Lusin approximation for horizontal curves in step 2 Carnot groups

A Carnot group G admits Lusin approximation for horizontal curves if for any absolutely continuous horizontal curve γ in G and ε> 0 , there is a C1horizontal curve Γ such that Γ = γ and Γ′= γ′outside a set of measure at most ε. We verify this property for free Carnot groups of step 2 and show that it is preserved by images of Lie group homomorphisms preserving the horizontal layer. Consequently, all step 2 Carnot groups admit Lusin approximation for horizontal curves.