−1 ···[휌11 , 휏11 ] 휏11 (휏11 휏12 ··· 휏1푏) = 퐴12 −1 −1 −1 −1 −1 −1 −1 [휌21 , 휏21] 휏21 [휌22 , 휏22] 휏22 ···[휌2푏 , 휏2푏] 휏2푏 (휏2푏 휏2 푏−1 ··· 휏21 ) = 퐴12

• Action of 휌1 푗 :

[휌1 푗 , 휌2푘 ] = 1 if 푗 < 푘 (7)

[휌1 푗 , 휌2 푗 ] = 1 −1 −1 −1 [휌1 푗 , 휌2푘 ] = 퐴12 휌2푘 휌2 푗 퐴12 휌2 푗 휌2푘 if 푗 > 푘

[휌1 푗 , 휏2푘 ] = 1 if 푗 < 푘 −1 [휌1 푗 , 휏2 푗 ] = 퐴12 −1 [휌1 푗 , 휏2푘 ] = [퐴12 , 휏2푘 ] if 푗 > 푘

−1 [휌1 푗 , 퐴12] = [휌2 푗 , 퐴12]

• Action of 휏1 푗 :

[휏1 푗 , 휌2푘 ] = 1 if 푗 < 푘 −1 [휏1 푗 , 휌2 푗 ] = 휏2 푗 퐴12 휏2 푗 −1 [휏1 푗 , 휌2푘 ] = [휏2 푗 , 퐴12] if 푗 > 푘

[휏1 푗 , 휏2푘 ] = 1 if 푗 < 푘 −1 [휏1 푗 , 휏2 푗 ] = [휏2 푗 , 퐴12] −1 −1 −1 −1 −1 [휏1 푗 , 휏2푘 ] = 휏2 푗 퐴12 휏2 푗 퐴12 휏2푘 퐴12 휏2 푗 퐴12 휏2 푗 휏2푘 if 푗 > 푘

−1 [휏1 푗 , 퐴12] = [휏2 푗 , 퐴12]

Remark 1 The inclusion map 휄: Σ푏 × Σ푏 − Δ −→ Σ푏 × Σ푏 induces a group epimorphism 휄∗ : P2 (Σ푏) −→ 휋1 (Σ푏 × Σ푏, 풫), whose kernel is the normal clo- sure of the subgroup generated by 퐴12. Thus, given any group homomorphism 휑 : P2 (Σ푏) −→ 퐺, it factors through 휋1 (Σ푏 × Σ푏, 풫) if and only if 휑(퐴12) is trivial. Tedious but straightforward calculations show that the presentation given in Proposition2 is invariant under the substitutions

−1 −1 퐴12 ←→ 퐴12 , 휏1 푗 ←→ 휏2 푏+1− 푗 , 휌1 푗 ←→ 휌2 푏+1− 푗 , where 푗 ∈ {1, . . . , 푏}. These substitutions correspond to the involution of P2 (Σ푏) induced by a reﬂection of Σ푏 switching the 푗-th handle with the (푏 + 1 − 푗)-th handle 8 Francesco Polizzi and Pietro Sabatino for all 푗. Hence we can exchange the roles of 푝1 and 푝2 in (3), and see P2 (Σ푏) as the middle term of a split short exact sequence of the form

1 −→ 휋1 (Σ푏 − {푝2}, 푝1) −→ P2 (Σ푏) −→ 휋1 (Σ푏, 푝2) −→ 1, (8) induced by the projection onto the second component

(Σ푏 × Σ푏 − Δ, 풫) −→ (Σ푏, 푝2).

Now the elements 휌11, . . . , 휌1푏, 휏11, . . . , 휏1푏, 퐴12 can be seen as generators of the kernel 휋1 (Σ푏 −{푝2}, 푝1) in (8), whereas the elements

휌21, . . . , 휌2푏, 휏21, . . . , 휏2푏 yield a complete system of coset representatives for 휋1 (Σ푏, 푝2). We can now deﬁne the objects studied in this paper.

Deﬁnition 2 Take positive integers 푏, 푛 ≥ 2. A ﬁnite group 퐺 is called a pure braid quotient of type (푏, 푛) if there exists a group epimomorphism

휑 : P2 (Σ푏) −→ 퐺 (9) such that 휑(퐴12) has order 푛.

Remark 2 Since we are assuming 푛 ≥ 2, the element 휑(퐴12) is non-trivial and so the epimorphism 휑 does not factor through 휋1 (Σ푏 × Σ푏, 풫), see Remark1. The geometrical relevance of this condition will be explained in Section3. The same condition also shows that a pure braid quotient is necessarily non-abelian, because 휑(퐴12) is a non-trivial commutator in 퐺, see (7).

Sometimes, by abuse of terminology, we will use the term pure braid quotient in order to indicate the full datum of the quotient homomorphism (9), instead of the quotient group 퐺 alone. If 퐺 is a pure braid quotient, then the two subgroups

퐾1 := h휑(휌11), 휑(휏11), . . . , 휑(휌1푏), 휑(휏1푏), 휑(퐴12)i (10) 퐾2 := h휑(휌21), 휑(휏21), . . . , 휑(휌2푏), 휑(휏2푏), 휑(퐴12)i are both normal in 퐺, and hence there are two short exact sequences

1 −→ 퐾1 −→ 퐺 −→ 푄2 −→ 1

1 −→ 퐾2 −→ 퐺 −→ 푄1 −→ 1, in which the elements 휑(휌21), 휑(휏21), . . . , 휑(휌2푏), 휑(휏2푏) yield a complete system of coset representatives for 푄2, whereas the elements 휑(휌11), 휑(휏11), . . . , 휑(휌1푏), 휑(휏1푏) yield a complete system of coset representatives for 푄1. Finite quotients of surface braid groups and double Kodaira ﬁbrations 9

Let us end this section with the following deﬁnition, whose geometrical meaning will become clear later, see Remark5 of Section3.

Deﬁnition 3 A pure braid quotient 휑 : P2 (Σ푏) −→ 퐺 is called strong if 퐾1 = 퐾2 = 퐺.

