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 $ϕ$ is equal to the positive Hofer distance between $ϕ$ and the subspace of Hamiltonian diffeomorphisms that can be expressed as a product of at most $g$ commutators.

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