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Using a probabilistic argument we show that the second bounded cohomology of a finitely-generated acylindrically hyperbolic group (e.g., a non-elementary hyperbolic or relatively hyperbolic group, non-exceptional mapping class group, Out embeds via the natural restriction maps into the inverse limit of the second bounded cohomologies of its virtually free subgroups, and in fact even into the inverse limit of the second bounded cohomologies of its hyperbolically embedded virtually free subgroups. This result is new and non-trivial even in the case where is a (non-free) hyperbolic group. The corresponding statement fails in general for the third bounded cohomology, even for surface groups.
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Tobias Hartnick, Alessandro Sisto, Bounded cohomology and virtually free hyperbolically embedded subgroups. Groups Geom. Dyn. 13 (2019), no. 2, pp. 677–694