  We study the action of a nilpotent group G with finite generating set S on
its horofunction boundary. We show that there is one finite orbit associated to
each facet of the polytope obtained by projecting S into the infinite component
of the abelianisation of G. We also prove that these are the only finite orbits
of Busemann points. To finish off, we examine in detail the Heisenberg group
with its usual generators.
