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arxiv: 2606.06122 · v1 · pith:V7WWUMM5new · submitted 2026-06-04 · 🧮 math.GR · math.GT

The stable Andrews-Curtis conjecture and thickenable presentations of the trivial group

Marc Lackenby This is my paper

Pith reviewed 2026-06-27 23:14 UTC · model grok-4.3

classification 🧮 math.GR math.GT
keywords Andrews-Curtis conjecturetrivial groupbalanced presentationsstable Andrews-Curtis movesthickenable presentationsgroup presentationsfree groups
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The pith

Thickenable balanced presentations of the trivial group can be reduced to the standard one-generator form by a bounded number of stable Andrews-Curtis moves, and they satisfy the full Andrews-Curtis conjecture.

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

The paper restricts attention to balanced presentations of the trivial group that are also thickenable. It proves that any such presentation can be transformed into the standard one-generator presentation by stable Andrews-Curtis moves and supplies an explicit upper bound on the number of moves required. It further shows that these same presentations satisfy the ordinary, non-stable version of the Andrews-Curtis conjecture. A reader would care because the Andrews-Curtis conjecture remains open in general, and the result settles both the stable and unstable versions inside this restricted but well-defined class.

Core claim

For every thickenable balanced presentation of the trivial group there is an explicit upper bound on the number of stable Andrews-Curtis moves needed to reach the standard one-generator presentation, and every such presentation satisfies the unstable Andrews-Curtis conjecture.

What carries the argument

Thickenable balanced presentations of the trivial group, which permit topological arguments that control the sequence of stable Andrews-Curtis moves.

If this is right

  • Every thickenable balanced presentation of the trivial group satisfies the stable Andrews-Curtis conjecture with a concrete move bound.
  • Every thickenable balanced presentation of the trivial group satisfies the ordinary Andrews-Curtis conjecture.
  • The reduction process for these presentations is effective and terminates after at most the given bound.
  • The topological thickening condition supplies enough control to replace the open general conjecture with a proved statement inside this subclass.

Where Pith is reading between the lines

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

  • If every balanced presentation of the trivial group turned out to be thickenable, the full Andrews-Curtis conjecture would follow.
  • The explicit bound could be used to algorithmically verify the conjecture for any concrete thickenable example by exhaustive search up to that length.
  • The same thickening technique might be adaptable to other classes of group presentations that admit geometric or topological realizations.

Load-bearing premise

The results are proved only for presentations that are both balanced and thickenable.

What would settle it

Exhibit one thickenable balanced presentation of the trivial group that cannot be reduced to the standard presentation by any finite sequence of stable Andrews-Curtis moves, or that requires more moves than the stated explicit bound.

Figures

Figures reproduced from arXiv: 2606.06122 by Marc Lackenby.

Figure 1
Figure 1. Figure 1: ) This provides the required 3-sphere triangulation. [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The Pachner moves for closed 3-manifolds The following key result of Mijatovic [14] will be central to our arguments. See also King’s paper [10], which contains a similar result using ‘bistellar moves’ rather than Pachner moves. Theorem 3.3. Any triangulation of the 3-sphere with t tetrahedra may be converted to the standard triangulation using at most at2 2 bt2 Pachner moves, where a ≤ 6 · 106 and b ≤ 5 ·… view at source ↗
Figure 3
Figure 3. Figure 3: The 3-ball B1 is shown with f and f2 in its boundary. A homotopy is performed, sliding the 2-cell across the annulus ∂B1\\(f ∪f2). A potential path in the middle of this homotopy is shown. 4. The bound on stable Andrews-Curtis moves In this section, we prove Theorem 2.6. As explained in Section 2.3, this result, combined with Proposition 2.5, immediately implies Theorem 1.2. We are given a balanced thicken… view at source ↗
Figure 4
Figure 4. Figure 4: The arrangement of tetrahedra after a 1-4 Pachner move Consider now a 4-1 Pachner move. The 1-skeleton of T has four edges meeting at the vertex that is to be removed. As above, label these w, x, y and z. Certainly at least one of these edges lies in X, w say. We can actually arrange that the other three edges do not lie in X, as follows. Suppose, for instance, that x lies in X. This implies that the 0-cel… view at source ↗
Figure 5
Figure 5. Figure 5: The meridian discs D′ 1 and D′ 2 shown in the right of [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗
read the original abstract

We establish an explicit upper bound on the number of stable Andrews-Curtis moves that convert thickenable balanced presentations of the trivial group to the standard one-generator presentation. We also present a proof that thickenable balanced presentations of the trivial group satisfy the (unstable) Andrews-Curtis conjecture.

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 / 0 minor

Summary. The manuscript claims to establish an explicit upper bound on the number of stable Andrews-Curtis moves converting thickenable balanced presentations of the trivial group to the standard one-generator presentation, and to prove that such presentations satisfy the unstable Andrews-Curtis conjecture.

Significance. If correct, the results would constitute a concrete partial advance on the Andrews-Curtis conjecture by supplying both a proof of the unstable form and an explicit move bound, but only inside the explicitly restricted class of thickenable balanced presentations. The restriction is stated at the outset and is not overclaimed.

major comments (1)
  1. Abstract: the manuscript asserts both an explicit upper bound and a complete proof, yet the provided text contains neither the derivation of the bound nor the proof itself, so it is impossible to verify correctness, completeness, or whether the stated claims follow from any argument.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We address the single major comment below.

read point-by-point responses

  1. Referee: [—] Abstract: the manuscript asserts both an explicit upper bound and a complete proof, yet the provided text contains neither the derivation of the bound nor the proof itself, so it is impossible to verify correctness, completeness, or whether the stated claims follow from any argument.

    Authors: The abstract is a concise summary of the main results. The full manuscript contains the explicit derivation of the upper bound on the number of stable Andrews-Curtis moves (in the sections following the introduction) together with the complete proof that thickenable balanced presentations of the trivial group satisfy the unstable Andrews-Curtis conjecture. If the version supplied to the referee consisted only of the abstract or an incomplete draft, we will gladly furnish the complete text. revision: no

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states direct proofs establishing an explicit upper bound on stable Andrews-Curtis moves and proving the unstable conjecture, both restricted explicitly to thickenable balanced presentations of the trivial group. No equations, definitions, or citations are provided that reduce a claimed derivation or prediction to a fitted input, self-definition, or load-bearing self-citation chain. The claims are scoped to a subclass and presented as independent mathematical arguments rather than renamings or ansatzes imported from prior work by the same author. This is the expected outcome for a self-contained proof paper in group theory with no visible internal reduction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities can be identified from the abstract alone.

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discussion (0)

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Reference graph

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