A metric boundary theory for Carnot groups

TL;DR

This paper investigates the horofunction boundaries of Carnot groups, revealing that they are piecewise defined by Pansu derivatives and discovering new examples where boundary dimensions deviate from the typical pattern.

Contribution

It establishes that all horofunctions in Carnot groups are piecewise using Pansu derivatives and identifies the first examples of Carnot groups with atypical boundary dimensions.

Findings

Horofunctions are piecewise-defined via Pansu derivatives.

Higher Heisenberg and filiform groups analyzed for boundary topology.

Existence of Carnot groups with non-standard boundary dimensions.

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 $\dim(G) - 1$.