  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.
