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 C¹ 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 C¹ horizontal approximation. [ABSTRACT FROM AUTHOR]