Kahler groups and subdirect products of surface groups

We present a construction that produces infinite classes of Miller 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 Kahler. We apply our construction to obtain new classes of irreducible, coabelian Kahler 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 >= 3 and 2 <= k <= r -1, our classes of subgroups contain Kahler 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 Kahler subdirect products of surface groups and, more generally, on homomorphisms from Miller groups to direct products of surface groups. We show that if a Miller subdirect product of r surface groups admits a classifying space with finite k-skeleton for k > 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.