Diagonal double Kodaira fibrations with minimal signature
DDIAGONAL DOUBLE KODAIRA FIBRATIONS WITH MINIMAL SIGNATURE
FRANCESCO POLIZZI AND PIETRO SABATINOA
BSTRACT . We study some special systems of generators on finite groups, introduced in theprevious work by the first author and called diagonal double Kodaira structures , in order to studynon-abelian, finite quotients of the pure braid group P (Σ b ) , where Σ b is a closed Riemannsurface of genus b . Our main result is that, if G is such a quotient, then | G | ≥ , and equalityholds if and only if G is extra-special. In the last section, as a geometrical application of ouralgebraic results, we construct two -dimensional families of double Kodaira fibrations havingsignature . C ONTENTS
0. Introduction 11. Group-theoretical preliminaries: CCT-groups and extra-special groups 52. Diagonal double Kodaira structures 103. Structures on groups of small order 143.1. Prestructures 143.2. The case | G | < | G | = 32 and G non-extra-special 163.4. The case | G | = 32 and G extra-special 174. Geometrical application: diagonal double Kodaira fibrations 20Acknowledgments 24Appendix. Non abelian groups of order and NTRODUCTION A Kodaira fibration is a smooth, connected holomorphic fibration f : S −→ B , where S is acompact complex surface and B is a compact closed curve, that is not isotrivial (this meansthat not all fibres are biholomorphic each other). The genus b := g ( B ) is called the basegenus of the fibration, whereas the genus g := g ( F ) , where F is any fibre, is called the fibregenus . A surface S that is the total space of a Kodaira fibration is called a Kodaira fibred surface .For every Kodaira fibration, we have b ≥ and g ≥ , see [Kas68, Theorem 1.1]. Since thefibration is smooth, the condition on the base genus implies that S contains no rational orelliptic curves; hence it is minimal and, by the sub-additivity of the Kodaira dimension, it isof general type, hence algebraic.An important topological invariant of a Kodaira fibred surface S is its signature σ ( S ) ,namely the signature of the intersection form on the middle cohomology group H ( S, R ) .Actually, the first examples of Kodaira fibrations (see [Kod67]) were constructed in order toshow that σ is not multiplicative for fibre bundles. In fact, σ ( S ) > for every Kodaira fibra-tion (see the introduction to [LLR17]), whereas σ ( B ) = σ ( F ) = 0 , hence σ ( S ) (cid:54) = σ ( B ) σ ( F ) ;by [CHS57], this in turn means that the monodromy action of π ( B ) on the rational cohomol-ogy ring H ∗ ( S, Q ) is non-trivial. Key words and phrases.
Surface braid groups, extra-special p -groups, Kodaira fibrations. a r X i v : . [ m a t h . AG ] F e b FRANCESCO POLIZZI AND PIETRO SABATINO
The original example by Kodaira, and its variants described in [At69, Hir69], are obtainedby taking ramified covers of products of curves, so they come with a pair of Kodaira fibra-tions. This leads to the following
Definition 0.1. A double Kodaira surface is a compact, complex surface S , endowed with a double Kodaira fibration , namely a surjective, holomorphic map f : S −→ B × B yielding, bycomposition with the natural projections, two Kodaira fibrations f i : S −→ B i , i = 1 , .Double Kodaira fibrations have been investigated by several authors, see [Zaal95,LeBrun00,BDS01, BD02, CatRol09, Rol10, LLR17]. Other authors considered Kodaira fibred surfacesas real, closed oriented -manifolds, using techniques from geometric topology such as theMeyer signature formula, the Birman-Hilden relations in the mapping class group and thesubtraction of Lefchetz fibrations, see [En98, EKKOS02, St02, L17]. In fact, Kodaira fibredsurface are a source of fascinating ad deep questions at the cross-road between the algebro-geometric properties of a complex surface and the topological properties of the underlying -manifold; we refer the reader to the survay paper [Cat17] and the references containedtherein for further details.Every Kodaira fibred surface S has the structure of a real surface bundle over a smooth realsurface, and so σ ( S ) is divisible by , see [Mey73]. If, in addition, S has a spin structure, i.e.its canonical class is -divisible in Pic( S ) , then σ ( S ) is a positive multiple of by Rokhlin’stheorem, and examples with σ ( S ) = 16 are constructed in [LLR17]. It is not known if thereexists a Kodaira fibred surface with σ ( S ) ≤ .Let us now describe our approach to the construction of double Kodaira fibrations, basedon the techniques introduced in [CaPol19, Pol20], and present our results. The main step isto “detopologize” the problem, by transforming it into a purely algebraic one. This will bedone in the particular case of diagonal double Kodaira fibrations, namely, Stein factorizationsof Galois covers(1) f : S −→ Σ b × Σ b , branched with order n > over the diagonal ∆ ⊂ Σ b × Σ b , where Σ b is a closed Riemannsurface of genus b . By Grauert-Remmert’s extension theorem together with Serre’s GAGA,the existence of a cover f as in (1), up to cover isomorphisms, is equivalent to the existenceof a group epimorphism(2) ϕ : π (Σ b × Σ b − ∆) −→ G, up to automorphisms of G . Furthermore, the condition that f is branched of order n over ∆ is rephrased by asking that ϕ ( γ ∆ ) has order n in G , where γ ∆ is the homotopy class in Σ b × Σ b − ∆ of a loop in Σ b × Σ b that “winds once” around ∆ .Now, the group π (Σ b × Σ b − ∆) is isomorphic to P (Σ b ) , the pure braid group of genus b on two strands; such a group admit a geometric presentation with g + 1 generators ρ , τ , . . . , ρ b , τ b , A , where A corresponds to γ ∆ , subject to the set of relations written in Section 2, see [GG04,Theorem 7]. Taking the images of these generators via the group epimorphism (2), we get anordered sets S = ( r , t , . . . , r b , t b , r , t , . . . , r b , t b , z ) of g + 1 generators of G , such that o ( z ) = n . This will be called a diagonal double Kodairastructure of type ( b, n ) on G , see Definition 2.1. From the previous considerations, we seethat the geometric problem of constructing a G -cover f as in (1) is now translated into thecombinatorial-algebraic problem of finding a diagonal double Kodaira structure S of type ( b, n ) in G . OUBLE KODAIRA STRUCTURES 3
Furthermore, the G -cover f is a diagonal double Kodaira fibration (namely, the two sur-jective maps f i : S −→ Σ b , obtained as composition with the natural projections, have con-nected fibres) if and only if S is strong , an additional condition introduced in Definition 2.8;moreover, the algebraic signature σ ( S ) , see Definition 2.7, equals the geometric signature σ ( S ) .Summing up, we are led to the problem of classifying those finite groups admitting adiagonal double Kodaira structure. Our first main result is the following, see Proposition 3.7,Proposition 3.9 and Theorem 3.13. Theorem A.
Assume that G is a finite group admitting a diagonal double Kodaira structure. Thenwe have | G | ≥ , with equality if and only if G is extra-special ( see Section 1 for the definition ) . Inthis case, the following holds. (1) G admits · diagonal double Kodaira structures of type (2 , . Everysuch a structure S is strong and satisfies σ ( S ) = 16 . (2) If G = G (32 ,
49) = H ( Z ) , these structures form orbits under the action of Aut( G ) . (3) If G = G (32 ,
50) = G ( Z ) , these structures form orbits under the action of Aut( G ) . Theorem A should be compared with previous results, obtained by the first author in col-laboration with A. Causin, regarding the construction of diagonal double Kodaira structureson some extra-special groups of order at least = 128 , see [CaPol19, Pol20]. It turns outthat the examples presented here are really new, in the sense that they cannot be obtained asquotients of structures on extra-special groups of bigger order, see Remark 3.15.We can now state the geometrical counterpart of Theorem A, concerning diagonal doubleKodaira fibrations, see Theorem 4.7. Theorem B.
Let G be a finite group and f : S −→ Σ b × Σ b be a Galois cover, with Galois group G ,branched on the diagonal ∆ with branching order n . Then | G | ≥ , with equality if and only if G isextra-special. In this case, the following holds. (1) There exist · distinct G -covers f : S −→ Σ × Σ , and all of them arediagonal double Kodaira fibrations such that b = b = 2 , g = g = 41 , σ ( S ) = 16 . (2) If G = G (32 ,
49) = H ( Z ) , these G -covers form equivalence classes up to coverisomorphisms. (3) If G = G (32 ,
50) = H ( Z ) , these G -covers form equivalence classes up to coverisomorphisms. As a consequence, we obtain a sharp lower bound for the signature of a diagonal doubleKodaira fibration or, equivalently, of a diagonal double Kodaira structure, see Corollary 4.8.
Theorem C.
Let f : S −→ Σ b × Σ b be a diagonal double Kodaira fibration, associated with adiagonal double Kodaira structure of type ( b, n ) on a finite group G . Then σ ( S ) ≥ , and equalityholds precisely when ( b, n ) = (2 , and G is an extra-special group of order . These results provide, in particular, new “double solutions” to a problem, posed by G.Mess, from Kirby’s problem list in low-dimensional topology [Kir97, Problem 2.18 A], askingwhat is the smallest number b for which there exists a real surface bundle over a real surfacewith base genus b and non-zero signature. We actually have b = 2 , also for double Kodairafibrations, as shown in [CaPol19, Proposition 3.19] and [Pol20, Theorem 3.6] by using doubleKodaira structures of type (2 , on extra-special groups of order . Those fibrations hadsignature and fibre genera ; we are now able to sensibly lower both these values, seeTheorem 4.9. Theorem D.
