---
title: "Gersten’s conjecture and virtual compact specialness of one-relator groups"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 258; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/Baumslag-Solitar-free-one-relator-groups-are-hyperbolic-September-25-2026/paper.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "group-theory"
  - "unformalized"
seed_rank: 2113
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "Baumslag-Solitar-free one-relator groups are hyperbolic"
    url: "https://github.com/openai/math/blob/main/preprints/Baumslag-Solitar-free-one-relator-groups-are-hyperbolic-September-25-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Virtual compact specialness of hyperbolic one-relator groups"
    url: "https://github.com/openai/math/blob/main/preprints/Virtual-compact-specialness-of-hyperbolic-one-relator-groups-September-25-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# Gersten’s conjecture and virtual compact specialness of one-relator groups

## One-sentence takeaway

OpenAI's result family 258 (Group theory) claims: 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.

## Why it matters here

Pure mathematics with no direct engine use; it is in the library as part of the openai/math corpus, which records what an AI research model claims to have proved. More broadly, every card in this batch is evidence for the `ai-mathematical-reasoning` thread: per the repo README, an unreleased internal OpenAI model produced it, using about three hours of ChatGPT Pro thinking compute per result on average, across roughly 4,000 posed problems.

## Key ideas

- Family summary (repo): It also proves that every word-hyperbolic one-relator group is virtually compact special.
- *Baumslag-Solitar-free one-relator groups are hyperbolic*: 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.
- *Virtual compact specialness of hyperbolic one-relator groups*: 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.

## Caveats

- Unreviewed: these are AI-generated manuscripts posted to GitHub with no peer review or arXiv deposit. The repo README says results are "at different stages of verification" and that "some of the unformalized results could have issues."
- This card restates the repo's family summary, manuscript abstracts and Lean scope notes. It is not an independent check of the proofs, and the scope of each theorem is exactly as stated there.
- Lean: not formalized. The repo has no Lean development for this family, so the claim rests on the prose manuscript(s) alone.

## Links

- Manuscript: [Baumslag-Solitar-free one-relator groups are hyperbolic](https://github.com/openai/math/blob/main/preprints/Baumslag-Solitar-free-one-relator-groups-are-hyperbolic-September-25-2026/paper.pdf)
- Manuscript: [Virtual compact specialness of hyperbolic one-relator groups](https://github.com/openai/math/blob/main/preprints/Virtual-compact-specialness-of-hyperbolic-one-relator-groups-September-25-2026/paper.pdf)
- Family entry: https://github.com/openai/math/blob/main/CONTENTS.md
- Overview PDF: https://github.com/openai/math/blob/main/overview.pdf
- Repo: https://github.com/openai/math