Kähler groups and subdirect products of surface groups

A Kähler group is a group which can be realised as fundamental group of a compact Kähler manifold. I shall begin by explaining why such groups are not arbitrary and then address Delzant-Gromov's question of which subgroups of direct products of surface groups are Kähler. We will give a new construction of Kähler subgroups of direct products of surface groups by mapping products of closed Riemann surfaces onto an elliptic curve. These groups have exotic finiteness properties: for every r at least three the construction produces Kähler groups which admit a classifying space with finite (r-1)-skeleton, but do not have any classifying space with finitely many r-cells. We will then explain how this construction can be generalised to higher dimensions.