Extending higher-dimensional quasi-cocycles

Let G be a group admitting a non-elementary acylindrical action on a Gromov hyperbolic space (for example, a non-elementary relatively hyperbolic group, or the mapping class group of a closed hyperbolic surface, or Out(F_n) for n ≥ 2). We prove that, in degree 3, the bounded cohomology of G with real coefficients is infinite-dimensional. Our proof is based on an extension to higher degrees of a recent result by Hull and Osin. Namely, we prove that if H is a hyperbolically embedded subgroup of G and V is any R[G]-module, then any n-quasi-cocycle on H with values in V may be extended to G. Also, we show that our extensions detect the geometry of the embedding of hyperbolically embedded subgroups in a suitable sense.