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Math-2714 · mathematics · Group theory

Virtual compact specialness of hyperbolic one-relator groups

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For any finite X and relator $r\in F(X)$, if $G=\langle X|r\rangle $ is word-hyperbolic, does some finite-index subgroup of G occur as the fundamental group of a finite compact special cube complex?

Short English statement

For any finite X and relator $r\in F(X)$, if $G=\langle X|r\rangle $ is word-hyperbolic, does some finite-index subgroup of G occur as the fundamental group of a finite compact special cube complex?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2714-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2714

For a word-hyperbolic one-relator group, the claimed finite-index subgroup has a compact nonpositively curved cube model admitting a local cubical isometry into a Salvetti model. This supplies compact specialness in the precise virtual sense requested. This is presented as a claimed affirmative answer.

Virtual compact specialness of hyperbolic one-relator groups

OpenAI's claim

OpenAI presents a proof 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

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