Let S be a diagonal double Kodaira surface, associated with a diagonal double Kodairastructure of strong type (2 , on an extra-special group G of order . Then the real manifold X FRANCESCO POLIZZI AND PIETRO SABATINO underlying S is a closed, orientable -manifold of signature that can be realized as a real surfacebundle over a real surface of genus , with fibre genus , in two different ways. In fact, we may ask whether and are the minimum possible values for the signatureand the fibre genus of a double Kodaira surface f : S −→ Σ × Σ , cf. Corollary 4.10.We believe that the results described above are significant for at least two reasons: ( i ) although we know that P (Σ b ) is residually p -finite for all prime number p ≥ , see[BarBel09, pp. 1481-1490], it can be tricky to explicitly exhibit some of its non-abelian,finite quotients. The results of A. Causin and the first author show that the extra-special groups of order p b +1 appear as quotients of P (Σ b ) for all b ≥ and all primenumbers p ≥ ; moreover, if p divides b + 1 , then the extra-special groups of order p b +1 appear as quotients, too. As we said before, the smallest groups that can beobtained in this way, corresponding to the case ( b, p ) = (3 , , have order = 128 .Our Theorem B sheds some new light on this problem, by providing a sharp lowerbound for the order of G : namely, if a non-abelian, finite group G appears as a quo-tient of P (Σ b ) for some b , then | G | ≥ , with equality if and only if G is extra-special.The extra-special groups of order actually appear as quotients of P (Σ ) ; more-over, for such groups, Theorem B also computes the total number of distinct groupepimorphisms ϕ : P (Σ ) −→ G , and the number of their equivalence classes up tothe natural action of Aut( G ) ; ( ii ) constructing (double) Kodaira fibrations with small signature is a rather difficultproblem. As far as we know, before the present work the only examples with sig-nature were the ones listed in [LLR17, Table 3, Cases 6.2, 6.6, 6.7 (Type 1), 6.9].Our examples in Theorem A are new, since both the base genera and the fibre generaare different from the ones in the aforementioned cases. Note that our results alsoshow that every curve of genus (and not only some special curve with non-trivialautomorphisms) is the base of a double Kodaira fibration with signature . Thus,we obtain two families of dimension of such fibrations that, to our knowledge, pro-vides the first examples of positive-dimensional families of double Kodaira fibrationswith small signature.Let us now describe how this paper is organized. In Section 1 we introduce some algebraicpreliminaries, in particular we discuss the so-called CCT-groups (Definition 1.1), namely,finite non-abelian groups in which commutativity is a transitive relation on the set of non-central elements. These groups are of historical importance in the context of classification offinite simple groups, see Remark 1.3, and they play a fundamental role in this paper, as wewill soon explain. It turns out that there are precisely eight groups G with | G | ≤ that arenot CCT-groups, namely S and seven groups of order , see Corollary 1.6, Proposition 1.7and Proposition 1.14.In Section 2 we define diagonal double Kodaira structures on finite groups and we explaintheir relation with their counterpart in geometric topology, namely group epimorphismsfrom pure surface braid groups.Section 3 is devoted to the study of diagonal double Kodaira structures in groups of orderat most . One crucial technical result is Proposition 3.4, stating that there are no suchstructures in CCT-groups. By the previous discussion, this implies that, in order to prove thefirst part of Theorem A, we only need to exclude the existence of diagonal double Kodairastructures on S and on five groups of order ; this is done in Proposition 3.7 and Proposition3.9, respectively. The second part of Theorem A, i.e. the computation of number of structuresin each case, is obtained by using the same techniques as in [Win72]; more precisely, weexploit the fact that V = G/Z ( G ) is a symplectic vector space of dimension over Z , andthat Out( G ) embeds in Sp(4 , Z ) as the orthogonal group associated with the quadratic form q : V −→ Z related to the symplectic form ( · , · ) by q ( x y ) = q ( x ) + q ( y ) + ( x , y ) . OUBLE KODAIRA STRUCTURES 5
Finally, Section 4 establishes the relation between our algebraic results and the geometricalframework of diagonal double Kodaira fibrations, and contains the proofs of Theorems B, Cand D.Tha paper ends with an Appendix, in which we collected the presentations for the non-abelian groups of order and used in our computations. Notations and conventions . All varieties, morphisms, etc. in this article are definedover C . If S is a projective, non-singular surface S then c ( S ) , c ( S ) denote the first andsecond Chern class of its tangent bundle T S , respectively.notation for groups: • Z n : cyclic group of order n . • D p, q, r = Z q (cid:111) Z p = (cid:104) x, y | x p = y q = 1 , xyx − = y r (cid:105) : split metacyclic group of order pq . The group D , n, − is the dihedral group of order n and will be denoted by D n . • If n is an integer greater or equal to , we denote by QD n the quasi-dihedral groupof order n , having presentation QD n := (cid:104) x, y | x = y n − = 1 , xyx − = y n − − (cid:105)• The generalized quaternion group of order n is denoted by Q n and is presented as Q n = (cid:104) x, y, z | x n = y = z = xyz (cid:105) For n = 2 we obtain the usual quaternion group Q , for which we adopt the classicalpresentation Q = (cid:104) i, j, k | i = j = k = ijk (cid:105) , and we denote by − the unique element of order . • S n , A n : symmetric, alternating group on n symbols. We write the composition ofpermutations from the right to the left; for instance, (13)(12) = (123) . • GL ( n, F q ) , SL ( n, F q ) , Sp ( n, F q ) : general linear group, special linear group and sym-plectic group of n × n matrices over a field with q elements. • The order of a finite group G is denoted by | G | . If x ∈ G , the order of x is denoted by o ( x ) and its centralizer in G by C G ( x ) . • If x, y ∈ G , their commutator is defined as [ x, y ] = xyx − y − . • The commutator subgroup of G is denoted by [ G, G ] , the center of G by Z ( G ) . • If S = { s , . . . , s n } ⊂ G , the subgroup generated by S is denoted by (cid:104) S (cid:105) = (cid:104) s , . . . , s n (cid:105) . • IdSmallGroup( G ) indicates the label of the group G in the GAP4 database of smallgroups, see [GAP4] For instance
IdSmallGroup( D ) = G (8 , means that D is thethird in the list of groups of order .1. G ROUP - THEORETICAL PRELIMINARIES : CCT-
GROUPS AND EXTRA - SPECIAL GROUPS
Definition 1.1.
A non-abelian, finite group G is said to be a center commutative-transitive group ( or a CCT- group , for short ) if commutativity is a transitive relation on the set on non-centralelements. In other words, if x, y, z ∈ G − Z ( G ) and [ x, y ] = [ y, z ] = 1 , then [ x, z ] = 1 . Proposition 1.2.
For a finite group G , the following properties are equivalent. (1) G is a CCT -group. (2)
For every pair x, y of non-central elements in G , the relation [ x, y ] = 1 implies C G ( x ) = C G ( y ) . (3) For every non-central element x ∈ G , the centralizer C G ( x ) is abelian.Proof. (1) ⇒ (2) Take two commuting elements x, y ∈ G − Z ( G ) and let z ∈ C G ( x ) . If z is central then z ∈ C G ( y ) by definition, otherwise [ x, y ] = [ x, z ] = 1 implies [ y, z ] = 1 bythe assumption that G is a CCT-group. This shows that C G ( x ) ⊆ C G ( y ) , and exchanging theroles of x, y we can deduce the reverse inclusion. FRANCESCO POLIZZI AND PIETRO SABATINO (2) ⇒ (3) Take any element x ∈ G − Z ( G ) ; is sufficient to check that [ y, z ] = 1 for everypair of non-central elements y, z ∈ C G ( x ) . By (2) we have C G ( y ) = C G ( z ) , in particular y ∈ C G ( z ) and we are done. (3) ⇒ (1) Let x, y, z ∈ G − Z ( G ) and suppose [ x, y ] = [ y, z ] = 1 , namely, x, z ∈ C G ( y ) .Since we are assuming that C G ( y ) is abelian, this gives [ x, z ] = 1 , showing that G is a CCT-group. (cid:3) Remark 1.3.
CCT-groups are of historical importance in the context of classification of finitesimple groups, see for instance [Suz61], where they are called CA- groups . Further referenceson the topic are [Schm70], [Reb71], [Rocke73], [Wu98].
Lemma 1.4. If G is a finite group such that G/Z ( G ) is cyclic, then G is abelian.Proof. Every element of G can be written as zy n , where y ∈ G is such that its image generates G/Z ( G ) , z ∈ Z ( G ) and n ∈ Z . It follows that any two elements of G commute. (cid:3) Proposition 1.5.
Let G be a non-abelian, finite group. (1) If | G | is the product of at most three prime factors ( non necessarily distinct ) , then G is a CCT -group. (2) If | G | = p , with p prime, then G is a CCT -group. (3) If G contains an abelian normal subgroup of prime index, then G is a CCT -group.Proof. (1)
Assume that | G | is the product of at most three prime factors, and take a non-central element y . Then the centralizer C G ( y ) has non-trivial center, because (cid:54) = y ∈ C G ( y ) ,and its order is the product of at most two primes. Therefore the quotient of C G ( y ) by itscenter is cyclic, hence C G ( y ) is abelian by Lemma 1.4. (2) Assume | G | = p and assume by contradiction that there exist x, y, z ∈ G − Z ( G ) such that [ x, y ] = [ y, z ] = 1 but [ x, z ] (cid:54) = 1 . They generate a subgroup N = (cid:104) x, y, z (cid:105) of G ,that is not the whole of G since y ∈ Z ( N ) but y / ∈ Z ( G ) . It follows that N has order p andso, since N is non-abelian, by Lemma 1.4 its center is cyclic of order p , generated by y . Thegroup G is a finite p -group, hence a nilpotent group; being a proper subgroup of maximalorder in a nilpotent group, N is normal in G (see [Mac12, Corollary 5.2]), so we have aconjugacy homomorphism G −→ Aut( N ) , that in turn induces a conjugacy homomorphism G −→ Aut( Z ( N )) (cid:39) Z p − . The image of such a homomorphism must have order dividingboth p and p − , hence it is trivial. In other words, the conjugacy action of G on Z ( N ) = (cid:104) y (cid:105) is trivial, hence y is central in G , contradiction. (3) Let N be an abelian subgroup of G such that G/N has prime order. As
G/N hasno proper subgroups, it follows that N is a maximal subgroup of G . Let now x be any non-central element of G , so that C G ( x ) is a proper subgroup of G ; then there are two possibilities: Case 1 : x ∈ N . Then N ⊆ C G ( x ) and so, by the maximality of N , we get C G ( x ) = N , that isabelian. Case 2 : x / ∈ N . Then the image of x generates G/N , and so every element y ∈ G can bewritten in the form y = ux r , where u ∈ N and ≤ r ≤ p − . In particular, if y ∈ C G ( x ) , the condition [ x, y ] = 1 yields [ x, u ] = 1 , namely u ∈ N ∩ C G ( x ) . Since N is abelian, from this it follows that C G ( x ) is abelian, too. (cid:3) As an immediate consequence of Parts (1) and (2) of Proposition 1.5, we have the follow-ing
Corollary 1.6.
Let G be a non-abelian, finite group such that | G | ≤ . If G is not a CCT -group,then either | G | = 24 or | G | = 32 . First of all, let us dispose of the case G = 24 . Proposition 1.7.
Let G be a non-abelian finite group such that | G | = 24 and G is not a CCT -group.Then G = S . OUBLE KODAIRA STRUCTURES 7
Proof.
We start by observing that S is not a CCT-group. In fact, (1234) commutes to itssquare (13)(24) , which commutes to (12)(34) , but (1234) and (12)(34) do not commute.Now we have to show that the remaining non-abelian groups of order are all CCT-groups; we will do a case-by-case analysis, referring the reader to the presentations given inTable 1 of the Appendix. Apart from G = G (24 ,
3) = SL (2 , F ) , for which we give an ad-hocproof, we will show that all of these groups contain an abelian subgroup N of prime index,so that we can conclude by using Part (3) of Proposition 1.5. • G = G (24 , . Take N = (cid:104) x y (cid:105) (cid:39) Z . • G = G (24 , . The action of
Aut( G ) has five orbits, whose representative elements are { , x, x , z, z } , see [SL(2,3)]. We have (cid:104) z (cid:105) = Z ( G ) and so, since C G ( x ) ⊆ C G ( x ) , itsuffices to show that the centralizers of x and z are both abelian. In fact, we have C G ( x ) = (cid:104) x (cid:105) (cid:39) Z , C G ( z ) = (cid:104) z (cid:105) (cid:39) Z . • G = G (24 , . Take N = (cid:104) x (cid:105) (cid:39) Z . • G = G (24 , . Take N = (cid:104) y (cid:105) (cid:39) Z . • G = G (24 , . Take N = (cid:104) y (cid:105) (cid:39) Z . • G = G (24 , . Take N = (cid:104) z, x y (cid:105) (cid:39) Z × Z . • G = G (24 , . Take N = (cid:104) y, z, w (cid:105) (cid:39) Z × Z . • G = G (24 , . Take N = (cid:104) z, y (cid:105) (cid:39) Z . • G = G (24 , . Take N = (cid:104) z, i (cid:105) (cid:39) Z . • G = G (24 , . Take N = (cid:104) z (cid:105) × V (cid:39) ( Z ) , where V (cid:67) A is the Klein subgroup. • G = G (24 , . Take N = (cid:104) z, w (cid:105) × (cid:104) (123) (cid:105) (cid:39) Z × Z .This completes the proof. (cid:3) Next, we want to give a description of all non-abelian, finite groups G of order | G | = 32 that are not CCT-groups; it will turn out that there are precisely seven of them, see Propo-sition 1.14. Before doing this, let us introduce the following classical definition, see for in-stance [Gor07, p. 183] and [Is08, p. 123]. Definition 1.8.
