On automorphisms, quasimorphisms, and coarse automorphisms of acylindrically hyperbolic groups

Ashot Minasyan; Alessandro Sisto; Federico Vigolo

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arXiv · 2509.04238

On automorphisms, quasimorphisms, and coarse automorphisms of acylindrically hyperbolic groups

Ashot Minasyan·Alessandro Sisto·Federico Vigolo

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Abstract

We investigate the action of the automorphism group of an acylindrically hyperbolic group G on its space of homogeneous quasimorphisms, and identify its kernel with the subgroup of "strongly commensurating" automorphisms. We deduce that if G has no non-trivial finite normal subgroups then it has sufficiently many quasimorphisms to recognize whether an automorphism is inner. As consequences, we show that Out(G) acts faithfully on the kernel of the comparison map in bounded cohomology and it embeds in (several) groups of coarse automorphisms.

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Ashot Minasyan, Alessandro Sisto, Federico Vigolo. 2025-09-04. On automorphisms, quasimorphisms, and coarse automorphisms of acylindrically hyperbolic groups. https://arxiv.org/abs/2509.04238

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