Math-2718 · mathematics · Group theory

Recognizing virtually polycyclic groups from quasi-isometry

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.

If a finitely generated group H is quasi-isometric to a finitely generated virtually polycyclic group P, must H itself be virtually polycyclic?

Short English statement

If a finitely generated group H is quasi-isometric to a finitely generated virtually polycyclic group P, must H itself be virtually polycyclic?

Sources

S1 Original question source or mathematical context. https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf.
S2 OpenAI, manuscript citation page. Source of the formulation motivating this local problem. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/quasi-isometric-recognition-of-virtually-polycyclic-groups-September-24-2026/README.md.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2718-A1 · Literature report · LLM-generated · unknown · 2026-10-06

OpenAI claim summary for Math-2718

The recognition theorem quantifies every finitely generated H and finitely generated virtually polycyclic P and has exactly the requested quasi-isometry implication. lean/docs/255.md documents the exact recognition statement. Within these assumptions, this is a claimed affirmative answer to the stated question.

Recognizing virtually polycyclic groups from quasi-isometry

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Virtual polycyclicity is claimed determined by the coarse geometry among groups with finite generators.

Mathematical statement of the claimed result

Result 1.

Let $P$ be finitely generated and virtually polycyclic, and let $H$ be any finitely generated group. The source claims \[ H\text{ quasi-isometric to }P\quad\Longrightarrow\quad H\text{ virtually polycyclic}. \] Quasi-isometry refers to word metrics from finite generating sets.

Outline of the argument

Cartan models and weighted boxes define canonical coarse height data. Path packing and recurrent switches force stationary slopes, and neighboring affine chart limits agree; the resulting solvable-lattice structure identifies virtual polycyclicity without fixing the original ambient Lie group.

Formal verification

For family 255, the documented theorem recognizes virtual polycyclicity by quasi-isometry and places a finite-index subgroup in a possibly different solvable Lie lattice; a height-control supplement is also listed. The listed Comparator files are PolycyclicRecognition.lean. The inventory does not supply a named declaration for these Comparator entries. 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

OpenAI, Quasi-isometric Recognition of Virtually Polycyclic Groups, 2026-09-24; citation key OAI:quasi-isometric-recognition-of-virtually-polycyclic-groups-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:recognition.
Family 255: Lean scope.
Comparator PolycyclicRecognition.json.
Comparator PolycyclicRecognition.lean.
OpenAI catalogue entry.

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Record details

Subjects: Group theory
Collection: Top Open Problems
Mathematics rank: 2718