Quasi-isometric rigidity of virtually polycyclic groups (the Eskin-Fisher-Whyte lattice-recognition conjecture)
Gromov's polynomial growth theorem implies that a finitely generated group quasi-isometric to a virtually nilpotent group is virtually nilpotent. For solvable groups of exponential growth, quasi-isometric rigidity was known only for special classes: Sol and lamplighters (Eskin-Fisher-Whyte), nondegenerate abelian-by-abelian groups (Peng), SOL-like groups (Dymarz; Dymarz-Fisher-Xie), and under stronger hypotheses such as commability (Le Boudec). Shalom showed such a group virtually has infinite abelianization. Eskin, Fisher and Whyte (2007) conjectured that a finitely generated group quasi-isometric to a lattice in a connected simply connected solvable Lie group is virtually a lattice in a possibly different such Lie group; equivalently: if a finitely generated group is quasi-isometric to a finitely generated virtually polycyclic group, must be virtually polycyclic?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Geometric group theory: quasi-isometric rigidity
- Posed by
- A. Eskin, D. Fisher and K. Whyte, Quasi-isometries and rigidity of solvable groups, Pure Appl. Math. Q. 3 (2007), Conjecture 1.2 and Remark (3), p. 929
- Year posed
- 2007
- Years open
- 19y
- Solved
- 2026-09-24
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Lean-checked, statement unaudited
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 50 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is virtually polycyclic; Corollary 1.2: 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 (possibly different) such Lie group. The key input is a uniform bounded-height theorem: every self quasi-isometry of a unimodular real-triangulable solvable Lie group preserves the exponential height map up to bounded error and one of finitely many linear symmetries. Amenability then gives elementary amenability, and finiteness and duality give virtual polycyclicity. It does not classify solvable groups up to quasi-isometry or describe their quasi-isometry groups in general.
What the AI did
The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author.
Verification
No independent mathematician has checked this yet. Checked here: the abstract, introduction, Theorem 1.1, Corollary 1.2 and Theorem 1.4 of the TeX source, read against Eskin-Fisher-Whyte's Conjecture 1.2 as the manuscript cites it; the proof was not refereed. Lean: the Comparator challenge PolycyclicRecognition (OAI.PolycyclicRecognition.group_recognition, lattice_recognition, main_uniform_bounded_height; solution module OAI/GroupTheory/PolycyclicRecognition/Recognition.lean) is not in the release's formalization catalogue, but its JSON and solution file exist at the pinned commit. Its statement was read here: group_recognition says that for finitely generated groups P and J joined by a coarse equivalence (maps with finite-set control of differences, the group form of a quasi-isometry), virtual polycyclicity of P gives virtual polycyclicity of J and a finite-index subgroup isomorphic to a uniform lattice in a simply connected solvable Lie group. This is the headline claim. Permitted axioms: propext, Quot.sound, Classical.choice. Not rebuilt here.