Commutator length of symplectomorphisms

Each element z of the commutator subgroup [G, G] of a group G can be represented as a product of commutators. The minimal number of factors in such a product is called the commutator length of x. The commutator length of G is defined as the supremum of commutator lengths of elements of [G, G]. We show that for certain closed symplectic manifolds (M, ω), including complex projective spaces and Grassmannians, the universal cover Ham (M, ω) of the group of Hamiltonian symplectomorphisms of (M, ωw) has infinite commutator length. In particular, we present explicit examples of elements in Ham (M, ω) that have arbitrarily large commutator length - the estimate on their commutator length depends on the multiplicative structure of the quantum cohomology of (M, ω). By a different method we also show that in the case c₁(M) = 0 the group Ham (M, ω) and the universal cover Symp₀ (M, ω) of the identity component of the group of symplectomorphisms of (M, ω) have infinite commutator length.