Commutator length of symplectomorphisms (Q1424240)

Each element \(x\) of the commutator subgroup \([G,G]\subset G\) of a group \(G\) can be represented as a product of commutators. The minimal number \(\text{cl}(x)\) of factors is called the commutator length of \(x\) and the commutator length \(\text{cl}(G)\) of \(G\) is defined as the supremum of all \(\text{cl}(x)\). Let \((M,\omega)\) be a closed connected symplectic manifold. A function \(H: S^1\times M\to \mathbb{R}\) (called Hamiltonian) determines a time-dependent Hamiltonian vector field \(X(t,x)\), \(t\in S^1\), \(x\in M\) such that \(dH(t,.)= \omega(X(t,.),.)\). The flow of this field preserves the symplectic form. A symplectomorphism of \(M\) that can be represented as the time-1 map of a Hamiltonian vector field is called a Hamiltonian symplectomorphism. All Hamiltonian symplectomorphisms form a subgroup \(\text{Ham}(M,\omega)\subset \text{Symp}_0(M,\omega)\) of the identity component of the group of symplectomorphisms. Main results: For certain particular cases (including complex projective spaces and Grassmann\-ians) the universal cover of \(\text{Ham}(M,\omega)\) has infinite commutator length and the estimate of the commutator length depends on the multiplicative structure of the quantum cohomology of \((M,\omega)\)). Moreover, in the case \(c_1(M)= 0\) both the universal covers of \(\text{Ham}(M,\omega)\) and \(\text{Symp}_0(M,\omega)\) have infinite commutator length. Rather advanced tools are employed (e.g., the rational Novikov ring, Floer cohomology, spectral numbers, Hofer metric, \(K\)-area, moduli spaces) and many references are involved.