Commutator length of symplectomorphisms

Abstract

Each element of the commutator subgroup of a group can be represented as a product of commutators. The minimal number of factors in such a product is called the commutator length of the element. The commutator length of a group is defined as the supremum of commutator lengths of elements of its commutator subgroup. We show that for certain closed symplectic manifolds, including complex projective spaces and Grassmannians, the universal cover of the group of Hamiltonian symplectomorphisms of the manifold has infinite commutator length. In particular, we present explicit examples of elements in that group that have arbitrarily large commutator length -- the estimate on their commutator length depends on the multiplicative structure of the quantum cohomology of the symplectic manifold. By a different method we also show that in the case when the first Chern class of the manifold is zero the universal covers of the group of Hamiltonian symplectomorphisms and of the identity component of the group of all symplectomorphisms both have infinite commutator length.