∑math20

Group theory · Result 258

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

2 papers

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.

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▶Introduction: the problem and why it mattersin preparation
▶The presented solutionin preparation

Papers 2

First page of Baumslag-Solitar-free one-relator groups are hyperbolic

Baumslag-Solitar-free one-relator groups are hyperbolic

OpenAI · September 25, 2026

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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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}
}
First page of Virtual compact specialness of hyperbolic one-relator groups

Virtual compact specialness of hyperbolic one-relator groups

OpenAI · September 25, 2026

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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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}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.