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

TL;DR

This paper investigates the action of nilpotent groups on their horofunction boundary, revealing a finite orbit structure linked to the group's abelianisation and providing detailed analysis for the Heisenberg group.

Contribution

It establishes a correspondence between finite orbits and facets of a polytope derived from the group's generators, identifying all finite orbits of Busemann points.

Findings

Finite orbits correspond to facets of a polytope from the group's generators.

Each facet of this polytope yields exactly one finite orbit.

The paper provides a detailed case study of the Heisenberg group.

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 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.