Result 258, Group theory

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 BS(m,n)\mathrm{BS}(m,n), with m,n≠0m,n\ne0, is word-hyperbolic. It also proves that every word-hyperbolic one-relator group is virtually compact special.

Proof

The bigger picture

Why it matters

Groups defined by a single equation can still have complicated geometry. Two unreviewed manuscripts report that excluding a specific family of subgroups forces negative-curvature behavior, and that this geometry brings additional structure.

What changes?

A one-relator group has a presentation with generators and one defining equation. The first manuscript claims that every finitely generated such group is word-hyperbolic if it contains no Baumslag-Solitar subgroup BS(m,n), for any nonzero integers m and n. Word-hyperbolicity means that triangles in the graph formed by generator steps are uniformly thin. A Baumslag-Solitar group has two generators and one relation: conjugation by one generator takes the other's mth power to its nth power.

What does that help mathematicians do?

The second manuscript claims that every word-hyperbolic one-relator group is virtually compact special. This means a subgroup with finitely many cosets can be represented by loops in a compact space assembled from cubes, subject to special restrictions on how its walls meet. Together, the claims let researchers pass from excluding Baumslag-Solitar subgroups to this structured geometric model. The subgroup condition therefore yields more than negative curvature alone.

Are there practical applications?

The immediate value is foundational: these claims connect subgroup exclusions, geometry and algebraic decomposition. The second manuscript states that, combined with work of Kielak-Linton, its result makes every hyperbolic one-relator group virtually free-by-cyclic. After passing to a subgroup with finitely many cosets, there is a normal free subgroup whose quotient is the integers. Crucially, that free subgroup may need infinitely many generators.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Baumslag-Solitar-free one-relator groups are hyperbolic

September 25, 2026 53 pages

We prove Gersten's conjecture: every finitely generated one-relator group containing no subgroup isomorphic to a Baumslag–Solitar group BS(m,n)\mathop{\mathrm{BS}}\nolimits (m,n), for nonzero integers m, n, is word-hyperbolic.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Virtual compact specialness of hyperbolic one-relator groups

September 25, 2026 145 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.