BRANCHED COVERS OF ELLIPTIC CURVES AND KÄHLER GROUPS WITH EXOTIC FINITENESS

ANNALES DE

L’INSTITUT FOURIER

LesAnnales de l’institut Fouriersont membres du

Claudio Llosa Isenrich

Branched covers of elliptic curves and Kähler groups with exotic finiteness properties

Tome 69, n^o1 (2019), p. 335-363.

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BRANCHED COVERS OF ELLIPTIC CURVES AND KÄHLER GROUPS WITH EXOTIC FINITENESS

PROPERTIES

by Claudio LLOSA ISENRICH (*)

Abstract. — We construct Kähler groups with arbitrary finiteness properties by mapping products of closed Riemann surfaces holomorphically onto an elliptic curve: for eachr>3, we obtain large classes of Kähler groups that have classifying spaces with finite (r−1)-skeleton but do not have classifying spaces with finitely manyr-cells. We describe invariants which distinguish many of these groups. Our construction is inspired by examples of Dimca, Papadima and Suciu.

Résumé. — Nous construisons des groupes kähleriens ayant des propriétés de finitude arbitraires en considérant des applications holomorphes de produits de surfaces de Riemann vers une courbe elliptique: pour tout r> 3, nous obte- nons une grande classe de groupes kähleriens qui ont un espace classifiant avec un (r−1)-squelette fini, mais n’ont aucun espace classifiant avec un nombre fini de r-cellules. Nous décrivons des invariants qui distinguent beaucoup de ces groupes.

Notre construction est inspirée par les exemples de Dimca, Papadima et Suciu.

1. Introduction

A Kähler group is a group which can be realised as fundamental group of a compact Kähler manifold. In particular, every Kähler group is finitely presented. A group G is of finiteness type F_r if there is a K(G,1) with finiter-skeleton; so typeF1 is equivalent to finite generation and typeF2

is equivalent to finite presentability.

Keywords:Kähler groups, Homological finiteness properties, Branched covers.

2010Mathematics Subject Classification:32J27, 20J05, 20F65.

(*) I am very grateful to my advisor Martin Bridson for his generous support and advice while writing this paper and for the very helpful discussions that we had in our numerous meetings. I am also very greatful to Mahan Mj for the very helpful discussions which allowed me to simplify the proof of Lemma 2.1 and for explaining to me how our examples can also be obtained using techniques from [2]. Moreover, I would like to thank the anonymous referee for helpful comments and suggestions that led to improvements to the exposition of this paper, and my office mates Alexander Betts and Giles Gardam for the many inspiring discussions about the content of this work.

In 2009, Dimca, Papadima and Suciu [8] constructed the first examples of Kähler groups that are of finiteness typeF_r−1, but not of typeFr, for allr>3. In fact their groups areprojective, that is, fundamental groups of smooth projective manifolds. In particular, these groups, although torsion- free, can not have compact Kähler manifolds as classifying spaces. Our main goal here is to construct broader classes of such groups.

We will show how to construct large classes of new examples of Kähler groups with arbitrary finiteness properties from surjective maps of a direct product of surface groups onto an elliptic curve. More precisely, we will prove

Theorem 1.1. — Letr>3 and letg1, . . . , gr>2. LetE be an elliptic curve and, fori = 1, . . . , r, let S_gi be a closed connected surface of genus gi together with a branched coveringfg_i :Sg_i →E. Assume that the map

f =

r

X

i=1

f_gi :S_g1× · · · ×S_gr −→E induces a surjective map on fundamental groups.

Then the generic fibre H off is connected andf induces a short exact sequence

1−→π1H−→π1Sg₁× · · · ×π1Sg_r −→π1E−→1

on fundamental groups. Moreover, the groupπ1H is a projective (and thus Kähler) group of typeF_r−1, but not of typeFr.

Under the additional assumption that all mapsfg_i arepurely branched, we will give the following complete classification of all Kähler groups that can be obtained using our construction. The notion of a purely branched covering will be defined in Definition 2.2. Roughly speaking a purely bran- ched covering is a branched covering which is obtained from the base space by alone branching.

Theorem 1.2. — LetEbe an elliptic curve. Letr, s>3, letgi, hj>2, letSg_i be a closed Riemann surface of genusgi>2and letRh_i be a closed Riemann surface of genush_i>2for16i6rand16j6s. Assume that there are purely branchedki-fold holomorphic covering mapspi:Sg_i →E and purely branched l_i-fold holomorphic covering maps q_i : R_hi → E.

Define p = Pr

i=1pi : Sg₁ × · · · ×Sg_r → E and q = Ps

j=1qj : Rh₁ ×

· · · ×R_hs →E, and denote by H_p and H_q the smooth generic fibres ofp, respectivelyq.

Then the Kähler groups π₁H_p and π₁H_q are isomorphic if and only if r=sand there is a permutation of theRh_i andqi such that gi=hi and k_i=l_i fori= 1, . . . , r.

