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

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 $C_H^1$-Lusin property fails in a strong sense in this group, and we deduce that such a curve must be purely $C_H^1$ 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.

26 pages