Seen from far away, still polycyclic?

0:00The question

Draw a group as a graph and step back. Two groups are quasi-isometric if they look the same from far away, up to bounded stretching and bounded error: the integer grid looks like the plane, and a subgroup of finite index looks like the whole group. Gromov showed in 1981 that groups of polynomial growth are virtually nilpotent, so this blurry view already recognizes virtually nilpotent groups. Our computer counted balls: the grid grows like r², the Heisenberg group like r⁴. Polycyclic groups are built as towers of cyclic layers. One example: pairs of integers, plus a height that acts by the matrix [[2,1],[1,1]]. It grows exponentially: 5 million elements within 17 steps. And it has shortcuts: the point (1597, 987) is 17 steps away, instead of 2005 hundred eighty-four steps on the grid. But not every solvable group is polycyclic, and Erschler showed that solvability itself is not visible from far away. Farb and Mosher had asked about this in 2000, and in 2005 Eskin, Fisher and Whyte conjectured, in an equivalent form, that being virtually polycyclic is visible: any finitely generated group quasi-isometric to a polycyclic group is virtually polycyclic. Gromov’s theorem covered the nilpotent case; Eskin, Fisher and Whyte proved it for lattices in the group Sol, and Peng, Dymarz and others for further families. In 2025, Le Boudec wrote that the general case was not known.

1:42The claim

Result 255 is one OpenAI preprint, 73 pages, dated September 24, 2026. It claims the conjecture: every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is virtually polycyclic, with no other assumption on the group. Equivalently, a group quasi-isometric to a lattice in a solvable Lie group is virtually a uniform lattice in some such group, possibly a different one. Not claimed: a classification of the quasi-isometries, or of which polycyclic groups are quasi-isometric to which. Our example: two Sol lattices, built from [[2,1],[1,1]] and from [[3,2],[1,1]], are quasi-isometric, but their abelianizations differ, the integers versus the integers plus a group of order 2. Also not claimed: anything about solvable groups that are not polycyclic.

2:40How the proof goes

The engine is a theorem about height. In these Lie groups, height is the direction whose motion stretches some directions and squeezes others. Paths whose height stays in a window reach exponentially many points, at a rate the paper computes, and a quasi-isometry cannot raise that rate. In our toy, letting the height range grow from {0} to [0, 6], with somewhat longer paths, raises the reachable points from 145 to about 163,000. So a quasi-isometry can change height only by one of finitely many symmetries, up to a bounded error: for Sol, keep it or flip it. In a rank-2 example we found 6 allowed symmetries, and the simple flip is not among them. Getting there takes four stages: averages over many quasi-isometries, using ergodic theory; rescaled local pictures, which turn out to be affine; agreement between neighbouring pieces; and an upgrade from a small error that grows slowly to a bounded one. Then algebra. The mystery group acts roughly on the Lie group, up to bounded error, and height becomes a homomorphism, after averaging with an invariant mean; the group is amenable because finite sets that barely move travel through quasi-isometries. Its kernel stays at bounded height, so its finitely generated pieces grow polynomially; our toy: paths at height 0 reach exactly two r² plus two r plus one points. Gromov’s theorem and published results of Kropholler, Martínez-Pérez and Nucinkis, of Li, and of Bieri then finish: the group is virtually polycyclic.

4:19Where it stands

The preprint is unrefereed and unchanged since release. In Lean, all three headline statements are formalized, about 233,000 lines with no unproved steps, but Gromov’s theorem comes from an outside formal package; the item is missing from the repository’s formalization list, and we did not build it. OSVerify has not reviewed it, and we found no expert comment on this result. If it holds, a group can be recognized as virtually polycyclic from far away: height, unlike most of a group’s structure, survives the blur. That is about 20 years after Eskin, Fisher and Whyte conjectured it.