The construction is inspired by the work of Dimca, Papadima and Su- ciu, but is more general. The construction given by Dimca, Papadima and Suciu is a topological construction and does not lead to an explicit fi- nite presentation. In [11], Llosa Isenrich derived an explicit presentation for these examples, answering a question of Suciu. The original proof that Dimca, Papadima and Suciu’s examples have arbitrary finiteness proper- ties consists of an involved argument making use of characteristic varieties.

Alternative proofs have been given since by Biswas, Mj and Pancholi [2]

and by Suciu [12]. We will give a shorter proof making use of the classifica- tion of the finiteness properties of subgroups of direct products of surface groups obtained by Bridson, Howie, Miller and Short [4].

Biswas, Mj and Pancholi [2] suggested a more general approach for arbi- trary irrational Lefschetz pencils over surfaces of positive genus with sin- gularities of Morse type. Although it is not explicit in [2], the class of examples constructed in our work can also be obtained as a consequence of their work. However, the techniques are quite different: their approach is based on topological Lefschetz fibrations which by definition have Morse type singularities and the result for non-Morse type singularities then fol- lows by a deformation argument. They do not develop any techniques to distinguish between different examples (cf. Theorem 1.2 above).

As an example of our construction we will produce a class of Kähler groups which to us seem the most natural analogue in the Kähler setting of a specific subclass of the Bestvina–Brady groups [1]. Using our classification we will then show that these groups are not isomorphic to Dimca, Papadima and Suciu’s groups. An interesting question that one can ask here is if one can generalise our construction further to produce natural Kähler analogues for all of the original Bestvina–Brady groups. Notice that Dimca, Papadima and Suciu showed that the only original Bestvina–Brady groups which are Kähler are free abelian groups of even rank [7, Corollary 1.3].

The structure of this work is as follows: in Section 2 we prove that the fibres of our maps are connected. In Section 3 we will prove an algebraic result which will allow us to deduce the finiteness properties of the groups arising from our construction. In Section 4 we explain our general construc- tion which we then apply in Section 5 to realise the algebraically described groups from Section 3 as Kähler groups. In Section 6 we explain how the

question about the existence of isomorphisms between Kähler groups ob- tained using our construction reduces to a question in Linear Algebra. In Section 7 we solve this question for maps arising from purely branched coverings and conclude that the groups in Section 5 are not isomorphic to Dimca, Papadima and Suciu’s groups.

2. Connectedness of fibres

In this section we prove that the fibres of the maps in Theorem 1.1 are connected.

Lemma 2.1. — Let r > 2, let E be an elliptic curve and letSg_i be a closed Riemann surface of genusg_i>2. Letf_gi :S_gi →E be holomorphic branched covers ofE.

The mapf =Pr

i=1f_gi :S_g1× · · · ×S_gr →E has connected fibres if and only if it induces a surjective map on fundamental groups.

Proof. — Every holomorphic maph:X →Y between compact complex manifoldsX andY with connected fibres induces a surjective map on fun- damental groups, since it is a locally trivial fibration over the complement of a complex codimension one subvariety ofY. Hence, if f has connected fibres then it induces a surjective map on fundamental groups.

Assume now that f induces a surjective map on fundamental groups. If f does not have connected fibres then Stein factorisation yields a closed Riemann surface S and holomorphic maps α : S → E and β : S_g1 ×

· · · ×Sg_r → S such that α is finite-to-one and β has connected fibres.

Since holomorphic finite-to-one maps between closed Riemann surfaces are branched covering maps, it follows thatαis a branched covering.

Choose a base point (p1, . . . , pr)∈Sg₁× · · · ×Sg_r and denote byβi the restriction ofβ to theith factor{(p₁, . . . , p_i−1)} ×S_gi × {(p_i+1, . . . , p_r)}.

Then there isei∈E such that α◦βi=ei+fg_i. Sinceβi is holomorphic, it is non-trivial and finite-to-one, and hence a finite-sheeted holomorphic branched covering map. It now follows that βi∗(π1Sg_i) 6 π1S is a finite index subgroup and therefore not cyclic fori= 1, . . . , r.

It then follows from [2, Lemma 7.1] thatSmust itself be an elliptic curve.

Since the argument is short we want to give it here. Chooseγ1∈π1Sg₁ and γ₂∈π₁S_g2 such that their imagesβ₁◦γ₁andβ₂◦γ₂do not lie in a common cyclic subgroup ofπ1S. Thenβ1◦γ1 andβ2◦γ2generate aZ²-subgroup of π₁S and the only closed Riemann surfaces withZ²-subgroups are elliptic curves (e.g. [10, Theorem 1]).

Using Euler characteristic, it follows that any branched covering map between 2-dimensional tori is an unramified covering map. By assumption the mapf =α◦βinduces a surjective map on fundamental groups. Hence, the mapα:S→E is an unramified holomorphic covering which is surjec- tive on fundamental groups and thereforeS =E andαis biholomorphic.

In particular, the mapf has connected fibres.

For a group G and a subset S ⊂ G denote by hhSii 6 G its normal closure inG.

