The action of a nilpotent group on its horofunction boundary has finite orbits

Abstract

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 torsion-free 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.