2 Extra-special groups as pure braid quotients

We know that P2 (Σ푏) is residually 푝-finite for all prime number 푝 ≥ 2, see [BarBel09, pp. 1481-1490]. This implies that, for every 푝, we can find a non-abelian finite 푝-group 퐺 that is a pure braid quotient of type (푏, 푞), where 푞 is a power of 푝. However, it can be tricky to explicitly describe some of these quotients. In this section we will present a number of results in this direction, obtained in the series of articles [CaPol19, Pol20, PolSab21]; our exposition here will closely follow the treatment given in these papers. Let us start by introducing the following classical definition, see for instance [Gor07, p. 183] and [Is08, p. 123].

Deﬁnition 4 Let 푝 be a prime number. A ﬁnite 푝-group 퐺 is called extra-special if its center 푍 (퐺) is cyclic of order 푝 and the quotient 푉 = 퐺/푍 (퐺) is a non-trivial, elementary abelian 푝-group.

An elementary abelian 푝-group is a finite-dimensional vector space over the field dim 푉 Z푝, hence it is of the form 푉 = (Z푝) and 퐺 fits into a short exact sequence

1 −→ Z푝 −→ 퐺 −→ 푉 −→ 1. (11)

Note that, 푉 being abelian, we must have [퐺, 퐺] = Z푝, namely the commutator subgroup of 퐺 coincides with its center. Furthermore, since the extension (11) is central, it cannot be split, otherwise 퐺 would be isomorphic to the direct product of the two abelian groups Z푝 and 푉, which is impossible because 퐺 is non-abelian. It can be also proved that, if 퐺 is extra-special, then dim푉 is even and so |퐺| = 푝dim 푉 +1 is an odd power of 푝. For every prime number 푝, there are precisely two isomorphism classes 푀(푝), 푁 (푝) of non-abelian groups of order 푝3, namely

푀(푝) = hr, t, z | r푝 = t푝 = 1, z푝 = 1, [r, z] = [t, z] = 1, [r, t] = z−1i 푁 (푝) = hr, t, z | r푝 = t푝 = z, z푝 = 1, [r, z] = [t, z] = 1, [r, t] = z−1i and both of them are in fact extra-special, see [Gor07, Theorem 5.1 of Chapter 5]. If 푝 is odd, then the groups 푀(푝) and 푁 (푝) are distinguished by their exponent, which equals 푝 and 푝2, respectively. If 푝 = 2, the group 푀(푝) is isomorphic to the dihedral group 퐷8, whereas 푁 (푝) is isomorphic to the quaternion group Q8. We can now provide the classiﬁcation of extra-special 푝-groups, see [Gor07, Section 5 of Chapter 5]. 10 Francesco Polizzi and Pietro Sabatino

Proposition 3 If 푏 ≥ 2 is a positive integer and 푝 is a prime number, there are exactly two isomorphism classes of extra-special 푝-groups of order 푝2푏+1, that can be described as follows.

• The central product H2푏+1 (Z푝) of 푏 copies of 푀(푝), having presentation

푝 푝 푝 H2푏+1 (Z푝) = h r1, t1,..., r푏, t푏, z | r 푗 = t 푗 = z = 1,

[r 푗 , z] = [t 푗 , z] = 1,

[r 푗 , r푘 ] = [t 푗 , t푘 ] = 1,

−훿 푗푘 [r 푗 , t푘 ] = z i.

If 푝 is odd, this group has exponent 푝. • The central product G2푏+1 (Z푝) of 푏 − 1 copies of 푀(푝) and one copy of 푁 (푝), having presentation

( ) h | 푝 푝 G2푏+1 Z푝 = r1, t1,..., r푏, t푏, z r푏 = t푏 = z, 푝 푝 푝 푝 푝 r1 = t1 = ... = r푏−1 = t푏−1 = z = 1, [r 푗 , z] = [t 푗 , z] = 1,

[r 푗 , r푘 ] = [t 푗 , t푘 ] = 1,

−훿 푗푘 [r 푗 , t푘 ] = z i.

If 푝 is odd, this group has exponent 푝2. We are now in a position to state our ﬁrst two results.

Theorem 1([CaPol19, Section 3], [Pol20, Theorem 2.10]) If 푏 ≥ 2 is an integer and 푝 ≥ 5 is a prime number, then both extra-special 푝-groups of order 푝4푏+1 are pure braid quotients of type (푏, 푝). All these quotients are non-strong, in fact 퐾1 2푏 and 퐾2 have index 푝 in 퐺.

Theorem 2([CaPol19, Section 3], [Pol20, Theorem 2.7]) If 푏 ≥ 2 is an integer and 푝 is a prime number dividing 푏 + 1, then both extra-special 푝-groups of order 푝2푏+1 are strong pure braid quotients of type (푏, 푝).

Theorems1 and2 were originally proved by the ﬁrst author and A. Causin in [CaPol19], but only in the case 퐺 = H4푏+1 (Z푝) and 퐺 = H2푏+1 (Z푝), respectively, by using some group-cohomological results related to the structure of the cohomology ∗ algebra 퐻 (Σ푏 × Σ푏 − Δ, Z푝). Let us give here a sketch of the argument, referring the reader to the aforementioned paper for full details. Assuming 푝 ≥ 3, we identiﬁed H4푏+1 (Z푝) with the symplectic Heisenberg group Heis(푉, 휔), where

4푏 푉 = 퐻1 (Σ푏 × Σ푏 − Δ, Z푝)' 퐻1 (Σ푏 × Σ푏, Z푝)'(Z푝) and 휔 is a symplectic form on 푉. This group is the central extension Finite quotients of surface braid groups and double Kodaira ﬁbrations 11

1 −→ Z푝 −→ Heis(푉, 휔) −→ 푉 −→ 1 (12) of the additive group 푉 given as follows: the underlying set of Heis(푉, 휔) is 푉 × Z푝, endowed with the group law

1 (푣 , 푡 )(푣 , 푡 ) = 푣 + 푣 , 푡 + 푡 + 휔(푣 , 푣 ) . (13) 1 1 2 2 1 2 1 2 2 1 2

4푏 By basic linear algebra, all symplectic forms on (Z푝) are equivalent to the standard symplectic form; thus, given two symplectic forms 휔1, 휔2 on 푉, the two Heisenberg groups Heis(푉, 휔1), Heis(푉, 휔2) are isomorphic. Moreover, the center of the Heisenberg group coincides with its commutator subgroup and is isomorphic to Z푝. Now, let 휙 : P2 (Σ푏) −→ 푉 be the group epimorphism given by the composition of the reduction mod 푝 map 퐻1 (Σ푏 × Σ푏 − Δ, Z) −→ 푉 with the abelianization map P2 (Σ푏) −→ 퐻1 (Σ푏 × Σ푏 − Δ, Z). We have a commutative diagram

