Extension problem of subset-controlled quasimorphisms

Abstract

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