Extension problem of subset-controlled quasimorphisms

Abstract

Let (G, H) be (Ham(R 2n ), Ham(B 2n )) or (B ∞ , B n ). We conjecture that any semi-homogeneous subset-controlled quasimorphism on [G, G] can be extended to a homogeneous subset-controlled quasimorphism on G and also give a theorem supporting this conjecture by using a Bavard-type duality theorem on conjugation invariant norms.

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

“…It is known that every homogeneous quasimorphism on Ĝ is Ĝ-invariant ( [Cale]). Thus, we see that Ĝ-invariance is necessary to extend φ : G → R to a homogeneous quasimorphism on Ĝ. Shtern and the first author also studied a similar topic [Sh,Ka18].…”

Extensions of quasi-morphisms to the symplectomorphism group of the disk

“…Let K be a subgroup of G. It is natural to ask whether given homogeneous quasi-morphism ψ : K → R can be extended to a homogeneous quasi-morphism on G. This extension problem of quasi-morphisms is studied in some papers ([11], [7], [8]). In this paper, we consider the extension problem of the Ruelle invariant and Gambaudo-Ghys quasi-morphisms on Symp(D, ∂D) to the group Symp(D).…”