arXiv:1704.07041v1 [math.GT] 24 Apr 2017

ON VIRTUAL PROPERTIES OF KÄHLER GROUPS
STEFAN FRIEDL AND STEFANO VIDUSSI
Abstract. This paper stems from the observation (arising from work of T. Delzant)
that “most” Kähler groups virtually algebraically fiber, i.e. admit a finite index subgroup that maps onto Z with finitely generated kernel. For the remaining ones, the
Albanese dimension of all finite index subgroups is at most one, i.e. they have virtual
Albanese dimension one. We show that the existence of (virtual) algebraic fibrations has implications in the study of coherence and of higher BNSR invariants of
the fundamental group of aspherical Kähler surfaces. The class of Kähler groups of
virtual Albanese dimension one contains groups commensurable to surface groups.
It is not hard to give further (albeit unsophisticated) examples; however, groups of
this class exhibit strong similarities with surface groups. In fact, we show that its
only virtually residually finite Q–solvable (vRFRS) elements are commensurable to
surface groups, and we show that their Green–Lazarsfeld sets (virtually) coincide
with those of surface groups.

1. Introduction
This paper is devoted to the study of some virtual properties of Kähler groups, i.e.
fundamental groups of compact Kähler manifolds. Recall that if P is a property of
groups, we say that a group G is virtually P if a finite index subgroup H ≤f G is P.
This paper stems from the desire to understand if some of the virtual properties
of fundamental groups of irreducible 3–manifolds with empty or toroidal boundary,
that have recently emerged from the work of Agol, Wise [Ag13, Wi09, Wi12] and
their collaborators, have a counterpart for Kähler groups. Admittedly, there is no
a priori geometric reason to expect any analogy. However this viewpoint seems to
be not completely fruitless: for example in [FV16] we investigated consequences of
the fact that both classes of groups satisfy a sort of “relative largeness” property,
namely that any epimomorphism φ : G → Z with infinitely generated kernel virtually
factorizes through an epimorphism to a free nonabelian group. (This is a property
that is nontrivial to prove for both classes.)
In this paper we study, in a sense, the opposite phenomenon, namely the existence
of epimorphisms φ : G → Z with finitely generated kernel. Or, stated otherwise,
whether G is an extension of Z by a finitely generated subgroup. This condition
has been conveniently referred to in [JNW17] by saying that G algebraically fibers
and here we will adhere to that terminology. Again, thanks to the recent results of
Date: April 25, 2017.
1

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STEFAN FRIEDL AND STEFANO VIDUSSI

Agol, Wise and collaborators (see [AFW15] for accurate statements and references)
the emerging picture is that “most” freely indecomposable 3–manifold groups (e.g.
hyperbolic groups) virtually algebraically fiber. (By Stallings Theorem [Sta62], this
is actually equivalent to the fact that the underlying 3–manifold is virtually a surface
bundle over S 1 .) This result has triggered recent interest in the study of (virtual)
algebraic fibration for various classes of groups, and relevant results have appeared,
including during the preparations of this manuscript, see [FGK17, JNW17].
We have tasked ourselves with the purpose of understanding virtual algebraic fibrations in the realm of Kähler groups.
The first result is little more than a rephrasing of Delzant’s results on the Bieri–
Neumann–Strebel invariants of Kähler groups ([De10]), and is possibly known at
least implicitly to those familiar with that result. It asserts that the “generic” Kähler
group virtually algebraically fibers, and more importantly gives a geometric meaning
to latter notion. To state this result, recall that the Albanese dimension a(X) ≥ 0 of
a Kähler manifold is defined as the complex dimension of the image of X under the
Albanese map Alb. We define the virtual Albanese dimension va(X) to be the supremum of the Albanese dimension of all finite covers of X. (This definition replicates
that of virtual first Betti number vb1 , that will be as well of use in what follows.) The
property of having Albanese dimension equal to zero or equal to one is determined by
the fundamental group G = π1 (X) alone (see e.g. Proposition 2.1); because of that, it
makes sense to talk of (virtual) Albanese dimension of a Kähler group as an element
of {0, 1, > 1}. With this in mind we have the following:
Theorem A. Let G be a Kähler group. Then either G virtually algebraically fibers,
or va(G) ≤ 1.
This statement is, in essence, an alternative: the only intersection are groups G
that have a finite index subgroup H ≤f G with b1 (H) = vb1 (G) = 2 such that the
commutator subgroup [H, H] is finitely generated. (Such a H appears as fundamental
group of a genus 1 Albanese pencil without multiple fibers.) We could phrase this
theorem as an alternative, but the form above comes naturally from its proof, and
fits well with what follows.
Theorem A kindles some interest in identifying the class of Kähler groups of virtual
Albanese dimension at most one, and in what follow we summarize what we know
about this class.
To start, a Kähler group has va(G) = 0 if and only if all its finite index subgroup
have finite abelianization, or equivalently vb1 (G) = 0. All finite groups fall in this
class, but importantly for us, there exist infinite examples as well: a noteworthy one
is Sp(2n, Z) for n ≥ 2 (see [To90]).
The class of Kähler groups with va(G) = 1 contains some obvious examples, namely
all surface groups (i.e. fundamental groups of compact Riemann surfaces of positive
genus); a moment’s thought shows that the same is true for all groups that are commensurable (in the sense of Gromov, see [Gr89] or Section 3) to surface groups. These