P2 (Σ푏) 휑 휙 x 1 / Z푝 / Heis(푉, 휔) / 푉 / 1

2 and we denote by 푢 ∈ 퐻 (푉, Z푝) the cohomology class corresponding to the bottom Heisenberg extension. Then a lifting 휑 : P2 (Σ푏) −→ Heis(푉, 휔) of 휙 exists if and ∗ 2 only if 휙 푢 = 0 ∈ 퐻 (P2 (Σ푏), Z푝). The next step is to provide an interpretation of the cohomological condition 휙∗푢 = 0 in terms of the symplectic form 휔, and this is achieved by using the following facts: • we have a natural identiﬁcation

2 2 ∨ ∨ 퐻 (푉, Z푝)' Λ (푉 ) ⊕ 푉 (14)

under which the extension class 푢 giving the Heisenberg central extension (12) corresponds to (휔, 휖). Here 휖 : 푉 −→ Z푝 stands for the linear functional on 푉 deﬁned by 휖 (푣) = 푤 푝, where 푤 is any preimage of 푣 in Heis(푉, 휔); • we have natural identiﬁcations

∨ 1 1 푉 ' 퐻 (Σ푏 × Σ푏 − Δ, Z푝)' 퐻 (Σ푏 × Σ푏, Z푝)

and there is a commutative diagram 12 Francesco Polizzi and Pietro Sabatino

2 2 ∨ 휉 2 Alt (푉) ' ∧ 푉 / 퐻 (Σ푏 × Σ푏, Z푝)

휂 ) 2 퐻 (Σ푏 × Σ푏 − Δ, Z푝)

where the vertical map is the quotient by the 1-dimensional vector subspace of 2 퐻 (Σ푏 × Σ푏, Z푝) generated by the class 훿 of the diagonal, whereas 휂 and 휉 stand for the cup product maps; • Σ푏 × Σ푏 − Δ is an aspherical space, namely all its higher homotopy group vanish, and so for all 푖 ≥ 1 there is a natural isomorphism

푖 푖 퐻 (Σ푏 × Σ푏 − Δ, Z푝)' 퐻 (P2 (Σ푏), Z푝) (15)

where Z푝 is endowed, as an abelian group, with the structure of trivial P2 (Σ푏)- module. Combining all this, we infer that there is a commutative diagram

∗ 2 ∨ ∨ ' 2 휙 2 ∧ 푉 ⊕ 푉 / 퐻 (푉, Z푝) / 퐻 (P2 (Σ푏), Z푝)

' 2 2 ∨ 휂 2 Alt (푉) ' ∧ 푉 / 퐻 (Σ푏 × Σ푏 − Δ, Z푝), where the isomorphism on the left is (14), the vertical map on the left is the projection onto the ﬁrst summand and the vertical map on the right is (15). Since the projection 2 2 of the extension class 푢 ∈ 퐻 (푉, Z푝) can be naturally identiﬁed with 휔 ∈ Alt (푉), we have proved the following

∗ 2 Proposition 4 The obstruction class 휙 푢 ∈ 퐻 (P2 (Σ푏), Z푝) can be naturally inter- 2 2 preted as the image 휂(휔) ∈ 퐻 (Σ푏 ×Σ푏 −Δ, Z푝) of the symplectic form 휔 ∈ Alt (푉) via the cup-product map 휂. As a consequence, we obtain the following lifting criterion, that we believe is of independent interest.

Proposition 5 A lifting 휑 : P2 (Σ푏) −→ Heis(푉, 휔) of 휙 : P2 (Σ푏) −→ 푉 exists if and only if 휂(휔) = 0. Furthermore, if 휑 exists, then 휑(퐴12) has order 푝 if and only 2 if 휉(휔) ∈ 퐻 (Σ푏 × Σ푏, Z푝) is a non-zero integer multiple of the diagonal class 훿. In this case, 휑 is necessarily surjective. Inspired by Proposition5, we say that a symplectic form 휔 ∈ Alt2 (푉) is of Heisenberg type if 휉(휔) is a non-zero integer multiple of 훿; equivalently, 휔 is of Heisenberg type if 휂(휔) = 0 and 휉(휔) ≠ 0. By the previous discussion it follows that, if 휔 is of Heisenberg type, Heis(푉, 휔) is a pure braid quotient of type (푏, 푝). We are therefore left with the task of constructing symplectic forms of Heisen- 1 berg type on 푉. We denote by 훼1, 훽1, . . . , 훼푏, 훽푏 the images in 퐻 (Σ푏, Z푝) = Finite quotients of surface braid groups and double Kodaira ﬁbrations 13

1 1 퐻 (Σ푏, Z) ⊗ Z푝 of the elements of a basis of 퐻 (Σ푏, Z) which is symplectic with respect to the cup product; then, we can choose for 푉 the ordered basis

푟11, 푡11, . . . , 푟1푏, 푡1푏, 푟21, 푡21, . . . , 푟2푏, 푡2푏 (16) where, under the isomorphism 푉 ' 퐻1 (Σ푏 × Σ푏, Z푝) induced by the inclusion 휄: Σ푏 × Σ푏 − Δ −→ Σ푏 × Σ푏, the elements 푟1 푗 , 푡1 푗 , 푟2 푗 , 푡2 푗 ∈ 푉 are the duals of 1 1 the elements 훼 푗 ⊗ 1, 훽 푗 ⊗ 1, 1 ⊗ 훼 푗 , 1 ⊗ 훽 푗 ∈ 퐻 (Σ푏 × Σ푏, Z푝)' 퐻 (Σ푏, Z푝) ⊗ 1 퐻 (Σ푏, Z푝), respectively. Since 푝 ≥ 5, we can ﬁnd non-zero scalars 휆1, . . . , 휆푏, 휇1, . . . , 휇푏 ∈ Z푝 such that 1 − 휆푖 휇1 ≠ 0 for all 푖 ∈ {1, . . . , 푏} and

푏 푏 ∑︁ ∑︁ 휆 푗 = 휇 푗 = 1. (17) 푗=1 푗=1

Then we consider the alternating form 휔 : 푉 × 푉 −→ Z푝 represented, with respect to the ordered basis (16), by the skew-symmetric matrix 퐿푏 퐽푏 Ω푏 = ∈ Mat(4푏, Z푝) (18) 퐽푏 푀푏 where the blocks are the elements of Mat(2푏, Z푝) given by

