We study the action of a nilpotent group with finite generating set on its horofunction boundary. We show that there is one finite orbit associated to each facet of the polytope obtained by projecting into the torsion-free component of the abelianisation of . 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.
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Cormac Walsh, The action of a nilpotent group on its horofunction boundary has finite orbits. Groups Geom. Dyn. 5 (2011), no. 1, pp. 189–206DOI 10.4171/GGD/122