  Starting from a given norm on the vector space of exact 1-forms of a compact
symplectic manifold, we produce pseudo-distances on its symplectomorphism group
by generalizing an idea due to Banyaga. We prove that in some cases (which
include Banyaga's construction), their restriction to the Hamiltonian
diffeomorphism group is equivalent to the distance induced by the initial norm
on exact 1-forms. We also define genuine "distances to the Hamiltonian
diffeomorphism group" which we use to derive several consequences, mainly in
terms of flux groups.
