We present a construction that produces infinite classes of Kähler groups that arise
as fundamental groups of fibres of maps to higher-dimensional tori. Following the
work of Delzant and Gromov, there is great interest in knowing which subgroups of
direct products of surface groups are Kähler. We apply our construction to obtain
new classes of irreducible, coabelian Kähler subgroups of direct products of
surface groups.
These cover the full range of possible finiteness properties of irreducible subgroups of direct
products of
surface
groups: for any
and
, our
classes of subgroups contain Kähler groups that have a classifying space with finite
–skeleton
while not having a classifying space with finitely many
–cells.
We also address the converse question of finding constraints on Kähler subdirect
products of surface groups and, more generally, on homomorphisms from Kähler groups
to direct products of surface groups. We show that if a Kähler subdirect product of
surface groups admits a classifying space with finite
–skeleton
for
,
then it is virtually the kernel of an epimorphism from a direct product of surface
groups onto a free abelian group of even rank.
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