Quasi-isometric rigidity of virtually polycyclic groups (the Eskin-Fisher-Whyte lattice-recognition conjecture)

Theorem 1.1: every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is virtually polycyclic; Corollary 1.2: a finitely generated group quasi-isometric to a lattice in a connected simply connected solvable Lie group has a finite-index subgroup that is a uniform lattice in some (possibly different) such Lie group. The key input is a uniform bounded-height theorem: every $(K,C)$ self quasi-isometry of a unimodular real-triangulable solvable Lie group preserves the exponential height map up to bounded error and one of finitely many linear symmetries. Amenability then gives elementary amenability, and finiteness and duality give virtual polycyclicity. It does not classify solvable groups up to quasi-isometry or describe their quasi-isometry groups in general.

Theorem 1.1: every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is virtually polycyclic; Corollary 1.2: a finitely generated group quasi-isometric to a lattice in a connected simply connected solvable Lie group has a finite-index subgroup that is a uniform lattice in some (possibly different) such Lie group. The key input is a uniform bounded-height theorem: every $(K,C)$ self quasi-isometry of a unimodular real-triangulable solvable Lie group preserves the exponential height map up to bounded error and one of finitely many linear symmetries. Amenability then gives elementary amenability, and finiteness and duality give virtual polycyclicity. It does not classify solvable groups up to quasi-isometry or describe their quasi-isometry groups in general.