#### Vol. 288, No. 1, 2017

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Bavard's duality theorem on conjugation-invariant norms

### Morimichi Kawasaki

Vol. 288 (2017), No. 1, 157–170
DOI: 10.2140/pjm.2017.288.157
##### 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 subset-controlled 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.

##### Keywords
Bavard's duality theorem, conjugation-invariant norm, quasimorphism, heavy subset, stable nondisplaceability
##### Mathematical Subject Classification 2010
Primary: 46B20, 53D35, 57M07, 57S05
Secondary: 57M27, 53D40, 51F99, 51K99