The large scale geometry of nilpotent-by-cyclic groups

A nonpolycyclic nilpotent-by-cyclic group $G$ 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 $f$ 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 $f,q$ 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 $Gf,Gq$ are quasi-isometric. Two quasi-isometry invariants are also found: (1) The set of “divergence rates” of vertical flow lines, $Df$; (2) The “growth spaces” $gn\subset g$. 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.