Let M∈operatornameSL(n,mathbfZ) have no eigenvalues on the unit circle, and let G be a finitely generated group quasi-isometric to the associated group Gamma_M. The quasi-isometric classification conjecture. There is a finite normal subgroup F triangleleft G such that G/F is abstractly commensurable to Gamma_N for some N∈operatornameSL(n,mathbfZ) with no eigenvalues on the unit circle. Together with the preceding theorem and conjecture, this would give a quasi-isometric classification of these polycyclic groups.

The quasi-isometric classification conjecture for polycyclic abelian-by-cyclic groups

Claimed by OpenAI. Related recognition progress: claims every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is virtually polycyclic. This does not determine the finite-normal quotient or commensurability classification requested for the abelian-by-cyclic target, and does not require the same ambient solvable Lie group in the conclusion.

The target asks for a finite-normal quotient commensurable with a hyperbolic abelian-by-cyclic lattice. The source claims only the recognition conclusion that groups quasi-isometric to virtually polycyclic groups are virtually polycyclic. This is relevant positive recognition progress in the target subclass, but no specific quotient, commensurability, matrix classification or same solvable Lie group is asserted.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/quasi-isometric-recognition-of-virtually-polycyclic-groups-September-24-2026/paper.pdf

The quasi-isometric classification conjecture for polycyclic abelian-by-cyclic groups

Primary source

Progress summary

Solutions 1