A Lipschitz curve in a Carnot group that is purely unrectifiable by smooth horizontal curves

TL;DR

We construct a Lipschitz curve in a specific Carnot group that is purely unrectifiable by smooth horizontal curves, demonstrating fundamental differences from Euclidean rectifiability properties.

Contribution

The paper provides a explicit example of a Lipschitz curve in a Carnot group that fails the $C^{1}_H$-Lusin property, highlighting the distinct nature of rectifiability in these groups.

Findings

The constructed curve meets every $C^{1}$ horizontal curve in a measure zero set.

The curve is purely $C^1_H$ 1-unrectifiable.

Rectifiability notions differ significantly between Carnot groups and Euclidean spaces.

Abstract

We construct a Lipschitz curve in the free Carnot group of step 3 with 2 generators that meets every $C^{1}$ horizontal curve in a set of measure zero. This shows that the $C^{1}_{H}$-Lusin property fails in a strong sense in this group, and we deduce that such a curve must be purely $C^1_H$ 1-unrectifiable. Hence 1-rectifiability in Carnot groups is wildly different to its counterpart in Euclidean spaces, wherein the Whitney Extension Theorem guarantees that Lipschitz rectifiability and $C^1$ rectifiability are equivalent.