Bounded cohomology is an invariant of groups and spaces developed by Gromov in the 80’s. Despite its purely topological definition, it turns out to have deep relations with geometric properties of the spaces or algebraic properties of the groups under consideration. In particular, its vanishing for a large family of coefficients modules allows to characterize amenability.

In joint work with Fournier-Facio, Lodha and Moraschini, we develop a new criterion for the vanishing of bounded cohomology for a subfamily of coefficients modules and apply it to a variety of examples of groups of topological, geometric and dynamical origin. We also remark an interesting relationship with homological stability of these families of groups.