Mathematics Physical Sciences Doctor of Philosophy 1 The Large Scale Geometry of Nilpotent-by-Cyclic Groups

A nonpolycyclic nilpotent-by-cyclic group Γ 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 φ 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 φ, θ 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 Γφ, Γθ are quasi-isometric. Two quasi-isometry invariants are also found: • The set of “divergence rates ” of vertical flow lines, Dφ • The “growth spaces ” gn ⊂ 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 non-polycyclic nilpotent-by-cyclic group is (virtually-nilpotent)-by-cyclic. Acknowledgements Chief thanks are due to my Heavenly Father, who “hum-bled me, causing me to hunger, and then fed me with manna, ” and who through Jesus has generously provided life and breath and the abilities and opportunities to complete this dissertation. I owe many thanks to Benson Farb for his outstanding mathematical mentor-ing and faithful encouragement. I also appreciate the mathematical assistanc