ON VIRTUAL PROPERTIES OF KÄHLER GROUPS

3

groups do not exhaust the class of groups with va(G) = 1: an easy method to build
further examples comes by taking the product of a surface group with Sp(2n, Z),
n ≥ 2, by which we obtain Kähler groups with va(G) = 1 but not commensurable
with surface groups. That said, we are not aware of any subtler construction, that
does not hinge on the existence of Kähler groups with vb1 = 0, and it would interesting to decide if such constructions exist. In particular we analyze in Section 3 the
implications of the existence of (virtual) algebraic fibrations in the context of aspherical Kähler surfaces. This allows us to partly recast and refine some results about
coherence of their fundamental group, which appear in [Ka98, Ka13, Py16]. (Recall
that a group is coherent if all its finitely generated subgroups are finitely presented.)
Combining these results with ours yields the following:
Theorem B. Let G be a group with b1 (G) > 0 which is the fundamental group of an
aspherical Kähler surface X; then G is not coherent, except for the case where it is
virtually the product of Z2 by a surface group, and perhaps for the case where X is
finitely covered by a Kodaira fibration of virtual Albanese dimension one.
(A Kodaira fibration is a smooth non-isotrivial pencil of curves.) We are not aware
of the existence of Kodaira fibrations of virtual Albanese dimension one (Question
3.2).
The proof of the theorem above actually entails the existence of Kähler groups
whose second Bieri-Neumann-Strebel-Renz invariant is strictly contained in the first
(see Lemma 3.4) and are not a direct product.
In fact, the properties of the Albanese map give a tight relation between Kähler
groups with va(G) = 1 and surface groups. In fact, the underlying Kähler manifold
(virtually) admits the following form: if a Kähler manifold X has Albanese dimension
one, the Albanese map f : X → Σ (after restricting its codomain to the image) is a
genus g = q(X) pencil. (Here q(X) := 12 b1 (X) denotes the irregularity of a Kähler
manifold X.) When va(G) = 1, the Albanese pencil lifts to an Albanese pencil
e →Σ
e for all finite covers X
e → X.
f˜: X
If we impose to G = π1 (X) residual properties that mirror those of 3–manifold
groups, we obtain a refinement to Theorem A that can be thought of as an analogue
(with much less work on our side) to Agol’s virtual fiberability result ([Ag08]) for
3–manifold groups that are virtually RFRS (see Section 3 for the definition).
Theorem C. Let G be a Kähler group that is virtually RFRS. Then either G virtually
algebraically fibers, or is commensurable to a surface group.
Turning this theorem on its head, examples of Kähler groups with va(G) ≤ 1 that
are not commensurable to surface groups are certified not to be virtually RFRS. To
date, the infinite Kähler groups known to be RFRS are subgroups of the direct product
of surface groups and abelian groups. This is a remarkable but not yet completely
understood class of Kähler groups: in [DG05, 7.5 sbc–Corollary] some conditions for
a Kähler group to be virtually of this type are presented.

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STEFAN FRIEDL AND STEFANO VIDUSSI

In general, the relation between X and Σ induced by the Albanese pencil f : X → Σ
turns out to be much stronger than the isomorphism of the first cohomology groups.
In fact, up to going to a finite index subgroup if necessary, the Green–Lazarsfeld
sets of their fundamental groups coincide. Given an Albanese pencil, we refer to the
induced map in homotopy f : G → Γ (where G := π1 (X) and Γ := π1 (Σ)) as Albanese
map as well. We have the following:
Theorem D. Let G be a group with va(G) = 1. After going to a finite index normal
subgroup if necessary, the Albanese map f : G → Γ induces an isomorphism of the
Green–Lazarsfeld sets
∼
=
fb: Wi (Γ) −→ Wi (G).

Structure of the paper. Section 2 discusses some preliminary results on the Albanese dimension of Kähler manifolds and groups, as well as the proof of Theorem
A. Section 3 is devoted to the study of groups of virtual Albanese dimension one, and
contains the proofs of Theorems B, C and D.
In order to keep the presentation reasonably self–contained, we included some fairly
classical results, for which we could not find a formulation in the literature suitable
for our purposes. For the same reason, we describe – hopefully with accurate and
appropriate attribution – more recent work that is germane to the purposes of this
paper.
Conventions. All manifolds are assumed to be compact, connected and orientable,
unless we say otherwise.

Acknowledgment. The authors want to thank Thomas Delzant, Dieter Kotschick,
and Mahan Mj for their comments and clarifications to previous versions of this paper. Also, they gratefully acknowledge the support provided by the SFB 1085 ‘Higher
Invariants’ at the University of Regensburg, funded by the Deutsche Forschungsgemeinschaft (DFG).
2. Albanese dimension and algebraic fibrations
We start with some generalities on pencils on Kähler manifolds and the properties
of their fundamental group. The reader can find in the monograph [ABCKT96] a
detailed discussion of Kähler groups, and the rôle they play in determining pencils
on the underlying Kähler manifolds. Given a genus g pencil on X (i.e. a surjective
holomorphic map with connected fibers f : X → Σ to a surface with g = g(Σ)) we
can consider the homotopy–induced epimorphism f : π1 (X) → π1 (Σ). In presence of
multiple fibers we have a factorization f : π1 (X) → π1orb (Σ) → π1 (Σ) through a further
epimorphism onto π1orb (Σ), the orbifold fundamental group of Σ associated to the
pencil f , with orbifold points and multiplicities corresponding to the multiple fibers
of the pencil. Throughout the paper we write G := π1 (X) and Γ := π1orb (Σ). The
factor epimorphism, that by slight abuse of notation we denote as well as f : G → Γ,

ON VIRTUAL PROPERTIES OF KÄHLER GROUPS

5

has finitely generated kernel so we have the short exact sequence of finitely generated
groups
f

1→K→G→Γ→1
(see e.g. [Cat03] for details of the above).
We have the following two results about Kähler manifolds of (virtual) Albanese
dimension one. These are certainly well–known to the experts (at least implicitly),
and we provide proofs for completeness.
Proposition 2.1. Let X be a Kähler manifold and let G = π1 (X) be its fundamental
group. If X has Albanese dimension a(X) ≤ 1, any Kähler manifold with isomorphic
fundamental group has the same Albanese dimension as X.
Proof. The case where a(X) = 0 corresponds to manifolds with vanishing irregularity
q(X) = 21 b1 (X) so it is determined by the fundamental group alone. Next, consider
a Kähler manifold X with positive irregularity. For any such X, the genus g(X) is
defined as the maximal rank of submodules of H 1 (X) isotropic with respect to the
cup product. In [ABCKT96, Chapter 2] it is shown that g(X) is in fact an invariant
of the fundamental group alone. As the cup product is nondegenerate, there is a
bound
1
1
g(X) ≤ q(X) = b1 (X) = b1 (G).
2
2
By Catanese’s version of the Castelnuovo–de Franchis theorem ([Cat91]), the case
where X has Albanese dimension one occurs exactly for g(X) = q(X). By the above,
this equality is determined by the fundamental group alone.

Based on the observations above, we will refer to the Albanese dimension of a
Kähler group as an element of the set {0, 1, > 1}.
Whenever G has Albanese dimension one, the kernel of the map f∗ : H1 (G) →
H1 (Γ) induced by the (homotopy) Albanese map, identified by the Hochschild–Serre
spectral sequence with a quotient of the coinvariant homology H1 (K)Γ by a torsion
group, is torsion (or equivalently f ∗ : H 1 (Γ) → H 1 (G) is an isomorphism). Note
that, by universality of the Albanese map, whenever a Kähler manifold X has a
pencil f : X → Σ such that the kernel of f∗ : H1 (G) → H1 (Γ) is torsion, the pencil is
Albanese.
e → X be a finite cover of X. Denote by H ≤f G the subgroup associLet π : X
ated to this cover; the regular cover of X determined by the normal core NG (H) =
T
−1
e
g∈G g Hg ≤f H is a finite cover of X as well. By universality of the Albanese map,
the Albanese dimension is nondecreasing when we pass to finite covers. Therefore (as
happens with virtual Betti numbers) we can define the virtual Albanese dimension
of X in terms of finite regular covers. Given an epimorphism onto a finite group

