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Math-1207 · mathematics · Group theory

Kourovka Notebook Problem 21.30

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

A finitely generated one-relator group is a group with a presentation \[ G=\langle x_1,\ldots,x_m\mid r=1\rangle =F_m/\langle\!\langle r\rangle\!\rangle, \] where $m\ge1$, $F_m$ is the free group on the displayed generators, and $\langle\!\langle r\rangle\!\rangle$ is the smallest normal subgroup containing the word $r$. The relator may be a proper power; no torsion-free assumption is made. A redundant trivial relator causes no change to the question.

A discrete group has the Haagerup property if it admits a metrically proper action by affine isometries on a real Hilbert space $\mathcal H$, possibly of infinite dimension. In detail, there are orthogonal operators $\pi(g)$ and vectors $b(g)$ with \[ \pi(gh)=\pi(g)\pi(h),\qquad b(gh)=b(g)+\pi(g)b(h), \] so that $g\cdot v=\pi(g)v+b(g)$, and for every $R\ge0$ the set $\{g\in G:\|b(g)\|\le R\}$ is finite. This is properness of the orbit map of the origin; choosing another basepoint gives the same condition.

Does every one-relator group have this property? The Hilbert space and action may depend on the group, and no uniform dimension or bound on displacement is requested. Properness controls group elements, not merely the set of orbit points. The affine formulation makes explicit the source's isometric-action convention; group actions by bijective Hilbert space isometries are affine in the real setting.

Short English statement

Does every group with a finite presentation containing one relator admit a proper affine isometric action on a Hilbert space?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-1207-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-1207

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.

Kourovka Notebook Problem 21.30

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

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