FLOER HOMOLOGY, DYNAMICS AND GROUPS

Abstract

We discuss some recent results on algebraic properties of the group of Hamil- tonian diffeomorphisms of a symplectic manifold. We focus on two topics which lie at the interface between Floer theory and dynamics: 1. Restrictions on Hamiltonian actions of finitely generated groups, including a Hamil- tonian version of the Zimmer program dealing with actions of lattices; 2. Quasi-morphisms on the group of Hamiltonian diffeomorphisms. The unifying theme is the study of distortion of cyclic and one-parameter subgroups with respect to various metrics on the group of Hamiltonian diffeomorphisms. In the present lectures we discuss some recent results on algebraic properties of the group of Hamiltonian diffeomorphisms Ham(M ,ω ) of a smooth connected symplectic manifold (M 2m ,ω ). We focus on two topics which lie at the interface between Floer theory and dynamics, where by dynamics we mean the study of asymptotic behavior of Hamiltonian diffeomorphisms under iterations: − Restrictions on Hamiltonian actions of finitely generated groups, and in par- ticular a Hamiltonian version of the Zimmer program dealing with actions of lattices (Polterovich, 2002);