Let p be a prime number. A finite p -group G is called extra-special if its center Z ( G ) is cyclic of order p and the quotient V = G/Z ( G ) is a non-trivial, elementary abelian p -group.An elementary abelian p -group is a finite-dimensional vector space over the field Z p , henceit is of the form V = ( Z p ) dim V and G fits into a short exact sequence(3) −→ Z p −→ G −→ V −→ . Note that, V being abelian, we must have [ G, G ] = Z p , namely the commutator subgroup of G coincides with its center. Furthermore, since the extension (3) is central, it cannot be split,otherwise G would be isomorphic to the direct product of the two abelian groups Z p and V ,which is impossible because G is non-abelian. It can be also proved that, if G is extra-special,then dim V is even, so | G | = p dim V +1 is an odd power of p .For every prime number p , there are precisely two isomorphism classes M ( p ) , N ( p ) ofnon-abelian groups of order p , namely M ( p ) = (cid:104) r , t , z | r p = t p = 1 , z p = 1 , [ r , z ] = [ t , z ] = 1 , [ r , t ] = z − (cid:105) N ( p ) = (cid:104) r , t , z | r p = t p = z , z p = 1 , [ r , z ] = [ t , z ] = 1 , [ r , t ] = z − (cid:105) and both of them are in fact extra-special, see [Gor07, Theorem 5.1 of Chapter 5].If p is odd, then the groups M ( p ) and N ( p ) are distinguished by their exponent, whichequals p and p , respectively. If p = 2 , the group M ( p ) is isomorphic to the dihedral group D , whereas N ( p ) is isomorphic to the quaternion group Q .The classification of extra-special p -groups is now provided by the result below, see [Gor07,Section 5 of Chapter 5]. FRANCESCO POLIZZI AND PIETRO SABATINO
Proposition 1.9. If b ≥ is a positive integer and p is a prime number, there are exactly twoisomorphism classes of extra-special p -groups of order p b +1 , that can be described as follows. • The central product H b +1 ( Z p ) of b copies of M ( p ) , having presentation H b +1 ( Z p ) = (cid:104) r , t , . . . , r b , t b , z | r pj = t pj = z p = 1 , [ r j , z ] = [ t j , z ] = 1 , [ r j , r k ] = [ t j , t k ] = 1 , [ r j , t k ] = z − δ jk (cid:105) (4) If p is odd, this group has exponent p . • The central product G b +1 ( Z p ) of b − copies of M ( p ) and one copy of N ( p ) , having presen-tation G b +1 ( Z p ) = (cid:104) r , t , . . . , r b , t b , z | r pb = t pb = z , r p = t p = . . . = r pb − = t pb − = z p = 1 , [ r j , z ] = [ t j , z ] = 1 , [ r j , r k ] = [ t j , t k ] = 1 , [ r j , t k ] = z − δ jk (cid:105) (5) If p is odd, this group has exponent p . Remark 1.10.
In both cases, from the relations above we deduce [ r − j , t k ] = z δ jk , [ r − j , t − k ] = z − δ jk Remark 1.11.
For both groups H b +1 ( Z p ) and G b +1 ( Z p ) , the center coincides with the derivedsubgroup and is equal to (cid:104) z (cid:105) (cid:39) Z p . Remark 1.12. If p = 2 , we can distinguish the two groups H b +1 ( Z p ) and G b +1 ( Z p ) by count-ing the number of elements of order . Remark 1.13.
The groups H b +1 ( Z p ) and G b +1 ( Z p ) are not CCT-groups. In fact, let us taketwo distinct indices j, k ∈ { , . . . , b } and consider the non-central elements r j , t j , t k . Thenwe have [ r j , t k ] = [ t k , t j ] = 1 , but [ r j , t j ] = z − .We can now dispose of the case | G | = 32 . Proposition 1.14.
Let G be a non-abelian, finite group such that | G | = 32 and G is not a CCT -group. Then G = G (32 , t ) , where t ∈ { , , , , , , } . Here G (32 ,
49) = H ( Z ) and G (32 ,
50) = G ( Z ) are the two extra-special groups of order , in particular they have nilpotenceclass , whereas the remaining five groups have nilpotence class .Proof. We first do a case-by case analysis showing that, if t / ∈ { , , , , , , } , then G = G (32 , t ) contains an abelian subgroup N of index , so that G is a CCT-group by Part (3) of Proposition 1.5. In every case, we refer the reader to the presentation given in Table 2of the Appendix. • G = G (32 , . Take N = (cid:104) x, y , z (cid:105) (cid:39) Z × ( Z ) . • G = G (32 , . Take N = (cid:104) x, y (cid:105) (cid:39) ( Z ) . • G = G (32 , . Take N = (cid:104) x, y (cid:105) (cid:39) Z × Z . • G = G (32 , . Take N = (cid:104) x, y (cid:105) (cid:39) Z × Z . • G = G (32 , . Take N = (cid:104) ix, k (cid:105) (cid:39) Z × Z . • G = G (32 , . Take N = (cid:104) x, y (cid:105) (cid:39) ( Z ) . • G = G (32 , . Take N = (cid:104) x , y (cid:105) (cid:39) ( Z ) . • G = G (32 , . Take N = (cid:104) x , y (cid:105) (cid:39) Z × Z . • G = G (32 , . Take N = (cid:104) x , y (cid:105) (cid:39) Z × Z . • G = G (32 , . Take N = (cid:104) x , y (cid:105) (cid:39) Z × Z . OUBLE KODAIRA STRUCTURES 9 • G = G (32 , . Take N = (cid:104) y (cid:105) (cid:39) Z . • G = G (32 , . Take N = (cid:104) y (cid:105) (cid:39) Z . • G = G (32 , . Take N = (cid:104) y (cid:105) (cid:39) Z . • G = G (32 , . Take N = (cid:104) x (cid:105) (cid:39) Z . • G = G (32 , . Take N = (cid:104) w (cid:105) × (cid:104) x, y (cid:105) (cid:39) Z × ( Z ) . • G = G (32 , . Take N = (cid:104) z (cid:105) × (cid:104) x, y (cid:105) (cid:39) Z × ( Z ) . • G = G (32 , . Take N = (cid:104) x, y (cid:105) (cid:39) ( Z ) . • G = G (32 , . Take N = (cid:104) z (cid:105) × (cid:104) y (cid:105) (cid:39) ( Z ) . • G = G (32 , . Take N = (cid:104) z (cid:105) × (cid:104) i (cid:105) (cid:39) ( Z ) . • G = G (32 , . Take N = (cid:104) x, y, a, b (cid:105) (cid:39) ( Z ) . • G = G (32 , . Take N = (cid:104) x, y, z (cid:105) (cid:39) Z × ( Z ) . • G = G (32 , . Take N = (cid:104) x, i, z (cid:105) (cid:39) Z × ( Z ) . • G = G (32 , . Take N = (cid:104) x, y, z (cid:105) (cid:39) Z × ( Z ) . • G = G (32 , . Take N = (cid:104) x, y (cid:105) (cid:39) ( Z ) . • G = G (32 , . Take N = (cid:104) y, z (cid:105) (cid:39) ( Z ) . • G = G (32 , . Take N = (cid:104) x, y (cid:105) (cid:39) ( Z ) . • G = G (32 , . Take N = (cid:104) x, y (cid:105) (cid:39) ( Z ) . • G = G (32 , . Take N = (cid:104) x, k (cid:105) (cid:39) ( Z ) . • G = G (32 , . Take N = (cid:104) x, y (cid:105) (cid:39) Z × Z . • G = G (32 , . Take N = (cid:104) x, y (cid:105) (cid:39) Z × Z . • G = G (32 , . Take N = (cid:104) z (cid:105) × (cid:104) y (cid:105) (cid:39) Z × Z . • G = G (32 , . Take N = (cid:104) z (cid:105) × (cid:104) y (cid:105) (cid:39) Z × Z . • G = G (32 , . Take N = (cid:104) w (cid:105) × (cid:104) x (cid:105) (cid:39) Z × Z . • G = G (32 , . Take N = (cid:104) x, y (cid:105) (cid:39) Z × Z . • G = G (32 , . Take N = (cid:104) z, w (cid:105) × (cid:104) y (cid:105) (cid:39) Z × ( Z ) . • G = G (32 , . Take N = (cid:104) z, w (cid:105) × (cid:104) i (cid:105) (cid:39) Z × ( Z ) . • G = G (32 , . Take N = (cid:104) x, y, z (cid:105) (cid:39) Z × ( Z ) .It remains to show that G = G (32 , t ) is not a CCT-group for t ∈ { , , , , , , } , andto compute the nilpotency class in each case. In the sequel, we will denote by G = Γ ⊇ Γ ⊇ Γ ⊇ . . . the lower central series of G . Recall that a group G has nilpotency class c if Γ c (cid:54) = { } and Γ c +1 = { } .For t = 49 and t = 50 we have the two extra-special cases, that are not CCT-groups byRemark 1.13; their nilpotency class is by Remark 1.11. Let us now deal with the remainingcases. For each of them, we exhibit three non-central elements for which commutativity isnot a transitive relation, and we show that Γ (cid:54) = { } and Γ = { } ; note that this means that Γ = [ G, G ] is not contained in Z ( G ) , whereas Γ = [Γ , G ] is contained in Z ( G ) . • G = G (32 , . The center of G is Z ( G ) = (cid:104) x (cid:105) (cid:39) Z . We have [ y, w ] = [ w , w ] = 1 , but [ y, w ] = x . The derived subgroup of G is Γ = [ G, G ] = (cid:104) x, y (cid:105) (cid:39) ( Z ) , and a shortcomputation gives Γ = Z ( G ) , so c = 3 . • G = G (32 , . The center of G is Z ( G ) = (cid:104) w (cid:105) (cid:39) Z . We have [ y, z ] = [ z, u ] = 1 , but [ y, u ] = w . The derived subgroup of G is Γ = [ G, G ] = (cid:104) w, z (cid:105) (cid:39) ( Z ) , and a shortcomputation gives Γ = Z ( G ) , so c = 3 . • G = G (32 , . The center of G is Z ( G ) = (cid:104) x (cid:105) (cid:39) Z . We have [ x, x ] = [ x , y ] = 1 , but [ x, y ] = z . The derived subgroup of G is Γ = [ G, G ] = (cid:104) x , y (cid:105) (cid:39) ( Z ) , and a shortcomputation gives Γ = Z ( G ) , so c = 3 . • G = G (32 , . The center of G is Z ( G ) = (cid:104) x (cid:105) (cid:39) Z . We have [ x, x ] = [ x , z ] = 1 ,but [ x, z ] = x . The derived subgroup of G is Γ = [ G, G ] = (cid:104) x (cid:105) (cid:39) Z , and a shortcomputation gives Γ = Z ( G ) , so c = 3 . • G = G (32 , . The center of G is Z ( G ) = (cid:104) i (cid:105) (cid:39) Z . We have [ x, xk ] = [ xk, z ] = 1 ,but [ x, z ] = i . The derived subgroup of G is Γ = [ G, G ] = (cid:104) k (cid:105) (cid:39) Z , and a shortcomputation gives Γ = Z ( G ) , so c = 3 .This completes the proof. (cid:3)
2. D
IAGONAL DOUBLE K ODAIRA STRUCTURES
For more details on the material contained in this section, we refer the reader to [CaPol19]and [Pol20]. Let G be a finite group and let b, n ≥ be two positive integers. Definition 2.1. A diagonal double Kodaira structure of type ( b, n ) on G is an ordered set of b +1 generators S = ( r , t , . . . , r b , t b , r , t , . . . , r b , t b , z ) , with o ( z ) = n , such that the following relations are satisfied. We systematically use thecommutator notation in order to indicate relations of conjugacy type, writing for instance [ x, y ] = zy − instead of xyx − = z . • Surface relations [ r − b , t − b ] t − b [ r − b − , t − b − ] t − b − · · · [ r − , t − ] t − ( t t · · · t b ) = z [ r − , t ] t [ r − , t ] t · · · [ r − b , t b ] t b ( t − b t − b − · · · t − ) = z − • Conjugacy action of r j [ r j , r k ] = 1 if j < k (6) [ r j , r j ] = 1[ r j , r k ] = z − r k r − j z r j r − k if j > k [ r j , t k ] = 1 if j < k [ r j , t j ] = z − [ r j , t k ] = [ z − , t k ] if j > k [ r j , z ] = [ r − j , z ] • Conjugacy action of t j [ t j , r k ] = 1 if j < k (7) [ t j , r j ] = t − j z t j [ t j , r k ] = [ t − j , z ] if j > k [ t j , t k ] = 1 if j < k [ t j , t j ] = [ t − j , z ][ t j , t k ] = t − j z t j z − t k z t − j z − t j t − k if j > k [ t j , z ] = [ t − j , z ] Remark 2.2.
