The Large Scale Geometry of Nilpotent-by-Cyclic Groups

Abstract

A nonpolycyclic nilpotent-by-cyclic group Gamma can be expressed as the HNN extension of a finitely-generated nilpotent group N. The first main result is that quasi-isometric nilpotent-by-cyclic groups are HNN extensions of quasi-isometric nilpotent groups. The nonsurjective injection defining such an extension induces an injective endomorphism phi of the Lie algebra g associated to the Lie group in which N is a lattice. A normal form for automorphisms of nilpotent Lie algebras--permuted absolute Jordan form-- is defined and conjectured to be a quasi-isometry invariant. We show that if phi, theta are endomorphisms of lattices in a fixed Carnot group G, and if the induced automorphisms of g have the same permuted absolute Jordan form, then Gamma_phi and Gamma_theta are quasi-isometric. Two quasi-isometry invariants are also found: the set of ``divergence rates'' of vertical flow lines: D_phi the ``growth spaces'': g_n subset g These do not establish that permuted absolute Jordan form is a quasi-isometry invariant, although they are major steps toward that conjecture. Furthermore, the quasi-isometric rigidity of finitely-presented nilpotent-by-cyclic groups is proven: any finitely-presented group quasi-isometric to a nonpolycyclic nilpotent-by-cyclic group is (virtually-nilpotent)-by-cyclic.