Quasi-morphisms on contactomorphism groups and contact rigidity

Abstract

We build homogeneous quasi-morphisms on the universal cover of the contactomorphism group for certain prequantizations of monotone symplectic toric manifolds. This is done using Givental's nonlinear Maslov index and a contact reduction technique for quasi-morphisms. We show how these quasi-morphisms lead to a hierarchy of rigid subsets of contact manifolds. We also show that the nonlinear Maslov index has a vanishing property, which plays a key role in our proofs. Finally we present applications to orderability of contact manifolds and Sandon-type metrics on contactomorphism groups.

40 pages, 1 figure; v3: added proof of C^0 continuity, minor corrections. To appear in Geometry & Topology

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Cited by in corpus (9)

• Contact Hamiltonian Systems
• Universal orderability of Legendrian isotopy classes
• Quasi-morphisms and quasi-states in symplectic topology
• Vanishing of Rabinowitz Floer homology on negative line bundles
• Givental's non-linear Maslov index on lens spaces
• Contactomorphism groups and Legendrian flexibility
• Bi-invariant metrics on contactomorphism groups
• Rabinowitz Floer homology of negative line bundles and Floer Gysin sequence
• Translated points for contactomorphisms of prequantization spaces over monotone symplectic toric manifolds