Stable commutator length of a relator is not a one-relator group invariant
Two explicit words give isomorphic one-relator groups whose relators have different stable commutator lengths, answering a question of Heuer and Loh in the negative.
- Lab
- Independent
- Model
- Claude Opus 5; Harmonic Aristotle
- Field
- Mathematics
- Date
- 2026-08-21
- Human collaborators
- Artem Semidetnov
- Problem posed
- 2019 · open 7 yrs
Sources
Original work
Independent commentary
What was found
For the two explicit words r = aabABabABBAbaabABBAb and r' = aabABabABabABBAbaBAb, both lying in the commutator subgroup of the free group on a and b, the one-relator groups presented by r = 1 and by r' = 1 are isomorphic, while r and r' have different stable commutator lengths. That combination is exactly the negative answer to the question of Heuer and Loh, since it shows the stable commutator length of the relator is not determined by the group it presents.
Novelty check
The question is attributed to Heuer and Loh and dated 2019 in the vibemathed record, and the counterexample is a pair of concrete words whose properties can be checked directly rather than an existence argument. The repository accompanying the paper reproduces the computation, so both halves of the claim, the isomorphism and the difference in stable commutator length, are independently re-runnable in principle. No prior counterexample appears.
Caveats and known objections
A short preprint, unrefereed, with no independent check on record. The formalization is partial and does not cover the result: the repository contains a Lean formalisation of the six identities used in the argument, produced with the assistance of Harmonic's Aristotle, not of the full claim, so the grade stays at author-verified rather than formal. Autonomy is graded ai-assisted, the weaker of the two readings the paper's own wording admits, since the body says Claude Desktop with Opus 5 designed and orchestrated an exhaustive search through orbits of words of bounded length while the acknowledgement says it assisted in designing and orchestrating that search.
Also recorded at
vibemathedthe-stable-commutator-length-of-a-relator-is-not-a-one-relator-group-invariant
Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.
Nobody outside the lab has checked this yet.
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Entry history (1 event)
- AddedEntered the registry graded Author verified and AI-assisted.
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Graded author verified for verification and ai-assisted for autonomy. What these mean.
Cite this entry
whataifound.org. (2026). Stable commutator length of a relator is not a one-relator group invariant. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-08-21-scl-relator-invarianceBibTeX
@misc{whataifound-independent-2026-invariance,
title = {Stable commutator length of a relator is not a one-relator group invariant},
author = {{whataifound.org}},
year = {2026},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Independent. Verification: Author verified. Autonomy: AI-assisted.},
url = {https://whataifound.org/finding/2026-08-21-scl-relator-invariance}
}