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 $L_n$ of dimension $n\geq 8$ provide the first-known examples of Carnot groups whose horofunction boundaries are not full-dimensional, i.e., of codimension 1.