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Quasi-isometric recognition of virtually polycyclic groups
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Abstract
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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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Lean scope notes for family 255: Quasi-isometric recognition of virtually polycyclic groups (lean/docs/255.md, release of 2026-10-06). The release's Lean scope note reports formalizations of Theorem 1.1 (virtually polycyclic recognition) with the finite-index uniform-lattice consequence and a linked bounded-height estimate for self quasi-isometries of the stated unimodular solvable Lie models; it links the comparator statement
PolycyclicRecognition.lean.
Possibly relevant literature (found by the agent, not checked)
- Alex Eskin and David Fisher, Quasi-isometric rigidity of solvable groups (Proceedings of the International Congress of Mathematicians, Vol. III, 2010). Conjecture 1.2 states the polycyclic recognition problem addressed by the paper's Theorem 1.1, and Section 2 surveys uniform height-preservation results for Sol and certain abelian-by-abelian models relevant to its bounded-height theorem.
- Tullia Dymarz and Irine Peng, Bilipschitz maps of boundaries of certain negatively curved homogeneous spaces (arXiv:0912.3591, December 2009). Extends Dymarz's boundary results from the diagonal case to non-diagonalizable extension matrices, completing rigidity for a class of solvable lattices; it belongs to the line of prior work surveyed in the paper's introduction but is not in its bibliography.
- Xiangdong Xie, Large scale geometry of negatively curved ℝⁿ ⋊ ℝ (arXiv:1001.0147, Geom. Topol. 18 (2014)). Studies quasi-isometries of these solvable Lie groups through quasisymmetric maps on their ideal boundaries, which relates to the paper's final step using boundary maps of contracting directions to bound the height error.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 255
Independent sites; their content is not reviewed here.
Cite
@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}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/255.md. The Lean statements have not been checked by people against the paper's claims.
Family 255: Quasi-isometric recognition of virtually polycyclic groups
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.
This is the only paper in this family.