Quasi-morphismes et invariant de Calabi

Abstract

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.Comment: 19 pages, juin 200

Py-calabi quasi-morphisms and quasi-states on orientable surfaces of higher genus

“…Similar methods have been used in [9]. We can now state Py's second result [18]. Let F : M → R be a generic…”

“…In this paper we show this connection for a recently discovered, due to P. Py [18], Calabi quasi-morphism on orientable surfaces of higher genus. The proof relies on hyperbolic geometry tools, surprisingly combined with combinatorial tools such as Hall's marriage theorem.…”

“…In this section we will examine the construction of Py's quasi-morphism as defined in [18] and show that it is continuous on time independent Hamiltonians, with respect to the C 2 -topology. Lets recall the following definitions.…”

“…Hence, the restriction of µ, defined in Theorem 1.6, on the subgroup Γ is a homomorphism. In [18] P. Py has proved the following formula for µ on Γ, assuming that the total area of M is equal to 2g − 2,…”

Проблема Каждана - Мильмана Для Полупростых Компактных Групп Ли

“…В частно-сти, отсюда следует утверждение о существовании представлений, близких к данному почти представлению аменабельной локально компактной группы в сопряженном банаховом пространстве. Условия существования гомоморфиз-мов, близких к данному квазигомоморфизму, изучались также с 40-х годов с различных точек зрения в [3]- [18], [20]- [22], [28]- [39], [41], [47], [54], [63]- [116]. Родственным задачам, в том числе -разделяющим отображениям, посвящены статьи [40], [117]- [122].…”