Quasi-morphisms via Local Action Data and Quasi-isometries

In this work we first present a principle which says that quasi-morphisms can be obtained via local data of group action on certain appropriate spaces. In a rough manner the principle says that instead of starting with a given group and trying to build or study its space of quasi-morphisms, we should start with a space with a certain structure, in such a way that groups acting on this space and respecting this structure will automatically carry quasi-morphisms, where these are supposed to be better understood. This principle plays an important role in the second result of this paper, which is a universal embedding of the projective space of the linear space of quasi-morphisms of any given countable group, into the space of quasi-isometrics of a certain universal metric space.