Inserted: 8 dec 2014
Last Updated: 16 may 2017
Journal: Revista Matematica Iberoamericana
21 pages. Corrected a technical error in the argument for the Heisenberg group and added previously omitted proofs. Improved the exposition and corrected typos as suggested by referees. To appear in Revista Matematica Iberoamericana
We show that, given an absolutely continuous horizontal curve $\gamma$ in the Heisenberg group, there is a $C^1$ horizontal curve $\Gamma$ such that $\Gamma=\gamma$ and $\Gamma'=\gamma'$ outside a set of small measure. Conversely, we construct an absolutely continuous horizontal curve in the Engel group with no $C^1$ horizontal approximation.