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Math-1272-A1 · revision r1 · on Math-1272

OpenAI claim summary for Math-1272

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Theorems 1.1–1.2 prove virtual compact specialness, hence the relevant cubical/RAAG consequences, in the hyperbolic one-relator or hyperbolic mapping-torus subcases. The record also includes nonhyperbolic groups. This entry records partial support; complete coverage of the question is not established by this catalogue.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Kourovka Notebook Problem 21.124
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Result 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Kourovka Notebook Problem 21.124

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. Every hyperbolic one-relator group is claimed virtually compact special and, by the cited theorem, virtually free-by-cyclic with possibly infinitely generated free kernel.

Mathematical statement of the claimed result

Result 1.
For a finite set $X$ and relator $r\in F(X)$, suppose \[ G=F(X)/\langle\!\langle r\rangle\!\rangle \] is word-hyperbolic. The claim supplies a finite-index subgroup $H\le G$, a finite nonpositively curved cube complex $C$ with $\pi_1C\cong H$, and a cubical local isometry from $C$ to a right-angled Artin Salvetti complex. This is virtual compact specialness. The second theorem applies to any injective endomorphism $\phi$ of a finitely generated free group $F$: \[ F*_{\phi}=\langle F,t\mid t^{-1}ft=\phi(f),\ f\in F\rangle \text{ word-hyperbolic}\quad\Longrightarrow\quad F*_{\phi}\text{ virtually compact special}. \] Word-hyperbolicity is assumed in both assertions.

Outline of the argument

Primitive-extension reduction and a filling criterion organize progressive characters and walls. The resulting cubulation and hierarchy meet the virtual-specialness criterion; the free-by-cyclic consequence is then imported from the established special-group result.

Formal verification

Family 258 has no Lean scope page in this release. Its selected result is not presented as formalized. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources