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Geometric Structures on Manifolds

Saiki, Sam (2017) Geometric Structures on Manifolds. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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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.


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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
Saiki, Samsys@pitt.edusys7
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairDeBlois,
Committee MemberMcReynolds,
Committee MemberHales,
Committee MemberSparling,
Date: 28 September 2017
Date Type: Publication
Defense Date: 4 May 2017
Approval Date: 28 September 2017
Submission Date: 7 August 2017
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Number of Pages: 98
Institution: University of Pittsburgh
Schools and Programs: Dietrich School of Arts and Sciences > Mathematics
Degree: PhD - Doctor of Philosophy
Thesis Type: Doctoral Dissertation
Refereed: Yes
Uncontrolled Keywords: projective, hyperbolic, geometric, structure, manifold, orbifold
Date Deposited: 29 Sep 2017 00:17
Last Modified: 29 Sep 2017 00:17

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