Quasi-morphisms on Free Groups

Abstract

Let F be the free group over a set of two or more generators. In [2] R. Brooks constructed an infinite family of quasi-morphisms F -> R such that an infinite subfamily gives rise to independent classes in the second bounded cohomology H2b(F,R), which proves that this space is infinite dimensional, cf. [7]. We give a simpler proof of this fact using a different type of quasi-morphisms. After computing the Gromov norm of the corresponding bounded classes, we generalize our example to obtain quasi-morphisms on free products, as well as quasi-morphisms into groups without small subgroups, also known as ε-representations.