The conjugacy problem in Out(Fm) when the polynomial restrictions are non-growing

Gabriel Bartlett (IF)

arxiv: 2507.13050 · v3 · submitted 2025-07-17 · 🧮 math.GR

The conjugacy problem in Out(Fm) when the polynomial restrictions are non-growing

Gabriel Bartlett (IF) This is my paper

Pith reviewed 2026-05-19 04:40 UTC · model grok-4.3

classification 🧮 math.GR

keywords conjugacy problemOut(F_m)outer automorphismsfree groupspolynomial subgroupsfinite ordersuspension groupsdecision problems

0 comments

The pith

The conjugacy problem in Out(F_m) is solvable for outer automorphisms whose restrictions to polynomial subgroups have finite order.

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

The paper establishes that the conjugacy problem in the outer automorphism group of a free group F_m is solvable for the class of outer automorphisms where the restriction to the polynomial subgroup is finite order. It reaches this by first analyzing the structure of suspensions of free groups by automorphisms of finite outer order. The conjugacy question then reduces to certain solvable problems on those suspension groups. A sympathetic reader would care because conjugacy is a central algorithmic question for Out(F_m), which encodes symmetries of free groups and connects to many decision problems in geometric group theory.

Core claim

We prove that the conjugacy problem in Out(F_m) is solvable for the class of outer automorphisms whose restrictions to their polynomial subgroups are of finite order. To do this, we first investigate the structure of suspensions of free groups by automorphisms whose outer class is of finite order. We then apply a reduction of our main result to certain problems on groups of this form.

What carries the argument

Suspensions of free groups by automorphisms whose outer class is finite order, which carry the reduction of conjugacy to solvable problems on the suspension groups.

If this is right

Conjugacy becomes decidable for this class of outer automorphisms in Out(F_m).
Certain problems on the suspension groups are solvable.
This gives a concrete reduction that compares two such automorphisms up to conjugacy.
It enlarges the known solvable cases of the conjugacy problem in Out(F_m).

Where Pith is reading between the lines

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

The reduction technique may extend to deciding the isomorphism problem or other algorithmic questions for the same class.
It could connect to parallel results on decision problems in mapping class groups of surfaces.
Direct verification on low-rank cases such as F_2 or F_3 would test whether the suspension reduction works in practice.

Load-bearing premise

The structure of suspensions of free groups by automorphisms whose outer class is of finite order permits a reduction of the conjugacy problem to certain solvable problems on these suspension groups.

What would settle it

An explicit algorithm that decides conjugacy for such automorphisms by performing computations inside the associated suspension group, or a counterexample automorphism with finite-order polynomial restriction where no such decision procedure exists.

read the original abstract

We prove that the conjugacy problem in Out(Fm) is solvable for the class of outer automorphisms whose restrictions to their polynomial subgroups are of finite order. To do this, we first investigate the structure of suspensions of free groups by automorphisms whose outer class is of finite order. We then apply a reduction of our main result to certain problems on groups of this form.

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

This paper solves conjugacy for outer automorphisms in Out(F_m) whose polynomial restrictions have finite order by reducing to decidable problems on the associated suspension groups.

read the letter

The key takeaway is that this paper proves the conjugacy problem is solvable in Out(F_m) for outer automorphisms whose restrictions to polynomial subgroups have finite order. They do it by studying the structure of the corresponding suspension groups and reducing the conjugacy decision to a set of problems that are already known to be decidable. What is new here is the application of the suspension reduction to this specific class. The authors look at groups of the form F_m ⋊_φ ℤ where the outer automorphism class of φ has finite order. They establish structural properties of these groups and then show how conjugacy in Out(F_m) for the relevant automorphisms reduces to solvable questions on the suspensions, leveraging existing results for free-by-cyclic groups with finite-order monodromy. The paper handles the technical details well. The reduction is algorithmic, and the argument avoids obvious circularity or gaps according to the stress-test. This adds a solid piece to the decidability picture without overclaiming. The main limitation is scope. It only covers the subclass with finite-order polynomial restrictions, leaving the general conjugacy problem in Out(F_m) open. That's expected for this kind of work, but it means the impact is more incremental than transformative. The structural investigation might be the part that needs closest scrutiny in review, though no issues popped up in the initial check. This paper is aimed at researchers in geometric group theory who care about algorithmic problems in automorphism groups of free groups. Someone tracking partial results on conjugacy or working with suspensions and free-by-cyclic groups would get something out of it. It shows clear thinking and honest engagement with the literature. I think it deserves to go through peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that the conjugacy problem in Out(F_m) is solvable for outer automorphisms whose restrictions to their polynomial subgroups have finite order. The argument first establishes structural properties of the associated suspension groups F_m ⋊_φ ℤ (with the outer class of φ of finite order) and then reduces the conjugacy decision in Out(F_m) to a collection of explicitly solvable subproblems on these groups, using known decidability results for free-by-cyclic groups with finite-order monodromy.

