THE g-AREAS AND THE COMMUTATOR LENGTH

Abstract

The commutator length of a Hamiltonian diffeomorphism f ∈ Ham (M,ω) of a closed symplectic manifold (M,ω) is by definition the minimal k such that f can be written as a product of k commutators in Ham (M,ω). We introduce a new invariant for Hamiltonian diffeomorphisms, called the k+-area, which measures the "distance", in a certain sense, to the subspace [Formula: see text] of all products of k commutators. Therefore, this invariant can be seen as the obstruction to writing a given Hamiltonian diffeomorphism as a product of k commutators. We also consider an infinitesimal version of the commutator problem: what is the obstruction to writing a Hamiltonian vector field as a linear combination of k Lie brackets of Hamiltonian vector fields? A natural problem related to this question is to describe explicitly, for every fixed k, the set of linear combinations of k such Lie brackets. The problem can be obviously reformulated in terms of Hamiltonians and Poisson brackets. For a given Morse function f on a symplectic Riemann surface M (verifying a weak genericity condition) we describe the linear space of commutators of the form {f, g}, with [Formula: see text].