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Group theory · Result 255

Quasi-isometric recognition of virtually polycyclic groups

1 paperLean

Proves that every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is virtually polycyclic, resolving the Eskin–Fisher–Whyte lattice-recognition conjecture. Equivalently, a group quasi-isometric to a lattice in a connected simply connected solvable Lie group is virtually a uniform lattice in some such Lie group, possibly a different one.

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▶Introduction: the problem and why it mattersin preparation
▶The presented solutionin preparation

Papers 1

First page of Quasi-isometric recognition of virtually polycyclic groups

Quasi-isometric recognition of virtually polycyclic groups

OpenAI · September 24, 2026

We prove that every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is virtually polycyclic. This resolves the lattice-recognition conjecture of Eskin, Fisher and Whyte, which allows the ambient solvable Lie group in the conclusion to differ from the one in the hypothesis.

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@misc{OAI:quasi-isometric-recognition-of-virtually-polycyclic-groups-September-24-2026,
  author = {{OpenAI}},
  title = {{Quasi-isometric Recognition of Virtually Polycyclic Groups}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/quasi-isometric-recognition-of-virtually-polycyclic-groups-September-24-2026/paper.pdf}{OAI:quasi-isometric-recognition-of-virtually-polycyclic-groups-September-24-2026}},
  year = {2026}
}

What was formalized in Lean

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization proves quasi-isometric recognition of virtually polycyclic groups: every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is itself virtually polycyclic. It also realizes a finite-index subgroup of the recognized group as a uniform lattice in a simply connected solvable Lie group, which may differ from the original ambient group.

The linked structural estimate controls the height-coordinate behavior of self quasi-isometries of the stated unimodular solvable Lie models, up to bounded error and one of finitely many linear height symmetries.

Comparator links

ResultComparator statement
Quasi-isometric recognition of virtually polycyclic groups and solvable latticesPolycyclicRecognition.lean

Scope note on GitHub →