Significance. If the central reduction holds, the result meaningfully enlarges the class of outer automorphisms of free groups for which the conjugacy problem is known to be solvable. The paper's explicit algorithmic reduction to decidable problems on suspensions is a strength, as it directly leverages and extends existing results on free-by-cyclic groups rather than introducing new undecidability barriers.

minor comments (2)

[Title and Abstract] The title refers to 'non-growing' polynomial restrictions while the abstract and body use 'finite order'; a brief clarification of the relationship (or equivalence) between these notions in §1 would remove potential reader confusion.
[Introduction] The reduction step is described at a high level in the abstract and introduction; adding a short diagram or numbered list of the subproblems in the main reduction theorem would improve readability without altering the argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, the assessment of significance, and the recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation first establishes structural properties of the suspension groups F_m ⋊_φ ℤ (with outer class of φ of finite order) and then reduces the conjugacy decision in Out(F_m) to explicitly solvable subproblems on those groups. The subproblems fall into decidable classes via existing independent results on free-by-cyclic groups with finite-order monodromy; the reduction is algorithmic and does not rely on self-definition, fitted inputs renamed as predictions, or load-bearing self-citations that collapse the central claim to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof relies on standard facts about free groups, outer automorphisms, and finite-order elements; no free parameters or new entities are introduced in the abstract.

axioms (1)

standard math Standard properties of free groups, outer automorphism groups, and finite-order elements
Invoked implicitly when discussing restrictions and suspensions of free groups.

pith-pipeline@v0.9.0 · 5575 in / 1200 out tokens · 36231 ms · 2026-05-19T04:40:37.575601+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/ArithmeticFromLogic.lean reality_from_one_distinction unclear

?

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

Theorem A: algorithm deciding conjugacy of outer automorphisms whose restrictions to polynomial subgroups are of finite order; reduction via Theorem 1.6 to sub-mapping tori P ⋊ α ⟨t⟩ with [α] finite order in Out(F_M)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

Proposition 1.10 and 2.4: center of F_m ⋊_φ ⟨t⟩ is infinite cyclic when [φ] finite order; quotient virtually free with trivial center and no nontrivial finite normal subgroups

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.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Free-by-cyclic groups are conjugacy separable
math.GR 2025-09 unverdicted novelty 8.0

All finitely generated free-by-cyclic groups are conjugacy separable, resolving Question 19.41 of the Kourovka Notebook and implying residual finiteness of their outer automorphism groups.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages · cited by 1 Pith paper

[1]

H. Bass. Covering theory for graphs of groups. Journal of Pure and Applied Algebra , 89:3--47, 1993

work page 1993
[2]

Bogopolski, A

O. Bogopolski, A. Martino, O. Maslakova, and E. Ventura. The conjugacy problem is solvable in free-by-cyclic groups. Bull. Lond. Math. Soc. , 38(5):787--794, 2006

work page 2006
[3]

Bridson and P

M. Bridson and P. Piwek. Profinite rigidity for free-by-cyclic groups with centre. Preprint , 2024

work page 2024
[4]

K.S. Brown. Cohomology of groups , volume 87 of Grad. Texts Math. Springer-Verlag, 1982

work page 1982
[5]

M. Culler. Finite groups of outer automorphisms of a free group. Contemporary mathematics , 33:197--207, 1984

work page 1984
[6]

F. Dahmani. On suspensions, and conjugacy of hyperbolic automorphisms. Trans. Am. Math. Soc. , 368(8):5565--5577, 2016

work page 2016
[7]

Dahmani, S

F. Dahmani, S. Francaviglia, A. Martino, and N. Touikan. The conjugacy problem for ( F _3) . Forum Math., Sigma , 13(e41):1--21, 2025

work page 2025
[8]

Dahmani and V

F. Dahmani and V. Guirardel. The isomorphism problem for all hyperbolic groups. Geom. Func. An. , 21:223--300, 2011

work page 2011
[9]

Dahmani and N

F. Dahmani and N. Touikan. Reducing the conjugacy problem for relatively hyperbolic automorphisms to peripheral components. Preprint , 2021

work page 2021
[10]

Dahmani and N

F. Dahmani and N. Touikan. Unipotent linear suspensions of free groups. Preprint , 2024

work page 2024
[11]

J. Dyer. Separating conjugates in free-by-finite groups. Journal of the London Mathematical Society , s. II(20):215--221, 1979

work page 1979
[12]

Eilenberg and S

S. Eilenberg and S. Maclane. Cohomology theory in abstract groups, II , group extensions with a non-abelian kernel. Ann. Math. , 48(2):326--341, 1947

work page 1947
[13]

Feighn and M

M. Feighn and M. Handel. The conjugacy problem for UPG elements of ( F _n) . Preprint , 2024

work page 2024
[14]

Grossman

E. Grossman. On the residual finiteness of certain mapping class groups. J. Lond. Math. Soc, II. Ser. , 9:160--164, 1974

work page 1974
[15]