6

STEFAN FRIEDL AND STEFANO VIDUSSI

α : π1 (X) → S we have the commutative diagram (with self–defining notation)
(1)

1

1
//

∆

//

Λ
//



K



α(K)

//



1
//

1



//

S/α(K)



1

Γ

α

S
//

//



G
//

α



//



H
//



1
1





1

1

//

1



1

1

e and Σ
e the induced covers of X and Σ respectively (so that π1 (X)
e =H
Denote by X
e = Λ). There exists a pencil f˜: X
e → Σ,
e which is a lift of f : X → Σ; in
and π1 (Σ)
e appearing in
homotopy, this corresponds to the epimorphism fe: H → Λ := π1orb (Σ)
(1) above.
In the next proposition, we illustrate the fact that when X has Albanese dimension one, its Albanese pencil f : X → Σ is the only irrational pencil of X, up to
holomorphic automorphisms of Σ.
Proposition 2.2. Let X be a Kähler manifold with a(X) = 1. Then the Albanese
pencil f : X → Σ is the unique irrational pencil on X up to holomorphic automorphism of the base. Moreover, if X satisfies va(X) = 1, any rational pencil has orbifold
base with finite orbifold fundamental group.
Proof. By assumption the Albanese pencil f : X → Σ factorizes the Albanese map
Alb. Let g : X → Σ′ be an irrational pencil, and compose it with the Jacobian map
j : Σ′ → Jac(Σ′ ). By universality of the Albanese map we have the commutative
diagram
X❃

f

//
❃❃
❃❃
❃
g ❃❃


Σ✤
✤

//

Alb(X)

h

✤

Σ′

j



//

Jac(Σ′ ).

The map h : Σ → Σ′ is well defined by injectivity of the Jacobian map, and is a
holomorphic surjection by universality of the Albanese map. Holomorphic surjections
of Riemann surfaces are ramified covers; however, unless the cover is one–sheeted, i.e.
h is a holomorphic isomorphism, the fibers of g : X → Σ′ will fail to be connected.
This argument above does not prevent X from having rational pencils. However, if
X has also virtual Albanese dimension one, this imposes constraints on the multiple
fibers of those pencils. Recall that orbifolds with infinite π1orb (Σ) are those that are flat
or hyperbolic, hence admit a finite index normal subgroup that is a surface group with
positive b1 (see [Sc83] for this result and a characterization of these orbifolds in terms

ON VIRTUAL PROPERTIES OF KÄHLER GROUPS

7

of the singular points). Therefore, if X were to admit a rational pencil g : X → Σ′
with infinite π1orb (Σ′ ), there would exist an irrational pencil without multiple fibers
e → Σ
e ′ covering g : X → Σ′ . We claim that the pencil e
e → Σ
e ′ cannot be
g: X
e
g: X
Albanese: building from the commutative diagram in (1) we have the commutative
diagram
1

1

K❋
❋❋
❋❋
""
∼
=
H1 (K)
//

//




//

K ●●
●●●
## 
H1 (K)

e
π1 (X)

//

ge

//

❑❑❑
❑%%
// H1 (X)
e

π1 (X)

g
e



//
▲▲▲
▲&&

// H1 (X)

e ′)
π1 (Σ

//

1

▼▼▼
▼&&
// H1 (Σ
e ′)

π1orb (Σ′ )
//

1

The subgroup im(H1 (K) → H1 (X)) ≤ H1 (X) has positive rank (it is a finite index
e ≤ H1 (X)
e has
subgroup), hence by commutativity the image im(H1 (K) → H1 (X))
e → H 1 (Σ
e ′ ) is
positive rank. It follows that the kernel of the epimorphism e
g : H1 (X)
e →Σ
e ′ is not Albanese. As X
e has Albanese dimension one, this
not torsion, so e
g: X
is inconsistent with the first part of the statement.

We are now in a position to prove Theorem A. In order to do so, it is both practical
and insightful to use the Bieri–Neumann–Strebel invariant of a finitely presented
group G (henceforth BNS), for which we refer to [BNS87] for definitions and properties
used here. This invariant is an open subset Σ1 (G) of the character sphere S(G) :=
(H 1 (G; R) \ {0})/R>0 of H 1 (G; R). For rational rays, the invariant can be described
as follows: a rational ray in S(G) is determined by a primitive class φ ∈ H 1 (G). Given
such φ, we can write G as HNN extension G = hA, t|t−1 Bt = Ci for some finitely
generated subgroups B, C ≤ A ≤ Ker φ with φ(t) = 1. The extension is called
ascending (descending) if A = B (resp. A = C). By [BNS87, Proposition 4.4] the
extension is ascending (resp. descending) if and only if the rational ray determined
by φ (resp. −φ) is contained in Σ1 (G).
We have the following theorem, which applied to the collection of finite index
subgroups of G, implies Theorem A:
Theorem 2.3. Let G be a Kähler group. The following are equivalent:
(1) G algebraically fibers;
(2) The BNS invariant Σ1 (G) ⊆ S(G) is nonempty;
(3) For any compact Kähler manifold X such that π1 (X) = G either the Albanese
map is a genus 1 pencil without multiple fibers, or X has Albanese dimension
greater than one.

8

STEFAN FRIEDL AND STEFANO VIDUSSI

Before proving this proposition, let us mention that (with varying degree of complexity) it is possible to verify directly that groups commensurable to nonabelian
surface groups cannot satisfy any of the three cases above.
Proof. We will first show (1) ⇔ (2), and then ¬(2) ⇔ ¬(3).
(1) ⇒ (2): If (1) holds we can write the short exact sequence
(2)

φ

1 → Ker φ → G → Z → 1

with Ker φ finitely generated. Hence G is both an ascending and descending HNN
extension. It follows that the rational rays determined by both ±φ ∈ H 1 (G) are
contained in Σ1 (G).
(2) ⇒ (1): Let us assume that Σ1 (G) is nonempty. As Σ1 (G) is open, we can assume
that there exists a primitive class φ ∈ H 1 (G) whose projective class is determined by
a rational ray in Σ1 (G). A remarkable fact at this point is that Kähler groups cannot
be written as properly ascending or descending extensions, i.e. ascending extensions
are also descending and viceversa. (This was first proven in [NR08]; see also [FV16]
for a proof much in the spirit of the present paper.) But this is to say that G has the
form of Equation (2) with Ker φ finitely generated.
To show the equivalence of ¬(2) and ¬(3), we start by recalling Delzant’s description of the BNS invariant of a Kähler group G. Let X be a Kähler manifold with
G = π1 (X). The collection of irrational pencils fα : X → Σα such that the orbifold
fundamental group Γα := π1orb (Σ) is a cocompact Fuchsian group, is finite up to
holomorphic automorphisms of the base (see [De08, Theorem 2]). (In the language
of orbifolds, these are the holomorphic orbifold maps with connected fibers from X
to hyperbolic Riemann orbisurfaces.) The pencil maps give, in homotopy, a finite
collection of epimorphisms with finitely generated kernel fα : G → Γa . Then [De10,
Théorème 1.1] asserts that the complement of Σ1 (G) in S(G) (i.e. the set of so–called
exceptional characters) is given by
S
(3)
S(G) \ Σ1 (G) = α [fα∗ H 1 (Γα ; R) − {0}],

