Groups, Geometry, and Dynamics


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Volume 5, Issue 1, 2011, pp. 189–206
DOI: 10.4171/GGD/122

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

Cormac Walsh[1]

(1) Centre de Mathématiques Appliquées, Ecole Polytechnique, Route de Saclay, 91128, PALAISEAU CEDEX, FRANCE

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.

Keywords: Group action, horoball, max-plus algebra, metric boundary, Busemann function

Walsh C. The action of a nilpotent group on its horofunction boundary has finite orbits. Groups Geom. Dyn. 5 (2011), 189-206. doi: 10.4171/GGD/122