Extension Problem of Subset-controlled Quasimorphisms

Let (G,H) be (Ham(R2n),Ham(B2n)) or (B∞, Bn). 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. 1. Problems and results To state our conjecture, we introduce the notion of subset-controlled quasimorphism (partial quasimorphism, pre-quasimorphism) which is a generalization of quasimorphism. Definition 1.1. Let G be a group and let H be a subset of G. We define the fragmentation norm qH with respect to H for an element f of G, qH(f)=min{k; ∃g1 . . . , gk ∈ G, ∃h1, . . . hk ∈ H such that f=g−1 1 h1g1 · · · g−1 k hkgk}. If there is no such decomposition of f as above, we put qH(f) = ∞. H c-generates G if such decomposition as above exists for any f ∈ G. Definition 1.2. Let H, G′ be subgroups of a group G. A function μ : G′ → R is called an H-quasimorphism on G′ if there exists a positive number C such that for any elements f , g of G′, |μ(fg)− μ(f)− μ(g)| < C ·min{qH(f), qH(g)}. μ is called homogeneous if μ(f) = nμ(f) holds for any element f of G′ and any n ∈ Z. μ is called semi-homogeneous if μ(f) = nμ(f) holds for any element f of G′ and any n ∈ Z≥0. Such generalization as above of quasimorphism appeared first in [EP06]. For a symplectic manifold (M,ω), let Ham(M) denote the group of Hamiltonian diffeomorphisms with compact support and (R, ω0), (B , ω0) denote the 2ndimensional Euclidean space, ball with the standard symplectic form, respectively. Let Bn denote the n-braid group and B∞ denote the infinite braid group ⋃ n Bn. We pose the following conjecture. For a group G, let [G,G] denote the commutator subgroup (the subgroup generated by the set {[a, b] = aba−1b−1|a, b ∈ G} of commutators). Received by the editors December 18, 2016, and, in revised form, April 19, 2017 and September 1, 2017. 2010 Mathematics Subject Classification. Primary 20J06, 53D22; Secondary 57M27. This work was supported by IBS-R003-D1. c ©2018 by the author under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)