Lusin Approximation and Horizontal Curves in Carnot Groups

We show that, given an absolutely continuous horizontal curve $γ$ in the Heisenberg group, there is a $C^1$ horizontal curve $Γ$ such that $Γ=γ$ and $Γ'=γ'$ outside a set of small measure. Conversely, we construct an absolutely continuous horizontal curve in the Engel group with no $C^1$ horizontal approximation.

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