Kähler groups and subdirect products of surface groups

Abstract

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 $r$ surface groups. These cover the full range of possible finiteness properties of irreducible subgroups of direct products of $r$ surface groups: for any $r\ge 3$ and $2\le k\le r-1$, our classes of subgroups contain Kähler groups that have a classifying space with finite $k$–skeleton while not having a classifying space with finitely many $\left(k+1\right)$–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 $r$ 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.