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

Quasi-morphisms on contactomorphism groups and contact rigidity