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Hofer & Zehnder  is dedicated to the implications of the existence of a non-trivial capacity. All in all the non-degeneracy of the Hofer distance (proved for example in the Section 5. Hofer & Zehnder ) is the cornerstone of symplectic rigidity theory. 6 Hausdorff dimension and Hofer distance We shall give in this section a metric proof of the non-degeneracy of the Hofer distance. A flow of symplectomorphisms t → φt with compact support in A is Hamiltonian if it is the projection onto R2n of a flow in Hom(H(n), vol, Lip)(A).

The purpose of Pansu paper  was to extend a result of Mostow , called Mostow rigidity. Although it is straightforward now to explain what Mostow rigidity means and how it can be proven, it is beyond the purposes of these notes. 3 The Heisenberg group Let us rest a little bit and look closer to an example. The Heisenberg group is the most simple non commutative Carnot group. We shall apply the achievements of the previous chapter to this group. 1 The group The Heisenberg group H(n) = R2n+1 is a 2-step nilpotent group with the operation: 1 (x, x¯)(y, y¯) = (x + y, x ¯ + y¯ + ω(x, y)) 2 where ω is the standard symplectic form on R2n .

Proof. Write the lifts φ˜t and φht , compute then the differential of the quantity φ˜t − φ˙ ht and show that it equals the differential of H. 3 THE HEISENBERG GROUP 39 Flows of volume preserving diffeomorphisms We want to know if there is any nontrivial smooth (according to Pansu differentiability) flow of volume preserving diffeomorphisms. 6 Suppose that t → φ˜t ∈ Dif f 2 (H(n), vol) is a flow such that - is C 2 in the classical sense with respect to (x, t), - is horizontal, that is t → φ˜t (x) is a horizontal curve for any x.