Quasi-morphisms on Free Groups

TL;DR

This paper simplifies the proof that the second bounded cohomology of free groups is infinite dimensional by introducing a new type of quasi-morphisms, and extends these results to free products and groups without small subgroups.

Contribution

It provides a simpler proof of the infinite dimensionality of bounded cohomology for free groups and generalizes the construction to free products and epsilon-representations.

Findings

Simplified proof of infinite-dimensional bounded cohomology for free groups.

Explicit computation of Gromov norms of the bounded classes.

Extension of quasi-morphisms to free products and groups without small subgroups.

Abstract

Let F be the free group over a set of two or more generators. R. Brooks constructed an infinite family of quasi-morphisms on F such that an infinite subfamily gives rise to independent classes in the second bounded cohomology of F, which proves that this space is infinite dimensional. 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 epsilon-representations.