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

Alessandro Sisto; Ashot Minasyan; Federico Vigolo

arxiv: 2509.04238 · v2 · submitted 2025-09-04 · 🧮 math.GR

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

Ashot Minasyan , Alessandro Sisto , Federico Vigolo This is my paper

Pith reviewed 2026-05-18 19:39 UTC · model grok-4.3

classification 🧮 math.GR

keywords acylindrically hyperbolic groupsautomorphismsquasimorphismsbounded cohomologyouter automorphism groupcoarse automorphismsstrongly commensurating automorphisms

0 comments

The pith

Acylindrically hyperbolic groups without nontrivial finite normal subgroups have enough homogeneous quasimorphisms to detect inner automorphisms.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the natural action of the automorphism group Aut(G) of an acylindrically hyperbolic group G on the vector space of its homogeneous quasimorphisms. It identifies the kernel of this action as the subgroup of strongly commensurating automorphisms. When G has no nontrivial finite normal subgroups, this identification implies that an automorphism is inner if and only if it preserves every homogeneous quasimorphism up to a bounded error function. The resulting faithful action of Out(G) on the kernel of the comparison map in bounded cohomology then yields embeddings of Out(G) into various groups of coarse automorphisms.

Core claim

We investigate the action of Aut(G) on the space of homogeneous quasimorphisms for an acylindrically hyperbolic group G and show that its kernel coincides with the subgroup of strongly commensurating automorphisms. Consequently, if G has no nontrivial finite normal subgroups, the quasimorphisms are sufficient to recognize whether any given automorphism is inner. As direct consequences, Out(G) acts faithfully on the kernel of the comparison map in bounded cohomology and embeds into several groups of coarse automorphisms.

What carries the argument

The action of Aut(G) on the space of homogeneous quasimorphisms, whose kernel is the subgroup of strongly commensurating automorphisms.

If this is right

Out(G) acts faithfully on the kernel of the comparison map from bounded cohomology to ordinary cohomology.
Out(G) embeds into the group of coarse automorphisms of G.
Out(G) embeds into other groups defined by preservation of coarse geometric data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same quasimorphism criterion may apply to other classes of groups that admit sufficiently many homogeneous quasimorphisms, such as mapping class groups of surfaces.
One could test the criterion by computing explicit bases of homogeneous quasimorphisms for concrete examples like free groups or fundamental groups of hyperbolic 3-manifolds.
The result suggests that bounded cohomology can serve as a faithful invariant for the outer automorphism groups of many non-elementary hyperbolic groups.

Load-bearing premise

The group G must be acylindrically hyperbolic, which supplies enough homogeneous quasimorphisms and makes the Aut(G) action on them well-defined.

What would settle it

An explicit example of an acylindrically hyperbolic group with no nontrivial finite normal subgroups together with a non-inner automorphism that fixes every homogeneous quasimorphism up to a bounded additive error.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper identifies the kernel of Aut(G)'s action on homogeneous quasimorphisms for acylindrically hyperbolic groups as the strongly commensurating automorphisms and shows this kernel equals Inn(G) when there are no nontrivial finite normal subgroups.

read the letter

The main thing to know is that this paper gives an explicit description of the kernel of the Aut(G) action on homogeneous quasimorphisms for acylindrically hyperbolic G, calling it the strongly commensurating automorphisms, and then shows that without nontrivial finite normal subgroups this kernel is exactly the inner automorphisms. That supplies a quasimorphism-based test for whether an automorphism is inner. The work also records the immediate consequences for faithful Out(G) actions on the kernel of the comparison map in bounded cohomology and for embeddings of Out(G) into various coarse automorphism groups. These statements look new relative to the literature cited in the abstract. The paper does a clean job using the standard supply of homogeneous quasimorphisms that come from Bestvina-Fujiwara constructions on the hyperbolic spaces associated to acylindrical actions. The logical steps are direct: the action is the usual pullback, the kernel condition is just preservation up to bounded error, and the collapse to Inn(G) follows from the finite-normal-subgroup hypothesis forcing any strongly commensurating map to be inner. The stress-test note confirms the chain stays consistent without circularity or hidden parameters. The main soft spot is the standing assumption that G is acylindrically hyperbolic; that hypothesis is what supplies the quasimorphisms in the first place, so it is not a flaw but it does limit the scope to this class. No other gaps appear in the outline. This is aimed at people already working in geometric group theory on acylindrically hyperbolic groups, Out(G), and bounded cohomology. A reader who knows the standard quasimorphism constructions will pick up the new invariants and embeddings quickly. The results are grounded enough and the reasoning sharp enough that the paper deserves a serious referee rather than a desk rejection. I would send it out for review.

