The primary aim of this thesis is to construct explicit examples of four dimensional hyperbolic link complements. Using the theory of Kirby diagrams and Kirby calculus we set up a general framework that one can use to attack such a problem. We use this framework to construct explicit examples in a smooth standard S⁴ and a smooth standard S² x S². We then characterise which homeomorphism types of smooth simply connected closed 4-manifolds can admit a hyperbolic link complement, along the way giving constructions of explicit examples.

Thesis (Ph.D.)--University of Melbourne, Dept. of Mathematics and Statistics, 2006. / Typescript. Includes bibliographical references (leaves 87-90).

Yiu, Fa Wai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 55-58). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Elementary properties and notations of Hyperbolic space --- p.9 / Chapter 3 --- Poisson kernel and Conformal densities --- p.16 / Chapter 3.1 --- Poisson kernel --- p.17 / Chapter 3.2 --- Conformal densities --- p.19 / Chapter 4 --- Patterson construction and decomposition --- p.27 / Chapter 4.1 --- Patterson construction --- p.27 / Chapter 4.2 --- Patterson decomposition --- p.33 / Chapter 5 --- Bonahon surfaces and Grided surfaces --- p.39 / Chapter 5.1 --- Bonahon surfaces --- p.40 / Chapter 5.2 --- Grided surfaces --- p.46 / Chapter 6 --- Margulis number of Hyperbolic Manifolds --- p.51 / Margulis Number for Hypcrbolic 3-manifolds --- p.5 / Chapter 6.1 --- Gcomertrically finite groups --- p.51 / Chapter 6.2 --- Margulis number of Closed Hyperbolic Manifolds --- p.53 / Bibliography --- p.55

Thesis (Ph. D.)--University of Texas at Austin, 2006. / Vita. Includes bibliographical references.

Not available / text

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 2000. / Includes bibliographical references. Also available on the Internet.

In this dissertation, we study etale correspondence of hyperbolic curves with unbounded dynamics. Mochizuki proved that over a field of characteristic 0, such curves are always Shimura curves. We explore variants of this question in positive characteristic, using graph theory, l-adic local systems, and Barsotti-Tate groups. Given a correspondence with unbounded dynamics, we construct an infinite graph with a large group of ”algebraic” automorphisms and roughly measures the ”generic dynamics” of the correspondence. We construct a specialization map to a graph representing the actual dynamics. Along the way, we formulate conjectures that etale correspondences with unbounded dynamics behave similarly to Hecke correspondences of Shimura curves. Using graph theory, we show that type (3,3) etale correspondences verify various parts of this philosophy. Key in the second half of this dissertation is a recent p-adic Langlands correspondence, due to Abe, which answers affirmatively the petites camarades conjecture of Deligne in the case of curves. This allows us the build a correspondence between rank 2 l-adic local systems with trivial determinant and Frobenius traces in Q and certain height 2, dimension 1 Barsotti-Tate groups. We formulate a conjecture on the fields of definitions of certain compatible systems of l-adic representations. Relatedly, we conjecture that the Barsotti-Tate groups over complete curves in positive characteristic may be ”algebraized” to abelian schemes.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1981. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Bibliography: leaf 80. / by Mark David Baker. / Ph.D.

Thesis (M.S.)--University of Delaware, 2006. / Principal faculty advisor: A.D. Kirwan, College of Marine and Earth Studies. Includes bibliographical references.

Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.