Geometric Structures on ManifoldsSaiki, Sam (2017) Geometric Structures on Manifolds. Doctoral Dissertation, University of Pittsburgh. (Unpublished) This is the latest version of this item.
AbstractIn 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. Share
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