0 휆 0 휇 1 0 1 0 ©− ª ©− ª 휆1 0 ® 휇1 0 ® . ® . ® 퐿푏 = . . ® 푀푏 = . . ® ® ® 0 휆 ® 0 휇 ® 0 푏® 0 푏® −휆 0 −휇 0 « 푏 ¬ « 푏 ¬ 0 −1 © 0 ª 1 0 ® . ® 퐽푏 = . . ® ® 0 −1® 0 ® 1 0 « ¬ Standard Gaussian elimination shows that

2 2 2 det Ω푏 = (1 − 휆1 휇1) (1 − 휆2 휇2) ···(1 − 휆푏 휇푏) > 0 and so 휔 is non-degenerate. Moreover, a direct computation yields 휉(휔) = 훿, that is, 휔 is of Heisenberg type. The calculation of the indices of 퐾1 and 퐾2 in 퐺 is now straightforward, and this completes the proof of Theorem1 in the case 퐺 = H4푏+1 (Z푝).

Now, let us assume that 푝 divides 푏 + 1, so that −푏 = 1 holds in Z푝, and take 14 Francesco Polizzi and Pietro Sabatino

휆1 = ... = 휆푏 = 휇1 = ... = 휇푏 = −1 ∈ Z푝.

Therefore relations (17) are satisﬁed and the same computations as in the previous case show that the corresponding alternating form 휔 satisﬁes 휉(휔) = 훿. However, 휔 is not symplectic, since its associate matrix 퐽푏 퐽푏 Ω푏 = ∈ Mat(4푏, Z푝) 퐽푏 퐽푏 has rank 2푏 and, subsequently, 휔 has a 2푏-dimensional kernel 푉0, namely

푉0 = h푟11 − 푟21, 푡11 − 푡21, . . . , 푟1푏 − 푟2푏, 푡1푏 − 푡2푏i. (19)

The set 푉 × Z푝, with the operation (13), is a group whose center equals 푉0 × Z푝 and that, with slight abuse of notation, we denote again by Heis(푉, 휔). Furthermore, the argument in Proposition5 still applies, providing the existence of a lifting P2 (Σ푏) −→ Heis(푉, 휔). Setting 푊 = 푉/푉0, the alternating form on 푉 descends to a symplectic form on 푊, that we denote it again by 휔; so Heis(푊, 휔) is a genuine Heisenberg group, endowed with a group epimorphism Heis(푉, 휔) −→ Heis(푊, 휔). Composing this epimorphism with the lifting P2 (Σ푏) −→ Heis(푉, 휔), we obtain a group epimorphism 휑 : P2 (Σ푏) −→ Heis(푊, 휔) such that 휑(퐴12) is non-trivial and central, hence of order 푝. Since 푊 is a Z푝-vector space of dimension 2푏, the group Heis(푊, 휔) is isomorphic to H2푏+1 (Z푝); ﬁnally, a simple computation based on the expression (19) for ker Ω푏 yields 퐾1 = 퐾2 = 퐺, and this shows Theorem2 in the case 퐺 = H2푏+1 (Z푝). The proof of Theorems1 and2 in full generality (i.e, for all extra-special groups) was given in [Pol20], using a completely algebraic technique that avoided the use of symplectic geometry and of group cohomology. It is based on the following

Deﬁnition 5 Let 퐺 be a ﬁnite group. A diagonal double Kodaira structure of type (푏, 푛) on 퐺 is an ordered set of 4푏 + 1 generators

픖 = (r11, t11,..., r1푏, t1푏, r21, t21,..., r2푏, t2푏, z), with 표(z) = 푛, that are images of the ordered set of generators

(휌11, 휏11, . . . , 휌1푏, 휏1푏, 휌21, 휏21, . . . , 휌2푏, 휏2푏, 퐴12) via a pure braid quotient 휑 : P2 (Σ푏) −→ 퐺 of type (푏, 푛). The structure is called strong if hr11, t11,..., r1푏, t1푏i = hr21, t21,..., r2푏, t2푏i = 퐺.

Therefore, checking whether 퐺 is a pure braid quotient of type (푏, 푛) is equivalent to checking whether it admits a diagonal double Kodaira structure 픖 of type (푏, 푛). Moreover, by deﬁnition, 휑 : P2 (Σ푏) −→ 퐺 is strong if and only if 픖 is. Let us refer now to the presentations for extra-special 푝-groups given in Propo- sition3. Assuming that 푝 divides 푏 + 1, in both cases 퐺 = H2푏+1 (Z푝) and Finite quotients of surface braid groups and double Kodaira ﬁbrations 15

퐺 = G2푏+1 (Z푝) we can obtain a strong diagonal double Kodaira structure 픖 on 퐺 by setting r1 푗 = r2 푗 = r 푗 , t1 푗 = t2 푗 = t 푗 for all 푗 = 1, . . . , 푏. The divisibility condition is necessary to ensure that the element of 픖 satisfy the two relations coming from the surface relations in (4). This proves Theorem2. In order to prove Theorem1, it is convenient to consider the following alter- native presentation of extra-special 푝-groups. Consider any non-degenerate, skew- symmetrix matrix 퐴 = (푎 푗푘 ) of order 2푏 over Z푝, and consider the ﬁnitely presented groups

( ) h | 푝 푝 푝 H 퐴 = x1,..., x2푏, z x1 = ... = x2푏 = z = 1, [x1, z] = ... = [x2푏, z] = 1, (20) 푎 푗푘 [x 푗 , x푘 ] = z i,

( ) h | 푝 푝 푝 G 퐴 = x1,..., x2푏, z x1 = ... = x2푏−2 = z = 1, 푝 푝 x = x = z, 2푏−1 2푏 (21) [x1, z] = ... = [x2푏, z] = 1, 푎 푗푘 [x 푗 , x푘 ] = z i,

푎 where the exponent in z 푗푘 stands for any representative in Z of 푎 푗푘 ∈ Z푝. Standard computations show that H(퐴)' H2푏+1 (Z푝) and G(퐴)' G2푏+1 (Z푝). Now we can take as 퐴 the matrix Ω푏 ∈ Mat(4푏, Z푝) given in (18). Setting 퐺 = H(Ω푏) or 퐺 = G(Ω푏), the group 퐺 is generated by a set of 4푏 + 1 elements

