Quasi-morphisms and symplectic quasi-states for convex symplectic manifolds

TL;DR

This paper extends the construction of symplectic quasi-states and quasi-morphisms to certain non-closed convex symplectic manifolds using quantum and Floer homology, building on prior results for closed manifolds.

Contribution

It introduces new methods to construct (partial) quasi-morphisms and quasi-states on non-closed convex symplectic manifolds, expanding the scope of previous work.

Findings

Constructed (partial) quasi-morphisms on Hamiltonian diffeomorphism groups.

Developed (partial) symplectic quasi-states on functions constant near infinity.

Extended results from closed to certain non-closed symplectic manifolds.

Abstract

We use quantum and Floer homology to construct (partial) quasi-morphisms on the universal cover of the group of compactly supported Hamiltonian diffeomorphisms for a certain class of non-closed strongly semi-positive symplectic manifolds $(M,\omega)$. This leads to construction of (partial) symplectic quasi-states on the space of continuous functions on $M$ that are constant near infinity. The work extends the results by Entov and Polterovich, which apply in the closed case.