| The group of diffeomorphisms of a symplectic manifold (M, o) preserving the symplectic form is called the symplectomorphism group of (M, o) and denoted by Symp(M, o). It has a very important subgroup Ham( M, o) called the Hamiltonian diffeomorphism group. The group Ham(M, o) admits a natural bi-invariant norm, known as the Hofer norm. In this thesis we study several aspects of bi-invariant norms on Symp(M, o), including the bounded isometry conjecture of Lalonde and Polterovich. In particular, we prove the conjecture for the Kodaira-Thurston manifold and for the 4-torus with all linear symplectic forms. Relatedly, we observe that there is an obstruction to extending the Hofer norm bi-invariantly to the identity component Symp0( M, o) of Symp(M, o). This obstruction is shown to be non-trivial in some cases. We also prove that no Finsler norm on Ham( T2n , o) satisfying a strong form of the invariance condition can extend to a bi-invariant norm on Symp0( T2n , o). Other bi-invariant norms on Symp0(M, o) are studied as well and the induced topologies are discussed. |
