Bounded Cohomology and Acylindrical Hyperbolicity for the Automorphism Group of Free Groups

This monograph investigates the structural interplay between the acylindrical hyperbolicity of the automorphism group of a free group, denoted Aut(Fn), and its second bounded cohomology group H2b(Aut(Fn); R). Leveraging the seminal constructions of hyperbolic complexes by Bestvina and Feighn, and the generalized acylindrical framework established by Osin, we provide a rigorous derivation of the infinite dimensionality of H2b(Aut(Fn); R) for n 2. We examine the action of Aut(Fn) on the complex of free factors and the associated weak proper discontinuity (WPD) of fully irreducible automorphisms. Furthermore, we discuss the implications of these geometric actions for the existence of Brooks-type quasi-morphisms, establishing a robust connection between the large-scale geometry of Outer space and the cohomological rigidity phenomena—or lack thereof—within the automorphism group. The results underscore the divergence of Aut(Fn) from higher-rank lattice rigidity, aligning it more closely with the mapping class groups of surfaces.