Growth of maps, distortion in groups and symplectic geometry.

*(English)*Zbl 1036.53064If \(f\: M \to M\) 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 \(M\).

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 \(M\).

Reviewer: John F. Oprea (Cleveland)

##### MSC:

53D35 | Global theory of symplectic and contact manifolds |

53D40 | Symplectic aspects of Floer homology and cohomology |