By Hopf H.

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Indeed, set A˜ to be the disjoint union of two isometric sets. Then we can keep one of these sets fixed and put on the top of it the other one, using only translations. The width will change. We can define other invariants though, using the heights and the number of connected components, for example. By using lifts of symplectic flows one can define symplectic invariants. Conversely, with any symplectic capacity comes an invariant of Hom(H(n), vol, Lip). Indeed, let c ˜ = c(A) is an invariant. be a capacity.

Therefore at the limit ε → 0 we get a Lie bracket. Moreover, it is straightforward to see from the definition of [x, y]n that δε is an algebra isomorphism. We conclude that (g, [·, ·]n ) is the Lie algebra of a Carnot group with dilatations δε . , Xp } of D. 2). Moreover the identiˆ i = Xi gives a Lie algebra isomorphism between (g, [·, ·]n ) and n(g, D). fication X Proof. , Xdim g} the Lie bracket on g looks like this: [Xi , Xj ] = Cijk Xk where cijk = 0 if l(Xi ) + l(Xj ) < l(Xk ). From here the first part of the proposition is straightforward.

This is because linear maps are not only group morphisms, but also commute with dilatations. 33 Let N , M be Carnot groups. We write N ≤ M if there is an injective group morphism f : N → M which commutes with dilatations. 34 Let X,Y be Lipschitz manifolds over the Carnot groups M, respectively N. If N ≤ M then there is no bi-Lipschitz embedding of X in Y . Rigidity in the sense of this section manifests in subtler ways. The purpose of Pansu paper [25] was to extend a result of Mostow [22], called Mostow rigidity.

### A New Proof of the Lefschetz Formula on Invariant Points by Hopf H.

by Ronald

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