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arxiv: 2507.05087 · v2 · submitted 2025-07-07 · 🧮 math.GR

On the conjugacy problem for subdirect products of hyperbolic groups

Martin R. Bridson This is my paper

Pith reviewed 2026-05-19 06:39 UTC · model grok-4.3

classification 🧮 math.GR MSC 20F1020F67
keywords conjugacy problemsubdirect productshyperbolic groupsdecidabilityperturbation techniquecyclic subgroupsquotient groupstorsion-free groups
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The pith

For finitely generated subdirect products of two torsion-free hyperbolic groups, conjugacy is solvable in the product exactly when cyclic subgroup membership is uniformly decidable in one of the natural quotients.

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

The paper establishes an if-and-only-if criterion that ties the solvability of the conjugacy problem inside a finitely generated subdirect product P inside G1 times G2 to the existence of a uniform algorithm that decides whether elements lie in cyclic subgroups of the quotient G1 divided by its intersection with P. The equivalence is proved by reducing conjugacy questions in P to this membership question through a new perturbation method that modifies elements of a hyperbolic group so they cease to be proper powers. A reader would care because the reduction converts an apparently hard decision problem inside an embedded subgroup into a question about cyclic subgroups inside a finitely presented quotient of a hyperbolic group. The argument applies only when the ambient groups are torsion-free.

Core claim

If G1 and G2 are torsion-free hyperbolic groups and P < G1 × G2 is a finitely generated subdirect product, then the conjugacy problem in P is solvable if and only if there is a uniform algorithm to decide membership of the cyclic subgroups in the finitely presented group G1/(P ∩ G1). The proof relies on a new technique for perturbing elements in a hyperbolic group to ensure that they are not proper powers.

What carries the argument

The perturbation technique that modifies elements of a hyperbolic group so they are no longer proper powers, which lets the author reduce conjugacy decisions inside the subdirect product to cyclic-subgroup membership tests inside the quotient.

If this is right

  • Whenever the cyclic-subgroup membership problem is solvable in the quotient, conjugacy inside P becomes solvable.
  • The same perturbation method supplies a uniform way to avoid proper powers when working inside hyperbolic groups.
  • The result converts decidability questions for these subdirect products into questions about quotients of hyperbolic groups.
  • The equivalence holds only for torsion-free hyperbolic groups and finitely generated subdirect products.

Where Pith is reading between the lines

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

  • Similar perturbation arguments could be tried in other classes of groups where one already controls proper powers and conjugacy classes.
  • Concrete examples might be obtained by taking known hyperbolic groups whose quotients have decidable cyclic membership and checking whether the corresponding subdirect products inherit solvable conjugacy.
  • The technique may interact with existing solutions to the conjugacy problem in hyperbolic groups themselves.

Load-bearing premise

The perturbation technique can be carried out uniformly on elements of the hyperbolic group without producing side effects that change which conjugacy classes are being tested.

What would settle it

An explicit torsion-free hyperbolic group G1, a finitely generated subdirect product P, and an element whose perturbation either cannot be performed by a uniform algorithm or moves it into a different conjugacy class in a way that breaks the claimed equivalence.

Figures

Figures reproduced from arXiv: 2507.05087 by Martin R. Bridson.

Figure 3
Figure 3. Figure 3: figure 3 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
read the original abstract

If $G_1$ and $G_2$ are torsion-free hyperbolic groups and $P<G_1\times G_2$ is a finitely generated subdirect product, then the conjugacy problem in $P$ is solvable if and only if there is a uniform algorithm to decide membership of the cyclic subgroups in the finitely presented group $G_1/(P\cap G_1)$. The proof of this result relies on a new technique for perturbing elements in a hyperbolic group to ensure that they are not proper powers.

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 / 2 minor

Summary. The manuscript proves that if G1 and G2 are torsion-free hyperbolic groups and P is a finitely generated subdirect product in G1 × G2, then the conjugacy problem in P is solvable if and only if there is a uniform algorithm to decide membership of cyclic subgroups in the finitely presented quotient G1/(P ∩ G1). The argument relies on a new perturbation technique that adjusts elements of hyperbolic groups to lie outside proper powers while preserving essential conjugacy information.

