Large Scale Geometry of Nilpotent-By-Cyclic Groups

We show quasi-isometric rigidity for a class of finitely generated, non-polycyclic nilpotent-by-cyclic groups. Specifically, let Γ₁, Γ₂ be ascending HNN extensions of finitely generated nilpotent groups N ₁ and N ₂, such that Γ₁ is irreducible (see Definition 1.1). If Γ₁ and Γ₂ are quasi-isometric to each other then N ₁ and N ₂ are virtual lattices in a common simply connected nilpotent Lie group (N)\tilde]{\tilde{N}}. As a consequence, we show the class of irreducible ascending HNN extensions of finitely generated nilpotent groups is quasi-isometrically rigid.