Karrass, A

A. Karrass, A. Pietrowski, and D. Solitar. Finite and infinite cyclic extensions of free groups. Journal of the Australian Mathematical Society , 16:458--466, 1973

work page 1973
[16]

G. Levitt. Couting growth types of automorphisms of free groups. Geometric and Functonal Analysis , 19(4):1119--1146, 2009

work page 2009
[17]

Magnus, A

W. Magnus, A. Karrass, and D Solitar. Combinatorial group theory . Dover Publications Inc. Minolia, NY, 2004

work page 2004
[18]

Minasyan and D

A. Minasyan and D. Osin. Normal automorphisms of relatively hyperbolic groups. Trans. Am. Math. Soc. , 362(11):6079--6103, 2010

work page 2010
[19]

Z. Sela. The isomorphism problem for hyperbolic groups I . Annales of Mathematics , 141(2):217--283, 1995

work page 1995
[20]

Zimmermann

B. Zimmermann. Uber homöomorphismen n-dimensionaler henkelkörper und endliche erweiterungen von schottky-gruppen. Commentarii Mathematici Helvetici , 58:474--486, 1981

work page 1981

[1] [1]

H. Bass. Covering theory for graphs of groups. Journal of Pure and Applied Algebra , 89:3--47, 1993

work page 1993

[2] [2]

Bogopolski, A

O. Bogopolski, A. Martino, O. Maslakova, and E. Ventura. The conjugacy problem is solvable in free-by-cyclic groups. Bull. Lond. Math. Soc. , 38(5):787--794, 2006

work page 2006

[3] [3]

Bridson and P

M. Bridson and P. Piwek. Profinite rigidity for free-by-cyclic groups with centre. Preprint , 2024

work page 2024

[4] [4]

K.S. Brown. Cohomology of groups , volume 87 of Grad. Texts Math. Springer-Verlag, 1982

work page 1982

[5] [5]

M. Culler. Finite groups of outer automorphisms of a free group. Contemporary mathematics , 33:197--207, 1984

work page 1984

[6] [6]

F. Dahmani. On suspensions, and conjugacy of hyperbolic automorphisms. Trans. Am. Math. Soc. , 368(8):5565--5577, 2016

work page 2016

[7] [7]

Dahmani, S

F. Dahmani, S. Francaviglia, A. Martino, and N. Touikan. The conjugacy problem for ( F _3) . Forum Math., Sigma , 13(e41):1--21, 2025

work page 2025

[8] [8]

Dahmani and V

F. Dahmani and V. Guirardel. The isomorphism problem for all hyperbolic groups. Geom. Func. An. , 21:223--300, 2011

work page 2011

[9] [9]

Dahmani and N

F. Dahmani and N. Touikan. Reducing the conjugacy problem for relatively hyperbolic automorphisms to peripheral components. Preprint , 2021

work page 2021

[10] [10]

Dahmani and N

F. Dahmani and N. Touikan. Unipotent linear suspensions of free groups. Preprint , 2024

work page 2024

[11] [11]

J. Dyer. Separating conjugates in free-by-finite groups. Journal of the London Mathematical Society , s. II(20):215--221, 1979

work page 1979

[12] [12]

Eilenberg and S

S. Eilenberg and S. Maclane. Cohomology theory in abstract groups, II , group extensions with a non-abelian kernel. Ann. Math. , 48(2):326--341, 1947

work page 1947

[13] [13]

Feighn and M

M. Feighn and M. Handel. The conjugacy problem for UPG elements of ( F _n) . Preprint , 2024

work page 2024

[14] [14]

Grossman

E. Grossman. On the residual finiteness of certain mapping class groups. J. Lond. Math. Soc, II. Ser. , 9:160--164, 1974

work page 1974

[15] [15]

Karrass, A

A. Karrass, A. Pietrowski, and D. Solitar. Finite and infinite cyclic extensions of free groups. Journal of the Australian Mathematical Society , 16:458--466, 1973

work page 1973

[16] [16]

G. Levitt. Couting growth types of automorphisms of free groups. Geometric and Functonal Analysis , 19(4):1119--1146, 2009

work page 2009

[17] [17]

Magnus, A

W. Magnus, A. Karrass, and D Solitar. Combinatorial group theory . Dover Publications Inc. Minolia, NY, 2004

work page 2004

[18] [18]

Minasyan and D

A. Minasyan and D. Osin. Normal automorphisms of relatively hyperbolic groups. Trans. Am. Math. Soc. , 362(11):6079--6103, 2010

work page 2010

[19] [19]

Z. Sela. The isomorphism problem for hyperbolic groups I . Annales of Mathematics , 141(2):217--283, 1995

work page 1995

[20] [20]

Zimmermann

B. Zimmermann. Uber homöomorphismen n-dimensionaler henkelkörper und endliche erweiterungen von schottky-gruppen. Commentarii Mathematici Helvetici , 58:474--486, 1981

work page 1981