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Virtual compact specialness of hyperbolic one-relator groups
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Abstract
We prove that every word-hyperbolic one-relator group is virtually compact special. Combined with the work of Kielak–Linton, this resolves Wise's virtual free-by-cyclic conjecture for hyperbolic one-relator groups: every such group is virtually free-by-cyclic, with the free kernel allowed to have infinite rank.
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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). Section 10 (including Theorem 10.2) assumes Theorem 1.1 of this paper to effectively construct virtual free-by-cyclic presentations of hyperbolic one-relator groups. 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)
- Daniel Groves, Jason F. Manning, Hyperbolic groups acting improperly (Geometry & Topology 27 (2023), 3387–3460). Its Theorem D generalizes both Agol's theorem and Wise's quasiconvex hierarchy theorem, the two results the paper combines in Proposition 2.1 to propagate virtual compact specialness.
- Marco Linton, The geometry of subgroups of mapping tori of free groups (arXiv:2510.03145, 2025). Studies relative hyperbolicity and quasiconvexity of subgroups in finitely generated mapping tori of free groups, the class in Theorem 1.2 (ascending mapping tori); the companion hyperbolicity paper cites it, but this paper does not.
- 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 non-hyperbolic one-relator groups that cannot act freely on a CAT(0) cube complex, a contrast to the hyperbolicity hypothesis of Theorem 1.1.
- 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:Virtual-compact-specialness-of-hyperbolic-one-relator-groups-September-25-2026,
author = {{OpenAI}},
title = {{Virtual compact specialness of hyperbolic one-relator groups}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Virtual-compact-specialness-of-hyperbolic-one-relator-groups-September-25-2026/paper.pdf}{OAI:Virtual-compact-specialness-of-hyperbolic-one-relator-groups-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:
- Baumslag-Solitar-free one-relator groups are hyperbolic · September 25, 2026