From (6) and (7) we can infer the corresponding conjugacy actions of r − j and t − j . We leave the cumbersome but standard computations to the reader. Remark 2.3.
Abelian groups admit no diagonal double Kodaira structures. Indeed, the rela-tion [ r j , t j ] = z − in (6) provides a non-trivial commutator in G , because o ( z ) = n . OUBLE KODAIRA STRUCTURES 11
Remark 2.4. If [ G, G ] ⊆ Z ( G ) , then the relations defining a diagonal double Kodaira struc-ture of type ( b, n ) assume the following simplified form. • Relations expressing the centrality of z [ r j , z ] = [ t j , z ] = [ r j , z ] = [ t j , z ] = 1 • Surface relations [ r − b , t − b ] [ r − b − , t − b − ] · · · [ r − , t − ] = z [ r − , t ] [ r − , t ] · · · [ r − b , t b ] = z − • Conjugacy action of r j [ r j , r k ] = 1 for all j, k [ r j , t k ] = z − δ jk • Conjugacy action of t j [ t j , r k ] = z δ jk [ t j , t k ] = 1 for all j, k where δ jk stands for the Kronecker symbol. Note that, being G non-abelian by Remark 2.3,the condition [ G, G ] ⊆ Z ( G ) is equivalent to G having nilpotency class , see [Is08, p. 22].The definition of diagonal double Kodaira structure can be motivated by means of somewell-known concepts in geometric topology. Let us denote by P (Σ b ) the pure braid group of genus b on two strands, which is isomorphic to the fundamental group π (Σ b × Σ b − ∆ , ( p , p )) of the configuration space of two ordered points on a real closed surface of genus b . By Gonc¸alves-Guaschi’s presentation of surface pure braid groups, see [GG04, Theorem7], [CaPol19, Theorem 1.7], we see that P (Σ b ) can be generated by g + 1 elements ρ , τ , . . . , ρ b , τ b , A subject to the following complete set of relations. • Surface relations [ ρ − b , τ − b ] τ − b [ ρ − b − , τ − b − ] τ − b − · · · [ ρ − , τ − ] τ − ( τ τ · · · τ b ) = A [ ρ − , τ ] τ [ ρ − , τ ] τ · · · [ ρ − b , τ b ] τ b ( τ − b τ − b − · · · τ − ) = A − • Conjugacy action of ρ j [ ρ j , ρ k ] = 1 if j < k [ ρ j , ρ j ] = 1[ ρ j , ρ k ] = A − ρ k ρ − j A ρ j ρ − k if j > k [ ρ j , τ k ] = 1 if j < k [ ρ j , τ j ] = A − [ ρ j , τ k ] = [ A − , τ k ] if j > k [ ρ j , A ] = [ ρ − j , A ] • Conjugacy action of τ j [ τ j , ρ k ] = 1 if j < k [ τ j , ρ j ] = τ − j A τ j [ τ j , ρ k ] = [ τ − j , A ] if j > k [ τ j , τ k ] = 1 if j < k [ τ j , τ j ] = [ τ − j , A ][ τ j , τ k ] = τ − j A τ j A − τ k A τ − j A − τ j τ − k if j > k [ τ j , A ] = [ τ − j , A ] Here the elements ρ ij and τ ij are the braids depicted in Figure 1, whereas A is the braiddepicted in Figure 2.F IGURE
1. The pure braids ρ j and ρ j on Σ b . If (cid:96) (cid:54) = i , the path correspondingto ρ ij and τ ij based at p (cid:96) is the constant path.F IGURE
2. The pure braid A on Σ b Remark 2.5.
Under the identificaion of P (Σ b ) with π (Σ b × Σ b − ∆ , ( p , p )) , the generator A ∈ P (Σ b ) represents the homotopy class γ ∆ ∈ π (Σ b × Σ b − ∆ , ( p , p )) of a loop in Σ b × Σ b that “winds once” around the diagonal ∆ .We can now state the following Proposition 2.6.
A finite group G admits a diagonal double Kodaira structure of type ( b, n ) if andonly if there is a surjective group homomorphism (8) ϕ : P (Σ b ) −→ G such that ϕ ( A ) has order n .Proof. If such a ϕ : P (Σ b ) −→ G exists, we can obtain a diagonal double Kodaira structureon G by setting(9) r ij = ϕ ( ρ ij ) , t ij = ϕ ( τ ij ) , z = ϕ ( A ) . Conversely, if G admits a diagonal double Kodaira structure, then (9) defines a group homo-morphism ϕ : P (Σ b ) −→ G with the desired properties. (cid:3) OUBLE KODAIRA STRUCTURES 13
The braid group P (Σ b ) is the middle term of two short exact sequences(10) −→ π (Σ b − { p i } , p j ) −→ P (Σ b ) −→ π (Σ b , p i ) −→ , where { i, j } = { , } , induced by the two natural projections of pointed topological spaces (Σ b × Σ b − ∆ , ( p , p )) −→ (Σ b , p i ) . Since we have π (Σ b − { p } , p ) = (cid:104) ρ , τ , . . . , ρ b , τ b , A (cid:105) π (Σ b − { p } , p ) = (cid:104) ρ , τ , . . . , ρ b , τ b , A (cid:105) , it follows that the two subgroups K := (cid:104) r , t , . . . , r b , t b , z (cid:105) K := (cid:104) r , t , . . . , r b , t b , z (cid:105) are both normal in G , and that there are two short exact sequences −→ K −→ G −→ Q −→ −→ K −→ G −→ Q −→ , such the elements r , t , . . . , r b , t b yield a complete system of coset representatives for Q ,whereas the elements r , t , . . . , r b , t b yield a complete system of coset representatives for Q .Let us now give a couple of definitions, whose geometrical meaning will become clear inSection 4, see in particular Proposition 4.3 and Remark 4.4. Definition 2.7.
The signature of a diagonal double Kodaira structure S of type ( b, n ) on afinite group G is defined as σ ( S ) = 13 | G | (2 b − (cid:18) − n (cid:19) Definition 2.8.
A diagonal double Kodaira structure on G is called strong if K = K = G .For later use, let us write down the special case consisting of a diagonal double Kodairastructure of type (2 , n ) . It is an ordered set of nine generators of G ( r , t , r , t , r , t , r , t , z ) , with o ( z ) = n , subject to the following relations. ( S1 ) [ r − , t − ] t − [ r − , t − ] t − ( t t ) = z ( S2 ) [ r − , t ] t [ r − , t ] t ( t − t − ) = z − ( R1 ) [ r , r ] = 1 ( R6 ) [ r , r ] = 1( R2 ) [ r , r ] = 1 ( R7 ) [ r , r ] = z − r r − z r r − ( R3 ) [ r , t ] = 1 ( R8 ) [ r , t ] = z − ( R4 ) [ r , t ] = z − ( R9 ) [ r , t ] = [ z − , t ]( R5 ) [ r , z ] = [ r − , z ] ( R10 ) [ r , z ] = [ r − , z ]( T1 ) [ t , r ] = 1 ( T6 ) [ t , r ] = t − z t ( T2 ) [ t , r ] = t − z t ( T7 ) [ t , r ] = [ t − , z ]( T3 ) [ t , t ] = 1 ( T8 ) [ t , t ] = [ t − , z ]( T4 ) [ t , t ] = [ t − , z ] ( T9 ) [ t , t ] = t − z t z − t z t − z − t t − ( T5 ) [ t , z ] = [ t − , z ] ( T10 ) [ t , z ] = [ t − , z ] (11) Remark 2.9.
When [ G, G ] ⊆ Z ( G ) , we have [ r , z ] = [ t , z ] = [ r , z ] = [ t , z ] = 1[ r , z ] = [ t , z ] = [ r , z ] = [ t , z ] = 1 and the previous relations become(12) ( S1 (cid:48) ) [ r − , t − ] [ r − , t − ] = z ( S2 (cid:48) ) [ r − , t ] [ r − , t ] = z − ( R1 (cid:48) ) [ r , r ] = 1 ( R6 (cid:48) ) [ r , r ] = 1( R2 (cid:48) ) [ r , r ] = 1 ( R7 (cid:48) ) [ r , r ] = 1( R3 (cid:48) ) [ r , t ] = 1 ( R8 (cid:48) ) [ r , t ] = z − ( R4 (cid:48) ) [ r , t ] = z − ( R9 (cid:48) ) [ r , t ] = 1( T1 (cid:48) ) [ t , r ] = 1 ( T6 (cid:48) ) [ t , r ] = z ( T2 (cid:48) ) [ t , r ] = z ( T7 (cid:48) ) [ t , r ] = 1( T3 (cid:48) ) [ t , t ] = 1 ( T8 (cid:48) ) [ t , t ] = 1( T4 (cid:48) ) [ t , t ] = 1 ( T9 (cid:48) ) [ t , t ] = 1
3. S
TRUCTURES ON GROUPS OF SMALL ORDER
Prestructures.Definition 3.1.
