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We study a class of quasimorphisms of the free group that can be expressed asinfinite sums of Brooks quasimorphisms with some nice properties. We thenreview Heuer's framework of decompositions developed in arXiv:1710.03193, andput these quasimorphisms in that context. This allows us to prove triviality ofa number of cup products in the bounded cohomology of the free group withtrivial real coefficients. Finally, we introduce free products ofdecompositions and relate them to Rolli's quasimorphisms of free products. Atechnical result concerning these, which remains an open question, implies astrong triviality result for the cup product.

Infinite sums of Brooks quasimorphisms and cup products in bounded cohomology