| 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 $\widetilde{\hbox{\rm Ham}\, (M,\omega)$ of the group of Hamiltonian
symplectomorphisms of $(M,\omega)$ has infinite
commutator length. In particular, we present explicit examples of
elements in $\widetilde{\hbox{\rm Ham}\, (M,\omega)$ 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
$\widetilde{\hbox{\rm Ham}\, (M,\omega)$ and the universal cover
${\widetilde{\Symp}}_0\, (M,\omega)$ of the identity
component of the group of symplectomorphisms of $(M,\omega)$
have infinite commutator length. |
