Bavard's duality theorem on conjugation-invariant norms

TL;DR

This paper extends Bavard's duality theorem to conjugation-invariant norms and subset-controlled quasimorphisms on stable groups, with applications to symplectic geometry and heavy subsets.

Contribution

It proves a Bavard-type duality theorem for conjugation-invariant (pseudo-)norms and subset-controlled quasimorphisms, generalizing previous dualities and linking to symplectic geometry.

Findings

Established duality between conjugation-invariant norms and subset-controlled quasimorphisms.

Applied the duality to symplectic geometry, relating to heavy subsets.

Proposed a generalization implying non-displaceability of certain subsets.

Abstract

Bavard proved a duality theorem between commutator length and quasimorphisms. Burago, Ivanov and Polterovich introduced the notion of a conjugation-invariant norm which is a generalization of commutator length. Entov and Polterovich proved that Oh-Schwarz spectral invariants are subset-controlled quasimorphisms which are geralizations of quasimorphisms. In the present paper, we prove a Bavard-type duality theorem between conjugation-invariant (pseudo-)norms and subset-controlled quasimorphisms on stable groups. %We also give an application of its generalization to symplectic geometry related to Entov-Polterovich's theory on heavy subsets. We also pose a generalization of our main theorem and prove that "stably non-displaceable subsets of symplectic manifolds are heavy" in a rough sense if that generalization holds.