OpenAI · 255-01 · 2026-09-24 · Group theory
Quasi-isometric recognition of virtually polycyclic groups
preprints/quasi-isometric-recognition-of-virtually-polycyclic-groups-September-24-2026
Family 255. Quasi-isometric rigidity of virtually polycyclic groups.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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.
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.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 255: Quasi-isometric recognition of virtually polycyclic groups. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.