A Lipschitz curve in the free 3-step Carnot group with 2 generators is purely unrectifiable by C^1 horizontal curves.
On conditions for unrectifiability of a metric space
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A Lipschitz curve in a Carnot group that is purely unrectifiable by smooth horizontal curves
A Lipschitz curve in the free 3-step Carnot group with 2 generators is purely unrectifiable by C^1 horizontal curves.