Bavard's duality theorem on conjugation-invariant norms

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 Oh-Schwarz spectral invariants are subsetcontrolled quasimorphisms, which are generalizations of quasimorphisms. We prove a Bavard-type duality theorem between subset-controlled quasimorphisms on stable groups and conjugation-invariant (pseudo)norms. We also pose a generalization of our main theorem and prove stably nondisplaceable subsets of symplectic manifolds are heavy in a rough sense if that generalization holds. © 2017 Mathematical Sciences Publishers