We introduce a special class of branched covering maps of tori. Let Y be a torus of real dimensionk >2, letX be a closed connected manifold of real dimensionk and letD⊂Y be a real subvariety of codimension at least two (we will be interested in the case when X and Y are complex manifolds andD is a complex subvariety). Let f :X →Y be a branched covering map with branching locusD, that is,f⁻¹(D) is a nowhere dense set inX mapping onto D and the restriction f :X\f⁻¹(D)→Y \D is an unramified covering.

Assume that for a base point z₀ ∈Y \D there are simple closed loops µ1, . . . , µk, b1, . . . , bl: [0,1]→Y \D based atz0and points p1, . . . , pl∈D with the following properties:

• π1(Y \D) =h[µ1], . . . ,[µk],[b1], . . . ,[bl]i;

• µi([0,1])∩µj([0,1]) ={z₀}fori6=j;

• the loopsµ1, . . . , µk generatesπ1Y; and

• for any choiceU1, . . . , Ul⊂Y of open neighbourhoods ofp1, . . . , pl

there are pathsδ₁, . . . , δ_l: [0,1]→Y \D starting atz₀ and ending in Ui \ {pi}, and loops νi : [0,1] → Ui\ {pi} such that for the concatenationβ_i =δ_i·ν_i·δ⁻¹_i , we have [β_i] = [b_i]∈π₁(Y \D).

Definition 2.2. — We call the map f purely branched if there exist loopsµ1, . . . , µk, b1, . . . , blas above which satisfy the condition

hh[µ1], . . . ,[µk]ii6f_∗(π1(X\f⁻¹(D)))6π1(Y \D), i.e. every lift toX of eachµ_i is a loop.

Note that for Y a 2-torus,X a closed connected surface of genus g>1 and f : X → Y a branched covering map with branching locus D = {p₁, . . . , p_r}, the map f is purely branched if and only if there are simple closed loopsµ1, µ2 : [0,1]→Y \D which generate π1Y, intersect only in µ₁(0) =µ₂(0) and have the property that every lift of µ₁and µ₂is a loop inX.

We want to remark that for a branched covering map being purely branched is not the same as being surjective on fundamental groups, as Example 7.9 at the end of Section 7 will show.

3. A Kähler analogue of Bestvina–Brady groups A large class of groups with prescribed finiteness type are the Bestvina–

Brady groups which were constructed using combinatorial Morse theory [1].

These groups arise as kernels of surjective maps from Right Angled Artin groups to the integers.

Given a finite simplicial graph Γ, denote by V(Γ) its vertex set and by E(Γ) its edge set. We define the Right Angled Artin group (RAAG) AΓ

for Γ as the group with the finite presentation

AΓ=hV(Γ)|[v, w] ifvw∈E(Γ)i.

Associated to every such presentation of a RAAG is a natural homomor- phism

φΓ : AΓ −→ Z=hti V(Γ)3v 7−→ t

Associated to the RAAGA_Γwe have the flag complexX_Γwith 1-skeleton Γ, that is, the simplicial complex obtained from Γ by requiring that when- ever a finite set of vertices in Γ is pairwise connected, there is a simplex in X_Γ with these vertices.

The Bestvina–Brady group BBΓ associated to Γ is defined as BBΓ = kerφ_Γ. The finiteness properties of the Bestvina–Brady groups are com- pletely understood:

Theorem 3.1 (Bestvina–Brady Theorem, [1]). — LetΓbe a finite sim- plicial graph. Then for every integer n > 0, the Bestvina–Brady group BBΓis of typeFn if and only if the spaceXΓis(n−1)-connected, that is, π_i(X_Γ) = 1 for all06i6n.

Dimca, Papdima and Suciu showed that the only Bestvina–Brady groups which are Kähler are the obvious ones in the following sense

Theorem 3.2([7, Corollary 1.3]). — LetΓbe a finite simplicial graph.

ThenBBΓ is Kähler if and only ifBBΓ is free abelian of even rank.

Free abelian groups are of typeF_∞, since their classifying space is a torus.

Hence, the Bestvina–Brady groups can not provide any examples of Kähler groups with arbitrary finiteness properties. However, Dimca, Papadima and

Suciu [8] observed that one can imitate the Bestvina–Brady construction to obtain Kähler groups with arbitrary finiteness properties.

From a group theoretic point of view, their groups arise as kernels of maps from a direct product of surface groups toZ²; We will describe the geomet- ric construction behind these maps later in this paper. Let Λg₁, . . . ,Λg_r, g1, . . . , gr>2 be surface groups with presentation

Λg_i=

aⁱ₁, . . . , aⁱ_gi, bⁱ₁, . . . , bⁱ_gi

aⁱ₁, bⁱ₁

· · ·

aⁱ_gi, bⁱ_gi

Although they did not describe them as such, Dimca, Papadima and Suciu’s groups are kernels of the surjective maps

(3.1)

φg₁,...,g_r : Λg₁× · · · ×Λg_r −→ Z²=ha, b|[a, b]i aⁱ₁, aⁱ₂ 7−→ a

bⁱ₁, bⁱ₂ 7−→ b aⁱ₃, . . . , aⁱ_g

i 7−→ 0

bⁱ₃, . . . , bⁱ_gi 7−→ 0,

as our explanation of their construction in Section 4 will show, where g1, . . . , gr>2 andr>3.

