Horofunctions on the Heisenberg and Cartan groups

TL;DR

This paper investigates the structure of horofunction boundaries in nilpotent groups, specifically the Heisenberg and Cartan groups, revealing new classifications of Busemann points and disproving existing conjectures.

Contribution

It classifies Busemann point orbits in Heisenberg groups, shows the uncountability of the horoboundary, and disproves conjectures regarding the Cartan group's horoboundary and group actions.

Findings

Finite set of Busemann point orbits in Heisenberg groups

Uncountable horoboundary via Lie group approximation

Continuum of Busemann points in the Cartan group

Abstract

We study the horofunction boundary of finitely generated nilpotent groups, and the natural group action on it. More specifically, we prove the followings results: For discrete Heisenberg groups, we classify the orbits of Busemann points. As a byproduct, we observe that the set of orbits is finite and the set of Busemann points is countable. Furthermore, using the approximation with Lie groups, we observe that the entire horoboundary is uncountable. For the discrete Cartan group, we exhibit an continuum of Busemann points, disproving a conjecture of Tointon and Yadin. As a byproduct, we prove that the group acts non-trivially on its reduced horoboundary, disproving a conjecture of Bader and Finkelshtein.