Aut(G) acts on homogeneous quasimorphisms of an acylindrically hyperbolic group G with kernel the strongly commensurating automorphisms; when G has no non-trivial finite normal subgroups, quasimorphisms recognize inner automorphisms, so Out(G) acts faithfully on the kernel of the comparison map in H
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On automorphisms, quasimorphisms, and coarse automorphisms of acylindrically hyperbolic groups
Aut(G) acts on homogeneous quasimorphisms of an acylindrically hyperbolic group G with kernel the strongly commensurating automorphisms; when G has no non-trivial finite normal subgroups, quasimorphisms recognize inner automorphisms, so Out(G) acts faithfully on the kernel of the comparison map in H