In particular, we will see that the maps constructed in Section 4 satisfy all the conditions of Theorem 1.1. This will provide us with a new proof that the groupsφg₁,...,g_r are Kähler groups of finiteness typeF_r−1, but not of typeFr. Thus:

Theorem 3.3. — If g1, . . . , gr > 2, then the kernel of φg₁,...,g_r is a Kähler group of finiteness typeF_r−1, but not of typeF_r.

The case when g1 = · · · =gr = 2 seems to be a completely analogous surface group version of the Bestvina–Brady group corresponding to the direct productF₂× · · · ×F₂ofrcopies of the free group on two generators.

More generally consider the direct product Λg₁×· · ·×Λg_r forg1, . . . , gr>

2. Then, in analogy to the mapφ_2,...,2, define the homomorphism

(3.2)

ψ_g1,...,gr : Λ_g1× · · · ×Λ_gr −→ Z²=ha, b|[a, b]i

aⁱ_j 7−→ a

bⁱ_j 7−→ b

To prove that all groups arising as kernel of one of theψg₁,...,g_r are not of typeFr, we use a result by Bridson, Miller, Howie and Short [4, Theorem B]

Theorem 3.4. — LetΛg₁, . . . ,Λg_nbe surface groups and letG6Λg₁×

· · · ×Λ_gnbe a subgroup of their direct product. Assume that each intersec- tionLi=G∩Λg_i is non-trivial and arrange the factors in such a way that

L₁, . . . , L_r are not finitely generated and L_r+1, . . . , L_n are finitely gener- ated.

If precisely r>1 of the L_i are not finitely generated, thenG is not of typeF Pr.

For finitely presented groups, being of type F Pr is equivalent to being of type F_r [6, p. 197]. Since all the groups we consider will be finitely presented, we do not dwell on the meaning of the homological finiteness conditionF Pr. As a consequence of Theorem 3.4 we can now prove

Corollary 3.5. — LetΛ_g1, . . . ,Λ_gr be surface groups withg_i>2and letr>3. Letk>1and letνi: Λg_i →Z^k be non-trivial homomorphisms.

Then the kernel of the map

ν=ν₁+· · ·+ν_r: Λ_g1× · · · ×Λ_gr −→Z^k is not of typeFr.

Proof. — Let G = ker(ν) < Λg₁ × · · · ×Λgr. Let Li = G∩Λgi :=

G∩1× · · · ×Λg_i× · · · ×1 and note thatLi= ker(νi). ThenLiis an infinite index normal subgroup of Λ_gi. Infinite index normal subgroups of a surface group are infinitely generated free groups (e.g. [10, Theorem 1] and [3, Theorem 3.1]). Hence, none of theL_i are finitely generated and therefore Gis not of type F Pr, by Theorem 3.4, and hence not of typeFr. In particular this gives an alternative proof of the finiteness properties of the Kähler groups constructed by Dimca, Papadima and Suciu. In fact, we immediately obtain

Corollary 3.6. — For all r > 3 and g₁, . . . , g_r > 2, the groups ker(φg₁,...,g_r)andker(ψg₁,...,g_r)are not of type Fr.

Remark 3.7. — Under the additional condition that the mapsfi: Λi→ Z^k are surjective all groups that arise in this way are group theoretic fibre products overZ^k; one can construct explicit finite presentations for them using the same methods as in [11].

4. Constructing large classes of examples

In this section we will show how to use our results to prove Theorem 1.1.

LetE be a closed Riemann surface of positive genus and letX be a closed connected complex analytic manifold of dimension r > 1. An irrational pencil h:X →E is a surjective holomorphic map such that the smooth generic fiberH is connected.

ForM,N complex manifolds, we say that a surjective holomorphic map f : M → N has isolated singularities if, for every y ∈ N, the set of singular points off inf⁻¹(y) is discrete. In the case of an irrational pencil h:X→E this is equivalent to the set of singular points off being finite.

The following result is due to Dimca, Papadima and Suciu

Theorem 4.1 ([8, Theorem C]). — Let h : X → E be an irrational pencil. Suppose thath has only isolated singularities. Then the following hold:

(1) The inclusionH ,→ X induces isomorphisms πi(H) ∼= πi(X) for 26i6r−2;

(2) hinduces a short exact sequence1→π1H →π1X→π1E→1.

A Stein manifold is a complex manifold that embeds biholomorphically as a closed submanifold in some affine complex space C^r. Theorem 4.1 allows us to prove

Theorem 4.2. — Letr>3 and letg1, . . . , gr>2. LetE be an elliptic curve and, fori = 1, . . . , r, let S_gi be a closed connected surface of genus gi together with a branched covering fg_i :Sg_i →E. Assume further that the map

f =

r

X

i=1

fgi :Sg₁× · · · ×Sgr −→E induces a surjective map on fundamental groups.

