Vanishing of linear combinations of link quasi-morphisms on Ham(S^2)

Determine whether every quasimorphism μ: Ham(S^2) → ℝ of the form μ = Σ_i a_i μ_{k_i,B_i}, where μ_{k,B} are the link quasi-morphisms on Ham(S^2) and the coefficients satisfy Σ_i a_i δ_{k_i,B_i} = 0 (with δ_{k,B} the averaged point-mass measure on L_{k,B}), vanishes identically on Ham(S^2).

Background

The paper reviews the link quasi-morphisms μ{k,B}: Ham(S2) → ℝ introduced in recent work and notes that certain linear combinations μ = Σ a_i μ{k_i,B_i}, under the constraint Σ a_i δ_{k_i,B_i} = 0, vanish on the subgroup generated by height functions and extend (by Hofer continuity) to the Hofer completion. Using the main symmetrization results, these combinations also vanish on autonomous Hamiltonian diffeomorphisms.

The authors pose whether all such constrained linear combinations vanish identically on Ham(S2). A non-vanishing example would imply that the Hofer distance from arbitrary elements to the autonomous subgroup is infinite, settling the last open case of a conjecture of Polterovich–Shelukhin for surfaces. If instead all such quasimorphisms vanish, the symmetrization map constructed in the paper would extend to all of Ham(S2), opening a path to further structural results.

References

In future work joint with P. Haim-Kislev we plan to attack the following question. Do all the quasimorphisms μ: (S2) \to R from (eq mu) vanish identically?

— A dichotomy for the Hofer growth of area preserving maps on the sphere via symmetrization  (2408.08854 - Buhovsky et al., 2024) in Discussion, Subsection “Quasi-morphisms and autonomous Hamiltonian diffeomorphisms,” Question \ref{question mu}