# Commutator length of symplectomorphisms

### Michael Entov

Technion - Israel Institute of Technology, Haifa, Israel

## 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 $(M,ω)$, 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 $widetildeSymp_0,(M,omega)$ of the identity component of the group of symplectomorphisms of $(M,omega)$ have infinite commutator length.

## Cite this article

Michael Entov, Commutator length of symplectomorphisms. Comment. Math. Helv. 79 (2004), no. 1, pp. 58–104

DOI 10.1007/S00014-001-0799-0