A metric boundary theory for Carnot groups

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 $\dim(G) - 1$.

💡 Research Summary

This paper investigates the horofunction boundaries of Carnot groups—simply connected nilpotent Lie groups equipped with left‑invariant, homogeneous metrics. The authors focus on a class of metrics they call “layered sup norms,” which are built by taking the maximum of norms on each layer of the grading (each layer may carry either a polyhedral or a smooth norm). By invoking Guivarc’h’s lemma, these sup‑quasi‑norms can be rescaled to genuine norms, thus yielding bona‑fide homogeneous distances.

The first major result (Theorem A) proves that for any Carnot group equipped with a layered sup norm of the described type, every horofunction is piecewise linear and depends only on the coordinates of the first layer of the grading. The proof uses Pansu differentiation: the horofunctions are shown to be Pansu‑differentiable, and the derivative lives entirely in the first layer, forcing the function to be linear on each region of a finite polyhedral decomposition.

To explore the second question—whether the horofunction boundary always has the “expected” dimension dim G − 1—the authors examine two infinite families of stratified groups that generalize the three‑dimensional Heisenberg group.

1. Higher Heisenberg groups H₂ₙ₊₁(ℝ) (dimension 2n + 1, step 2). For any layered sup norm built from polyhedral or smooth layer norms, the horofunction boundary is a 2n‑dimensional topological space homeomorphic to a “button‑pillow”: a 2n‑sphere with the north and south poles identified. In the generic (non‑separated) case the boundary is a smooth manifold; in the special symmetric case where the norm is non‑separated, distinct pieces of the boundary intersect, but the dimension remains 2n.

2. Filiform groups Lₙ (first kind, dimension n, step n − 1). These groups have a very simple Lie algebra: