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arxiv: 2509.22346 · v2 · submitted 2025-09-26 · 🧮 math.GR

Free-by-cyclic groups are conjugacy separable

Fran\c{c}ois Dahmani , Sam Hughes , Monika Kudlinska , Nicholas Touikan This is my paper

Pith reviewed 2026-05-18 12:46 UTC · model grok-4.3

classification 🧮 math.GR
keywords free-by-cyclic groupsconjugacy separabilityresidually finite groupsouter automorphism groupsgraph of groupsDehn fillingsrelatively hyperbolic groupsKourovka Notebook
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The pith

Every finitely generated free-by-cyclic group is conjugacy separable.

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

The paper establishes that if a finitely generated group G maps onto the integers with free kernel, then any two non-conjugate elements of G can be sent to non-conjugate images in some finite quotient of G. This property is called conjugacy separability. A sympathetic reader cares because it gives a concrete way to detect conjugacy via finite approximations, and the result settles an open question about these groups while also showing that their outer automorphism groups are residually finite.

Core claim

We show that all finitely generated free-by-cyclic groups are conjugacy separable: if a finitely generated group G surjects onto Z with free kernel, then for every pair of non-conjugate elements g,h in G, there exists a finite quotient alpha:G twoheadrightarrow Q such that alpha(g) is not conjugate to alpha(h). This resolves Question 19.41 of the Kourovka Notebook. We apply this to prove that the outer automorphism group of a finitely generated free-by-cyclic group is residually finite. Along the way we prove that if the monodromy of a finitely generated free-by-cyclic group is polynomially growing, then the double cosets of a cyclic subgroup are separable.

What carries the argument

Combination of vertex fillings in graph-of-groups decompositions with Dehn fillings in relatively hyperbolic groups, applied according to the different geometric regimes present in free-by-cyclic groups.

If this is right

  • The outer automorphism group of any finitely generated free-by-cyclic group is residually finite.
  • When the monodromy is polynomially growing, double cosets of a cyclic subgroup become separable.
  • Question 19.41 from the Kourovka Notebook is answered affirmatively for this class of groups.

Where Pith is reading between the lines

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

  • The same filling techniques may adapt to prove conjugacy separability for other extensions that admit similar graph-of-groups decompositions.
  • Residual finiteness of Out(G) for these G opens the possibility of algorithmic recognition of conjugacy classes via finite quotients in practice.
  • The polynomial-growth case for double-coset separability could be tested first on explicit examples such as mapping tori of free-group automorphisms with known growth rates.

Load-bearing premise

The argument requires the existence of suitable geometric regimes inside free-by-cyclic groups that permit vertex fillings and Dehn fillings to be combined effectively.

What would settle it

A single explicit example of a finitely generated free-by-cyclic group containing two non-conjugate elements that remain conjugate in every finite quotient would disprove the claim.

Figures

Figures reproduced from arXiv: 2509.22346 by Fran\c{c}ois Dahmani, Monika Kudlinska, Nicholas Touikan, Sam Hughes.

Figure 1
Figure 1. Figure 1: The three configurations for THK. The axes αh, αk are drawn thicker. k translate points along the axes αh and αk, respectively, we immediately see that in the first two configurations the set HK ⋅ v = {h mk n ⋅ v ∶ m, n ∈ Z} is in bijective correspondence with HK. In the third case, where the axes have an intersection that contains a non-trivial arc I, the only way for it to be possible that h m1 k n1 ⋅ v … view at source ↗
Figure 2
Figure 2. Figure 2: The configuration for THK when H is hyperbolic and K elliptic. Proof. Pick a vertex v ∈ Fix(H)∩Fix(K). Hence, H, K ≤ Gv. Then, since double cosets of cyclic subgroups are separable in Gv, it follows that HK is closed in the profinite topology on Gv. By assumption, G induces the full profinite topology on Gv and thus HK ⊆ G is separable. □ Lemma 6.5. Let G be a clean piecewise trivial suspension with free l… view at source ↗
read the original abstract

We show that all finitely generated free-by-cyclic groups are conjugacy separable: if a finitely generated group $G$ surjects onto $\mathbb{Z}$ with free kernel, then for every pair of non-conjugate elements $g,h\in G$, there exists a finite quotient $\alpha:G\twoheadrightarrow Q$ such that $\alpha(g)$ is not conjugate to $\alpha(h)$. This resolves Question 19.41 of the Kourovka Notebook. We apply this to prove that the outer automorphism group of a finitely generated free-by-cyclic group is residually finite. Along the way we prove that if the monodromy of a {finitely generated free}-by-cyclic group is polynomially growing, then the double cosets of a cyclic subgroup are separable. Our approach combines vertex fillings in graph-of-groups decompositions, and Dehn fillings in relatively hyperbolic groups, according to the different geometric regimes in free-by-cyclic groups.

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.

Referee Report

1 major / 3 minor

Summary. The manuscript proves that all finitely generated free-by-cyclic groups are conjugacy separable: if G surjects onto Z with free kernel, then for any non-conjugate g, h in G there is a finite quotient in which their images are not conjugate. The proof proceeds by case analysis on geometric regimes determined by the monodromy automorphism, combining vertex fillings in graph-of-groups decompositions with Dehn fillings in relatively hyperbolic groups. As corollaries, the outer automorphism group of such a G is residually finite, and cyclic double cosets are separable when the monodromy has polynomial growth. This resolves Question 19.41 of the Kourovka Notebook.

Significance. If the central claim holds, the result resolves a longstanding open question in geometric group theory and supplies a new application of filling techniques to separability problems. The consequence for residual finiteness of Out(G) is a concrete advance for automorphism groups of free-by-cyclic groups. The auxiliary result on double-coset separability for polynomially growing monodromy is independently useful and may extend to other classes.

major comments (1)
  1. [Geometric regimes / case division (likely §3–5)] The division of monodromy automorphisms into geometric regimes (described in the abstract and presumably detailed in the section on case analysis) must be shown to be exhaustive. It is unclear whether every automorphism of a free group—including reducible actions or those with mixed polynomial/exponential growth—falls into a regime where at least one of the two filling techniques produces a finite quotient that separates the given conjugacy classes. This exhaustiveness is load-bearing for the claim that the construction works for arbitrary finitely generated free-by-cyclic groups.
minor comments (3)
  1. [Abstract] In the abstract, the parenthetical phrase 'finitely generated free-by-cyclic group' is slightly inconsistent with the earlier wording; standardize the terminology.
  2. [Main proof sections] Add explicit cross-references when the vertex-filling and Dehn-filling arguments are invoked in later sections so that the reader can trace which regime is being used for each case.
  3. [Preliminaries] Verify that all background results on relatively hyperbolic groups and graph-of-groups decompositions are cited with precise theorem numbers rather than general references.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for recognizing the significance of the result in resolving Question 19.41 of the Kourovka Notebook. We address the major comment below and will revise the manuscript accordingly to strengthen the exposition.

read point-by-point responses

  1. Referee: The division of monodromy automorphisms into geometric regimes (described in the abstract and presumably detailed in the section on case analysis) must be shown to be exhaustive. It is unclear whether every automorphism of a free group—including reducible actions or those with mixed polynomial/exponential growth—falls into a regime where at least one of the two filling techniques produces a finite quotient that separates the given conjugacy classes. This exhaustiveness is load-bearing for the claim that the construction works for arbitrary finitely generated free-by-cyclic groups.

    Authors: We agree that explicit verification of exhaustiveness is essential for the argument. The geometric regimes in the manuscript are determined by the standard classification of automorphisms of free groups (Bestvina–Handel train-track theory and the theory of free factor systems). Every automorphism φ of a finitely generated free group F is either irreducible or reducible. If irreducible, it admits a train-track representative whose growth is either polynomial or exponential. If reducible, there exists a φ-invariant free factor system, and φ restricts to automorphisms on the complementary factors; the classification then applies recursively to each factor. Our case analysis covers: (i) irreducible exponential growth via Dehn fillings in the relatively hyperbolic mapping torus, (ii) polynomial growth (including fully reducible polynomial cases) via vertex fillings in the associated graph-of-groups decomposition, and (iii) mixed-growth reducible cases by applying the appropriate technique componentwise on the invariant factors. To make this mapping fully explicit and address the referee’s concern, we will insert a short subsection (or expanded paragraph) in Section 2 that recalls the classification and states which filling technique applies in each subcase. This addition will confirm there are no gaps for arbitrary monodromy. revision: yes

Circularity Check

0 steps flagged

No circularity: proof combines independent filling techniques via case analysis

full rationale

The paper establishes conjugacy separability for finitely generated free-by-cyclic groups by case division into geometric regimes determined by the monodromy automorphism, then applying vertex fillings in graph-of-groups decompositions or Dehn fillings in relatively hyperbolic groups as appropriate. No load-bearing step reduces by definition, by fitted parameter, or by self-citation chain to the target result itself. The cited filling methods are standard tools from geometric group theory whose validity is independent of the present theorem, and the case analysis is presented as exhaustive without circular redefinition of the regimes. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger records the background assumptions explicitly invoked by the described method.

axioms (2)
  • domain assumption Relatively hyperbolic groups admit Dehn fillings that produce finite quotients separating given elements.
    The abstract states that Dehn fillings are used according to the geometric regime.
  • domain assumption Graph-of-groups decompositions of free-by-cyclic groups admit vertex fillings that detect conjugacy.
    The abstract states that vertex fillings in graph-of-groups decompositions are combined with the Dehn fillings.

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