Then the generic fibreH of the mapf is a connected(r−1)-dimensional smooth projective variety such that

(1) The homotopy groupsπiH are trivial for26i6r−2 andπ_r−1H is nontrivial;

(2) The universal coverHe ofH is a Stein manifold;

(3) The fundamental groupπ₁His a projective (and thus Kähler) group of finiteness typeF_r−1, but not of finiteness typeFr;

(4) The fundamental group π₁H is not commensurable (up to finite kernels) to any group having a classifying space of finite type;

(5) The mapf induces a short exact sequence

1→π₁H →π₁S_g1× · · · ×π₁S_gr →π₁E→1 on fundamental groups.

Proof. — It is well-known that there is a unique complex structure on S_gi with respect to which the mapf_gi is holomorphic, since f_gi is a finite- sheeted branched covering map. In particular, fg_i has critical points the

finite preimageC_i=f_g⁻¹

i (D_i), where D_i⊂E is the finite set of branching points offg_i.

This equipsX =S_g1×· · ·×S_grwith a projective structure with respect to whichf :X=Sg₁×· · ·×Sg_r →Eis a surjective holomorphic map. The set of singular points off is then the set of points (x₁, . . . , xr)∈Sg₁× · · · ×Sg_r

such that 0 = df(x1, . . . , xr) = (dfg₁(x1), . . . ,dfgr(xr)). It follows that the set of singular points off is the finite setC1× · · · ×Cr. In particular, f has isolated singularities.

By assumption all of thefg_i are branched covering maps andf is surjec- tive on fundamental groups. Thus, by Lemma 2.1, the mapfhas connected fibers.

It follows thatf is an irrational pencil with isolated singularities. Hence, by Theorem 4.1, we obtain thatf induces a short exact sequence

1−→π1H −→π1X −→^f∗ π1E −→1

on fundamental groups proving assertion (5). Furthermore, we obtain that πiH ∼=πiX ∼= 0, for 26i6r−2, where the last equality follows sinceX is aK(π₁X,1). This implies the first part of assertion (1).

The group π1H is projective, since the generic smooth fibreH off is a complex submanifold of the compact projective manifoldX.

Because π1H = 0 for 26i6r−2, we obtain aK(π1H,1) from H by attaching cells of dimension>r. Since H is a compact complex manifold, it follows thatH has a finite cell structure. Thus, the group ker(f∗) =π1H is of finiteness typeF_r−1 and, by Corollary 3.5, it is not of type Fr. This implies assertion (3) and the second part of assertion (1), since if π_r−1H were trivial we could construct aK(G,1) with finiter-skeleton.

Assertion (4) is an immediate consequence of the well-known [8, Propo- sition 2.7].

Assertion (1) follows similarly as in the proof of [8, Theorem A]. Namely, the universal coveringX,e q:Xe →XofX =Sg₁×· · ·×Sg_r is a contractible Stein manifold and the pair (X, H) is (r−1)-connected by Theorem 4.1.

Hence, the preimageq⁻¹(H) ofH inXe is a closed complex submanifold of the Stein manifoldXe which is biholomorphic to the universal coveringHe

ofH. Thus,He is Stein.

Theorem 1.1 is now a direct consequence of Theorem 4.2. Note that Theorem 4.2 is a generalisation of [8, Theorem A].

5. Constructing a Bestvina–Brady type class of examples We will now proceed to explain how one can realise the maps φg1,...,gr

in (3.1) andψg₁,...,g_rin (3.2) geometrically. More precisely, they will arise as the induced maps on fundamental groups of maps satisfying the conditions of Theorem 1.1.

The map realisingφ_g1,...,gr is the map used for the original construction of Kähler groups with arbitrary finiteness properties by Dimca, Papadima and Suciu. Since our construction imitates the construction by Dimca, Pa- padima and Suciu, we want to give a brief description of their construction first.

Let E be an elliptic curve, that is, a complex torus of dimension one, letg >2, and let B ={b1, . . . , b_2g−2} ⊂E be a finite subset of even size.

Choose a set of generators α, β, γ₁, . . . , γ_2g−2 of the first homology group H1(E\B,Z) such that γi is the boundary of a small disc centred at bi, i= 1, . . . ,2g−2, andα, β are generators ofπ1E which have simple closed representatives intersecting in a single point.

Then the map H1(E\B,Z)→ Z/2Z defined by γi 7→1 and α, β 7→0 induces a 2-fold normal covering ofπ₁(E\B) which extends continuously to a 2-fold branched coveringfg:Sg→E from a topological surfaceSg of genusg >2 onto E. It is well-known there is a unique complex structure onSg such that the mapfg is holomorphic.

By looking at a suitable concrete realisation of the mapfg (see [11] for more details), it is not hard to see that it induces the map

f_g∗ : Λ_g=π₁S_g −→ π₁E=Z² a1, a2 7−→ a

b₁, b₂ 7−→ b

ai, bi 7−→ 0 ∀i= 3, . . . , g

with respect to presentations Λ_g = ha₁, b₁, . . . , a_g, b_g|[a₁, b₁]· · ·[a_g, b_g]i andπ1E=ha, b|[a, b]iof the fundamental groups.

Then for g₁, . . . , g_r >2 andr >3, use addition in the elliptic curve to define the mapf =Pr

i=1fg_i :Sg₁× · · · ×Sg_r →E.

