Quasi-isometric recognition of virtually polycyclic groups

Any finitely generated group that looks like a virtually polycyclic group from far away is virtually polycyclic.

1 paper September 24, 2026 Explained by AI

Some groups can be built in finitely many steps, each step adding a cycle; these are polycyclic. Viewed from far away, a group's word distances form a shape. This paper proves that any group with the same far-away shape as such a group is itself built that way, up to a finite-index part. Eskin, Fisher and Whyte conjectured this. A Lean formalization is listed.

First step

1981

Mikhail Gromov, whose polynomial growth theorem settled the virtually nilpotent case

Significance

Major

It resolves the lattice-recognition conjecture of Eskin, Fisher and Whyte for the full class of virtually polycyclic groups.

Trust

Formal proof listed

Field

Group theory

1 paper

The explanation

Stand at the corner (0, 0) of a city grid and walk to (3, 4). Along streets you walk 3 + 4 = 7 blocks. A bird flies √(3² + 4²) = 5 blocks. The ratio is 7/5 = 1.4. For any two grid points, the street distance is between 1 and √2 ≈ 1.414 times the bird distance. Also every point of the plane is within √2/2 ≈ 0.707 of a grid point. So the integer grid and the flat plane look the same from far away. This is a quasi-isometry: a map that distorts distances by at most a fixed factor K plus a fixed error CSource 1. Every point of the target must also lie within C of the imageSource 1.

A group is polycyclic if it has a finite chain of subgroups, each normal in the next, with cyclic stepsSource 2. The grid group Z² is polycyclic: Z² contains Z, which contains the trivial group. Virtually polycyclic means a polycyclic subgroup has finite indexSource 2. Finite index means finitely many shifted copies of the subgroup cover the group.

Eskin, Fisher and Whyte conjectured that this property can be read from large-scale geometrySource 3. Gromov's theorem on polynomial growth settled the virtually nilpotent caseSource 4. Shalom showed a group quasi-isometric to an infinite polycyclic group has a finite-index subgroup with infinite abelianizationSource 5. Eskin, Fisher and Whyte proved recognition for the three-dimensional group SolSource 6, and Peng extended it to some split modelsSource 7. The general case stayed open.

The paper proves the full statement. If a finitely generated group H is quasi-isometric to a finitely generated virtually polycyclic group, then H is virtually polycyclicSource 8. H needs no algebraic assumption beyond finite generationSource 9. An equivalent form uses uniform lattices, discrete subgroups with compact quotient in a solvable Lie group. A group quasi-isometric to such a lattice has a finite-index subgroup that is a lattice in some, possibly different, solvable Lie groupSource 10.

The obstacle is exponential growth. There, distances grow fast and local pictures can twist. The key step is uniform height control. A height map projects the Lie group onto a flat space. The paper shows every self quasi-isometry moves height by one of finitely many linear symmetries, up to a bounded errorSource 11. With amenability, this control gives elementary amenability of the group; finiteness and duality then give virtual polycyclicitySource 12.

How sure is this? A computer proof checker called Lean has a formal proof of the recognition theorem and of the lattice form. It also covers the height estimate for the stated unimodular models. No specialist review of the paper has been published.

What it doesn't show. This does not say the new group is a lattice in the same Lie group as the original; the ambient Lie group may change.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major if it holds, and the recognition theorem has a Lean formalization. No specialist review is published.

Words used

Quasi-isometry: A map between spaces that distorts distances by at most a fixed factor plus a fixed error, and nearly covers the target.
Polycyclic group: A group with a finite chain of subgroups, each normal in the next, whose successive quotients are cyclic.
Virtually polycyclic: Containing a polycyclic subgroup of finite index.
Finite index: A subgroup whose shifted copies, finitely many of them, cover the whole group.
Lattice: A discrete subgroup of a Lie group with finite covolume; uniform when the quotient is compact.
Solvable Lie group: A smooth group whose repeated commutators eventually become trivial.
Word metric: The distance between group elements given by the fewest generators needed to get from one to the other.

The original paper

Quasi-isometric recognition of virtually polycyclic groups (PDF, opens in a new tab)
September 24, 2026

Lean notes

All quoted lines (12)

A $(K,C)$ -quasi-isometry $F:X\to Y$ , with $K\geq1$ and $C\geq0$ , satisfies $K^{-1}d_X(x,x')-C\leq d_Y(F(x),F(x'))\leq Kd_X(x,x')+C$ for all $x,x'\in X$ , and every point of $Y$ is within distance $C$ of $F(X)$ .
Quasi-isometric recognition of virtually polycyclic groups Back to text
A group is polycyclic if it has a finite subnormal series with cyclic factors.
Quasi-isometric recognition of virtually polycyclic groups Back to text
Eskin, Fisher and Whyte formulated the lattice-recognition conjecture and its equivalent polycyclic formulation in.
Quasi-isometric recognition of virtually polycyclic groups Back to text
Gromov's theorem on groups of polynomial growth settles the virtually nilpotent case.
Quasi-isometric recognition of virtually polycyclic groups Back to text
For the full class, Shalom proved that a finitely generated group quasi-isometric to an infinite polycyclic group has a finite-index subgroup with infinite abelianization.
Quasi-isometric recognition of virtually polycyclic groups Back to text
Eskin, Fisher and Whyte established recognition for $\mathrm{Sol}$ by coarse differentiation.
Quasi-isometric recognition of virtually polycyclic groups Back to text
Peng extended this analysis to nondegenerate unimodular split abelian-by-abelian Lie groups.
Quasi-isometric recognition of virtually polycyclic groups Back to text
If $H$ and $P$ are quasi-isometric, then $H$ is virtually polycyclic.
Quasi-isometric recognition of virtually polycyclic groups Back to text
The group $H$ in Theorem … is subject to no algebraic hypothesis beyond finite generation.
Quasi-isometric recognition of virtually polycyclic groups Back to text
If a finitely generated group $H$ is quasi-isometric to a lattice in a connected simply connected solvable Lie group, then a finite-index subgroup of $H$ is a uniform lattice in some connected simply connected solvable Lie group.
Quasi-isometric recognition of virtually polycyclic groups Back to text
There is a finite subgroup $\A_G\leq\GL(W)$ such that, for every $K\geq1$ and $C\geq0$ , there is a constant $B=B(G,K,C)$ with the following property.
Quasi-isometric recognition of virtually polycyclic groups Back to text
Combined with amenability, this control yields elementary amenability of the comparison group; finiteness and duality then yield virtual polycyclicity.
Quasi-isometric recognition of virtually polycyclic groups Back to text

Other sources

openai/math, Lean scope note for family 255 every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is itself virtually polycyclic
openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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