Gersten's conjecture and virtual specialness of one-relator groups

Gersten's conjecture and virtual specialness of one-relator groups concerns the geometry of groups defined by a single relation: Gersten's conjecture that a finitely generated one-relator group is word-hyperbolic whenever it contains no Baumslag–Solitar subgroup, and Wise's questions on whether hyperbolic one-relator groups are virtually compact special and virtually free-by-cyclic. Two manuscripts dated 25 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claim Gersten's conjecture and the virtual compact specialness of every hyperbolic one-relator group, and deduce, with a theorem of Kielak and Linton, that every such group is virtually free-by-cyclic. The collection has no Lean formalization of these results. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

Background

• As recounted in the introductions of the manuscripts:
  • The Baumslag–Solitar groups BS(m, n) are the basic obstructions to hyperbolicity among one-relator groups, and Gersten asked whether they are the only subgroup obstructions. Magnus's Freiheitssatz and the Magnus–Moldavanskii hierarchy make inductive arguments possible, Newman's spelling theorem gives hyperbolicity for proper-power relators, and Linton developed the geometric form of the hierarchy.
  • Louder and Wilton characterized negative immersions for one-relator complexes by primitivity rank greater than two and proved coherence results; Linton proved hyperbolicity and virtual specialness for groups with negative immersions, characterized the hyperbolic and locally quasiconvex one-relator groups as those whose relator has primitivity rank different from two, gave a hierarchy criterion, and reduced the general problem to primitive extension groups. The combination theorem of Bestvina and Feighn, Brinkmann's theorem on atoroidal automorphisms and Mutanguha's criterion for ascending HNN extensions of free groups are recalled as precedents.
  • For specialness, Baumslag conjectured residual finiteness of one-relator groups with torsion, and Wise's quasiconvex hierarchy theory proved virtual compact specialness in that case; Haglund and Wise connected special cube complexes with right-angled Artin groups, and Agol's theorem covers hyperbolic groups acting properly and cocompactly on CAT(0) cube complexes. Kielak and Linton recall Wise's conjecture that every hyperbolic one-relator group is virtually free-by-cyclic and prove it under the additional hypothesis of virtual compact specialness.

The original results

• Claim (hyperbolicity). Theorem 1.1 of Baumslag–Solitar-free one-relator groups are hyperbolic states that if X is a finite set, r an element of the free group on X, and the one-relator group ⟨X | r⟩ contains no subgroup isomorphic to BS(m, n) for nonzero integers m, n, then the group is word-hyperbolic; the manuscript states that this resolves Gersten's conjecture and includes trivial relators and proper powers. Its Corollary 9.1, which uses the companion manuscript, states that such a group has a finite-index subgroup embedding in a free-by-cyclic group F_d ⋊ Z with finitely generated free kernel.
• Claim (specialness). Theorem 1.1 of Virtual compact specialness of hyperbolic one-relator groups states that every word-hyperbolic one-relator group has a finite-index subgroup that is the fundamental group of a finite nonpositively curved cube complex with a cubical local isometry to the Salvetti complex of a right-angled Artin group. With the corollary of Kielak and Linton, the manuscript states that this answers Wise's virtual free-by-cyclic conjecture for hyperbolic one-relator groups, the free kernel being allowed to have infinite rank.
• Method, as described: the hyperbolicity manuscript proves a combination theorem for HNN extensions of torsion-free hyperbolic groups with quasiconvex free edge groups satisfying conjugate-intersection and strong-inertia conditions, showing that the absence of Baumslag–Solitar subgroups forces annular flaring; the flaring is proved by contradiction using free-group currents, carriers of persistent path stabilizers and Brinkmann's expansion theorem, and a geometric Magnus induction then gives the theorem. The specialness manuscript treats the primitive extension groups identified by Linton, constructing virtually compact special quotients by filling, recovering the structure through selective covers, and propagating it up a quasiconvex tower; it uses the hyperbolicity manuscript at two specified points.
• Sources. The manuscripts Baumslag–Solitar-free one-relator groups are hyperbolic and Virtual compact specialness of hyperbolic one-relator groups at commit 1581d4c.
• Formalization. The repository has no Lean scope note for this family.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 20 October 2026)
2026-10-16	J. Jardine, Lattice24, referee reviews of the hyperbolicity manuscript and the specialness manuscript in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995)	Review (AI-assisted audit)	Both reviews report no gap. The review of the hyperbolicity manuscript states that the reviewer read all of the LaTeX and checked the preliminary lemmas, the tree computations, the window rank bound, the annulus-to-current limits and the exclusion of atoms in detail. It notes that the manuscript uses the specialness companion only for the free-by-cyclic corollary, so that the main theorem does not depend on it. The review of the specialness manuscript has two minor notes, which it states are not defects. Sections 4 to 9, about 5,000 lines constructing the terminal peripheral family and the intersection transfer on which the primitive case depends, were not audited line by line, so that the review states it cannot certify the primitive case. In the filling lemmas, the uniformity of Coulon's constants and the position of the sheets are sketched rather than computed, which the reviewer describes as plausible and standard.	Part of an audit of the collection's manuscripts without Lean that the audit describes as complete. The reviews are published under CC BY 4.0.
2026-10-18	O. Kharlampovich, A. Miasnikov, Model Theory and Algorithms for One-Relator Groups, arXiv:2610.25305v1	Use	Studies the elementary theories and algorithmic properties of one-relator groups, including a first-order sentence characterizing Baumslag–Solitar-freeness in this class and undecidability of the elementary theories of one-relator groups containing a non-unimodular Baumslag–Solitar subgroup. The abstract states that, using "the recent Open AI result that BS-free one-relator groups are hyperbolic", it obtains a terminating algorithm recognizing this class, and that the virtual compact specialness result of the second manuscript gives an effective construction of virtual free-by-cyclic presentations for hyperbolic one-relator groups, the free kernel possibly of infinite rank.	arXiv preprint, version 1 of 18 October 2026.

Notes

• The earlier results are due to the authors named in the Background section, as recounted by the manuscripts.
• The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.