The maps fg_i are branched coverings inducing surjective maps on fun- damental groups. They restrict to unramified normal coverings ofE\B_i which correspond to kernels of homomorphisms mapping a set of generators α, β: [0,1]→E\B_i ofπ₁E to zero. It follows thatf_gi is purely branched fori= 1, . . . , r. In particular they are surjective on fundamental groups.

Hence, by Theorem 1.1, the kernel of the induced map f_∗: Λ_g1× · · · × Λg_r →Z²is projective (and thus Kähler) of finiteness typeFr−1, but not

Fr. Observe that the map f_∗ is indeed identical with the map φ_g1,...,gr defined in (3.1).

We will imitate the construction of these groups in order to produce holomorphic mapsf_h1, . . . , f_hr :Sh_i →E for allh1, . . . , hr>2 andr>3 such thatf =Pr

i=1f_hi realises the mapψh₁,...,h_r defined in (3.2). We will present two different constructions, each of which has advantages.

Construction 1

This is the more natural construction. It has the advantage that the mapsf_hi are normal branched coverings, but it comes at the cost that the singularities off are not quadratic and thereforef is not a Morse function.

As above, let E be an elliptic curve and letB ={d1, d₂} ⊂E be two arbitrary points. Let α, β, γ1, γ2 be generators for the homology group H₁(E \ B,Z) of the form described in the paragraph preceding Defini- tion 2.2,α, βgenerateπ1Eand have simple closed representatives inE\B intersecting positively with respect to the orientation induced by the com- plex structure onE, andγ1, γ2 are the positively oriented boundary loops of small discs aroundb1, respectively b2. For h>2, the surjective homo- morphismH₁(E\B,Z)→Z/hZ defined byα, β 7→0,γ₁ 7→1,γ₂7→ −1, defines a h-fold normal branched coveringf_h :Sh → E from a topologi- cal surface of genushwith branching locusB. We may assume that, after connecting the generators ofH1(E\B,Z) to a base point, the fundamental group ofE\B is

π₁(E\B) =hα, β, γ1, γ₂|[α, β]γ₁γ₂i.

It is well-known that there is a unique complex structure on Sh such that the map f_h is holomorphic. Denote by C = {c1 = f⁻¹_h (d1), c2 = f⁻¹_h (d₂)} ⊂S_h the set of critical points off_h.

In analogy to the DPS groups, we define the map f using the additive structure onE,

f =

r

X

i=1

f_hi :Sh₁× · · · ×Sh_r −→E for allr>3 andh1, . . . , hr>2.

The maps f_h

i are branched coverings induced by the surjective com- position of homomorphismsπ1(E\B)→ H1(E\B,Z)→ Z/hiZdefined by α, β 7→ 0, γ₁ 7→ 1, γ₂ 7→ −1. In particular, all of the f_h

i are purely branched, sinceαandβ are elements of the kernel of this homomorphism,

which is a normal subgroup ofπ₁(E\B). It follows that all of thef_h

i are surjective on fundamental groups.

Hence, Theorem 4.2 can be applied to f. This implies that the funda- mental group π1H of the generic fibre H of f is a projective (and thus Kähler) group of finiteness typeF_r−1, but not of finiteness typeFr.

Since all of the f_hi are purely branched, any lift of a generator α, β : [0,1] → E \Bi of π1E to Sh_i is a loop. Choose a fundamental domain F⊂S_hifor theZ/h_iZ-action such that the images ofαandβare contained in the imagef_hi(U) of an open subset U ⊂F on whichf_hi restricts to a homeomorphism.

Denote theh_ilifts ofαbya⁽ⁱ⁾₁ , . . . , a⁽ⁱ⁾_h

i and theh_ilifts ofβbyb⁽ⁱ⁾₁ ,· · ·b⁽ⁱ⁾_h

i, where we choose lifts so thata⁽ⁱ⁾_j andb⁽ⁱ⁾_j are in the interior of the fundamen- tal domainj·F forj∈Z/hiZ. In particular, the loopsa⁽ⁱ⁾₁ , b⁽ⁱ⁾₁ , . . . , a⁽ⁱ⁾_h

i, b⁽ⁱ⁾_h

i

form a standard symplectic basis for the (symplectic) intersection form on H1(Shi,Z).

It is then well-known that we can find generators α₁⁽ⁱ⁾, β₁⁽ⁱ⁾,· · ·α⁽ⁱ⁾_h

i, β⁽ⁱ⁾_h

i

of π₁S_hi such that the abelianisation is given by α⁽ⁱ⁾_j 7→ a⁽ⁱ⁾_j , β_j⁽ⁱ⁾ 7→ b⁽ⁱ⁾_j and

π1Shi =D

α⁽ⁱ⁾₁ , β₁⁽ⁱ⁾,· · ·α⁽ⁱ⁾_h

i, β_h⁽ⁱ⁾

i

h

α⁽ⁱ⁾₁ , β⁽ⁱ⁾₁ i

· · ·h α⁽ⁱ⁾_h

i, β_h⁽ⁱ⁾

i

iE .

