A cocycle on the group of symplectic diffeomorphisms

TL;DR

This paper introduces a new cocycle on symplectic diffeomorphisms, explores its properties, and applies it to analyze symplectic group actions, providing an alternative proof of a known theorem about subgroup distortion.

Contribution

It defines a novel cocycle on symplectic diffeomorphisms and applies it to study group actions, offering new insights and an alternative proof of existing results.

Findings

Defined a cocycle on symplectic diffeomorphisms

Analyzed properties of the cocycle

Provided an alternative proof of Polterovich's theorem

Abstract

We define a cocycle on the group of symplectic diffeomorphisms of a symplectic manifold and investigate its properties. The main applications are concerned with symplectic actions of discrete groups. For example, we give an alternative proof of the Polterovich theorem about the distortion of cyclic subgroups in finitely generated groups of Hamiltonian diffeomorphisms.