  We construct K\"ahler groups with arbitrary finiteness properties by mapping
products of closed Riemann surfaces holomorphically onto an elliptic curve: for
each $r\geq 3$, we obtain large classes of K\"ahler groups that have
classifying spaces with finite $(r-1)$-skeleton but do not have classifying
spaces with finitely many $r$-cells. We describe invariants which distinguish
many of these groups. Our construction is inspired by examples of Dimca,
Papadima and Suciu.