This is for instance an easy consequence of Theorem 6.5.

With respect to this presentation, the map on fundamental groups in- duced byf_hi is given by

f_hi∗: π1Shi −→ π1E α⁽ⁱ⁾_j 7−→ α β_j⁽ⁱ⁾ 7−→ β

For an illustration of the mapf_hand the generatorsαj, βj, see Figure 5.1.

As a direct consequence, we obtain that f induces the map f_∗: π1Sh₁× · · · ×π1Sh_r −→ π1E

α⁽ⁱ⁾_j 7−→ α

β_j⁽ⁱ⁾ 7−→ β

on fundamental groups. Thus, the induced mapf_∗ on fundamental groups is indeed the mapψh₁,...,h_r in (3.2).

. . .

. .

α a_h-1

a₁

a₂

a₃

β b_h-1

b₁

b₂

b₃

d₁ d₂ c₁

c₂ S

E

h

f_h

Figure 5.1. Theh-fold branched normal coveringf_hofEin Construc- tion 1

Construction 2

We will now give an alternative construction which realises ψ_h1,...,hr as the fundamental group of the generic fibre of a fibration over an elliptic curve with Morse type singularities only. This is at the expense of the maps being regular branched coverings rather than normal branched coverings.

LetE be an elliptic curve, leth>2, letd_1,1, d_1,2, d_2,1, . . . , d_h−1,1, d_h−1,2 be 2(h−1) points inEand lets1, . . . , s_h−1: [0,1]→E be simple, pairwise non-intersecting paths with starting points_i(0) =d_i,1and endpoints_i(1) = di,2 fori= 1, . . . , h−1. TakehcopiesE0,E1, . . . , Eh−1 ofE, cutE0 open along all of the pathssiand cutEiopen along the pathsifor 16i6h−1.

This produces surfacesF0, . . . , Fh−1 with boundary.

Glue the boundary ofFito the boundary component ofF0corresponding to the cut produced by the paths_i where we identify opposite edges with respect to the identity homeomorphismE0→Ei fori= 1, . . . , h−1. This yields a closed genus hsurface R_h together with a (h−1)-fold branched covering mapf_h⁰ : Rh → E with critical points c1,1, c1,2, c2,1, . . . , ch−1,1, c_h−1,2 ∈Rh, f⁰(ci,j) =di,j of order two. EndowRh with the unique com- plex structure that makes the mapf_h⁰ holomorphic.

It is clear that the map f_h⁰ is purely branched and surjective on funda- mental groups: in analogy to Construction 1 we find standard generating setsα, β of π1E and α1, . . . , αh, β1, . . . , βh ofπ1Rh with respect to which the induced map on fundamental groups is given byα_i 7→αandβ_i7→β. See Figure 5.2 for an illustration of the mapf_h⁰.

For h₁, . . . , hr >2,r >3, the holomorphic mapf⁰ =Pr i=1f_h⁰

i : Rh₁×

· · · ×Rhr → E induces the map ψh₁,...,hr on fundamental groups. The mapf⁰ has isolated singularities and connected fibres and in fact we can see, by considering local coordinates around the singular points, that all singularities off⁰ are of Morse type.

6. Reducing the isomorphism type of our examples to Linear Algebra

We will now show that our groups are not isomorphic to Dimca, Pa- padima and Suciu’s groups and thus provide genuinely new examples rather than being their examples in disguised form.

As before, let Λg be the fundamental group of a closed orientable surface of genusg. Forr, k>1, consider epimorphismsφ_gi : Λ_gi →Z^k andψ_hi : Λh_i → Z^k, where gi, hi > 2 and 1 6 i 6 r. Recall that ker(φg_i) and ker(ψ_hi) are infinitely generated free groups, since they are infinite index normal subgroups of surface groups.

Define maps φg₁,...,g_r = φg₁ +· · ·+φg_r : Λg₁ × · · · ×Λg_r → Z^k and ψh1,...,hr =ψh1+· · ·+ψhr: Λh1× · · · ×Λhr →Z^k and let

Li:= Λg_i∩ker(φg₁,...,g_r) = ker(φg_i), Ki:= Λh_i∩ker(ψh₁,...,h_r) = ker(ψh_i).

. . .

. .

α a_h-1

a₁

a₂

a₃

β b_h-1

b₁

b₂

b₃ c_h-1,1

c_h-1,2

R

E

h

c_1,1 c_1,2 c_2,1

c_2,2 c_3,1

c_3,2

d_h-1,1

d_h-1,2

d_1,1 d_1,2 d_2,1 d_2,2 d_3,1

d_3,2

. . .

. .. .

. . . . .

.

.

. . .

f'h

a_h b_h

Figure 5.2. The h-fold branched covering f_h⁰ of E with Morse type singularities in Construction 2

Then the following Lemma is a special case of [5, Theorem C (3), (4)]

Lemma 6.1. — Every isomorphism

θ: ker(φ_g1,...,gr)−→ker(ψ_h1,...,hr)

satisfiesθ(L_i) =K_iup to reordering of the factors. In particular, we obtain thatθrestricts to an isomorphismL1× · · · ×Lr∼=K1× · · · ×Kr.

