Proof of the main conjecture on g-areas

TL;DR

This paper proves a long-standing conjecture relating the g-area of Hamiltonian diffeomorphisms to the positive Hofer distance, establishing a key geometric measure in symplectic topology.

Contribution

It provides a rigorous proof of the main conjecture on g-areas, connecting algebraic and geometric properties of Hamiltonian diffeomorphisms.

Findings

g-area equals the positive Hofer distance to g-commutator subgroup

Confirms the conjecture announced in 2004

Advances understanding of Hamiltonian diffeomorphism metrics

Abstract

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.