픖 = {r11, t11,..., r1푏, t1푏, r21, t21,..., r2푏, t2푏, z} subject to the relations (20) or (21), respectively. One can check that 픖 provides a diagonal double Kodaira structure of type (푏, 푝) on 퐺, and so a diagonal double Kodaira structure of the same type on the isomorphic group H4푏+1 (Z푝) or G4푏+1 (Z푝). This proves Theorem1. Remark 3 In particular, the pure braid quotients of smallest order detected by the methods detailed so far are the extra-special groups of order 27 = 128, corresponding to the case (푏, 푝) = (3, 2) in Theorem2. Recently, in the paper [PolSab21], we were able to signiﬁcantly lower the value of |퐺|, actually providing a sharp lower bound for the order of a pure braid quotient. Theorem 3([PolSab21, Theorem A]) Assume that 퐺 is a ﬁnite group that is a pure braid quotient. Then |퐺| ≥ 32, with equality if and only if 퐺 is extra-special. In this case, the following holds. • There are precisely 2211840 = 1152 · 1920 distinct group epimorphisms 휑 : P2 (Σ2) −→ 퐺, and all of them make 퐺 a strong pure braid quotient of type (2, 2). 16 Francesco Polizzi and Pietro Sabatino

• if 퐺 = 퐺(32, 49) = H5 (Z2), these epimorhisms form 1920 orbits under the natural action of Aut(퐺). • if 퐺 = 퐺(32, 50) = G5 (Z2), these epimorhisms form 1152 orbits under the natural action of Aut(퐺).

The proof of Theorem3 is obtained again by looking at the diagonal double Kodaira structures on 퐺, see Deﬁnition5.

Remark 4 A key observation is that if 퐺 is a CCT-group, namely 퐺 is not abelian and commutativity is a transitive relation on the set of the non-central elements, then 퐺 admits no diagonal double Kodaira structures and, subsequently, it cannot be a pure braid quotient.

A long but straightforward analysis shows that there are precisely eight non- CCT groups with 퐺 ≤ 32, namely 퐺 = S4 and 퐺 = 퐺(32, 푡) with 푡 ∈ {6, 7, 8, 43, 44, 49, 50}. These case are handled separately, and a reﬁned analysis proves that only 퐺(32, 49) and 퐺(32, 50), i.e. the two extra-special groups, admit diagonal double Kodaira structures. Finally, the number of such structures in each case is computed by using the same techniques as in [Win72]; more precisely, we exploit the fact that 푉 = 퐺/푍 (퐺) can be endowed with a natural structure of 4-dimensional symplectic vector space over Z2, and that Out(퐺) embeds in Sp(4, Z2) as the orthogonal group associated with the quadratic form 푞 on 푉 related to the symplectic form (· , ·) by 푞(x y) = 푞(x) + 푞(y) + (x, y).

3 Geometrical application: diagonal double Kodaira ﬁbrations

Recall that a Kodaira fibration is a smooth, connected holomorphic fibration 푓1 : 푆 −→ 퐵1, where 푆 is a compact complex surface and 퐵1 is a compact complex curve, which is not isotrivial (this means that not all fibres are biholomorphic each other). The genus 푏1 := 푔(퐵1) is called the base genus of the fibration, whereas the genus 푔 := 푔(퐹), where 퐹 is any fibre, is called the fibre genus. Definition 6 A double Kodaira surface is a compact complex surface 푆, endowed with a double Kodaira fibration, namely a surjective, holomorphic map 푓 : 푆 −→ 퐵1 × 퐵2 yielding, by composition with the natural projections, two Kodaira fibrations 푓푖 : 푆 −→ 퐵푖, 푖 = 1, 2.

With a slight abuse of notation, in the sequel we will use the symbol Σ푏 to indicate both a closed Riemann surface of genus 푏 and its underlying real surface. If a ﬁnite group 퐺 is a pure braid quotient of type (푏, 푛) then, by using Grauert- Remmert’s extension theorem together with Serre’s GAGA, the group epimorphism 휑 : P2 (Σ푏) −→ 퐺 yields the existence of a smooth, complex, projective surface 푆 endowed with a Galois cover Finite quotients of surface braid groups and double Kodaira ﬁbrations 17

f : 푆 −→ Σ푏 × Σ푏 with Galois group 퐺 and branched precisely over Δ with branching order 푛, see [CaPol19, Proposition 3.4]. Composing the group monomorphisms 휋1 (Σ푏 − {푝푖 }, 푝 푗 ) −→ P2 (Σ푏) with 휑 : P2 (Σ푏) −→ 퐺, we get two homomorphisms

휑1 : 휋1 (Σ푏 − {푝2}, 푝1) −→ 퐺, 휑2 : 휋1 (Σ푏 − {푝1}, 푝2) −→ 퐺, whose images are the normal subgroups 퐾1 and 퐾2 defined in (10). By construction, these are the homomorphisms induced by the restrictions f푖 : Γ푖 −→ Σ푏 of the Galois cover f : 푆 −→ Σ푏 × Σ푏 to the fibres of the two natural projections 휋푖 : Σ푏 × Σ푏 −→ Σ푏. Since Δ intersects transversally at a single point all the fibres of the natural projections, it follows that both such restrictions are branched at precisely one point, and the number of connected components of the smooth curve Γ푖 ⊂ 푆 equals the index 푚푖 := [퐺 : 퐾푖] of 퐾푖 in 퐺. So, taking the Stein factorizations of the compositions 휋푖 ◦ f : 푆 −→ Σ푏 as in the diagram below 휋푖 ◦f 푆 Σ푏 푓 푖 휃푖 (22)

Σ푏푖 we obtain two distinct Kodaira ﬁbrations 푓푖 : 푆 −→ Σ푏푖 , hence a double Kodaira ﬁbration by considering the product morphism

푓 = 푓1 × 푓2 : 푆 −→ Σ푏1 × Σ푏2 .

Definition 7 We call 푓 : 푆 −→ Σ푏1 × Σ푏2 the diagonal double Kodaira fibration associated with the pure braid quotient 휑 : P(Σ푏) −→ 퐺. Conversely, we will say that a double Kodaira fibration 푓 : 푆 −→ Σ푏1 × Σ푏2 is of diagonal type (푏, 푛) if there exists a pure braid quotient 휑 : P(Σ푏) −→ 퐺 of the same type such that 푓 is associated with 휑.

