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Abstract

We present a construction that produces infinite classes of K\"ahler 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 K\"ahler. We apply our
construction to obtain new classes of irreducible, coabelian K\"ahler subgroups
of direct products of $r$ surface groups. These cover the full range of
possible finiteness properties of irreducible subgroups of direct products of
$r$ surface groups: For any $r\geq 3$ and $2\leq k \leq r-1$, our classes of
subgroups contain K\"ahler groups that have a classifying space with finite
$k$-skeleton while not having a classifying space with finitely many
$(k+1)$-cells.
We also address the converse question of finding constraints on K\"ahler
subdirect products of surface groups and, more generally, on homomorphisms from
K\"ahler groups to direct products of surface groups. We show that if a
K\"ahler subdirect product of $r$ surface groups admits a classifying space
with finite $k$-skeleton for $k>\frac{r}{2}$, then it is virtually the kernel
of an epimorphism from a direct product of surface groups onto a free abelian
group of even rank.