where we use the brackets [·] to denote the image of a subset of (H 1(G; R) \ {0}) in
(H 1 (G; R) \ {0})/R>0 . (Note, instead, that genus 1 pencils without multiple fibers
do not induce exceptional characters.)
¬(3) ⇒ ¬(2): The negation of (3) asserts that X has either Albanese dimension
zero, in which case S(G) is empty, or it has an Albanese pencil f : X → Σ with
Γ := π1orb (Σ) cocompact Fuchsian, in which case f ∗ H 1 (Γ; R) = H 1 (G; R). In either
case Σ1 (G) is empty, i.e. ¬(2) holds.
¬(2) ⇒ ¬(3): If the set Σ1 (G) ⊆ S(G) is empty, either S(G) is empty (i.e. b1 (G) =
0) whence G has Albanese dimension zero, or by Equation (3) there exists an irrational
pencil f : X → Σ, with Γ := π1orb (Σ) cocompact Fuchsian, inducing an isomorphism
f ∗ : H 1 (Γ; R) → H 1 (G; R). (The union of finitely many proper vector subspaces of
H 1 (G; R) cannot equal H 1 (G; R).) Such a pencil is then the Albanese pencil of X. 

ON VIRTUAL PROPERTIES OF KÄHLER GROUPS

9

In our understanding, Kähler manifolds whose Albanese dimension is smaller than
their dimension are “nongeneric”, and their study should reduce, through a sort of
dimensional reduction induced by the Albanese map, to the study of lower dimensional
spaces (see e.g. [Cat91]). In that sense, we think of groups that, together with their
finite index subgroups, fail to satisfy the equivalent conditions (1) to (3) of Theorem
2.3 as nongeneric.
Examples of Kähler groups with a(G) > 1 abound. It is less obvious to provide
examples of Kähler groups that have a jump in Albanese dimension, i.e. a(G) = 1 but
va(G) > 1. Before doing so, we can make an observation about the geometric meaning
of such occurrence. Given an irrational pencil f : X → Σ, its relative irregularity is
defined as
qf = q(X) − q(Σ)
(where the irregularity of Σ equals its genus). The Albanese pencil occurs exactly
when qf = 0. The notion of relative irregularity allows us to tie the notion of virtual
Albanese dimension larger than one with the more familiar notion of virtual positive
Betti number (or more properly, in Kähler context, virtual irregularity): a Kähler
e →Σ
e of the
group with a(G) = 1 has va(G) > 1 if and only if there is a lift f˜: X
Albanese pencil which is irregularly fibered, i.e. qf˜ > 0.
A fairly simple class of examples comes from groups of type G = πg × K where
πg is the fundamental group of a genus g > 1 surface and K the fundamental group
of a hyperbolic orbisurface of genus 0, so that b1 (K) = 0. The group K is Kähler
(e.g. it is the fundamental group of an elliptic surface with enough multiple fibers
and multiplicity), hence so is G. As H 1 (G; Z) = H 1 (πg ; Z), the Albanese dimension
of G must be one. On the other hand, K has a finite index subgroup that is the
fundamental group of a genus h > 1 surface, hence G is virtually πg × πh . The
algebraic surface Σg × Σg has Albanese dimension 2, so by Proposition 2.1 the virtual
Albanese dimension of G is greater than one. (Group theoretically, this can be seen
as consequence of [BNS87, Theorem 7.4], which asserts that for cartesian products of
groups with positive b1 the BNS invariant is nonempty.)
Less trivial examples with a(G) = 1 but va(G) > 1 come from bielliptic surfaces. These possess an Albanese pencil of genus 1 without multiple fibers, but their
fundamental groups are virtually Z4 , hence have virtual Albanese dimension 2 (see
[BHPV04, Section V.5]). More sophisticated examples of Kähler surfaces (hence
groups) with a(G) = 1 that are finitely covered by the product of curves of genera
bigger than one are discussed in [Cat00, Theorem F].
The most interesting class of examples we are aware of, however, comes from the
recent paper of Stover ([Sto15, Theorem 3]) that gives examples of Kähler groups
that arise as cocompact arithmetic lattices in P U(n, 1) for which a(G) = 1; Stover
proves that those groups are virtually extensions of Z by a finitely generated group,
and application of Theorem 2.3 yields to the conclusion that va(G) > 1.

10

STEFAN FRIEDL AND STEFANO VIDUSSI

3. Groups with Virtual Albanese Dimension One
In this section, we will discuss groups with va(G) = 1. Familiar examples of Kähler
groups with va(G) = 1 are given by surface groups, and other simple examples arise
as follows. Following Gromov ([Gr89]) we say that a group G is commensurable to
a surface group if it admits a (normal) finite index subgroup H ≤f G that admits
an epimorphism f : H → Γ to a surface group Γ with finite kernel, namely for which
there exists an exact sequence
1→F →H→Γ→1
with Γ a surface group and F finite. The map f : H1 (H) → H1 (Γ) is then an epimorphism with torsion kernel, i.e. f : H → Γ represents, in homotopy, an Albanese
map. Quite obviously, each finite index subgroup of G will also be commensurable to
a surface group, hence the virtual Albanese dimension of G equals one.
We want to analyze the picture so far in comparison with the situation for 3–
manifold groups. The class of irreducible 3–manifolds that are not virtually fibered
is limited (it is composed entirely by graph manifolds). One may contemplate that,
similarly, the Kähler counterpart to that class contains only the obvious candidates,
namely manifolds whose fundamental group is commensurable to a surface group.
Proposition 3.1 below guarantees that this is not quite the case. The starting point is
the existence of infinite Kähler groups (such as Sp(2n, Z), n > 1) with vb1 (G), hence
va(G), equal to zero. These do not have a counterpart in dimension 3.
Proposition 3.1. Let Γ be the fundamental group of a genus g > 0 surface and let
K be an infinite Kähler groups with va(K) = 0; then va(K × Γ) = a(K × Γ) = 1 and
K × Γ is not commensurable to a surface group.
Proof. The projection map f : K × Γ → Γ is the Albanese map, hence a(K × Γ) = 1.
We claim that as vb1 (K) = 0, the virtual Albanese dimension of K × Γ is one. In
fact, for any normal subgroup H ≤f K × Γ we have from (1)
1
//

∆ ■■

//

■■
$$

H ◆◆◆
◆◆◆
''

H1 (∆)
1
//



//

K



//

K×Γ

fe

Λ ❍❍
//

H1 (H)
f

//

1

❍❍
##
// H1 (Λ)



//

Γ
//

1.

