UC Santa Barbara

This thesis contains two separate but related projects. Part I produces explicit conjugacy paths for the product geometries, Hyperbolic 2-space cross the Real Line and Spherical 2-space cross the Real Line, that limit on Nil.Part II gives a new definition of convex projective cone manifolds, that is stable under small deformations. Deforming from cusps produces new discrete groups that arise as the holonomies of properly convex projective orbifolds.