Proof of the main conjecture on g-areas

In this paper, we prove the main conjecture on $g$-areas that was announced by the first author in 2004. It states that the $g$-area of any Hamiltonian diffeomorphism $\phi$ is equal to the positive Hofer distance between $\phi$ and the subspace of Hamiltonian diffeomorphisms that can be expressed as a product of at most $g$ commutators.