Commutator length of symplectomorphisms

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Abstract

Each element z 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 Ham (M, ω) of the group of Hamiltonian symplectomorphisms of (M, ωw) has infinite commutator length. In particular, we present explicit examples of elements in Ham (M, ω) that have arbitrarily large commutator length - the estimate on their commutator length depends on the multiplicative structure of the quantum cohomology of (M, ω). By a different method we also show that in the case c1(M) = 0 the group Ham (M, ω) and the universal cover Symp0 (M, ω) of the identity component of the group of symplectomorphisms of (M, ω) have infinite commutator length.

Original languageEnglish
Pages (from-to)58-104
Number of pages47
JournalCommentarii Mathematici Helvetici
Volume79
Issue number1
DOIs
StatePublished - 2004

Keywords

  • Commutator length
  • Floer homology
  • Hamiltonian symplectomorphism
  • Quantum cohomology
  • Quasimorphism
  • Symplectic manifold

ASJC Scopus subject areas

  • General Mathematics

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