Let G be a finite group. A prestructure on G is an ordered set of nine elements ( r , t , r , t , r , t , r , t , z ) , with o ( z ) = n > , subject to the relations (R1) , . . . , (R10) , (T1) , . . . , (T10) in (11). OUBLE KODAIRA STRUCTURES 15
In other words, the nine elements must satisfy all the relations defining a diagonal doubleKodaira structure of type (2 , n ) , except the surface relations. Note that we are not requiringthat these elements generate the group G . Proposition 3.2.
If a finite group G admits a diagonal double Kodaira structure of type ( b, n ) , thenit admits a prestructure with o ( z ) = n .Proof. Consider the ordered set of nine elements ( r , t , r , t , r , t , r , t , z ) in Defi-nition (2.1) and the relations satisfied by them, with the exception of the surface relations. (cid:3) Remark 3.3.
Let G be a finite group that admits a prestructure. Then z and all its conjugatesare non-trivial elements of G . By the definition of prestructure, in particular by relations (R4) , (R8) , (T2) , (T6) , it follows that r , r , r , r and t , t , t , t are non-central elementsof G . Proposition 3.4. If G is a CCT -group, then G admits no prestructures and, subsequently, no diago-nal double Kodaira structures.Proof. By Proposition 3.2, the second statement is a direct consequence of the first one, itsuffices then to check that G admits no prestructures. Keeping in mind Remark 3.3, then (R6) and (T1) imply [ r , t ] = 1 . From this last relation and (T3) we get [ r , t ] = 1 , thatcontradicts (T8) . (cid:3) Given a finite group G , we define the socle of G , denoted by soc( G ) , as the intersection ofall non-trivial, normal subgroups of G . Note that G is simple if and only if soc( G ) = G . Definition 3.5.
A finite group G is called monolithic if soc( G ) (cid:54) = { } . Equivalently, G ismonolithic if it contains precisely one minimal non-trivial, normal subgroup. Proposition 3.6.
The following holds. (1)
Assume that G admits a prestructure, whereas no proper quotient of G does. Then G ismonolithic and z ∈ soc( G ) . (2) Assume that G admits a prestructure, whereas no proper subgroup of G does. Then theelements of the prestructure generate G .Proof. (1) Let S = ( r , t , r , t , r , t , r , t , z ) be a prestructure in G . Assume thatthere is a proper normal subgroup N of G such that z / ∈ N . Then ¯z ∈ G/N is non-trivial,and so ¯ S = ( ¯r , ¯t , ¯r , ¯t , ¯r , ¯t , ¯r , ¯t , ¯z ) is a prestructure in the quotient group G/N ,contradiction. Therefore we must have z ∈ soc( G ) , in particular, G is monolithic. (2) Obvious, because every prestructure S in G is also a prestructure in the subgroup (cid:104) S (cid:105) . (cid:3) The case | G | < .Proposition 3.7. If | G | < , then G admits no diagonal double Kodaira structures.Proof. By Corollary 1.6, Proposition 1.7 and Proposition 3.4, it remains only to check that thesymmetric group S admits no prestructures. We start by observing that soc( S ) = V = { (1) , (1 2)(3 4) , (1 3)(2 4) , (1 4)(2 3) } and so, by part (1) of Proposition 3.6, if S is a prestructure on S then z ∈ V . Let x, y ∈ S besuch that [ x , y ] = z . Examining the tables of subgroups of S given in [S4], by straightforwardcomputations and keeping in mind that the cycle type determines the conjugacy class, wededuce that either x , y ∈ C S ( z ) (cid:39) D or x , y ∈ A . Every pair in A includes at least a -cycle;so, if [ x , y ] = z and both x , y have even order, it follows that both x , y centralize z .If x ∈ S is a -cycle, then C S ( x ) = (cid:104) x (cid:105) (cid:39) Z . So, from relations (R1) , (R2) , (R3) , (R6) ,it follows that, if one of the elements r , r , r , r , t is a -cycle, then all these elements generate the same cyclic subgroup. This contradicts (R8) , hence r , r , r , r , t all haveeven order.Let us look at relation (R8) . Since r , t have even order, from the previous remark weinfer r , t ∈ C S ( z ) . Let us consider now r . If r belongs to A , being an element of evenorder and since cycle type determines conjugacy class, it must be conjugate to z , it followsthat r commutes with z ; otherwise, by (R4) , both r and t commute with z . Summing up,in any case we have r ∈ C S ( z ) .Relation (R5) can be rewritten as r r ∈ C S ( z ) , hence r ∈ C S ( z ) . Analogously, relation (R10) can be rewritten as r r ∈ C S ( z ) , hence r ∈ C S ( z ) .Using relation (R9) , we get r z ∈ C S ( t ) . Since r and z commute and have both order , it follows o ( r z ) ≤ . Therefore t cannot be a -cycle, otherwise C S ( t ) (cid:39) Z and so r = z − = z and [ r , t ] = 1 , contradicting (R8) . It follows that t has even order andthen, since r has even order as well, by (R4) we infer t ∈ C S ( z ) .Now we can rewrite (T2) as [ t , r ] = z . If t were a -cycle, from (T1) we would get r ∈ C S ( t ) (cid:39) Z , a contradiction since r has even order. Thus t has even order andso it belongs to C S ( z ) , because r has even order, too. Analogously, by using (T6) and (T7) we infer t ∈ C S ( z ) .Summarizing, if S were a prestructure on S we should have (cid:104) S (cid:105) = C S ( z ) (cid:39) D , contradicting part (2) of Proposition 3.6. (cid:3) The case | G | = 32 and G non-extra-special. We start by proving the following resultthat, when H = Z ( G ) , gives a special case of Proposition 3.4. Proposition 3.8.
Let G be a non-abelian finite group, containing an abelian subgroup H such that • for any g ∈ G , the centralizer C G ( g ) is non-abelian if and only if g ∈ H • Z ( G ) is contained in H and [ H : Z ( G )] ≤ .Then G admits no prestructures with z ∈ Z ( G ) .Proof. By contradiction, assume that S = ( r , t , r , t , r , t , r , t , z ) is a prestruc-ture on G , with z ∈ Z ( G ) . Then the elements of S satisfy relations (R1 (cid:48) ) , . . . , (R9 (cid:48) ) , (T1 (cid:48) ) , . . . , (T9 (cid:48) ) in (12). As H is abelian, (R4 (cid:48) ) implies that at least one between r , t does not belong to H .First of all, let us assume r / ∈ H . Thus C G ( r ) is abelian, and so (R2 (cid:48) ) and (R3 (cid:48) ) yield [ r , t ] = 1 . From this, using (T2 (cid:48) ) and (T3 (cid:48) ) we infer that C G ( t ) is non-abelian. Similarconsiderations show that C G ( r ) and C G ( r ) are non-abelian, and so we have r , r , t ∈ H . Using (T2 (cid:48) ) , (T6 (cid:48) ) , (R8 (cid:48) ) , together with the fact that H is abelian, we deduce t , t , r / ∈ H . In particular, C G ( r ) is abelian, so (R7 (cid:48) ) and (R9 (cid:48) ) yield [ r , t ] = 1 ; therefore (T2 (cid:48) ) and (T4 (cid:48) ) imply that C G ( t ) is non-abelian, and so t ∈ H . Summing up, we have provedthat the four elements r , t , r , t belong to H ; since they are all non-central, it fol-lows that they give four non-trivial elements in the quotient group H/Z ( G ) . On the otherhand, we have [ H : Z ( G )] ≤ , and so H/Z ( G ) contains at most three non-trivial elements;it follows that (at least) two among the elements r , t , r , t have the same image in H/Z ( G ) . This means that these two elements are of the form g, gz , with z ∈ Z ( G ) , andso they have the same centralizer. But this is impossible: in fact, relations (12) show thateach element in the set { r , t , r , t } fails to commute with exactly one element in the set { r , t , r , t } , and no two elements in { r , t , r , t } fail to commute with the sameelement in { r , t , r , t } .The remaining case, namely t / ∈ H , can be dealt with in an analogous way. Indeed, inthis situation { r , t , r , t } ⊆ H , that leads to a contradiction as above. (cid:3) We can now rule out the non-extra-special groups of order . Proposition 3.9.
Let G be a finite group of order which is not extra-special. Then G admits nodiagonal double Kodaira structures. OUBLE KODAIRA STRUCTURES 17
Proof. If G is a CCT-group, then the result follows from Proposition 3.4. Then, by Proposi-tion 1.14, we must only consider the cases where G = G (32 , t ) , where t ∈ { , , , , } .Standard computations using the presentations in Table 2 of the Appendix show that allthese groups are monolithic, and that for all of them soc( G ) = Z ( G ) (cid:39) Z , cf. the proof ofProposition 1.14. Since no proper quotients of G admit diagonal double Kodaira structures(Proposition 3.7), it follows from Proposition 3.6 that every diagonal double Kodaira struc-ture on G is such that z is the generator of Z ( G ) . Thus, if in every case we are able to find asubgroup H of G as in Proposition 3.8, we are done. For each group that we are considering,let H be the subset of elements with non-abelian centralizer. It turns out that this subset isactually a subgroup, that we write down explicitly in each situation. The straightforwardcomputations needed to show that H has the desired properties are left to the reader. • G = G (32 , . In this case soc( G ) = Z ( G ) = (cid:104) x (cid:105) and H = (cid:104) x, y, w (cid:105) . Then H (cid:39) ( Z ) and [ H : Z ( G )] = 4 . • G = G (32 , . In this case soc( G ) = Z ( G ) = (cid:104) w (cid:105) and H = (cid:104) z, u, w (cid:105) . Then H (cid:39) Z × Z and [ H : Z ( G )] = 4 . • G = G (32 , . In this case soc( G ) = Z ( G ) = (cid:104) x (cid:105) and H = (cid:104) x , y, z (cid:105) . Then H (cid:39) Z × Z and [ H : Z ( G )] = 4 . • G = G (32 , . In this case soc( G ) = Z ( G ) = (cid:104) x (cid:105) and H = (cid:104) x , z (cid:105) . Then H (cid:39) Z × Z and [ H : Z ( G )] = 4 . • G = G (32 , . In this case soc( G ) = Z ( G ) = (cid:104) i (cid:105) and H = (cid:104) x, k (cid:105) . Then H (cid:39) Z × Z and [ H : Z ( G )] = 4 .The proof is now complete. (cid:3) The case | G | = 32 and G extra-special. We are now ready to address the case where | G | = 32 and G is extra-special. Let us first recall some results on extra-special -groups,referring the reader to [Win72] for more details.Let G be an extra-special p -group of order p b +1 and x , y ∈ G . Setting (¯ x , ¯ y ) = ¯ a where [ x , y ] = z a , the quotient group V = G/Z ( G ) becomes a non-degenerate symplectic vectorspace over F p . Looking at (4) and (5), we see that in both cases G = H b +1 ( Z p ) and G = G b +1 ( Z p ) we have (¯ r j , ¯ r k ) = 0 , (¯ t j , ¯ t k ) = 0 , (¯ r j , ¯ t k ) = − δ jk for all j, k ∈ { , . . . , b } , so that(13) ¯ r , ¯ t , . . . , ¯ r b , ¯ t b is an ordered symplectic basis for V (cid:39) ( Z p ) b . If p = 2 , we can also set q (¯ x ) = ¯ c , where x = z c ( c = 0 or ); this is a quadratic form on V . If ¯ x ∈ G/Z ( G ) is expressed in coordinatesby the vector ( ξ , ψ , . . . , ξ b , ψ b ) ∈ ( Z p ) b with respect to the symplectic basis (13), then astraightforward computation shows that(14) q (¯ x ) = (cid:40) ξ ψ + · · · + ξ b ψ b , if G = H b +1 ( Z ) ξ ψ + · · · + ξ b ψ b + ξ b + ψ b if G = G b +1 ( Z ) These are the two possible normal forms of non-degenerate quadratic forms over Z . More-over, in both cases the symplectic and the quadratic form are related by q (¯ x ¯ y ) = q (¯ x ) + q (¯ y ) + (¯ x , ¯ y ) for all ¯ x , ¯ y ∈ V. Note that, if φ ∈ Aut( G ) , then φ induces a linear map ¯ φ ∈ End( V ) ; moreover, if p = 2 , then φ acts trivially on Z ( G ) = [ G, G ] and this in turn implies that φ preserves the symplecticform on V . In other words, if we identify V with ( Z ) b via the symplectic basis (13), then ¯ φ ∈ Sp (2 b, Z ) .We are now in a position to describe the structure of Aut( G ) , see [Win72, Theorem 1]. Proposition 3.10.