Furthermore, with the same reordering of factors, we haveθ((Λ_gi

1× · · · × Λg_ik)∩ker(φg₁,...,g_r)) = (Λh_i1 × · · · ×Λh_ik)∩ker(ψh₁,...,h_r)for 16k6r and16i₁<· · ·< i_k 6r.

Remark 6.2. — It also follows from [5, Theorem C (3), (4)] that if we have an isomorphism between direct products ofrsurface groups Λg₁× · · · ×Λg_r

and Λh₁× · · · ×Λh_r, then after reordering of the factors it is induced by isomorphisms Λgi ∼= Λhi and, in particular,g_i=h_i fori= 1, . . . , r.

We obtain the following consequence

Proposition 6.3. — There is an isomorphism of the short exact se- quences

1−→ker(φg₁,...,g_r)−→Λg₁× · · · ×Λg_r

φ_g1,...,gr

−−−−−−→Z^k−→1 and

1−→ker(ψh₁,...,h_r)−→Λh₁× · · · ×Λh_r

ψ_h1,...,hr

−−−−−−→Z^k−→1 if and only if (up to reordering factors)g₁=h₁, . . . , g_r=h_r and there are isomorphisms of the short exact sequences

1−→ker(φg_i)−→Λg_i φ_gi

−−→Z^k−→1 and

1−→ker(ψh_i)−→Λh_i ψ_hi

−−→Z^k −→1

for alli= 1, . . . , rsuch that the isomorphismA:Z^k−→Z^k is independent ofi.

Proof. — The if direction follows immediately by taking the Cartesian product of the automorphisms θ_i : Λ_gi →Λ_gi to be the automorphism of Λg₁× · · · ×Λg_r and the identity map to be the automorphism ofZ^k.

For the only if direction we use that, by Remark 6.2, the isomorphism Λg₁× · · ·Λg_r →Λh₁× · · · ×Λh_r is realised by a direct product of isomor- phisms θi : Λg_i → Λh_i, after possibly reordering factors. Restricting to factors then implies that the automorphismθi of Λgi and the identity on Z^k induce an isomorphism of short exact sequences fori= 1, . . . , r.

Proposition 6.4. — Let r > 2. The groups H1 = ker(φg₁,...,g_r) and H₂ = ker(ψ_h1,...,hs) are isomorphic if and only if there is an isomorphism of the short exact sequences

1−→ker(φg₁,...,g_r)−→Λg₁× · · · ×Λg_r

φ_g1,...,gr

−−−−−−→Z^k−→1

and

1−→ker(ψh₁,...,h_s)−→Λh₁× · · · ×Λh_s

ψ_h1,...,hs

−−−−−−→Z^k −→1.

Proof. — Let θ : H₁ → H₂ be an abstract isomorphism of groups and letLi 6H1,Ki 6H2 be as above. By Lemma 6.1, we may assume that after reordering factorsθ(Li)6Ki and thatθ(M1) =M2 for M1 =H1∩ (1×Λg2× · · · ×Λgr),M₂=H₂∩(1×Λh2× · · · ×Λhs).

Since φg_i,ψh_j are surjective for all 16i6r, 16j6s, we obtain that H₁/M₁∼= Λg1, H₁/L₁∼= Λg2× · · · ×,Λgr

H2/M2∼= Λh₁, H2/K1∼= Λh₂× · · · ×Λh_s, where the isomorphisms are induced by the projection maps.

In particular, the mapθinduces an isomorphism of short exact sequences

1 //K1 //

∼= θ

H1/M1×H1/L1∼=

∼= θ

Λg₁×· · ·×Λg_r //Z^k //

∼= θ

1

1 //K2 //H2/M2×H2/K1∼= Λh1×· · ·×Λ_hs //Z^k //1 proving the only if direction. The if direction is trivial.

Note that Proposition 6.4 is well-known (see for instance [5, Section 7.5]).

Observe that the result does not hold forr = 1: every infinite index nor- mal subgroup of a hyperbolic surface group is a non-finitely generated free group; thus we get isomorphisms between the kernels of any two epimor- phismsφg₁ : Λg₁ →Z^k andψh₁ : Λh₁ →Z^k, withg1, h1 >2, while clearly the corresponding short exact sequences are not isomorphic forg₁6=h₁.

It follows that in order to understand abstract isomorphisms of the ker- nels of maps of the form φ_g1,...,gr and ψ_h1,...,hr with r > 2, it suffices to understand isomorphisms of short exact sequences of the form

1−→N −→Λg νg

−→Z^k−→1.

The latter reduces to Linear Algebra. We will explain this fact briefly.

A detailed exposition of the subject can be found in [9] (see in particular Chapter 6). In the following letS_g be a closed surface of genusg and let Λg=π1(Sg) be its fundamental group.

Let MCG^±(S_g) be the(extended) mapping class groupofS_g, that is, the group of homeomorphisms ofSgup to homotopy equivalences, where by the extended mapping class group we mean that we allow orientation reversing homeomorphisms. Let Inn(Λg) be the group ofinner automorphismsof Λg,