For the groups G(F,F'), if F' is 2-transitive then the group is boundedly acyclic; otherwise its second bounded cohomology is infinite dimensional.
Trivial Massey Product in Bounded Cohomology
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abstract
We show that in the bounded cohomology of non-abelian free groups the Massey triple product is always trivial when the second factor is represented by the coboundary of a decomposable quasi-morphism. We also show that in the bounded cohomology of a negatively curved compact Riemannian manifold the Massey triple product is always trivial when the second factor is represented by an exact differential form.
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Bounded cohomology of groups acting on trees with almost prescribed local actions
For the groups G(F,F'), if F' is 2-transitive then the group is boundedly acyclic; otherwise its second bounded cohomology is infinite dimensional.