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 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 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.

$\hat{G}$-invariant quasimorphisms and symplectic geometry of surfaces

“…Some parts of the proof go through in the same way as the arguments in [Ka17]. Moreover, some parts are much easier than the ones in [Ka17] because a technical lemma corresponding to [Ka17,Lemma 2.6] follows immediately in our situation. Thus, we often omit such parts of the proof.…”

“…These operators are well-defined [Ka17, Proposition 2.2] and (A, • ) is a normed vector space [Ka17,Proposition 2.3]. By the Hahn-Banach theorem, we obtain the following proposition.…”

“…One side follows from Lemma 2.1, thus we prove the other side. For this purpose, we use the strategy of Calegari-Zhuang [CZ] (see also [Ka17]). Some parts of the proof go through in the same way as the arguments in [Ka17].…”

“…For this purpose, we use the strategy of Calegari-Zhuang [CZ] (see also [Ka17]). Some parts of the proof go through in the same way as the arguments in [Ka17]. Moreover, some parts are much easier than the ones in [Ka17] because a technical lemma corresponding to [Ka17,Lemma 2.6] follows immediately in our situation.…”

Extension problem of subset-controlled quasimorphisms

“…([Ka2]). Let (G, H) be (Ham(R 2n ), Ham(B 2n )) or (B ∞ , B n ) and let ν be a conjugation-invariant norm on G. Then, for any element g of G such that sν(g) > 0, there exists a homogeneous H-quasimorphism μ : G → R such that μ(g) > 0.3.…”