a Quasi - morphism ?

208 NOTICES OF THE AMS VOLUME 51, NUMBER 2 As far as I know, the notion of a quasi-morphism does not have much to do with category theory. This very natural idea underlies several interesting recent developments at the crossroads of algebra, topology, geometry, and dynamics. Other, perhaps better, names currently in use for the same concept are quasi-homomorphism and pseudo-character. Let G be a group. A map f :G → R is called a quasi-morphism if its deviation from being a homomorphism is bounded; in other words, there exists a constant D(f ), called the defect of f, such that |f (xy)− f (x)− f (y)| ≤ D(f ) for all x, y ∈ G. The most obvious examples of quasi-morphisms are of course homomorphisms and arbitrary bounded maps. To avoid trivialities associated with the latter and to make subsequent arguments neater, one usually passes to homogeneous quasi-morphisms. Every quasi-morphism can be homogenized by defining