The Horofunction boundary of the Heisenberg Group: The Carnot-Carathéodory metric

We find the horofunction boundary of the $(2n+1)$ -dimensional Heisenberg group with the Carnot-Carathéodory distance and show that it is homeomorphic to a $2n$ -dimensional disk and that the Busemann points correspond to the $(2n-1)$ -sphere boundary of this disk. We also show that the compactified Heisenberg group is homeomorphic to a $(2n+1)$ -dimensional sphere. As an application, we find the group of isometries of the Carnot-Carathéodory distance.