Since the morphism 휃푖 : Σ푏푖 −→ Σ푏 is étale of degree 푚푖, by using the Hurwitz formula we obtain

푏1 − 1 = 푚1 (푏 − 1), 푏2 − 1 = 푚2 (푏 − 1). (23)

Moreover, the ﬁbre genera 푔1, 푔2 of the Kodaira ﬁbrations 푓1 : 푆 −→ Σ푏1 , 푓2 : 푆 −→

Σ푏2 are computed by the formulae |퐺| |퐺| 2푔1 − 2 = (2푏 − 2 + 픫), 2푔2 − 2 = (2푏 − 2 + 픫) , 푚1 푚2 where 픫 := 1 − 1/푛. Finally, the surface 푆 ﬁts into a diagram 18 Francesco Polizzi and Pietro Sabatino

f 푆 Σ푏 × Σ푏 푓 ×휃2 휃1

Σ푏1 × Σ푏2

so that the diagonal double Kodaira ﬁbration 푓 : 푆 −→ Σ푏1 × Σ푏2 is a ﬁnite cover of degree |퐺 | , branched precisely over the curve 푚1푚2

−1 (휃1 × 휃2) (Δ) = Σ푏1 ×Σ푏 Σ푏2 .

Such a curve is always smooth, being the preimage of a smooth divisor via an étale morphism. However, it is reducible in general, see [CaPol19, Proposition 3.11].

Remark 5 By definition, the pure braid quotient 휑 : P2 (Σ푏) −→ 퐺 is strong (see Definition3) if and only if 푚1 = 푚2 = 1, that in turn implies 푏1 = 푏2 = 푏, i.e., 푓 = f. In other words, 휑 is strong if and only if no Stein factorization as in (22) is needed or, equivalently, if and only if the Galois cover f : 푆 −→ Σ푏 × Σ푏 induced by 휑 is already a double Kodaira fibration, branched on the diagonal Δ ⊂ Σ푏 × Σ푏.

We can now compute the invariants of 푆 as follows, see [CaPol19, Proposition 3.8].

Proposition 6 Let 푓 : 푆 −→ Σ푏1 × Σ푏2 be a diagonal double Kodaira ﬁbration, associated with a pure braid quotient 휑 : P2 (Σ푏) −→ 퐺 of type (푏, 푛). Then we have

2 2 푐1 (푆) = |퐺| (2푏 − 2)(4푏 − 4 + 4픫 − 픫 ) 푐2 (푆) = |퐺| (2푏 − 2)(2푏 − 2 + 픫) where 픫 = 1 − 1/푛. As a consequence, the slope and the signature of 푆 can be expressed as

푐2 (푆) 2픫 − 픫2 휈(푆) = 1 = 2 + 푐 (푆) 2푏 − 2 + 픫 2 (24) 1 1 1 휎(푆) = 푐2 (푆) − 2푐 (푆) = |퐺| (2푏 − 2) 1 − . 3 1 2 3 푛2

Remark 6 Not all double Kodaira ﬁbrations are of diagonal type. In fact, if√푆 is of diagonal type, then its slope satisﬁes 휈(푆) = 2 + 푠, where 0 < 푠 < 6 − 4 2, see [Pol20, Proposition 3.12 and Remark 3.13].

We can now specialize these results, by taking as 퐺 an extra-special 푝-group and using what we have proved in Section2. Fix 푏 = 2 and let 푝 ≥ 5 be a prime number. Then every extra-special 푝-group 퐺 of order 푝4푏+1 = 푝9 is a non-strong pure braid quotient of type (2, 푝) and such 2푏 0 4 that 푚1 = 푚2 = 푝 , see Theorem1. Setting 푏 := 푝 + 1, cf. equations (23), by [CaPol19, Proposition 3.11] the associated diagonal double Kodaira ﬁbration Finite quotients of surface braid groups and double Kodaira ﬁbrations 19

푓 : 푆 −→ Σ푏0 × Σ푏0 is a cyclic cover of degree 푝, branched over a reduced, smooth divisor 퐷 of the form ∑︁ 퐷 = 퐷푐 2푏 푐∈(Z푝) where the 퐷푐 are pairwise disjoint graphs of automorphisms of Σ푏0 . By using Proposition6, we can now construct inﬁnitely many double Kodaira ﬁbrations with slope strictly higher than 2 + 1/3.

Theorem 4([CaPol19, Proposition 3.12], [Pol20, Theorem 3.8]) Let 푓 : 푆 푝 −→ Σ푏0 × Σ푏0 be the diagonal double Kodaira ﬁbration associated with a non-strong pure braid quotient 휑 : P2 (Σ2) −→ 퐺 of type (2, 푝), where 퐺 is an extra-special 9 푝-group 퐺 of order 푝 . Then the maximum slope 휈(푆 푝) is attained for precisely two values of 푝, namely 12 휈(푆 ) = 휈(푆 ) = 2 + . 5 7 35

Furthermore, 휈(푆 푝) > 2 + 1/3 for all 푝 ≥ 5. More precisely, if 푝 ≥ 7 the function 휈(푆 푝) is strictly decreasing and 1 lim 휈(푆 푝) = 2 + . 푝→+∞ 3 Remark 7 The original examples by Atiyah, Hirzebruch and Kodaira have slope lying in the interval (2, 2 + 1/3], see [BHPV03, p. 221]. Our construction provides an infinite family of Kodaira fibred surfaces such that 2 + 1/3 < 휈(푆) ≤ 2 + 12/35, maintaining at the same time a complete control on both the base genus and the signature. By contrast, the ingenious “tautological construction" used in [CatRol09] yields a higher slope than ours, namely 2 + 2/3, but it involves an étale pullback “of sufficiently large degree”, that completely loses control on the other quantities. Note that [LLR20] gives (at least in principle) an effective version of the pullback construction. If 푝 is a prime number dividing 푏 + 1, by Theorem2 every extra-special 푝-group 퐺 of order 푝2푏+1 is a strong pure braid quotient of type (푏, 푝), and this gives in turn a diagonal double Kodaira fibration 푓 : 푆 −→ Σ푏 ×Σ푏, see Remark5. If 훚: N −→ N stands for the arithmetic function counting the number of distinct prime factors of a positive integer, see [HarWr08, p. 335], we obtain

