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 C1 horizontal curve in a set of measure zero. This shows that the C1H-Lusin property fails in a strong sense in this group, and we deduce that such a curve must be purely C1H 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 C1 rectifiability are equivalent.
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