Construction of quasi-morphisms on groups of measure-preserving homeomorphisms of non-orientable surfaces

A quasi-morphism on a group G​​ is a real-valued function which satisfies the homomorphism equation up to a bounded error. Let N_g​​ be a non-orientable surface of genus g≥ 3​​ and let Homeo₀(N_g,μ)​​ be the identity component of the group of measure-preserving homeomorphisms of N_g​​.
We prove that the space of homogeneous quasi-morphisms on the group Homeo₀(N_g,μ)​​ is infinite-dimensional. This project is part of an M.Sc. thesis under the supervision of Dr. Brandenbursky.