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A metric boundary theory for Carnot groups

Nate Fisher

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 $\dim(G) - 1$.

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 provide the first-known examples of Carnot groups whose horofunction boundaries are not of dimension .
Paper Structure (24 sections, 17 theorems, 69 equations, 8 figures, 3 tables)

This paper contains 24 sections, 17 theorems, 69 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background and notation
  3. Graded Lie algebras
  4. Exponential coordinates and the Baker--Campbell--Hausdorff formula
  5. Examples
  6. Homogeneous structures on graded groups
  7. Homogeneous distances and quasi-norms
  8. Convex geometry
  9. Horofunction boundary of a metric space
  10. Pansu differentiability and horofunctions
  11. Blow-ups and horofunctions
  12. Blow-ups of functions
  13. Horofunctions in general Carnot groups
  14. Horofunction boundaries of higher Heisenberg groups
  15. Blow-ups at interior ceiling/floor points
  16. ...and 9 more sections

Key Result

Proposition 2.3

(fischer-ruzhansky Prop. 3.1.24). For any $j = 1, \ldots, n,$ we have where $\alpha$ and $\beta$ are multi-indices in ${\mathbb N}_0^n\setminus\{\vec{0}\}$ and $c_{j, \alpha, \beta}$ are real coefficients.

Figures (8)

  • Figure 1: Example of convex body, polar dual, and exposed dual.
  • Figure 2: Example of smooth convex body with extreme points and their exposed duals.
  • Figure 3: On the left we have the unit metric spheres in $H(\mathbb{R})$ equipped with a Euclidean layered sup norm (above) and the homogeneous sup norm (below). On the right, we have the corresponding horofunction boundaries. Colors give correspondence between points on the sphere and blow-ups of the norm at those points.
  • Figure 4: Unit metric sphere (left) and horofunction boundary (right) of $H(\mathbb{R})$ where $\|\cdot\|_{V_1}= 2\|\cdot\|_{\text{Eucl}}$. In this non-separated example, the boundary has a different homeomorphism type.
  • Figure 5: Example of a principal blow-up function in two variables with two linear partial functions (center) along with two of its translates (left and right), where $f(e)$ always equals 0.
  • ...and 3 more figures

Theorems & Definitions (34)

  • Example 2.1
  • Example 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Theorem 2.8
  • Proposition 3.1
  • proof
  • ...and 24 more