Symplectic, Hofer and sub-Riemannian geometry

The purpose of this note is to make some connection between the sub-Riemannian geometry on Carnot-Caratheodory groups and symplectic geometry. We shall concentrate here on the Heisenberg group, although it is transparent that almost everything can be done on a general Carnot-Caratheodory group. Such generalisations will be the subject of a forthcoming note.