Large Scale Geometry of Nilpotent-By-Cyclic Groups

Abstract

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