Growth of maps, distortion in groups and symplectic geometry

In the present paper we study two sequences of real numbers associated to a symplectic diffeomorphism: the uniform norm of the differential of its n-th iteration and the word length of its n-th iteration. In the latter case we assume that our diffeomorphism lies in a finitely generated group of symplectic diffeomorphisms. We find lower bounds on the growth rates of these sequences in a number of situations. These bounds depend on the symplectic geometry of the manifold rather than on the specific choice of a diffeomorphism. They are obtained by using recent results of Schwarz on Floer homology. Applications to the Zimmer program are presented. We prove non-existence of certain non-linear symplectic representations for finitely generated groups including some lattices and Baumslag-Solitar groups.