Let G be an extra-special -group of order b +1 . Then the kernel of the grouphomomorphism Aut( G ) −→ Sp (2 b, Z ) given by φ (cid:55)→ ¯ φ is the subgroup Inn( G ) of inner automor-phisms of G . Therefore Out( G ) = Aut( G ) / Inn( G ) embeds in Sp (2 b, Z ) . More precisely, Out( G ) coincides with the orthogonal group O (cid:15) (2 b, Z ) , of order (15) | O (cid:15) (2 b, Z ) | = 2 b ( b − (2 b − (cid:15) ) b − (cid:89) i =1 (2 i − , associated with the quadratic form (14) . Here (cid:15) = 1 if G = H b +1 ( Z ) and (cid:15) = − if G = G b +1 ( Z ) . Corollary 3.11.
Let G be an extra-special -group of order b +1 . We have (16) | Aut( G ) | = | O (cid:15) (2 b, Z ) | = 2 b ( b +1)+1 (2 b − (cid:15) ) b − (cid:89) i =1 (2 i − Proof.
By Proposition 3.10 we get | Aut( G ) | = | Inn( G ) | · | O (cid:15) (2 b, Z ) | . Since Inn( G ) (cid:39) G/Z ( G ) has order b , the claim follows from (15). (cid:3) In particular, applying (16) with b = 2 , we can compute the orders of automorphismgroups of extra-special -groups of order , namely(17) | Aut( H ( Z )) | = 1152 , | Aut( G ( Z )) | = 1920 Assume now that S = ( r , t , r , t , r , t , r , t , z ) is a diagonal double Kodairastructure of type (2 , on an extra-special -group G of order . Then(18) ¯ S = ( ¯r , ¯t , ¯r , ¯t , ¯r , ¯t , ¯r , ¯t ) is an ordered set of generators of the symplectic Z -vector space V = G/Z ( G ) (cid:39) ( Z ) , and(12) yields the relations ( ¯r , ¯t ) + ( ¯r , ¯t ) = 1( ¯r , ¯t ) + ( ¯r , ¯t ) = 1( ¯r j , ¯t k ) = δ jk , ( ¯r j , ¯r k ) = 0( ¯t j , ¯r k ) = δ jk , ( ¯t j , ¯t k ) = 0 (19)Conversely, given any set of generators ¯ S of V as in (18), whose elements satisfy (19), adiagonal double Kodaira structure of type (2 , on G inducing ¯ S is necessarily of the form S = ( r z a , t z b , r z a , t z b , r z a , t z b , r z a , t z b , z ) , where r ij , t ij are any fixed lifts in G of ¯r ij , ¯t ij and a ij , b ij ∈ Z . This proves the following Lemma 3.12.
The total number of diagonal double Kodaira structures of type (2 , on an extra-special -group G of order is obtained multiplying by the number of ordered sets of generators ¯ S of V as in (18) , whose elements satisfy (19) . In particular, such a number does not depend on G . We are now ready to state the main result of this section.
Theorem 3.13.
A finite group G of order admits a diagonal double Kodaira structure if and onlyif G is extra-special. More precisely, the following holds. (1) G admits · distinct diagonal double Kodaira structures of type (2 , .Every such a structure S is strong and satisfies σ ( S ) = 16 . (2) If G = G (32 ,
49) = H ( Z ) , these structures form orbits under the action of Aut( G ) . (3) If G = G (32 ,
50) = G ( Z ) , these structures form orbits under the action of Aut( G ) .Proof. Looking at the first two relations in (19), we see that we must consider four cases: ( a ) ( ¯r , ¯t ) = 0 ( ¯r , ¯t ) = 1 ( ¯r , ¯t ) = 0 ( ¯r , ¯t ) = 1 ( b ) ( ¯r , ¯t ) = 0 ( ¯r , ¯t ) = 1 ( ¯r , ¯t ) = 1 ( ¯r , ¯t ) = 0 ( c ) ( ¯r , ¯t ) = 1 ( ¯r , ¯t ) = 0 ( ¯r , ¯t ) = 1 ( ¯r , ¯t ) = 0 OUBLE KODAIRA STRUCTURES 19 ( d ) ( ¯r , ¯t ) = 1 ( ¯r , ¯t ) = 0 ( ¯r , ¯t ) = 0 ( ¯r , ¯t ) = 1 Observe that exchanging the indices i and j we exchange ( a ) with ( c ) and ( b ) and ( d ) . So,it suffices to consider only cases ( a ) and ( b ) . Case ( a ) . In this case the vectors ¯r , ¯t , ¯r , ¯t are a symplectic basis of V , whereasthe subspace W = (cid:104) ¯r , ¯t , ¯r , ¯t (cid:105) is isotropic, namely the symplectic form is identicallyzero on it. Since V is a hyperbolic vector space of dimension , the Witt index of V , i.e.the dimension of a maximal isotropic subspace of V , is dim( V ) = 2 , see [Ar62, Th´eor`eme3.10, 3.11]. On the other hand, we have ( ¯r , ¯t ) = 1 and ( ¯t , ¯t ) = 0 , so ¯r , ¯t arelinearly independent and so they must generate a maximal isotropic subspace; it followsthat W = (cid:104) ¯r , ¯t (cid:105) .Let us set now ( ¯r , ¯r ) = a ( ¯r , ¯t ) = b ( ¯r , ¯t ) = c ( ¯t , ¯t ) = d ( ¯r , ¯r ) = e ( ¯r , ¯t ) = f ( ¯r , ¯t ) = g ( ¯t , ¯t ) = h where a, b, c, d, e, f, g, h ∈ Z , and let us express the remaining vectors of ¯ S in terms of thesymplectic basis. Standard computations yield ¯r = c ¯r + a ¯t + ¯r ¯t = d ¯r + b ¯t + ¯t (20) ¯r = ¯r + f ¯r + e ¯t ¯t = ¯t + h ¯r + g ¯t Now recall that W is isotropic; then, using the expressions in (20) and imposing the relations ( ¯r , ¯t ) = 0 ( ¯r , ¯r ) = 0 ( ¯r , ¯t ) = 0( ¯r , ¯t ) = 0 ( ¯t , ¯t ) = 0 ( ¯r , ¯t ) = 0 we get bc + ad = 1 a + e = 0 c + g = 0 b + f = 0 d + h = 0 eh + f g = 1 Summing up, the elements ¯r , ¯t , ¯r , ¯t can be determined from the symplectic basis viathe relations ¯r = c ¯r + a ¯t + ¯r ¯t = d ¯r + b ¯t + ¯t (21) ¯r = ¯r + b ¯r + a ¯t ¯t = ¯t + d ¯r + c ¯t where a, b, c, d ∈ Z and ad + bc = 1 . Conversely, given any symplectic basis ¯r , ¯t , ¯r , ¯t of V and elements ¯r , ¯t , ¯r , ¯t as in (21), with ad + bc = 1 , we get an ordered basis ¯ S whose elements satisfy (19). Therefore we infer that the total number of such ¯ S in Case ( a ) ,and equivalently in Case ( c ) , is given by | Sp (4 , Z ) | · | GL (2 , Z ) | = 720 · and so, by Lemma 3.12, the corresponding number of diagonal double Kodaira structures is · . All these structures are strong; in fact, we have K = (cid:104) r , t , r , t (cid:105) = (cid:104) r , t , r c t a r , r d t b t (cid:105) = (cid:104) r , t , r , t (cid:105) = GK = (cid:104) r , t , r , t (cid:105) = (cid:104) r r b t a , t r d t c , r , t (cid:105) = (cid:104) r , t , r , t (cid:105) = G, the last equality following in both cases because (cid:104) ¯r , ¯t , ¯r , ¯t (cid:105) = V and [ r , t ] = z . Case ( b ) . We will show that this case do not occur, and so Case ( d ) does not occur, either. Infact, in this situation the subspace W = (cid:104) ¯r , ¯t , ¯r (cid:105) is isotropic. Take a linear combinationof its generators giving the zero vector, namely a ¯r + b ¯t + c ¯r = 0 . Taking the intersection (with respect to the symplectic form) with ¯t , ¯t , ¯r , respectively,we get c = a = b = 0 . Thus, these generators are linearly independent, hence we found anisotropic subspace of dimension on the -dimensional hyperbolic space V , contradiction.Summarizing, we have found diagonal double Kodaira structures in cases ( a ) and ( c ) and no structure at all in cases ( b ) and ( d ) . So the total number of diagonal double Kodairastructures on G is , and this concludes the proof of part (1) of our theorem.Now observe that, since any diagonal double Kodaira structure S generates G , every au-tomorphism φ of G fixing S elementwise must be the identity. This means that Aut( G ) actsfreely on the set of diagonal double Kodaira structures, hence the number of orbits is ob-tained dividing by | Aut( G ) | . Part (2) and (3) now follow from (17), and we aredone. (cid:3) Example 3.14.
Let us give an explicit example of diagonal double Kodaira structure on anextra-special group G of order , by using the construction described in the proof of part ( ) of Theorem 4.7. Referring to the presentations for H ( Z ) and G ( Z ) given in Proposition 1.9,we start by choosing in both cases the following elements, whose images give a symplecticbasis for V : r = r , t = t , r = r , t = t . Choosing a = d = 1 and b = c = 0 in (21), we find the remaining elements, obtaining thediagonal double Kodaira structure r = r , t = t , r = r t , t = r t r = r t , t = r t , r = r , t = t . Remark 3.15.
