Quasi-morphisms on contactomorphism groups and contact rigidity

TL;DR

This paper constructs new homogeneous quasi-morphisms on contactomorphism groups of certain contact manifolds, revealing contact rigidity phenomena and applications to orderability and metrics.

Contribution

It introduces a method to build quasi-morphisms using Givental's index and contact reduction, advancing understanding of contact rigidity and group structures.

Findings

Existence of homogeneous quasi-morphisms on specific contactomorphism groups

Identification of rigid subsets within contact manifolds

Applications to contact manifold orderability and metrics

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.