As vb1 (K) = 0, H1 (∆) is torsion, hence Ker(H1 (H) → H1 (Λ)) = Im(H1 (∆) →
H1 (H)) is torsion as well. It follows that the map fe: H → Λ is, in homotopy, the
Albanese map, hence a(H) = 1.
Assume by contradiction that K × Γ is commensurable to a surface group, and
denote by H ≤f K × Γ a normal subgroup such that
1→F →H→π→1

ON VIRTUAL PROPERTIES OF KÄHLER GROUPS

11

with F finite and π a surface group. As F is finite, Ker(H1 (H) → H1 (π)) is torsion,
hence the epimorphism H → π is, in homotopy, the Albanese map. But then it
must coincide with the epimorphism fe: H → Λ as described above (perhaps up to an
automorphism of Λ). This is not possible, as the former has a finite kernel F , while the
latter has kernel ∆, which is infinite as ∆ ≤f K and K is infinite by assumption. 
Proposition 3.1 guarantees that the class of Kähler groups that do not virtually
admit an epimorphism to Z with finitely generated kernel is more variegated than its
counterpart in the 3–manifold world.
We should, however, qualify this result. The examples of Proposition 3.1 build on
the existence of infinite Kähler groups with vb1 = 0, and leverage on the fact that
we can take products of finitely presented groups, as the class of Kähler manifolds
is closed under cartesian product (and, more generally, holomorphic fiber bundles).
Neither of these phenomena has a counterpart in the realm of 3–manifolds. It is
perhaps not too greedy to ask for examples of Kähler groups with va(G) = 1 in a
realm where simple constructions as the one of Proposition 3.1 are tuned out.
As we are about to see, an instance of this occurs in the case of aspherical surfaces:
Question 3.2. Does there exist a group G with b1 (G) > 0 that is the fundamental
group of an aspherical Kähler surface and does not virtually algebraically fiber?
The reason why an example like the one we are after in Question 3.2 would be
appealing comes from the fact that, perhaps going to a finite index subgroup, the fundamental group G of an aspherical Kähler surface with va(G) = 1 is a 4–dimensional
Poincaré duality group whose Albanese pencil f : X → Σ determines a short exact
sequence of finitely generated groups
(4)

1 → K → G → Γ → 1,

with Γ a surface group and K finitely generated. As first remarked by Kapovich
in [Ka98], it is a theorem of Hillman that either K is itself a (nontrivial) surface
group, or it is not finitely presented (see e.g. [Hil02, Theorem 1.19]). In either case,
a construction like the one in Proposition 3.1 (or even a twist thereof) is excluded.
Constructions of this type may not be easy to find. As we mentioned before, the
examples of Stover virtually algebraically fiber. The same is true for the Cartwright–
Steger surface ([CS10]), another ball quotient with b1 (G) = 2 and whose Albanese
map (as shown in [CKY15, Corollary 5.3]) has no multiple fiber: by the discussion
in the proof of Theorem 2.3 G itself has BNS invariant Σ1 (G) equal to the entire
S(G) and all epimorphisms to Z have finitely generated kernel. (According to the
introduction to [Sto15], this fact was known also to Stover and collaborators.)
We want now to show how the question above ties with the study of coherence
of fundamental groups of aspherical Kähler surfaces, which was initiated in [Ka98,
Ka13] using the aforementioned result of Hillman, and further pursued in [Py16].
The outcome of these articles is that, with the obvious exceptions, most aspherical

12

STEFAN FRIEDL AND STEFANO VIDUSSI

Kähler surfaces can be shown to have non–coherent fundamental group. (See [Py16,
Theorem 4] for a detailed statement, which uses a notation slightly different from
ours.) Drawing from the same circle of ideas of these references (as well as a minor
extension of the work in [Ko99]) we can prove the following result, which improves
on the existing results insofar as it further narrows possible coherent fundamental
groups to finite index subgroups of the fundamental group of Kodaira fibrations with
virtual Albanese dimension one. (A pencil on a Kähler surface is called a Kodaira
fibration if it is smooth and not isotrivial.)
Theorem 3.3. Let G be a group with b1 (G) > 0 which is the fundamental group of
an aspherical Kähler surface X; then G is not coherent, except for the case where it
is virtually the product of Z2 by a surface group, or perhaps for the case where X is
finitely covered by a Kodaira fibration of virtual Albanese dimension one.
Proof. If G has va(G) > 1, let H ≤f G be a subgroup, corresponding to a finite
e of X, which algebraically fibers. Let
n–cover X

(5)

φ

1 → Ker φ → H → Z → 1

with Ker φ finitely generated represent an algebraic fibration. By [Hil02, Theorem
4.5(4)] the finitely generated group Ker φ has type F P2 if and only if the Euler
e = ne(X) = 0. A finitely presented group has type F P2 . It
characteristic e(X)
follows that Ker φ ≤ G is finitely generated but not finitely presented, hence G is
not coherent, unless e(X) = 0. If e(X) = 0 the classification of compact complex
surfaces entails that X admits an irrational pencil with elliptic fibers. As e(X) = 0
the Zeuthen–Segre formula (see e.g. [BHPV04, Proposition III.11.4]) implies that the
only singular fibers can be multiple covers of an elliptic fiber, hence X is finitely
covered by a torus bundle. An holomorphic fibration with smooth fibers of genus 1
is also isotrivial (i.e. all fibers are isomorphic), namely a holomorphic fiber bundle,
see [BHPV04, Section V.14]. We can invoke then [BHPV04, Sections V.5 and V.6] to
deduce that some finite cover of X is a product T 2 × Σg . In this case, the fundamental
group is virtually Z2 ×Γ, with Γ a surface group. By Theorem B of [BHMS02] a finitely
generated subgroup L ≤ Z2 × Γ has a finite index subgroup that is the product of
finitely generated subgroups of each factor. As surface groups are coherent, L must
be finitely presented, hence in this case the fundamental group of X is coherent.
If G has va(G) = 1, perhaps going to a finite index subgroup, G is a 4–dimensional
Poincaré duality group whose Albanese pencil f : X → Σ determines as usual the
short exact sequence of finitely generated groups 1 → K → G → Γ → 1, with
Γ a surface group and K finitely generated. To deal with this case, we relay on
the strategy of [Ka98, Ka13]: by the aforementioned theorem of Hillman [Hil02,
Theorem 1.19] either K is not finitely presented (and G is not coherent) or it is itself
a nontrivial surface group. In the latter case, we can follow the path of [Ko99] (see
also [Hil00]) to complete the proof of the statement. We first observe that the surface
X, being aspherical, is homotopy equivalent to a smooth 4–manifold M 4 , a surface