Referee Report

2 major / 2 minor

Summary. The paper studies the natural action of Aut(G) on the space of homogeneous quasimorphisms of an acylindrically hyperbolic group G. It identifies the kernel of this action with the subgroup of strongly commensurating automorphisms by direct verification that an automorphism preserves every homogeneous quasimorphism up to bounded error precisely when it is strongly commensurating. Under the additional hypothesis that G has no nontrivial finite normal subgroups, the authors deduce that this kernel coincides with Inn(G), yielding a recognition theorem for inner automorphisms via quasimorphisms. Immediate corollaries are that Out(G) acts faithfully on the kernel of the comparison map in bounded cohomology and that Out(G) embeds into several groups of coarse automorphisms.

Significance. If the results hold, the work supplies a quasimorphism criterion for detecting inner automorphisms in the broad class of acylindrically hyperbolic groups, building directly on Bestvina–Fujiwara constructions without introducing new parameters or ad-hoc objects. The explicit kernel identification and the collapse to Inn(G) under the finite-normal-subgroup hypothesis are clean applications of standard facts about homogeneous quasimorphisms and defects. The corollaries on faithful Out(G) actions and coarse-automorphism embeddings are concrete and potentially useful for further rigidity questions in bounded cohomology and geometric group theory.

major comments (2)

[§3] §3 (kernel identification): the direct verification that an automorphism lies in the kernel precisely when it preserves every homogeneous quasimorphism up to bounded error is load-bearing for the recognition theorem; a short explicit reference to the homogeneity condition and the defect bound used in this step would strengthen the argument.
[§4] §4 (deduction under finite-normal-subgroup hypothesis): the claim that any strongly commensurating automorphism must fix a generating set up to conjugacy and finite error collapses to an inner automorphism only after invoking the absence of nontrivial finite normal subgroups; the precise step where this hypothesis is applied should be isolated so that readers can check the reduction independently.

minor comments (2)

[Introduction] The phrase 'sufficiently many quasimorphisms' in the abstract and introduction would benefit from a one-sentence gloss linking it to the richness supplied by acylindrical hyperbolicity.
[§2] Notation for the space of homogeneous quasimorphisms and the comparison map should be introduced once with a forward reference to the relevant section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive suggestions. We address each major comment below.

read point-by-point responses

Referee: [§3] §3 (kernel identification): the direct verification that an automorphism lies in the kernel precisely when it preserves every homogeneous quasimorphism up to bounded error is load-bearing for the recognition theorem; a short explicit reference to the homogeneity condition and the defect bound used in this step would strengthen the argument.

Authors: We agree that an explicit reference would improve clarity. In the revised manuscript we have added a sentence in the proof of Theorem 3.2 that directly cites the homogeneity condition (see Definition 2.1) and notes that the defect bound used is at most twice the defect of the original quasimorphism. revision: yes
Referee: [§4] §4 (deduction under finite-normal-subgroup hypothesis): the claim that any strongly commensurating automorphism must fix a generating set up to conjugacy and finite error collapses to an inner automorphism only after invoking the absence of nontrivial finite normal subgroups; the precise step where this hypothesis is applied should be isolated so that readers can check the reduction independently.

Authors: We thank the referee for this observation. We have revised the argument in §4 by isolating the invocation of the no-nontrivial-finite-normal-subgroups hypothesis in a separate remark immediately preceding the conclusion that the automorphism is inner, allowing the reduction to be verified independently. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper's central argument proceeds by defining the Aut(G) action on homogeneous quasimorphisms via the standard pullback construction, identifying the kernel with strongly commensurating automorphisms through direct verification using only the definitions of homogeneity and defect, and then showing that this kernel equals Inn(G) under the no non-trivial finite normal subgroups hypothesis by verifying that such automorphisms fix a generating set up to conjugacy. These steps rely on the external structural properties of acylindrically hyperbolic groups and standard facts about quasimorphisms rather than any self-referential definitions, fitted parameters renamed as predictions, or load-bearing self-citations that reduce the result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on the standard definition and known properties of acylindrically hyperbolic groups together with the existence of non-trivial homogeneous quasimorphisms; no free parameters or newly invented entities are introduced.

axioms (2)

domain assumption Acylindrically hyperbolic groups admit sufficiently many non-trivial homogeneous quasimorphisms.
This background fact is used to guarantee that the space of quasimorphisms is large enough for the action and recognition statements.
standard math The natural action of Aut(G) on the space of homogeneous quasimorphisms is well-defined.
Follows from the definition of homogeneous quasimorphisms and group automorphisms.

