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Growth of maps, distortion in groups and symplectic geometry. (English) Zbl 1036.53064
If $$f\: M \to M$$ is a diffeomorphism of a smooth compact connected manifold, its growth sequence is defined by, $\Gamma_n(f) = \max\left(\max_{x \in M} | d_xf^n| , \max_{x \in M} | d_xf^{-n}| \right).$ Here, $$d_xf$$ denotes the differential and $$| \cdot|$$ denotes the operator norm with respect to a Riemannian metric. If $$\Gamma_n(f)$$ grows exponentially, then $$f$$ is said to be hyperbolic; if $$\Gamma_n(f)$$ is bounded, then $$f$$ is said to be elliptic; and if neither of these cases holds, then $$f$$ is parabolic. The dynamical (or geometric) qualities of diffeomorphisms are sometimes reflected in the growth sequence. For instance, if $$f$$ can be included in the action of a compact group, then it is elliptic. In this paper, the author finds lower bounds for the growth sequence of symplectic diffeomorphisms satisfying certain conditions.
A main result states that, for a closed symplectic manifold $$(M,\omega)$$ with $$\pi_2(M) = 0$$ and $$f$$ a symplectic diffeomorphism with a fixed point of contractible type, the growth sequence is bounded below by what the author calls a symplectic filling function associated to the symplectic manifold $$M$$. (For example, a standard symplectic torus has symplectic filling function on the order of $$\sqrt{s}$$. Also, Hamiltonian diffeomorphisms always have such fixed points.)
This whole approach allows the author to obtain interesting results about the group $$\text{Ham}(M,\omega)$$ of Hamiltonian diffeomorphisms of $$M$$ (with the same assumptions on $$M$$ as above). In particular, if $$G \subset \text{Ham}(M,\omega)$$ is a finitely generated subgroup and $$f \in G$$ is a non-identity element, the growth sequence of the word length (with respect to a fixed finite set of generators of $$G$$) of powers $$f^n$$ is bounded below by the symplectic filling function of $$M$$.

##### MSC:
 53D35 Global theory of symplectic and contact manifolds 53D40 Symplectic aspects of Floer homology and cohomology
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