Quasi-morphismes et invariant de Calabi

Pierre Py (1)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

In this paper, we give two elementary constructions of homogeneous quasi-morphisms defined on the group of Hamiltonian diffeomorphisms of certain closed connected symplectic manifolds (or on its universal cover). The first quasi-morphism, denoted by $\calabi_{S}$, is defined on the group of Hamiltonian diffeomorphisms of a closed oriented surface $S$ of genus greater than $1$. This construction is motivated by a question of M. Entov and L. Polterovich. If $U\subset S$ is a disk or an annulus, the restriction of $\calabi_{S}$ to the subgroup of diffeomorphisms which are the time one map of a Hamiltonian isotopy in $U$ equals Calabi's homomorphism. The second quasi-morphism is defined on the universal cover of the group of Hamiltonian diffeomorphisms of a symplectic manifold for which the cohomology class of the symplectic form is a multiple of the first Chern class.

Mots clés: Hamiltonian diffeomorphism; quasi-morphism; bounded cohomology

Domaines: Géométrie symplectique [math.SG]; Théorie des groupes [math.GR]

Pierre Py. Quasi-morphismes et invariant de Calabi. Annales Scientifiques de l'École Normale Supérieure, 2006, 39 (no.1), pp.177--195. ⟨10.1016/j.ansens.2005.11.003⟩. ⟨hal-00005151v2⟩