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Baumslag-Solitar-free one-relator groups are hyperbolic
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Abstract
We prove Gersten's conjecture: every finitely generated one-relator group containing no subgroup isomorphic to a Baumslag–Solitar group $\mathop{\mathrm{BS}}\nolimits (m,n)$, for nonzero integers m, n, is word-hyperbolic.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Olga Kharlampovich, Alexei Miasnikov, Model Theory and Algorithms for One-Relator Groups (arXiv:2610.25305, 18 October 2026). Uses Theorem 1.1 of this paper, which they call (H), for a terminating algorithm that recognizes BS-free one-relator groups and for several corollaries (isomorphism, conjugacy, elementary theory). The authors state that these results are conditional: "We have read both papers, but have not verified all details of their proofs." [family-level]
Possibly relevant literature (found by the agent, not checked)
- Giovanni Italiano, Bruno Martelli, Matteo Migliorini, Hyperbolic 5-manifolds that fiber over S^1 (arXiv:2105.14795, 2021). Constructs a finite-type subgroup of a hyperbolic group that is not hyperbolic, which is relevant to whether the one-relator hypothesis of Theorem 1.1 can be weakened.
- Giles Gardam, Daniel J. Woodhouse, The geometry of one-relator groups satisfying a polynomial isoperimetric inequality (Proc. Amer. Math. Soc. 147 (2019), 125–129). Gives one-relator groups that contain no BS(m,n) with m ≠ ±n but are not automatic or CAT(0). This bears on the paper's remark that its hypothesis must also exclude BS(m,±m), including Z².
- Robert D. Gray and Marco Linton, Finite complete rewriting systems for graphs of free groups with applications to free-by-cyclic, one-relator, and three-manifold groups (arXiv:2607.07496, 2026-07-08). Its theorem on finite complete rewriting systems for finitely generated virtually free-by-cyclic groups may provide algorithmic consequences of this family’s claimed virtual free-by-cyclic corollaries. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 258
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Baumslag-Solitar-free-one-relator-groups-are-hyperbolic-September-25-2026,
author = {{OpenAI}},
title = {{Baumslag--Solitar-free one-relator groups are hyperbolic}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Baumslag-Solitar-free-one-relator-groups-are-hyperbolic-September-25-2026/paper.pdf}{OAI:Baumslag-Solitar-free-one-relator-groups-are-hyperbolic-September-25-2026}},
year = {2026}
}Family 258: Gersten’s conjecture and virtual compact specialness of one-relator groups
Proves Gersten's conjecture: every finitely generated one-relator group containing no Baumslag–Solitar subgroup $\mathrm{BS}(m,n)$, with $m,n\ne0$, is word-hyperbolic. It also proves that every word-hyperbolic one-relator group is virtually compact special.
Other papers in this family:
- Virtual compact specialness of hyperbolic one-relator groups · September 25, 2026