On automorphisms, quasimorphisms, and coarse automorphisms of acylindrically hyperbolic groups
On automorphisms, quasimorphisms, and coarse automorphisms of acylindrically hyperbolic groups
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.
Quasimorphisms, Outer Automorphisms, Coarse Automorphisms, Acylindrical Hyperbolicity, (Strongly) Commensurable Elements
Minasyan, Ashot
3de640f5-d07b-461f-b130-5b1270bfdb3d
Sisto, Alessandro
6ff3aa0e-7627-4918-bc1c-416da947dc8c
Vigolo, Federico
914e52c5-0729-42c2-b487-52f2292f1712
Minasyan, Ashot
3de640f5-d07b-461f-b130-5b1270bfdb3d
Sisto, Alessandro
6ff3aa0e-7627-4918-bc1c-416da947dc8c
Vigolo, Federico
914e52c5-0729-42c2-b487-52f2292f1712
Minasyan, Ashot, Sisto, Alessandro and Vigolo, Federico
(2026)
On automorphisms, quasimorphisms, and coarse automorphisms of acylindrically hyperbolic groups.
Commentarii Mathematici Helvetici.
(In Press)
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.
Text
Out_and_Quasimorphisms--April_20_2026
- Accepted Manuscript
More information
Accepted/In Press date: 18 April 2026
Keywords:
Quasimorphisms, Outer Automorphisms, Coarse Automorphisms, Acylindrical Hyperbolicity, (Strongly) Commensurable Elements
Identifiers
Local EPrints ID: 512020
URI: http://eprints.soton.ac.uk/id/eprint/512020
ISSN: 0010-2571
PURE UUID: d60a91e3-12c7-46d9-b525-e12f1d7d5cf7
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Date deposited: 12 Jun 2026 16:31
Last modified: 19 Aug 2026 06:09
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Contributors
Author:
Alessandro Sisto
Author:
Federico Vigolo
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