A metric boundary theory for Carnot groups

Abstract

In this paper, we study characteristics of horofunction boundaries of Carnot groups. In particular, we show that for Carnot groups, i.e., stratified nilpotent Lie groups equipped with certain left-invariant homogeneous metrics, all horofunctions are piecewise-defined using Pansu derivatives. For higher Heisenberg groups and filiform Lie groups, two families which generalize the standard 3-dimensional real Heisenberg group, we study the dimensions and topologies of their horofunction boundaries. In doing so, we find that filiform Lie groups of dimension $n\geq 8$ provide the first-known examples of Carnot groups $G$ whose horofunction boundaries are not of dimension $\dim(G) - 1$.