  Let X be a finite CW-complex of dimension q. If its fundamental group
$\pi_{1}(X)$ is polycyclic of Hirsch number h>q we show that at least one of
the homotopy groups $\pi_{i}(X)$ is not finitely generated. If h=q or h=q-1 the
same conclusion holds unless X is an Eilenberg-McLane space $K(\pi_{1}(X),1)$.
