Commutator length of symplectomorphisms

Abstract

Each element $x$ 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 $(M,\omega )$, including complex projective spaces and Grassmannians, the universal cover $\widetilde{\mathrm{Ham}}(M,\omega )$ of the group of Hamiltonian symplectomorphisms of $(M,\omega )$ has infinite commutator length. In particular, we present explicit examples of elements in $\widetilde{\mathrm{Ham}}(M,\omega )$ that have arbitrarily large commutator length – the estimate on their commutator length depends on the multiplicative structure of the quantum cohomology of $(M,\omega )$. By a different method we also show that in the case $c_1(M)=0$ the group $\widetilde{\mathrm{Ham}}(M,\omega )$ and the universal cover ${\widetilde{\mathrm{Symp}}}_0(M,\omega )$ of the identity component of the group of symplectomorphisms of $(M,\omega )$ have infinite commutator length.