Growth of maps, distortion in groups and symplectic geometry

By L. Polterovich. 2001

135 citations recorded by the source corpus.

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;¶• The word length of its n-th iteration, where we assume that our diffeomorphism lies in a finitely generated group of symplectic diffeomorphisms.¶We find lower bounds for 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. As an application, we prove non-existence of certain non-linear symplectic representations for finitely generated groups.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8ebcf46ec8a3b9241a8853662c790ad1c1be2af9

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Stable commutator length in word-hyperbolic groups (2006, background): “From another point of view, there are relations between symplectic geometry and stable commutator length which are more intimately connected with invariants like Hofer length and subgroup distortion; see work of Polterovich, especially [33].”
Distortion in transformation groups (2005, background): “Sn/ , thought of as a discrete group.”
Distortion elements in group actions on surfaces (2004, background): “C of [18]) proved that if S is a closed surface of genus at least two and ω is a smooth measure on S, then finitely generated subgroups of Hamω(S), the group of Hamiltonian diffeomorphisms on S, do not contain distortion elements.”