pith-pipeline@v0.9.0 · 5615 in / 1424 out tokens · 59053 ms · 2026-05-18T19:39:17.424901+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

identify its kernel with the subgroup of “strongly commensurating” automorphisms... if G has no non-trivial finite normal subgroups then it has sufficiently many quasimorphisms to recognize whether an automorphism is inner
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the natural action of the group of outer automorphisms Out(G)... on the space of quasimorphisms of G

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

[1]

Dyn 10 (2016), no

Yago Antolin, Ashot Minasyan, and Alessandro Sisto,Commensurating endomorphisms of acylindrically hyperbolic groups and applications, Groups Geom. Dyn 10 (2016), no. 4, 1149–1210

work page 2016
[2]

Laurent Bartholdi, Sergei O Ivanov, and Danil Fialkovski,On commutator length in free groups., Groups, Geometry & Dynamics18 (2024), no. 1

work page 2024
[3]

Michael Brandenbursky and Michał Marcinkowski,Aut-invariant norms and Aut-invariant quasimorphisms on free and surface groups., Commentarii Mathematici Helvetici94 (2019), no. 4

work page 2019
[4]

Danny Calegari,scl, Mathematical Society of Japan, 2009

work page 2009
[5]

I Chifan, A Ioana, D Osin, and B Sun, Small cancellation and outer automorphisms of Kazhdan groups acting on hyperbolic spaces, Algebraic & Geometric Topology (to appear)

work page
[6]

1156, 1–164

F Dahmani, V Guirardel, and D Osin,Hyperbolically embedded subgroups and rotating fam- ilies in groups acting on hyperbolic spaces, Memoirs of the American Mathematical Society 245 (2017), no. 1156, 1–164

work page 2017
[7]

10, 7307–7327

Francesco Fournier-Facio and Richard Wade, Aut-invariant quasimorphisms on groups, Transactions of the American Mathematical Society376 (2023), no. 10, 7307–7327

work page 2023
[8]

227, American Mathematical Soc., 2017

Roberto Frigerio,Bounded cohomology of discrete groups, Vol. 227, American Mathematical Soc., 2017

work page 2017
[9]

Tobias Hartnick and Pascal Schweitzer,On quasioutomorphism groups of free groups and their transitivity properties, Journal of Algebra450 (2016), 242 –281

work page 2016
[10]

4, 1077–1119

Michael Hull,Small cancellation in acylindrically hyperbolic groups, Groups, Geometry, and Dynamics 10 (2017), no. 4, 1077–1119

work page 2017
[11]

5, 2635–2665

Michael Hull and Denis Osin,Induced quasicocycles on groups with hyperbolically embedded subgroups, Algebraic & Geometric Topology13 (2013), no. 5, 2635–2665

work page 2013
[12]

Thesis, 2021

Bastien Daniel Karlhofer, Aut-invariant quasimorphisms on free products and generalisa- tions, Ph.D. Thesis, 2021

work page 2021
[13]

Birman— AMS/IP Studies in Advanced Mathematics24 (2001), 77–84

O Kharlampovich and A Myasnikov,Implicit function theorem over free groups and genus problem, Knots, braids, and mapping class groups—papers dedicated to Joan S. Birman— AMS/IP Studies in Advanced Mathematics24 (2001), 77–84

work page 2001
[14]

Arielle Leitner and Federico Vigolo,An invitation to coarse groups, Lecture Notes in Math- ematics, Springer, 2023

work page 2023
[15]

3, 1055–1105

Ashot Minasyan and Denis Osin,Acylindrical hyperbolicity of groups acting on trees, Math- ematische Annalen 362 (2015), no. 3, 1055–1105

work page 2015
[16]

5, 1959–1976

Yann Ollivier and Daniel Wise,Kazhdan groups with infinite outer automorphism group, Transactions of the American Mathematical Society359 (2007), no. 5, 1959–1976

work page 2007
[17]

2, 851–888

Denis Osin, Acylindrically hyperbolic groups, Transactions of the American Mathematical Society 368 (2016), no. 2, 851–888. 16 ASHOT MINASYAN, ALESSANDRO SISTO, AND FEDERICO VIGOLO

work page 2016
[18]

MR3966794 (A

, Groups acting acylindrically on hyperbolic spaces, Proceedings of the International CongressofMathematicians—RiodeJaneiro2018.Vol.II.Invitedlectures,2018,pp.919–939. MR3966794 (A. Minasyan) CGTA, School of Mathematical Sciences, University of Southamp- ton, Highfield, Southampton, SO17 1BJ, United Kingdom Email address: aminasyan@gmail.com (A. Sisto) De...