Theorem 5([CaPol19, Corollary 3.18], [Pol20, Theorem 3.5]) Let Σ푏 be any closed Riemann surface of genus 푏. Then there exists a double Kodaira fibration 푓 : 푆 −→ Σ푏 × Σ푏. Moreover, denoting by 휅(푏) the number of such fibrations, we have 휅(푏) ≥ 훚(푏 + 1). In particular, lim sup 휅(푏) = +∞. 푏→+∞ As far as we know, this is the first construction showing that all curves of genus 푏 ≥ 2 (and not only some special curves with extra automorphisms) appear in the 20 Francesco Polizzi and Pietro Sabatino base of at least one double Kodaira fibration 푓 : 푆 −→ Σ푏 × Σ푏. In addition, two Kodaira fibred surfaces corresponding to two distinct prime divisors of 푏 + 1 are non-homeomorphic, because the corresponding signatures are different: just use (24) with 푛 = 푝 and |퐺| = 푝2푏+1 and note that, for fixed 푏, the function expressing 휎(푆) is strictly increasing in 푝. This shows that the number of topological types of 푆, for a fixed base Σ푏, can be arbitrarily large. Let us now consider Theorem3, whose geometrical translation is

Theorem 6([PolSab21, Theorem B]) Let 퐺 be a ﬁnite group and f : 푆 −→ Σ푏 ×Σ푏 be a Galois cover, with Galois group 퐺, branched over the diagonal Δ with branching order 푛. Then |퐺| ≥ 32, and equality holds if and only if 퐺 is extra-special. In this case, the following holds.

(1) There exist 2211840 = 1152 · 1920 distinct 퐺-covers f : 푆 −→ Σ2 × Σ2, and all of them are diagonal double Kodaira ﬁbrations such that

푏1 = 푏2 = 2, 푔1 = 푔2 = 41, 휎(푆) = 16.

(2) If 퐺 = 퐺(32, 49) = H5 (Z2), these 퐺-covers form 1920 equivalence classes up to cover isomorphisms. (3) If 퐺 = 퐺(32, 50) = H5 (Z2), these 퐺-covers form 1152 equivalence classes up to cover isomorphisms. As a consequence, we obtain a sharp lower bound for the signature of a diagonal double Kodaira ﬁbration. In fact, the second equality in (24) together with Theorem 6 imply that, for every such ﬁbration, we have

1 1 1 1 휎(푆) = |퐺| (2푏 − 2) 1 − ≥ · 32 ·(2 · 2 − 2) 1 − = 16, 3 푛2 3 22 and this in turn establishes the following result. Theorem 7([PolSab21, Theorem C]) Let 푆 be a double Kodaira surface, associated with a pure braid quotient 휑 : P2 (Σ푏) −→ 퐺 of type (푏, 푛). Then 휎(푆) ≥ 16, and equality holds precisely when (푏, 푛) = (2, 2) and 퐺 is an extra-special group of order 32. Remark 8 If 푆 is a double Kodaira fibration, corresponding to a pure braid quotient 휑 : P2 (Σ푏) −→ 퐺 of type (푏, 푛), then, using the terminology in [CatRol09], it is very simple. Let us denote by 픐푆 the connected component of the Gieseker moduli space of surfaces of general type containing the class of 푆, and by M푏 the moduli space of smooth curves of genus 푏. Thus, by applying [Rol10, Thm. 1.7] and using the fact that Δ ⊂ Σ푏 × Σ푏 is the graph of the identity id: Σ푏 −→ Σ푏, we infer that every surface in 픐푆 is still a very simple double Kodaira fibration and that there is a natural map of schemes M푏 −→ 픐푆, which is an isomorphism on geometric points. Roughly speaking, since the branch locus Δ ⊂ Σ푏 × Σ푏 is rigid, all the deformations of 푆 are realized by deformations Finite quotients of surface braid groups and double Kodaira fibrations 21 of Σ푏 × Σ푏 preserving the diagonal, hence by deformations of Σ푏, cf. [CaPol19, Proposition 3.22]. In particular, this shows that 픐푆 is a connected and irreducible component of the Gieseker moduli space.

Every Kodaira fibred surface 푆 has the structure of a real surface bundle over a real surface, and so 휎(푆) is divisible by 4, see [Mey73]. If, in addition, 푆 has a spin structure, i.e. its canonical class is 2-divisible in Pic(푆), then 휎(푆) is a positive multiple of 16 by Rokhlin’s theorem, and examples with 휎(푆) = 16 are constructed in [LLR20]. It is not known if there exists a Kodaira fibred surface with 휎(푆) ≤ 12. Constructing (double) Kodaira fibrations with small signature is a rather difficult problem. As far as we know, before the present work the only examples with signature 16 were the ones listed in [LLR20, Table 3, Cases 6.2, 6.6, 6.7 (Type 1), 6.9]. The examples in Theorem7 are new, since both the base genera and the fibre genera are different from the ones in the aforementioned cases. Our results also show that every curve of genus 2 is the base of a double Kodaira fibration with signature 16. Thus, we obtain two families of dimension 3 of such fibrations that, to our knowledge, provides the first examples of positive-dimensional families of double Kodaira fibrations with small signature. Theorem7 also provide new “double solutions” to a problem, posed by G. Mess and included in Kirby’s problem list in low-dimensional topology, see [Kir97, Prob- lem 2.18 A], asking what is the smallest number 푏 for which there exists a real surface bundle over a real surface with base genus 푏 and non-zero signature. We actually have 푏 = 2, also for double Kodaira fibrations.

Theorem 8([PolSab21, Theorem D]) Let 푆 be a double Kodaira surface, associated with a pure braid quotient 휑 : P2 (Σ푏) −→ 퐺 of type (2, 2), where 퐺 is an extra- special group of order 32. Then the real manifold 푋 underlying 푆 is a closed, orientable 4-manifold of signature 16 that can be realized as a real surface bundle over a real surface of genus 2, with ﬁbre genus 41, in two diﬀerent ways.

It is an interesting question whether 16 and 41 are the minimum possible values for the signature and the ﬁbre genus of a (non necessary diagonal) double Kodaira ﬁbration 푓 : 푆 −→ Σ2 × Σ2; however, this topic exceeds the scope of this paper.

