View source for Quasi-isometric recognition of virtually polycyclic groups

'''Quasi-isometric recognition of virtually polycyclic groups''' asks whether a finitely generated group that is quasi-isometric to a virtually polycyclic group must itself be virtually polycyclic, so that this algebraic property can be seen from the large-scale geometry of a word metric. Eskin, Fisher and Whyte conjectured that it can, in an equivalent form about lattices in solvable Lie groups. A manuscript in OpenAI's public mathematics collection, dated 24 September 2026, claims the conjecture with no hypothesis on the comparison group beyond finite generation. The collection's Lean formalization states the main theorem and its lattice form.

== Background ==
* As recounted in the manuscript's introduction:
** ''The conjecture.'' Eskin, Fisher and Whyte formulated the lattice-recognition conjecture and its equivalent polycyclic form. Gromov's theorem on groups of polynomial growth settles the virtually nilpotent case, and Shalom proved that a finitely generated group quasi-isometric to an infinite polycyclic group has a finite-index subgroup with infinite abelianization. In general the comparison group is not assumed solvable.
** ''Height rigidity.'' Farb and Mosher formulated horizontal-preservation conjectures for abelian-by-cyclic models and the recognition conjecture for Sol. Eskin, Fisher and Whyte proved recognition for Sol by coarse differentiation, and Peng extended the analysis to nondegenerate unimodular split abelian-by-abelian Lie groups, including nontrivial Jordan blocks.
** ''Further cases.'' Dymarz treated diagonalizable models through boundary maps; Dymarz, Fisher and Xie proved a conjugation theorem for suitable nilpotent boundaries and recognition for SOL-like groups under structural and amenability hypotheses; Grayevsky and Pallier obtained rigidity of Sol<sub>5</sub> and its lattices; and Le Boudec proved recognition under the stronger hypothesis of commability.

== The original result ==
* '''Claim, 24 September 2026.''' The main theorem of ''Quasi-isometric recognition of virtually polycyclic groups'' states that if a finitely generated group ''H'' is quasi-isometric to a finitely generated virtually polycyclic group, then ''H'' is virtually polycyclic. An equivalent corollary states that a finitely generated group quasi-isometric to a lattice in a connected simply connected solvable Lie group has a finite-index subgroup that is a uniform lattice in some such Lie group, which may differ from the original one.
* ''Scope, as stated:'' the comparison group is subject to no algebraic hypothesis beyond finite generation. The result does not classify quasi-isometries or decide which polycyclic groups are quasi-isometric to which.
* ''Method, as described:'' the manuscript proves uniform height control for real-triangulable solvable Lie groups: a quasi-isometry changes the exponential height coordinate only by one of finitely many linear symmetries, up to bounded error. Combined with amenability, this control yields elementary amenability of the comparison group, and finiteness and duality results then give virtual polycyclicity, with Gromov's theorem used for the kernel of the height homomorphism.
* '''Sources.''' Commit <code>adc7f12</code> of the openai/math repository: [https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/quasi-isometric-recognition-of-virtually-polycyclic-groups-September-24-2026 ''Quasi-isometric recognition of virtually polycyclic groups''].
* '''Formalization.''' The [https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/lean/docs/255.md scope note], present since the repository's first commit of 6 October 2026, states that the Lean development proves the recognition theorem and realizes a finite-index subgroup of the recognized group as a uniform lattice in a simply connected solvable Lie group, possibly a different one, and that a linked structural estimate controls the height coordinate of self quasi-isometries of the stated unimodular solvable Lie models. Comparator statement: [https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/lean/ComparatorChallenges/PolycyclicRecognition.lean <code>PolycyclicRecognition.lean</code>].

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 1 November 2026)
|-
| 2026-10-10 || [https://github.com/carlostoledo1891/cert-machine/blob/e35cd57a5fbd5b870c66d88697297a33f64f0eb2/certs/openai-math-kernel.json cert-machine], openai/math lane K, run 38051470892 || Verification, two kernels || Records the challenge <code>PolycyclicRecognition</code> as certified: Lean's default kernel and the nanoda kernel both accept the solution, in 7,353 s. C. Toledo's [https://carlostoledo.co/reports/openai-math.html report] lists the lane's results. || Repository. The run concerns the formalized statement only.
|-
| 2026-10-10 to 2026-10-14 || [https://github.com/leanprover/con-leche/blob/90577d9e49ee3aed45da3d421bdf5dc780192304/DESIGN.md con-leche development journal], task #348 || Verification, second kernel || In its run over all Comparator challenges of the collection at commit <code>fd4aeeb</code>, the journal's table of exceptions does not list this family's challenge, so it was accepted by con-leche and by Lean's official kernel. || Development journal.
|-
| 2026-10-13 || [https://github.com/mathvet/mathvet MathVet], a GitHub review of openai/math, tables at repository commit <code>8cc0dcf</code> of 30 October 2026 || Review, fidelity of the formal statement || Rates this family "full": the Lean states the recognition theorem and the lattice form, with quasi-isometry encoded as a coarse equivalence of groups, which for finitely generated groups is equivalent, and polycyclicity as an explicit cyclic subnormal series. Among the definitions to check it lists the hand-built coarse geometry, the definition of a uniform lattice and the packaging of the solvable Lie model, which it calls plausible and standard but outside Mathlib, and it notes that an external package for Gromov's theorem is in the import cone. It records a sandboxed Comparator pass on 13 October 2026, with 232,617 lines in the dependency cone. || Review repository. Read on 1 November 2026.
|-
| 2026-10-17 || ''AI Proved What?'', episode 255 [https://aiprovedwhat.com/episodes/255-virtually-polycyclic-qi-recognition/ Seen from far away, still polycyclic?], remade version of 22 October 2026 || Reading, exposition || Video explainer of the 73-page preprint. It recalls Gromov's theorem, Erschler's result that solvability itself is not a quasi-isometry invariant, the question of Farb and Mosher and the conjecture of Eskin, Fisher and Whyte, and states that Le Boudec wrote in 2025 that the general case was not known. It states that the paper classifies neither the quasi-isometries nor which polycyclic groups are quasi-isometric, and illustrates this with two quasi-isometric Sol lattices whose abelianizations differ by its own computation. It reports its own growth counts in small examples. || Episode status, last checked 30 October 2026: posted as a preprint, not revised; under expert review, not yet; peer reviewed, not yet; checked in Lean, yes, with Gromov's theorem from an outside package, about 233,000 lines, the item missing from the repository's formalization list; no expert comment found. The site states that it is made with Anthropic's Claude. Read on 1 November 2026.
|-
| 2026-10-18 || ''AI Proof Watch'', record [https://aiproofwatch.com/updates/openai-reports-quasi-isometric-recognition-of-virtually-polycyclic-gro-f817188f OpenAI reports quasi-isometric recognition of virtually polycyclic groups] || Catalogue record, assessment || Rates the importance as major, as a resolution of the lattice-recognition conjecture of Eskin, Fisher and Whyte in its full virtually polycyclic form, and the AI involvement as AI-led. It lists as limitations that the comparison group need only be finitely generated and that the lattice realization may use a different Lie group, and it rates the technical verification as formal, on OpenAI's reported Lean proof, and the human understanding as few, the default when no named explanation is documented. || Curated public tracker; record published 18 October 2026, assessment reviewed 31 October 2026. Read on 1 November 2026.
|}

== Notes ==
* The background summarizes the manuscript's own account of earlier work.
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]