Discreteness of the Calabi period group Λ_ω for general symplectic manifolds

Determine whether the subgroup Λ_ω—defined as the image of π_1(Ham_c(M, ω)) under the lifted Calabi homomorphism—is discrete for general symplectic manifolds (M, ω), possibly with boundary.

Background

The Calabi homomorphism for non-closed or boundary cases descends to Ham_c(M, ω) with target R/Λω, where Λω is the image of π1(Ham_c(M, ω)) under the lifted Calabi map. Beyond exact cases without boundary (where Λω vanishes), the structure of Λ_ω is largely unknown. Its discreteness affects the topology that can be naturally placed on Ham_c(M, ω) and its kernel Ham_c^0(M, ω).