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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Quasimorphisms on contactomorphism groups and contact rigidity

Matthew Strom Borman and Frol Zapolsky

Geometry & Topology 19 (2015) 365–411
Abstract

We build homogeneous quasimorphisms 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 quasimorphisms 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.

Keywords
quasimorphism, contactomorphism, contact rigidity, nonlinear Maslov index, prequantization, toric
Mathematical Subject Classification 2010
Primary: 53D35
Secondary: 53D12, 53D20
References
Publication
Received: 15 August 2013
Revised: 21 January 2014
Accepted: 30 January 2014
Published: 27 February 2015
Proposed: Yasha Eliashberg
Seconded: Danny Calegari, Walter Neumann
Authors
Matthew Strom Borman
Department of Mathematics
Stanford University
Building 380
Stanford, CA 94305
USA
Frol Zapolsky
Department of Mathematics
University of Haifa
Mount Carmel
Haifa 31905
Israel