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Math-1272 · mathematics · Group theory

Kourovka Notebook Problem 21.124

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

A free-by-cyclic group here is a semidirect product $G=F\rtimes_\varphi\mathbb Z$, with $F$ a free group and $\varphi\in\operatorname{Aut}(F)$. A CAT(0) space is a geodesic metric space in which distances between points on any geodesic triangle are at most the corresponding distances in its Euclidean comparison triangle. A CAT(0) group acts by isometries on such a space properly discontinuously and cocompactly: compact subsets meet only finitely many of their translates, and the quotient is compact.

For a graph $\Gamma$, its right-angled Artin group is $A_\Gamma=\langle v\in V\Gamma\mid uv=vu\text{ for }\{u,v\}\in E\Gamma\rangle$. Its Salvetti cube complex has one vertex, one oriented loop for each generator, and the corresponding cubical torus for every finite clique, glued along common coordinate subtori.

A cube complex is built by isometrically gluing faces of unit Euclidean cubes. It is special in the Haglund--Wise sense if it admits a cubical local isometry to a Salvetti complex. A cubical local isometry maps the link of every vertex injectively onto a full subcomplex of the target link; full means containing every target simplex all of whose vertices it contains. A group is virtually special if a finite-index subgroup is the fundamental group of a special cube complex.

(a) Is every CAT(0) free-by-cyclic group virtually special? (b) Does every such group have a finite-index subgroup admitting an injective homomorphism into a right-angled Artin group?

Each question permits the subgroup, cube complex, or target graph to depend on $G$. Neither part prescribes a particular action or an embedding compatible with the semidirect-product coordinates.

Short English statement

Must a CAT(0) free-by-cyclic group be virtually special, or at least virtually embed into a right-angled Artin group?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-1272-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-1272

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 21.124

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

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