Calabi quasimorphism and quantum homology

Abstract

We prove that the group of area-preserving diffeomorphisms of the 2-sphere admits a non-trivial homogeneous quasimorphism to the real numbers with the following property. Its value on any diffeomorphism supported in a sufficiently small open subset of the sphere equals to the Calabi invariant of the diffeomorphism. This result extends to more general symplectic manifolds: If the symplectic manifold is monotone and its quantum homology algebra is semi-simple we construct a similar quasimorphism on the universal cover of the group of Hamiltonian diffeomorphisms.

Revised and considerably extended version; links to Seidel representation and combinatorics added

References in corpus (2)

• The Verlinde Algebra And The Cohomology Of The Grassmannian
• Commutator length of symplectomorphisms

Cited by in corpus (16)

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• Growth of maps, distortion in groups and symplectic geometry
• Persistent homology and Floer-Novikov theory
• On the classification of rank two representations of quasiprojective fundamental groups
• Hamiltonian Pseudo-rotations of Projective Spaces
• Quasi-morphisms on contactomorphism groups and contact rigidity
• Quasi-states, quasi-morphisms, and the moment map
• Quasi-morphisms and symplectic quasi-states for convex symplectic manifolds
• On bi-invariant word metrics
• Descent and C^0-rigidity of spectral invariants on monotone symplectic manifolds
• New energy-capacity-type inequalities and uniqueness of continuous Hamiltonians
• Quantitative Heegaard Floer cohomology and the Calabi invariant
• Unboundedness of the Lagrangian Hofer distance in the Euclidean ball
• Hofer's length spectrum of symplectic surfaces
• The Nonlinear Maslov index and the Calabi homomorphism
• A Family of Quasimorphism Constructions