Theorem 3.13 should be compared with previous results of [CaPol19] and[Pol20], regarding the construction of diagonal double Kodaira structures on some extra-special groups of order at least = 128 . The examples on extra-special groups of order presented here are really new, in the sense that they cannot be obtained by taking the epi-morphic image of structures on extra-special groups of bigger order: in fact, an extra-specialgroup admits no non-abelian proper quotients, see [Rob96, Exercise 9 p. 146]. Remark 3.16.
Although we know that P (Σ b ) is residually p -finite for all prime number p ≥ ,see [BarBel09, pp. 1481-1490], it can be tricky to explicitly describe some of its non-abelian,finite quotients. The extra-special examples of order at least cited in Remark 3.15 were theoutcome of the first (as far as we know) systematic investigation of this matter. Our approachin the present work sheds some new light on this problem, providing a sharp lower boundfor the order of a non-abelian quotient G of P (Σ b ) . More precisely, we have | G | ≥ , withequality if and only if G is extra-special, and the extra-special groups of order actuallyappear as quotients of P (Σ ) . Moreover, for such groups, Theorem 3.13 also compute thetotal number of distinct group epimorphisms ϕ : P (Σ ) −→ G , and the number of theirequivalence classes up to the natural action of Aut( G ) .4. G EOMETRICAL APPLICATION : DIAGONAL DOUBLE K ODAIRA FIBRATIONS
Recall that a
Kodaira fibration is a smooth, connected holomorphic fibration f : S −→ B ,where S is a compact complex surface and B is a compact complex curve, which is notisotrivial. The genus b := g ( B ) is called the base genus of the fibration, whereas the genus g := g ( F ) , where F is any fibre, is called the fibre genus . Definition 4.1. A double Kodaira surface is a compact complex surface S , endowed with a double Kodaira fibration , namely a surjective, holomorphic map f : S −→ B × B yielding, bycomposition with the natural projections, two Kodaira fibrations f i : S −→ B i , i = 1 , . OUBLE KODAIRA STRUCTURES 21
The aim of this section is to show how the existence of diagonal double Kodaira structuresis equivalent to the existence of some special double Kodaira fibrations, that we call diagonaldouble Kodaira fibrations .With a slight abuse of notation, in the sequel we will use the symbol Σ b to indicate botha smooth complex curve of genus b and its underlying real surface. By using Grauert-Remmert’s extension theorem together with Serre’s GAGA, the group epimorphism ϕ : P (Σ b ) −→ G described in Proposition 2.6 gives the existence of a smooth, complex, projective surface S endowed with a Galois cover f : S −→ Σ b × Σ b , with Galois group G and branched precisely over ∆ with branching order n , see [CaPol19,Proposition 3.4]. Composing the left homomorphism in (10) with ϕ : P (Σ b ) −→ G , we gettwo homomorphisms ϕ : π (Σ b − { p } , p ) −→ G, ϕ : π (Σ b − { p } , p ) −→ G, whose image equals K and K , respectively. By construction, these are the homomorphismsinduced by the restrictions f i : Γ i −→ Σ b of the Galois cover f : S −→ Σ b × Σ b to the fibresof the two natural projections π i : Σ b × Σ b −→ Σ b . Since ∆ intersects transversally at asingle point all the fibres of the natural projections, it follows that both such restrictionsare branched at precisely one point, and the number of connected components of the smoothcurve Γ i ⊂ S equals the index m i := [ G : K i ] of K i in G .So, taking the Stein factorizations of the compositions π i ◦ f : S −→ Σ b as in the diagrambelow(22) S Σ b Σ b i π i ◦ f f i θ i we obtain two distinct Kodaira fibrations f i : S −→ Σ b i , hence a double Kodaira fibration byconsidering the product morphism f = f × f : S −→ Σ b × Σ b . Definition 4.2.
We call f : S −→ Σ b × Σ b the diagonal double Kodaira fibration associatedwith the diagonal double Kodaira structure S on the finite group G . Conversely, we will saythat a double Kodaira fibration f : S −→ Σ b × Σ b is of diagonal type ( b, n ) if there exists afinite group G and a diagonal double Kodaira structure S of type ( b, n ) on it such that f isassociated with S .Since the morphism θ i : Σ b i −→ Σ b is ´etale of degree m i , by using the Hurwitz formula weobtain b − m ( b − , b − m ( b − . Moreover, the fibre genera g , g of the Kodaira fibrations f : S −→ Σ b , f : S −→ Σ b arecomputed by the formulae(23) g − | G | m (2 b − n ) , g − | G | m (2 b − n ) , where n := 1 − /n . Finally, the surface S fits into a diagram S Σ b × Σ b Σ b × Σ b f f θ × θ so that the diagonal double Kodaira fibration f : S −→ Σ b × Σ b is a finite cover of degree | G | m m , branched precisely over the curve ( θ × θ ) − (∆) = Σ b × Σ b Σ b . Such a curve is always smooth, being the preimage of a smooth divisor via an ´etale mor-phism. However, it is reducible in general, see [CaPol19, Proposition 3.11]. The invariants of S can be now computed as follows, see [CaPol19, Proposition 3.8]. Proposition 4.3.
Let f : S −→ Σ b × Σ b be a diagonal double Kodaira fibration, associated with adiagonal double Kodaira structure S of type ( b, n ) on a finite group G . Then we have c ( S ) = | G | (2 b − b − n − n ) c ( S ) = | G | (2 b − b − n ) where n = 1 − /n . As a consequence, the slope and the signature of S can be expressed as ν ( S ) = c ( S ) c ( S ) = 2 + 2 n − n b − n σ ( S ) = 13 (cid:0) c ( S ) − c ( S ) (cid:1) = 13 | G | (2 b − (cid:18) − n (cid:19) = σ ( S ) (24) Remark 4.4.
By definition, the diagonal double Kodaira structure S is strong if and only if m = m = 1 , that in turn implies b = b = b , i.e., f = f . In other words, S is strong type ifand only if no Stein factorization as in (22) is needed or, equivalently, if and only if the Galoiscover f : S −→ Σ b × Σ b induced by (8) is already a double Kodaira fibration, branched on thediagonal ∆ ⊂ Σ b × Σ b . Remark 4.5.
Every Kodaira fibred surface S satisfies σ ( S ) > , see the introduction to[LLR17]; moreover, since S is a differentiable -manifold that is a real surface bundle, itssignature is divisible by , see [Mey73]. In addition, if S is associated with a diagonal doubleKodaira structure of type ( b, n ) , with n odd, then K S is -divisible in Pic ( S ) and so σ ( S ) is apositive multiple of by Rokhlin’s theorem, see [CaPol19, Remark 3.9]. Remark 4.6.
Not all double Kodaira fibration are of diagonal type. In fact, if S is of diagonaltype then its slope satisfies ν ( S ) = 2 + s , where s is rational and < s < − √ , see [Pol20,Proposition 3.12 and Remark 3.13].We are now ready to give a geometric interpretation of Proposition 3.7, Proposition 3.9and Theorem 3.13 in terms of double Kodaira fibrations. Theorem 4.7.
Let G be a finite group and (25) f : S −→ Σ b × Σ b be a Galois cover with Galois group G , branched over the diagonal ∆ . Then the following hold. (1) We have | G | ≥ , with equality precisely when G is extra-special. (2) If G = G (32 ,
49) = H ( Z ) and b = 2 , there are G -covers of type (25) , up to coverisomorphisms. (3) If G = G (32 ,
50) = G ( Z ) and b = 2 , there are G -covers of type (25) , up to coverisomorphisms. OUBLE KODAIRA STRUCTURES 23
Finally, in both cases (2) and (3) , each cover f is a double Kodaira fibration such that b = b = 2 , g = g = 41 , σ ( S ) = 16 . Proof.
By the result of Section 4, a cover as in (25), branched over ∆ with order n , exists if andonly if G admits a double Kodaira structure of type (2 , n ) , and the number of such covers,up to cover isomorphisms, equals the number of structures up the natural action of Aut( G ) .Then, (1) , (2) and (3) can be deduced from the corresponding statements in Theorem 3.13.The same theorem tells us that all double Kodaira structures on an extra-special group oforder are strong, so m = m and the cover f is already a double Kodaira fibration, noStein factorization is needed. The fibre genera, the slope and the signature of S can be nowcomputed by using (23) and (24). (cid:3) As a consequence, we obtain a sharp lower bound for the signature of a diagonal doubleKodaira fibration or, equivalently, of a diagonal double Kodaira structure.
Corollary 4.8.
Let f : S −→ Σ b × Σ b be a diagonal double Kodaira fibration, associated with adiagonal double Kodaira structure of type ( b, n ) on a finite group G . Then σ ( S ) ≥ , and equalityholds precisely when ( b, n ) = (2 , and G is an extra-special group of order .Proof. Theorem 3.13 implies | G | ≥ . Since b ≥ and n ≥ , from (24) we get σ ( S ) = 13 | G | (2 b − (cid:18) − n (cid:19) ≥ · · (2 · − (cid:18) − (cid:19) = 16 , and equality holds if and only if we are in the situation described in Theorem 4.7, namely, b = n = 2 and G an extra-special group of order . (cid:3) These results provide, in particular, new “double solutions” to a problem, posed by G.Mess, from Kirby’s problem list in low-dimensional topology [Kir97, Problem 2.18 A], askingwhat is the smallest number b for which there exists a real surface bundle over a real surfacewith base genus b and non-zero signature. We actually have b = 2 , also for double Kodairafibrations, as shown in [CaPol19, Proposition 3.19] and [Pol20, Theorem 3.6] by using doubleKodaira structures of type (2 , on extra-special groups of order . Those fibrations hadsignature and fibre genera ; we are now able to sensibly lower both these values. Theorem 4.9.
Let S be a diagonal double Kodaira surface, associated with a diagonal double Kodairastructure of strong type (2 , on an extra-special group G of order . Then the real manifold X underlying S is a closed, orientable -manifold of signature that can be realized as a real surfacebundle over a real surface of genus , with fibre genus , in two different ways. Theorem 4.7 also implies the following partial answer to [CaPol19, Question 3.20].
Corollary 4.10.
Let g min and σ min be the minimal possible fibre genus and signature for a doubleKodaira fibration f : S −→ Σ × Σ . Then we have g min ≤ , σ min ≤ . In fact, it is an interesting question whether and are the minimum possible valuesfor the signature and the fibre genus of a (non necessarily diagonal) double Kodaira fibration f : S −→ Σ × Σ , but we will not address this problem here. Remark 4.11.