ON VIRTUAL PROPERTIES OF KÄHLER GROUPS

13

bundle over a surface F ֒→ M → Σ, where π1 (F ) = K, π1 (Σ) = Γ. Note that M
is uniquely determined by the short exact sequence of its fundamental group, see
[Hil02, Theorem 5.2]. Next, as X and M are homotopy equivalent, they have the
same Euler characteristic e(X) = e(M) = e(F ) · e(Σ). At this point, the Zeuthen–
Segre formula entails that the only nonsmooth fibers of the Albanese pencil could be
multiple covers of an elliptic fiber (in particular, g(F ) = 1). The assumption that K
is finitely generated excludes the presence of multiple fibers ([Cat03, Lemma 4.2]).
This means that Albanese pencil f : X → Σ is smooth (i.e. a holomorphic fibration of
maximal rank), namely X is actually a surface bundle over a surface, in particular it
is diffeomorphic to M. If the pencil was isotrivial (i.e. all fibers isomorphic), it would
be a holomorphic fiber bundle, and we would conclude as above that some finite cover
of X is a product, whence va(X) = 2 and va(G) > 1. The statement follows.

In summary, the existence of non-obvious examples of coherent fundamental groups
of aspherical Kähler surfaces hinges on an affirmative answer to Question 3.2.
Remarks.
(1) Note that the proof of the first case of Theorem 3.3 applies verbatim also in the case of aspherical surfaces with a(G) = 1 that algebraically
fiber; in particular, this entails the Cartwright–Steger surface and the surfaces
described in [Sto15, Theorem 2], [DiCS17, Theorem 1.2] have non–coherent
fundamental groups. Those groups are torsion–free lattices G ≤ P U(2, 1),
with the surfaces appearing as ball quotient BC2 /G, i.e. complex hyperbolic
surfaces. This was implicitly known (for slightly different reasons) also from
[Ka98]; we point out that for the argument above we don’t need to invoke
[Liu96].
(2) The first examples of Kodaira fibrations, due to Kodaira and Atiyah, actually
carry two inequivalent structures of Kodaira fibrations, hence are guaranteed
to have Albanese dimension two. By the above, their fundamental group is not
coherent. The same result applies for the doubly fibered Kodaira fibrations
constructed in [CR09]. It is not difficult to prove the existence of Kodaira
fibrations of Albanese dimension one (which, by the Hochschild–Serre spectral sequence, are surface bundles whose coinvariant homology of the fiber
H1 (K; Z)Γ has rank zero): in fact, the “generic” Kodaira fibration arising
from a holomorphic curve in a moduli space of curves has Albanese dimension
one. However, this seems to have no obvious consequences for the discussion
above: we ignore if there exist Kodaira fibration of virtual Albanese dimension
one. We can add that, even if such surfaces did exist, we cannot decide if their
fundamental groups are coherent. For instance, it is not obvious whether they
may contain F2 × F2 as subgroup.
It is perhaps interesting to flesh out one consequence of the proof of Theorem 3.2
that gives some information on higher BNS–type invariants of some Kähler groups.
Precisely, we will consider the homotopical BNSR invariant Σ2 (G) ⊆ Σ1 (G) ⊆ S(G)

14

STEFAN FRIEDL AND STEFANO VIDUSSI

introduced in [BR88]. We will not need the definition of these invariant and we will
limit ourselves to mention the well–known facts (see e.g. [BGK10, Section 1.3]) that,
using the notation preceding Theorem 2.3, given a primitive class φ ∈ H 1 (G), the
kernel Ker φ ≤ G is of type F2 (namely, finitely presented) if and only if the rational
rays determined by both ±φ ∈ H 1 (G) are contained in Σ2 (G).
While we know no way to get complete information on the full invariant Σ2 (G),
the ingredients of the proof of Theorem 3.2 is sufficient to entail the following lemma,
that per se refines the previous result of non–coherence, and is possibly one of the
first results on higher invariants of Kähler groups, besides the case of direct products:
Lemma 3.4. The fundamental group G of an aspherical Kähler surface X of strictly
positive Euler characteristic with Albanese dimension two has BNSR invariants satisfying the inclusions
Σ2 (G) ( Σ1 (G) ⊆ S(G).
Proof. The point of this statement is that the first inclusion is strict. For sake of
clarity, we review the argument we used in the proof of Theorem 3.2: the condition
on the Albanese dimension implies that G algebraically fibers, for some primitive
class φ ∈ H 1 (G). As the Euler characteristic of X is strictly positive, Hillman’s
Theorem ([Hil02, Theorem 4.5(4)]) entails that Ker φ is not F P2 , nor a fortiori
finitely presented.

Note that the same conclusion of the lemma holds, even when a(G) = 1, as long as
the fundamental group group algebraically fibers, e.g. for the Cartwright–Steger surface. Perhaps more importantly, the corollary applies to the aforementioned Kodaira
fibrations defined by Kodaira and Atiyah. Topologically, these are surface bundles
over a surface, so that their fundamental groups are nontrivial extension of a surface
group by a surface group. Higher BNSR invariants of direct products of surfaces
groups are (to an extent) well understood by purely group theoretical reasons. Instead, Lemma 3.4 seems to be the first result of that type for nontrivial extensions,
and uses in crucial manner the fact that the group is Kähler: we are not aware of any
other means to show that Σ1 (G) is nonempty.
Remark. The reader familiar with BNSR invariants may notice that we are just shy
of being able to conclude that the fundamental group of aspherical Kähler surfaces
of positive Euler characteristic has empty Σ2 (G). We conjecture that this is true.
(The conjecture holds true whenever Σ2 (G) = −Σ2 (G).) We mention also that the
statement of Lemma 3.4 remains true if we consider the homological BNSR invariant
Σ2 (G; Z) of [BR88].
We want to discuss now a result that may prevent the existence of new simple
examples of Kähler groups with va(G) = 1, under the assumption that the group satisfies residual properties akin to those holding for most irreducible 3–manifold groups,
namely being virtually RFRS (in particular, that holds for the fundamental group of

ON VIRTUAL PROPERTIES OF KÄHLER GROUPS

15

irreducible manifolds that have hyperbolic pieces in their geometric decomposition).
This class of groups was first introduced by Agol in the study of virtual fibrations of
3–manifold groups: A group G is RFRS if there
T exists a filtration {Gi |i ≥ 0} of finite
index normal subgroups Gi Ef G0 = G with i Gi = {1} whose successive quotient
maps αi : Gi → Gi /Gi+1 factorize through the maximal free abelian quotient:
1
//