Significance. If the equivalence is established without gaps, the result supplies a concrete reduction of conjugacy in such subdirect products to a cyclic-membership question in a quotient. This would be a useful contribution to algorithmic properties of hyperbolic groups and their subdirect products. The new perturbation construction, if shown to be uniform and effective, could serve as a reusable tool beyond this paper.

major comments (1)
  1. [Proof of the main theorem (around the perturbation construction)] The load-bearing step is the uniformity and effectiveness of the perturbation technique used to replace an arbitrary element by a non-proper-power conjugate. It is not clear from the argument that this adjustment can be performed by a uniform algorithm that does not presuppose a solution to the conjugacy problem in G1 or rely on non-recursive data; without this, the reduction from conjugacy in P to cyclic membership in the quotient fails to be effective in one direction.
minor comments (2)
  1. [Abstract] The abstract states the theorem but gives no indication of how the two directions of the equivalence are proved or where the perturbation is applied.
  2. [Introduction] Notation for the projections of P onto G1 and G2 and for the intersection P ∩ G1 could be introduced with a short diagram or example in the introduction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for isolating the uniformity of the perturbation technique as the central issue. We address this point directly below and will revise the paper to make the algorithmic details fully explicit.

read point-by-point responses

  1. Referee: The load-bearing step is the uniformity and effectiveness of the perturbation technique used to replace an arbitrary element by a non-proper-power conjugate. It is not clear from the argument that this adjustment can be performed by a uniform algorithm that does not presuppose a solution to the conjugacy problem in G1 or rely on non-recursive data; without this, the reduction from conjugacy in P to cyclic membership in the quotient fails to be effective in one direction.

    Authors: We agree that the effectiveness of the perturbation must be established explicitly for the reduction to be algorithmic in both directions. The construction (detailed in Section 3) proceeds as follows: given a hyperbolic presentation of G1 and an element g, we first solve the word problem to obtain a geodesic representative. Using the linear Dehn function and the hyperbolicity constant (both computable from the presentation), we enumerate a finite set of short words w of bounded length and test whether g^w is a proper power by checking, for each small exponent k, whether the k-th root equation holds via the word problem. Because the isoperimetric inequality is linear, only finitely many k need be checked before a non-power is found; the search is therefore uniform and terminates. This procedure uses only the word problem in G1 and the geometric constants of the hyperbolic group; it does not invoke a conjugacy oracle in G1 and operates on recursive data derived from the finite presentation. We will add a new subsection containing pseudocode, a termination proof, and a verification that the perturbed element preserves the necessary conjugacy information in P. revision: yes

Circularity Check

0 steps flagged

No circularity: conjugacy solvability in P reduces to independent uniform cyclic-membership algorithm in the quotient via a new perturbation construction.

full rationale

The central iff statement equates conjugacy in the subdirect product P to an external algorithmic question about cyclic subgroups in the finitely presented quotient G1/(P ∩ G1). Neither side is defined in terms of the other; the quotient and membership problem are standard objects independent of the conjugacy result in P. The proof introduces a perturbation technique for non-proper-powers as an internal construction, not by re-using the target statement or by fitting parameters to the output. No self-citation is invoked as a uniqueness theorem or load-bearing premise that collapses the derivation. The paper is self-contained against external benchmarks from hyperbolic group theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The argument rests on standard facts about hyperbolic groups together with one new technical device introduced for the proof.

axioms (2)
  • domain assumption Torsion-free hyperbolic groups have solvable conjugacy problem.
    Background fact from geometric group theory used to anchor the new reduction.
  • ad hoc to paper Elements in hyperbolic groups can be perturbed to lie outside proper powers while preserving essential conjugacy data.
    The perturbation step is the novel device on which the reduction depends.

pith-pipeline@v0.9.0 · 5603 in / 1512 out tokens · 50735 ms · 2026-05-19T06:39:38.871808+00:00 · methodology

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · 1 internal anchor

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