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.
Infinite sums of Brooks quasimorphisms and cup products in bounded cohomology
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abstract
We study a class of quasimorphisms of the free group that can be expressed as infinite sums of Brooks quasimorphisms with some nice properties. We then review Heuer's framework of decompositions developed in arXiv:1710.03193, and put these quasimorphisms in that context. This allows us to prove triviality of a number of cup products in the bounded cohomology of the free group with trivial real coefficients. Finally, we introduce free products of decompositions and relate them to Rolli's quasimorphisms of free products. A technical result concerning these, which remains an open question, implies a strong triviality result for the cup product.
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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.