Growth of maps, distortion in groups and symplectic geometry

Abstract

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.

Latex, 45 pages, revised version: new results on the Zimmer program added; proofs of the distortion bounds rewritten in a more geometric language; section on maps with the slow growth removed and will appear separately

References in corpus (4)

• Calabi quasimorphism and quantum homology
• On the Flux Conjectures
• A linear isoperimetric inequality for the punctured Euclidean plane
• A growth gap for diffeomorphisms of the interval

Cited by in corpus (46)

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• Distortion in transformation groups
• Hamiltonian Pseudo-rotations of Projective Spaces
• Area preserving group actions on surfaces
• Lectures on Groups of Symplectomorphisms
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• Invariant measures and measurable projective factors for actions of higher-rank lattices on manifolds
• Continuation homomorphism in Rabinowitz Floer homology for symplectic deformations
• Entropy zero area preserving diffeomorphisms of $S^2$
• Bounded cohomology via partial differential equations, I
• Symplectic $(-2)$-spheres and the symplectomorphism group of small rational 4-manifolds, II
• Slow entropy and symplectomorphisms of cotangent bundles
• Volume growth in the component of fibered twists
• Distortion elements in $Dif{f}^{\infty }(R/Z)$
• Symplectic deformations of Floer homology and non-contractible periodic orbits in twisted disc bundles
• Topological complexity of symplectic manifolds
• On hyperbolic cohomology classes
• Distortion elements for surface homeomorphisms
• Hamiltonian no-torsion
• On the generic behavior of the spectral norm
• Zimmer's conjecture for actions of $\mathrm{SL}(m,\mathbb{Z})$
• A two-cocycle on the group of symplectic diffeomorphisms
• A cocycle on the group of symplectic diffeomorphisms
• Distortion in Groups of Circle and Surface Diffeomorphisms
• Stable commutator length in word-hyperbolic groups
• Distortion Elements in Group actions on surfaces
• Symplectically hyperbolic manifolds
• A generalization of classical action of Hamiltonian diffeomorphisms to Hamiltonian homeomorphisms on fixed points
• The growth sequence of symplectomorphisms on symplectically hyperbolic manifolds
• Some properties of Hamiltonian homeomorphisms on closed aspherical surfaces
• Matrix group actions on product of spheres and Zimmer's program
• On distortion in groups of homeomorphisms
• Piecewise linear actions and Zimmer's program
• Non-symplectic smooth circle actions on symplectic manifolds
• Hamiltonian dynamics on convex symplectic manifolds
• Approximate Identities and Lagrangian Poincaré Recurrence
• Real-analytic, volume-preserving actions of lattices on 4-manifolds
• Symplectically aspherical manifolds
• Fiberwise volume growth via Lagrangian intersections
• Holomorphic curves into infinite dimensional almost Kähler manifolds and Hamiltonian dynamics
• Enlarging the Hamiltonian group
• No $C^1$-recurrence of iterations of symplectomorphisms
• Divergence, Undistortion and Hölder Continuous Cocycle Superrigidity for Full Shifts
• Representations of some lattices into the group of analytic diffeomorphisms of the sphere ${\mathbb{S}}^2$
• Remarks on symplectic circle actions, torsion and loops
• Slow Diffeomorphisms of a Manifold with Two Dimensions Torus Action