Symplectic diffeomorphisms as isometries of Hoferʹs norm

In this paper, we extend the Hofer norm to the group of symplectic diffeomorphisms of a manifold. This group acts by conjugation on the group of Hamiltonian diffeomorphisms, so each symplectic diffeomorphism induces an isometry of the group Ham(M) with respect to the Hofer norm. The C0-norm of this isometry, once restricted to the ball of radius α of Ham(M) centered at the identity, gives a scale of norms rα on the group of symplectomorphisms. We conjecture that the subgroup of the symplectic diffeomorphisms which are isotopic to the identity and whose norms rα remain bounded when α → ∞ coincide with the group of Hamiltonian diffeomorphisms. We prove this conjecture for products of surfaces.