Commutator length of symplectomorphisms

  • Michael Entov

    Technion - Israel Institute of Technology, Haifa, Israel


Each element xx of the commutator subgroup [G,G][G, G] of a group GG can be represented as a product of commutators. The minimal number of factors in such a product is called the commutator length of xx. The commutator length of GG is defined as the supremum of commutator lengths of elements of [G,G][G, G]. We show that for certain closed symplectic manifolds (M,ω)(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)(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)(M,\\omega). By a different method we also show that in the case c_1(M)=0c\_1 (M) = 0 the group \\widetilde{\\hbox{\\rm Ham}\\, (M,\\omega) and the universal cover widetildeSymp_0,(M,omega){\\widetilde{\\Symp}}\_0\\, (M,\\omega) of the identity component of the group of symplectomorphisms of (M,omega)(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