Gi+1

αi

// Gi /Gi+1
Gi ❖❖❖
❖❖''
❦
❦
❦ 55
H1 (Gi )/Tor
//

//

1

Subgroups of the direct product of surface groups and abelian groups are virtually
RFRS. The largest source of virtually RFRS group we are aware of is given by subgroups of right–angled Artin groups (RAAGs), that are virtually RFRS by [Ag08].
However, this class does not give us new examples, as Py proved in [Py13, Theorem
A] that all Kähler groups that are subgroups of RAAGs are in fact virtually subgroups
of the product of surface groups and abelian groups.
We have the following, that combined with Theorem 2.3 gives Theorem C:
Theorem 3.5. Let G be a virtually RFRS Kähler group. Then for any Kähler mane of X with Albanese
ifold X such that π1 (X) = G either there exists a finite cover X
dimension greater than one, or G is virtually a surface group.
Proof. After going to a suitable finite cover can assume that X has RFRS fundamental
group G, with associated sequence {Gi }. A nontrivial RFRS group has positive first
Betti number. In light of this, it is sufficient to show that if G is a RFRS group with
Albanese map f : G → Γ and virtual Albanese dimension one, then it is a surface
group. (This implies, because of the initial cover to get G RFRS, the theorem as
stated.) Recall that we have a short exact sequence
1
//

K
//

G

f

//

Γ
//

1

where Γ can be assumed to be a surface group and K is finitely generated. We
claim that if X has virtual Albanese dimension one, then K is actuallyTtrivial, i.e.
f is injective. Let γ ∈ G be a nontrivial element; the assumption that i Gi = {1}
implies that there exist an index j such that γ ∈ Gj \Gj+1. Consider now the diagram
(6)

1
//

K ∩ Gj

fj

//
Gj ❙❙❙❙
♦ Γj
❙❙))
ww♦ ♦
αj
H1 (Gj )/Tor
//



Gj /Gj+1


//

1

uu❦❦❦

1
where fj : Gj → Γj is the restriction epimorphism between finite index subgroups
of G and Γ respectively determined by Gj E G as in the commutative diagram of

16

STEFAN FRIEDL AND STEFANO VIDUSSI

(1). This map represents, in homotopy, the pencil fj : Xj → Σj of the cover of X
associated to Gj . By assumption, this pencil is Albanese. This entails that there is
an isomorphism
∼
=

(fj )∗ : H1 (Gj )/Tor −→ H1 (Γj ).
Composing the inverse of this isomorphism with the maximal free abelian quotient of
Γj gives the map denoted with a dashed arrow in the diagram of Equation (6). By the
commutativity of that diagram we deduce that the quotient map αj : Gj → Gj /Gj+1
factors through fj : Gj → Γj . As γ ∈ Gj \ Gj+1 , the image αj (γ) ∈ Gj /Gj+1 is
nontrivial, hence so is fj (γ). This implies that f (γ) = fj (γ) ∈ Γ is nontrivial, i.e.
f : G → Γ is injective.

The conclusion of this theorem asserts that virtually RFRS Kähler groups of virtual
Albanese dimension one are virtually (and not just commensurable to) surface groups.
In fact, it is a simple exercise to verify that a residually finite group commensurable
to a surface group is virtually a surface group, hence the result is exactly what we
should expect.
We will finish this section with a result that further ties groups with virtual Albanese dimension one and surface groups, asserting that the Green–Lazarsfeld sets
of such groups coincide (up to going to a finite index subgroup) with those of their
Albanese image.
The Green–Lazarsfeld sets of a Kähler manifold X (and, by extension, of its fundamental group G) are subsets of the character variety of G, the complex algebraic
b := H 1 (G; C∗ ) . The Green–Lazarsfeld sets Wi (G) are defined as
group defined as G
the collection of cohomology jumping loci of the character variety, namely
b | rk H 1 (G; Cξ ) ≥ i},
Wi (G) = {ξ ∈ G

b
nested by the depth i: Wi (G) ⊆ Wi−1 (G) ⊆ . . . ⊆ W0 (G) = G.
For Kähler groups the structure of W1 (G) is well–understood. The projective case
appeared in [Si93] (that refined previous results of [GL87, GL91]); this result was
then extended to the Kähler case in [Cam01] (see also [De08]). Briefly, W1 (G) is
the union of a finite set of isolated torsion characters and the inverse image of the
Green–Lazarsfeld set of hyperbolic orbisurfaces under the finite collection of pencils
of X with hyperbolic base.
If X has Albanese dimension one, the Albanese map f : G → Γ induces an epimorphism f∗ : H1 (G) → H1 (Γ). Therefore, we have an induced isomorphism of the
connected components of the character varieties containing the trivial character
(7)

∼
= b
b −→
G1̂
fb: Γ
1̂

where for a group G, we denote the connected component of the character variety
b .
containing the trivial character 1̂ : G → C∗ as G
1̂

ON VIRTUAL PROPERTIES OF KÄHLER GROUPS

17

The next theorem, a restatement of Theorem D, shows that if X has virtual Albanese dimension one, then after perhaps going to a cover, the map fb restricts to an
isomorphism of the Green–Lazarsfeld sets.
Theorem 3.6. Let X be a Kähler manifold with va(X) = 1. Up to going to a
finite index normal subgroup if necessary, the Albanese map f : G → Γ induces an
isomorphism
∼
=
fb : Wi (Γ) −→ Wi (G)

of the Green–Lazarsfeld sets.

Proof. Up to going to a finite cover, we can assume that X admits an Albanese pencil
f : X → Σ. Moreover, as every cocompact Fuchsian group of positive genus admits
a finite index normal subgroup which is a honest surface group, we can also assume
that, after going to a further finite cover if necessary, the Albanese pencil doesn’t
contain any multiple fibers. In particular, H1 (Σ) will be torsion–free. Without loss
of generality, by going to the normal core of the associated finite index subgroup, we
can always assume that the cover is regular. Summing up, after possibly going to
a finite cover the Albanese map, in homotopy, is an epimorphism f : G → Γ where
Γ = π1 (Σ) is a genus g(Γ) surface group.
The Green–Lazarsfeld sets Wi (Γ) for a surface group are determined in [Hir97], and
are given by

b if 1 ≤ i ≤ 2g(Γ) − 2,
 Γ
(8)
Wi (Γ) =
1̂ if 2g(Γ) − 1 ≤ i ≤ 2g(Γ),

∅ if i ≥ 2g(Γ) + 1.
b surjectivity of f : G → Γ implies by general arguments (see e.g. [Hir97,
Given ρ ∈ Γ,
Proposition 3.1.3]) that f ∗ : H 1 (Γ; Cρ ) → H 1(G; Cf ∗ (ρ) ) is a monomorphism. But we
will actually need more, namely that by [Br02, Theorem 1.1] or [Br03, Theorem 1.8]
b is a torsion character.
f ∗ is an isomorphism, except perhaps when ρ ∈ Γ
b
This implies that f : Wi (Γ) → Wi (G) is an injective map, and it will fail to preserve
the depth (i.e. dimension of the twisted homology) only for torsion characters.
Consider the short exact sequence of groups
t
b −→ G
b −→
1 −→ G
Hom(TorH1 (G); C∗ ) −→ 1,
1̂

b connected to the trivial character 1̂.
b refers as above to the component of G
where G
1̂
By [GL91, Theorem 0.1] all irreducible positive dimensional components of Wi (G)
are inverse images of the Green–Lazarsfeld set of the hyperbolic orbisurfaces. By
b ⊆ Wi (G) (for
Proposition 2.2, the Albanese pencil is unique, hence fb(Wi (Γ)) = G
1̂
1 ≤ i ≤ 2q(G) − 2) is the only positive dimensional component, that will occur if
(and only if ) q(G) = g(Γ) ≥ 2.
We now have the following claim.

