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Math-2210 · mathematics · Group theory

Questions in Geometric Group Theory — Q 1.1

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Suppose $G$ admits a finite $K(G,1)$. If $G$ does not contain any Baumslag-Solitar subgroups $BS(m,n)$, is $G$ necessarily hyperbolic? If $G$ embeds in a hyperbolic group, is it hyperbolic?

Formulation references: [S1], [S2].

Short English statement

Suppose $G$ admits a finite $K(G,1)$. If $G$ does not contain any Baumslag-Solitar subgroups $BS(m,n)$, is $G$ necessarily hyperbolic? If $G$ embeds in a hyperbolic group, is it hyperbolic?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2210-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2210

The main theorem settles the one-relator subcase of the first implication, with no extra power/torsion restriction. The record includes arbitrary finite-classifying-space groups and a separate subgroup question. This entry records partial support; complete coverage of the question is not established by this catalogue.

Questions in Geometric Group Theory — Q 1.1

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. A finitely generated one-relator group without any nontrivial-parameter Baumslag--Solitar subgroup is claimed hyperbolic.

Mathematical statement of the claimed result

Result 1.
Given finite $X$ and $r\in F(X)$, form $G=\langle X\mid r\rangle$. Define \[ \operatorname{BS}(m,n)=\langle a,t\mid ta^mt^{-1}=a^n\rangle \qquad(m,n\in\mathbb Z\setminus\{0\}). \] The source implication is \[ \left(\forall m,n\ne0,\ \operatorname{BS}(m,n)\not\hookrightarrow G\right) \quad\Longrightarrow\quad G\text{ word-hyperbolic}. \] It is restricted to finitely generated one-relator groups.

Outline of the argument

Intersection and commensurator bounds reduce the problem to a restricted HNN setting. Annular diagrams generate a stationary current with finite carriers; failure of expansion would give periodic line trajectories and a forbidden Baumslag--Solitar configuration, completing the hyperbolicity argument.

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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