Abstract: The horofunction compactification of a proper metric space (X,d), also known as the Busemann compactification, is obtained by using the distance function d to embed X into the space of continuous real valued functions on X and taking the closure. The horofunction boundary of X is the complement of the image of X in its horofunction compactification.

We explicitly find the horofunction boundary of the (2n+1)-dimensional Heisenberg group with the Carnot-Caratheodory metric and show that it is homeomorphic to a 2n-dimensional disk. We also show that the Busemann points correspond to the (2n-1)-sphere boundary of this disk and that the compactified Heisenberg group is homeomorphic to a (2n+1)-dimensional sphere. As an application, we find all isometries of the Carnot-Caratheodory metric. This is joint work with Tom Klein.