Ergodicity of algebraic actions of nilpotent groups

An algebraic $\Gamma$-action is an action of a countable group $\Gamma$ on a compact abelian group $X$ by continuous automorphisms of $X$. We prove that any expansive algebraic action of a finitely generated nilpotent group $\Gamma$ on a connected group $X$ is ergodic. We also show that this result does not hold for actions of polycyclic groups.