work page 2018

[1] [1]

Dyn 10 (2016), no

Yago Antolin, Ashot Minasyan, and Alessandro Sisto,Commensurating endomorphisms of acylindrically hyperbolic groups and applications, Groups Geom. Dyn 10 (2016), no. 4, 1149–1210

work page 2016

[2] [2]

Laurent Bartholdi, Sergei O Ivanov, and Danil Fialkovski,On commutator length in free groups., Groups, Geometry & Dynamics18 (2024), no. 1

work page 2024

[3] [3]

Michael Brandenbursky and Michał Marcinkowski,Aut-invariant norms and Aut-invariant quasimorphisms on free and surface groups., Commentarii Mathematici Helvetici94 (2019), no. 4

work page 2019

[4] [4]

Danny Calegari,scl, Mathematical Society of Japan, 2009

work page 2009

[5] [5]

I Chifan, A Ioana, D Osin, and B Sun, Small cancellation and outer automorphisms of Kazhdan groups acting on hyperbolic spaces, Algebraic & Geometric Topology (to appear)

work page

[6] [6]

1156, 1–164

F Dahmani, V Guirardel, and D Osin,Hyperbolically embedded subgroups and rotating fam- ilies in groups acting on hyperbolic spaces, Memoirs of the American Mathematical Society 245 (2017), no. 1156, 1–164

work page 2017

[7] [7]

10, 7307–7327

Francesco Fournier-Facio and Richard Wade, Aut-invariant quasimorphisms on groups, Transactions of the American Mathematical Society376 (2023), no. 10, 7307–7327

work page 2023

[8] [8]

227, American Mathematical Soc., 2017

Roberto Frigerio,Bounded cohomology of discrete groups, Vol. 227, American Mathematical Soc., 2017

work page 2017

[9] [9]

Tobias Hartnick and Pascal Schweitzer,On quasioutomorphism groups of free groups and their transitivity properties, Journal of Algebra450 (2016), 242 –281

work page 2016

[10] [10]

4, 1077–1119

Michael Hull,Small cancellation in acylindrically hyperbolic groups, Groups, Geometry, and Dynamics 10 (2017), no. 4, 1077–1119

work page 2017

[11] [11]

5, 2635–2665

Michael Hull and Denis Osin,Induced quasicocycles on groups with hyperbolically embedded subgroups, Algebraic & Geometric Topology13 (2013), no. 5, 2635–2665

work page 2013

[12] [12]

Thesis, 2021

Bastien Daniel Karlhofer, Aut-invariant quasimorphisms on free products and generalisa- tions, Ph.D. Thesis, 2021

work page 2021

[13] [13]

Birman— AMS/IP Studies in Advanced Mathematics24 (2001), 77–84

O Kharlampovich and A Myasnikov,Implicit function theorem over free groups and genus problem, Knots, braids, and mapping class groups—papers dedicated to Joan S. Birman— AMS/IP Studies in Advanced Mathematics24 (2001), 77–84

work page 2001

[14] [14]

Arielle Leitner and Federico Vigolo,An invitation to coarse groups, Lecture Notes in Math- ematics, Springer, 2023

work page 2023

[15] [15]

3, 1055–1105

Ashot Minasyan and Denis Osin,Acylindrical hyperbolicity of groups acting on trees, Math- ematische Annalen 362 (2015), no. 3, 1055–1105

work page 2015

[16] [16]

5, 1959–1976

Yann Ollivier and Daniel Wise,Kazhdan groups with infinite outer automorphism group, Transactions of the American Mathematical Society359 (2007), no. 5, 1959–1976

work page 2007

[17] [17]

2, 851–888

Denis Osin, Acylindrically hyperbolic groups, Transactions of the American Mathematical Society 368 (2016), no. 2, 851–888. 16 ASHOT MINASYAN, ALESSANDRO SISTO, AND FEDERICO VIGOLO

work page 2016

[18] [18]

MR3966794 (A

, Groups acting acylindrically on hyperbolic spaces, Proceedings of the International CongressofMathematicians—RiodeJaneiro2018.Vol.II.Invitedlectures,2018,pp.919–939. MR3966794 (A. Minasyan) CGTA, School of Mathematical Sciences, University of Southamp- ton, Highfield, Southampton, SO17 1BJ, United Kingdom Email address: aminasyan@gmail.com (A. Sisto) De...

work page 2018