Quasimorphisms 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 quasimorphisms. 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.

Givental’s Non-linear Maslov Index on Lens Spaces

“…The proof of the following result is a direct imitation of the proof of the similar statement for real projective spaces that is given in [BZ15].…”

“…For the next result we also follow [BZ15]. Recall that a subset U of a contact manifold (V, ξ) is said to be displaceable if there exists a contactomorphism ψ contact isotopic to the identity such that U ∩ ψ(U) = ∅.…”

“…Weinstein conjecture. Consider a contact manifold (V, ξ), and suppose that it is equipped with a non-trivial monotone quasimorphism ν : Cont 0 (V, ξ) → R. Following [BZ15] we say that a subset Y of V is subheavy with respect to ν if ν vanishes on all elements that can be represented by a contact isotopy generated by an autonomous Hamiltonian that vanishes on Y . Suppose now that (V, ξ) is also equipped with a contact T n -action, and denote by f α : V → R n the moment map with respect to a T n -invariant contact form α for ξ.…”

“…Ben Simon [BeS07]). While quasimorphisms on Hamiltonian groups were studied by several authors, starting with Biran, Entov and Polterovich [BEP04] and Entov and Polterovich [EnP03], Givental's non-linear Maslov index and its reductions studied by Borman and Zapolsky [BZ15] are the only known non-trivial quasimorphisms on contactomorphism groups.…”

“…In [BZ15] Borman and Zapolsky showed that, in analogy with the symplectic case [Bo13], in certain situations monotone quasimorphisms descend under contact reduction; starting from Givental's non-linear Maslov index on RP 2n−1 they thus obtained induced quasimorphisms on those contact toric manifolds that can be written in a certain way as contact reductions of RP 2n−1 . Moreover, it is proved in [BZ15] that if a contact manifold admits a non-trivial monotone quasimorphism then it is orderable, and if the prequantization of a symplectic toric manifold admits a non-trivial monotone quasimorphism with the vanishing property (a property that is preserved under contact reduction) then it has a non-displaceable pre-Lagrangian toric fibre. As already observed in [BZ15, Remark 1.5], our generalization to lens spaces of Givental's non-linear Maslov index allows us to extend the class of contact toric manifolds that inherit, by contact reduction, a quasimorphism.…”