| In this thesis I will introduce three questions that involve hyperbolic and projective structures on manifolds and present my progress toward their solution.;I prove that the Hilbert length spectrum (a natural generalization of the marked length spectrum) determines the projective structure on certain non compact properly convex orbifolds up to duality, generalizing a result of Daryl Cooper and Kelly Delp ("The marked length spectrum of a projective manifold or orbifold") in the compact case.;I develop software that computes the complex volume of a boundary unipotent representation of a 3-manifold's fundamental group into PSL(2,C) and SL(2,C). This extends the Ptolemy module software of Matthias Goerner and uses the theory of Stavros Garoufalidis, Dylan Thurston, and Christian Zickert found in "The complex volume of SL(n,C)-representations of 3-manifolds". I apply my software to a census of Carlo Petronio and find non-trivial representations from non torus boundary manifolds. I also find numerical examples of Neumann's conjecture.;I develop theory and software which describes a deformation variety of projective structures on a fixed manifold. In particular, I compute the tangent space of the variety at the complete hyperbolic structure for the figure-eight knot complement. This is a philosophical continuation of Thurston's deformation variety in the hyperbolic setting, which is implemented in the 3-manifold software SnapPea. |