Constructing (double) Kodaira fibrations with small signature is a rather dif-ficult problem. As far as we know, before our work the only examples with signature were the ones listed in [LLR17, Table 3, Cases 6.2, 6.6, 6.7 (Type 1), 6.9]. Our examples inTheorem 4.7 are new, since both the base genera and the fibre genera are different. Notethat our results also show that every curve of genus (and not only some special curve withextra automorphisms) is the base of a double Kodaira fibration with signature . Thus, weobtain two families of dimension of such fibrations that, at least to our knowledge, providethe first examples of a positive-dimensional families of double Kodaira fibrations with smallsignature. A CKNOWLEDGMENTS
F. Polizzi was partially supported by GNSAGA-INdAM. He thanks Andrea Causin fordrawing the figures. Furthermore, he is very grateful to Ian Agol, Yves de Cornulier, “Jonathan”,Derek Holt, Max Horn, Moishe Kohan, Roberto Pignatelli, “Primoz”, Geoff Robinson, JohnShareshian, Remy van Dobben de Bruyn for their precious answers and comments in theMathOverflow threads https://mathoverflow.net/questions/357453https://mathoverflow.net/questions/366044https://mathoverflow.net/questions/366771https://mathoverflow.net/questions/368628https://mathoverflow.net/questions/371181https://mathoverflow.net/questions/379272https://mathoverflow.net/questions/380292 A PPENDIX . N
ON ABELIAN GROUPS OF ORDER AND G ) G Presentation G (24 , D , , − (cid:104) x, y | x = y = 1 , xyx − = y − (cid:105) G (24 , SL (2 , F ) (cid:104) x, y, z | x = y = z = xyz (cid:105) G (24 , Q (cid:104) x, y, z | x = y = z = xyz (cid:105) G (24 , D , , (cid:104) x, y | x = y = 1 , xyx − = y (cid:105) G (24 , D (cid:104) x, y | x = y = 1 , xyx − = y − (cid:105) G (24 , Z × D , , − (cid:104) z | z = 1 (cid:105) × (cid:104) x, y | x = y = 1 , xyx − = y − (cid:105) G (24 ,
8) (( Z ) × Z ) (cid:111) Z (cid:104) x, y, z, w | x = y = z = w = 1 , [ y, z ] = [ y, w ] = [ z, w ] = 1 ,xyx − = y, xzx − = zy, xwx − = w − (cid:105) G (24 , Z × D (cid:104) z | z = 1 (cid:105) × (cid:104) x = y = 1 , xyx − = y − (cid:105) G (24 , Z × Q (cid:104) z | z = 1 (cid:105) × (cid:104) i, j, k | i = j = k = ijk (cid:105) G (24 , S (cid:104) x, y | x = (12) , y = (1234) (cid:105) G (24 , Z × A (cid:104) z | z = 1 (cid:105) × (cid:104) x, y | x = (12)(34) , y = (123) (cid:105) G (24 ,
14) ( Z ) × S (cid:104) z, w | z = w = [ z, w ] = 1 (cid:105)×(cid:104) x, y | x = (12) , y = (123) (cid:105) T ABLE
1. Nonabelian groups of order .Source: groupprops.subwiki.org/wiki/Groups_of_order_24 OUBLE KODAIRA STRUCTURES 25
IdSmallGroup( G ) G Presentation G (32 ,
2) ( Z × Z ) (cid:111) Z (cid:104) x, y, z | x = y = z = 1 , [ x, y ] = z, [ x, z ] = [ y, z ] = 1 (cid:105) G (32 , D , , (cid:104) x, y | x = y = 1 , xyx − = y (cid:105) G (32 ,
5) ( Z × Z ) (cid:111) Z (cid:104) x, y, z | x = y = z = 1 , [ x, y ] = 1 ,zxz − = x y, zyz − = y (cid:105) G (32 ,
6) ( Z ) (cid:111) Z (cid:104) x, y, z, w | x = y = z = w = 1 , [ x, y ] = 1 , [ x, z ] = 1 , [ y, z ] = 1 ,wxw − = x, wyw − = xy, wzw − = yz (cid:105) G (32 ,
7) ( Z (cid:111) Z ) (cid:111) Z (cid:104) x, y, z, u, w | y = z = w = 1 ,u = w − , x = u, ( yz ) = 1 , ( yu − ) = 1 ,uzu − = z − , xyzx − = y − (cid:105) G (32 ,
8) ( Z ) . ( Z × Z ) (cid:104) x, y, z | x = y = 1 , z = x ,xy = yx , [ y, z ] = 1 , xz = zxy − (cid:105) G (32 ,
9) ( Z × Z ) (cid:111) Z (cid:104) x, y, z | x = y = z = 1 , [ x, y ] = 1 ,zxz − = x y, zyz − = y (cid:105) G (32 , Q (cid:111) Z (cid:104) i, j, k, x | i = j = k = ijk, x = 1 ,xix − = j, xjx − = i, xkx − = k − (cid:105) G (32 ,
11) ( Z ) (cid:111) Z (cid:104) x, y, z | x = y = [ x, y ] = 1 , z = 1 ,zxz − = y, zyz − = x (cid:105) G (32 , D , , (cid:104) x, y | x = y = 1 , xyx − = y (cid:105) G (32 , D , , (cid:104) x, y | x = y = 1 , xyx − = y (cid:105) G (32 , D , , − (cid:104) x, y | x = y = 1 , xyx − = y − (cid:105) G (32 , Z . D (cid:104) x, y, z, u, w | w = 1 , z = u = w − ,x = u, y = z, xzx − = z − , [ y, u ] = 1 , xyxu = y − (cid:105) G (32 , D , , (cid:104) x, y | x = y = 1 , xyx − = y (cid:105) G (32 , D (cid:104) x, y | x = y = 1 , xyx − = y − (cid:105) G (32 , QD (cid:104) x, y | x = y = 1 , xyx − = y (cid:105) G (32 , Q (cid:104) x, y, z | x = y = z = xyz (cid:105) G (32 , Z × (( Z × Z ) (cid:111) Z ) (cid:104) w | w = 1 (cid:105)×(cid:104) x, y, z | x = y = z = 1 , [ x, y ] = 1 ,zxz − = xy, zyz − = y (cid:105) G (32 , Z × D , , (cid:104) z | z = 1 (cid:105) × (cid:104) x, y | x = y = 1 , xyx − = y (cid:105) G (32 ,
24) ( Z ) (cid:111) Z (cid:104) x, y, z | x = y = z = 1 , [ x, y ] = 1 , zxz − = x, zyz − = x y (cid:105) G (32 , Z × D (cid:104) z | z = 1 (cid:105) × (cid:104) x, y | x = y = 1 , xyx − = y − (cid:105) G (32 , Z × Q (cid:104) z | z = 1 (cid:105) × (cid:104) i, j, k | i = j = k = ijk (cid:105) G (32 ,
27) ( Z ) (cid:111) ( Z ) (cid:104) x, y, z, a, b | x = y = z = a = b = 1 , [ x, y ] = [ y, z ] = [ x, z ] = [ a, b ] = 1 ,axa − = x, aya − = y, aza − = xz,bxb − = x, byb − = y, bzb − = yz (cid:105) G (32 ,
28) ( Z × ( Z ) ) (cid:111) Z (cid:104) x, y, z, w | x = y = z = w = 1 , [ x, y ] = [ x, z ] = [ y, z ] = 1 ,wxw − = x − , wyw − = z, wzw − = y (cid:105) G (32 ,
29) ( Z × Q ) (cid:111) Z (cid:104) x, i, j, k, z | x = z = 1 , i = j = k = ijk, [ x, i ] = [ x, j ] = [ x, k ] = 1 ,zxz − = x, ziz − = i, zjz − = xj − (cid:105) G (32 ,
30) ( Z × ( Z ) ) (cid:111) Z (cid:104) x, y, z, w | x = y = z = w = 1 , [ x, y ] = [ x, z ] = [ y, z ] = 1 ,wxw − = xy, wyw − = y, wzw − = x z (cid:105) IdSmallGroup( G ) G Presentation G (32 ,
31) ( Z ) (cid:111) Z (cid:104) x, y, z | x = y = [ x, y ] = 1 , z = 1 ,zxz − = xy , zyz − = x y (cid:105) G (32 ,
32) ( Z ) . ( Z ) (cid:104) x, y, z, u, w | u = w = 1 ,u = z , u = x − , w = y − ,yxy − = x − , [ y, z ] = 1 , xzxwz = 1 (cid:105) G (32 ,
33) ( Z ) (cid:111) Z (cid:104) x, y, z | x = y = [ x, y ] = 1 , z = 1 ,zxz − = xy , zyz − = x y − (cid:105) G (32 ,
34) ( Z ) (cid:111) Z (cid:104) x, y, z | x = y = [ x, y ] = 1 , z = 1 ,zxz − = x − , zyz − = y − (cid:105) G (32 , Z (cid:111) Q (cid:104) x, i, j, k | x = 1 , i = j = k = ijk,ixi − = x − , jxj − = x − , kxk − = x (cid:105) G (32 ,
37) ( Z × Z ) (cid:111) Z (cid:104) x, y, z | x = y = z = 1 , [ x, y ] = 1 ,zxz − = x , zyz − = y (cid:105) G (32 ,
38) ( Z × Z ) (cid:111) Z (cid:104) x, y, z | x = y = z = 1 , [ x, y ] = 1 ,zxz − = x, zyz − = x y (cid:105) G (32 , Z × D (cid:104) z | z = 1 (cid:105) × (cid:104) x, y | x = y = 1 , xyx − = y − (cid:105) G (32 , Z × QD (cid:104) z | z = 1 (cid:105) × (cid:104) x, y | x = y = 1 , xyx − = y (cid:105) G (32 , Z × Q (cid:104) w | w = 1 (cid:105) × (cid:104) x, y, z | x = y = z = xyz (cid:105) G (32 ,
42) ( Z × Z ) (cid:111) Z (cid:104) x, y, z | x = y = z = 1 , [ x, y ] = 1 ,zxz − = x , zyz − = x y (cid:105) G (32 , Z (cid:111) ( Z ) (cid:104) x, y, z | x = 1 , y = z = [ y, z ] = 1 ,yxy − = x − , zxz − = x (cid:105) G (32 ,
44) ( Z × Q ) (cid:111) Z (cid:104) x, i, j, k, z | x = z = 1 , i = j = k = ijk, [ x, i ] = [ x, j ] = [ x, k ] = 1 ,zxz − = xi , ziz − = j, zjz − = i (cid:105) G (32 ,
46) ( Z ) × D (cid:104) z, w | z = w = [ z, w ] = 1 (cid:105)×(cid:104) x, y | x = y = 1 , xyx − = y − (cid:105) G (32 ,
47) ( Z ) × Q (cid:104) z, w | z = w = [ z, w ] = 1 (cid:105)×(cid:104) i, j, k | i = j = k = ijk (cid:105) G (32 ,
48) ( Z × ( Z ) ) (cid:111) Z (cid:104) x, y, z, w | x = y = z = w = 1 , [ x, y ] = [ x, z ] = [ y, z ] = 1 ,wxw − = x, wyw − = y, wzw − = x z (cid:105) G (32 , H ( Z ) (cid:104) r , t , r , t , z | r j = t j = z = 1 , [ r j , z ] = [ t j , z ] = 1 , [ r j , r k ] = [ t j , t k ] = 1 , [ r j , t k ] = z − δ jk (cid:105) , see (4) G (32 , G ( Z ) (cid:104) r , t , r , t , z | , r = t = z = 1 , r = t = z [ r j , z ] = [ t j , z ] = 1 , [ r j , r k ] = [ t j , t k ] = 1 , [ r j , t k ] = z − δ jk (cid:105) , see (5)T ABLE
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RCAVACATA DI R ENDE , C
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