4 Beyond |푮| = 32

This last section contains the new result of this article. As we already observed, so far we have detailed a rather clear picture regarding pure braid quotient groups 퐺 and the relative diagonal double Kodaira fibrations for |퐺| ≤ 32. Indeed, there is no pure braid quotient of order strictly less than 32 and for |퐺| = 32 we have only the two extra-special groups, see Theorem3. Furthermore, Theorem2 provides examples of pure braid quotients starting with order equal to 128, see Remark3, and they are extra-special groups, too. It seems then natural to investigate this matter further for |퐺| > 32, for instance, in order to look for non extra-special examples. In this 22 Francesco Polizzi and Pietro Sabatino direction we have obtained the following result that highlights the existence of a gap between orders 32 and 64. Theorem 9 If 퐺 is a finite group with 32 < |퐺| < 64, then 퐺 is not a pure braid quotient. Here we just give a sketch of the proof, while the full details will appear elsewhere. We know that the group 퐺 cannnot be abelian, see Remark2, and we also mentioned that CCT-groups cannot be pure braid quotients, see Remark4. On the other hand, by [PolSab21, Section 3] we know that, if 퐺 is a pure braid quotient and admits no proper quotients that are pure braid quotients, then 퐺 must be monolithic, i.e., the intersection soc(퐺) of its non-trivial normal subgroups is non-trivial. In fact, consider the epimorphism 휑 : P2 (Σ푏) −→ 퐺 and assume that there is a non-trivial, normal subgroup 푁 of 퐺 such that 휑(퐴12) ∉ 푁. Then, composing the projection 퐺 −→ 퐺/푁 with 휑, we obtain a pure braid quotient 휑¯ : P2 (Σ푏) −→ 퐺/푁, which leads to a contradiction. It follows that 휑(퐴12) ∈ soc(퐺), in particular 퐺 is monolithic. By Theorem3 this implies that, if a pure braid quotient 퐺 satisfies our assumptions on the order, then 퐺 must be monolithic. A straightforward computer calculation with GAP4 now shows that there are precisely two non-abelian groups 퐺 with 32 < |퐺| < 64 that are both non-CCT and monolithic, namely 퐺(54, 5) and 퐺(54, 6); by the remarks above, they are the only possible candidates to be pure braid quotients in that range for |퐺|. Finally, a brute force check (again by using GAP4) shows that these groups admit no diagonal double Kodaira structure, proving our assertion.

References

At69. M. F. Atiyah: The signature of fibre bundles, in Global Analysis (Papers in honor of K. Kodaira) 73–84, Univ. Tokyo Press (1969). (Cited on p.2) BarBel09. V. G. Bardakov, P. Bellingeri: On residual properties of pure braid groups of closed surfaces, Comm. Alg. 37 (2009), 1481–1490. (Cited on p.2,9) BHPV03. W. Barth, K. Hulek, C.A.M. Peters, A. Van de Ven, Compact Complex Surfaces. Grundlehren der Mathematischen Wissenschaften, Vol 4, Second enlarged edition, Springer- Verlag, Berlin, 2003. (Cited on p. 19) Bel04. P. Bellingeri: On presentations of surface braid groups, J. Algebra 274 (2004), 543–563. (Cited on p.5) Bir69. J. Birman: On braid groups, Comm. Pure Appl. Math. 22 (1969), 41–72. (Cited on p.5) CaPol19. A. Causin, F. Polizzi: Surface braid groups, finite Heisenberg covers and double Kodaira fibrations, e-print arXiv : 1905.03170v3 (2019), to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci. (Cited on p.1,2,6,9, 10, 17, 18, 19, 21) CatRol09. F. Catanese, S. Rollenske: Double Kodaira fibrations, J. Reine Angew. Math. 628 (2009), 205–233. (Cited on p. 19, 20) GAP4. The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.11.0; 2020. (Cited on p.4) GG04. D. L. Gonçalves, J. Guaschi: On the structure of surface pure braid groups, J. Pure Appl. Algebra 186 (2004), 187–218. (Cited on p.5,6) Finite quotients of surface braid groups and double Kodaira fibrations 23

Gor07. D. Gorenstein: Finite groups, reprinted edition by the AMS Chelsea Publishing, 2007. (Cited on p.9) HarWr08. G. H. Hardy, E. M. Wright: An introduction to the theory of numbers, sixth edition, Oxford University Press 2008. (Cited on p. 19) Kas68. A. Kas: On deformations of a certain type of irregular algebraic surface, Amer. J. Math. 90 (1968), 789–804. (Cited on p.1) Kir97. R. Kirby (editor): Problems in low-dimensional topology, AMS/IP Stud. Adv. Math., 2.2, Geometric topology (Athens, GA, 1993), 35–473, Amer. Math. Soc., Providence, RI, 1997. (Cited on p.4, 21) Kod67. K. Kodaira: A certain type of irregular, algebraic surfaces, J. Anal. Math. 19 (1967), 207–215. (Cited on p.2) Is08. M. Isaacs: Finite groups theory, Graduate Studies in Mathematics 92, American Mathemat- ical Society 2008. (Cited on p.9) LLR20. J. A Lee, M. Lönne, S. Rollenske: Double Kodaira fibrations with small signature, Internat. J. Math. 31 (2020), no. 7, 2050052, 42 pp. (Cited on p.2, 19, 21) Mey73. W. Meyer: Die Signatur von Flächenbündeln, Math. Ann. 201 (1973), 239–264. (Cited on p. 21) Pol20. F. Polizzi: Diagonal double Kodaira structures on finite groups, e-print arXiv : 2002.01363 (2020). To appear in the Proceedings of the 2019 ISAAC Congress (Aveiro, Portugal). (Cited on p.1,2,9, 10, 14, 18, 19) PolSab21. F. Polizzi, P. Sabatino: Diagonal double Kodaira fibrations with minimal signature, e-print arXiv : 2102.04963 (2021). (Cited on p.1,2,9, 15, 20, 21, 22) Rol10. S. Rollenske: Compact moduli for certain Kodaira fibrations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. IX (2010), 851-874. (Cited on p. 20) S70. G. P. Scott: Braid groups and the groups of homeomorphisms of a surface, Proc. Camb. Phil. Soc. 68 (1970), 605–617. (Cited on p.5) Win72. D. L. Winter: The automorphism group of an extra-special 푝-group, Rocky Mountain J. Math. 2 (1972), 159–168. (Cited on p. 16)