18

STEFAN FRIEDL AND STEFANO VIDUSSI

Claim. For all i ≥ 1, Wi (G) \ fb(Wi (Γ)) is composed of torsion characters.

If q(G) = 1, this follows immediately from [Cam01, Théorème 1.3], as in this case
W1 (Γ) is torsion. If q(G) ≥ 2, we need a bit more work: again by [Cam01, Théorème
1.3] and the above, we have that
S
b
W1 (G) = G
Z
1̂

where Z is a finite collection of torsion characters, that we will assume to be disjoint
b . By definition Wi (G) ⊆ W1 (G). This implies, by the aforementioned result
from G
1̂
of Brudnyi, that:
– if 1 ≤ i ≤ 2q(G) − 2 the isolated points of Wi (G) are contained in Z;
b .
– if i ≥ 2q(G) − 1, they are either contained in Z, or are torsion characters in G
1̂
In either case, Wi (G) \ fb(Wi (Γ)) is composed of torsion characters as claimed. This
concludes the proof of the claim.
All this, so far, is a consequence of the fact that X has Albanese dimension one.
Now we will make use of the assumption on the virtual Albanese dimension to show
that Wi (G) \ fb(Wi (Γ)) is actually empty.
In order to prove this, recall the formula for the first Betti number for finite regular
abelian covers of X, as determined in [Hir97]: Given an epimorphism α : G → S to
a finite abelian group, and following the notation from the diagram in (1), the finite
cover H of X determined by α has first Betti number
X
b
(9)
b1 (H) =
|Wi (G) ∩ α
b(S)|.
i≥1

This formula says that a character ξ : G → C∗ such that ξ ∈ Wi (G) contributes with
multiplicity equal to its depth to the Betti number of the cover defined by α : G → S
whenever it factorizes via α. Similarly, the corresponding cover Λ of Σ has first Betti
number
X
\
b S/α(K))|.
(10)
b1 (Λ) =
|Wi (Γ) ∩ β(
i≥1

\
b S/α(K)).
Consider a character ρ ∈ Wi (Γ) ∩ β(
We have a commutative diagram
G

α

f

//

Γ❃

❃❃
❃❃
❃❃

// S/α(K) ❃❃❃ρ
◆◆◆ ❃❃❃
◆◆◆ ❃❃
◆◆◆ ❃
◆'' 



β

S

C∗

whence f ∗ (ρ) factorizes via α. As fb(Wi (Γ)) ⊆ Wi (G) this implies that f ∗ (ρ) ∈
b We deduce that for any α : G → S any contribution to the right hand
Wi (G) ∩ α
b(S).

ON VIRTUAL PROPERTIES OF KÄHLER GROUPS

19

side of Equation (10) is matched by an equal contribution to the right-hand-side of
Equation (9).
Now assume that, for some i ≥ 1, there is a nontrivial character ξ : G → C∗ such
that ξ ∈ Wi (G) \ fb(Wi (Γ)). (The case of the trivial character 1̂ : G → C∗ is dealt
with in the same way; we omit the details to avoid repetition.) As shown in the claim
above, such character must be torsion. Because of that, its image is a finite abelian
subgroup S ≤ C∗ , i.e. ξ factors through an epimorphism αξ : G → S. This character
either lies in Z or it is of the form ξ = f ∗ (ρ) for ρ ∈ W2g(Γ)−2 (Γ) (in which case
i > 2g(Γ) − 2). These two cases are treated in slightly different ways:
– If the case ξ ∈ Z holds, ξ will give a positive contribution to b1 (Hξ ) that is not
matched by any term in the right-hand-side of Equation (10).
– If the case ξ = f ∗ (ρ) holds, it is immediate to verify that ρ factors through
βξ : Γ → S/αξ (K) ∼
= S. However, the contribution of ρ to b1 (Λξ ) is at least i −
2g(Γ) + 2 > 0 short of the contribution of ξ = f ∗ (ρ) to b1 (Hξ ), as
dim H 1(G; Cξ ) ≥ i > dim H 1 (Γ; Cρ ) = 2g(Γ) − 2.
In either case, the outcome is that b1 (Hξ ) > b1 (Λξ ), which violates the assumption
that X has virtual Albanese dimension one.
Summing up, fb: Wi (Γ) → Wi (G) is a bijection. These sets coincide therefore with
the connected components of the respective character variety (for i ≤ i ≤ 2g(Γ) − 2),
the trivial character (for 2g(Γ) − 1 ≤ i ≤ 2g(Γ)), and are empty otherwise. Whenever
nonempty, both sets inherit a group structure as subsets of the respective character
varieties. The map fb is the restriction of an homomorphism between these character
varieties, hence an isomorphism as stated.

Example. The bielliptic surfaces mentioned in Section 2 are a clean example of the fact
that we need more than Albanese dimension one to get the isomorphism of Theorem
3.6. These surfaces admit genus one Albanese pencils without multiple fibers, and
there exists an epimorphism α : G → S with S abelian and H = Ker α ∼
= Z4 (see
[BHPV04, Section V.5]). This entails that, for some i ≥ 1, Wi (G) is strictly larger
than fb(Wi (Z2 )) (the difference being torsion characters, contributing to the first Betti
number of H).
By Theorem 3.6 the only characters of G that contribute to the rational homology
of finite abelian covers of X are those that descend to Γ, i.e. restrict trivially to
im(H1 (K) → H1 (G)). In particular, the finite abelian cover of X determined by the
image, in H1 (G), of the homology of the fiber of the Albanese map has the same
first Betti number as X. Note that such an image can be nonzero: infinitely many
examples arise from Proposition 3.1 by taking K equal to any finite index subgroups
of Sp(2n, Z), n ≥ 2.

20

STEFAN FRIEDL AND STEFANO VIDUSSI

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22

STEFAN FRIEDL AND STEFANO VIDUSSI

Fakultät für Mathematik, Universität Regensburg, Germany
E-mail address: sfriedl@gmail.com
Department of Mathematics, University of California, Riverside, CA 92521, USA
E-mail address: svidussi@ucr.edu

