According to the work of Dennis Sullivan, there exists a smooth flow on the 5-sphere all of whose orbits are periodic although there is no uniform bound on their periods. The question addressed in this thesis is whether such an example can occur in the partially hyperbolic context. That is, does there exist a partially hyperbolic diffeomorphism of a compact manifold such that all the leaves of its center foliation are compact although there is no uniform bound for their volumes. We will show that the answer to the previous question under the very mild hypothesis of dynamical coherence is no. The thesis is organized as follows. In the first chapter we give the necessary background and results in partially hyperbolic dynamics needed for the rest of the work, studying in particular the geometry of the center foliation. Chapter two is devoted to a general discussion of compact foliations. We give proof or sketches of all the relevant results used. Chapter three is the core of the thesis, where we establish the non existence of Sullivan's type of examples in the partially hyperbolic domain, and generalize to diffeomorphisms whose center foliation has arbitrary dimension. The last chapter is devoted to applications of the results of chapter three, where in particular it is proved that if the center foliation of a dynamically coherent partially hyperbolic diffeomorphism is compact and without holonomy, then it is plaque expansive.

This thesis is an investigation of simple closed geodesics, or geodesic knots, in hyperbolic 3-manifolds. / Adams, Hass and Scott have shown that every orientable finite volume hyperbolic 3-manifold contains at least one geodesic knot. The first part of this thesis is devoted to extending this result. We show that all cusped and many closed orientable finite volume hyperbolic 3-manifolds contain infinitely many geodesic knots. This is achieved by studying infinite families of closed geodesics limiting to an infinite length geodesic in the manifold. In the cusped manifold case the limiting geodesic runs cusp-to-cusp, while in the closed manifold case its ends spiral around a short geodesic in the manifold. We show that in the above manifolds infinitely many of the closed geodesics in these families are embedded. / The second part of the thesis is an investigation into the topology of geodesic knots, and is motivated by Thurston’s Geometrization Conjecture relating the topology and geometry of 3-manifolds.We ask whether the isotopy class of a geodesic knot can be distinguished topologically within its homotopy class. We derive a purely topological description for infinite subfamilies of the closed geodesics studied previously in cusped manifolds, and draw explicit projection diagrams for these geodesics in the figure-eight knot complement. This leads to the result that the figure-eight knot complement contains geodesics of infinitely many different knot types in the3-sphere when the figure-eight cusp is filled trivially. / We conclude with a more direct investigation into geodesic knots in the figure-eight knot complement. We discuss methods of locating closed geodesics in this manifold including ways of identifying their isotopy class within a free homotopy class of closed curves. We also investigate a specially chosen class of knots in the figure-eight knot complement, namely those arising as closed orbits in its suspension flow. Interesting examples uncovered here indicate that geodesics of small tube radii may be difficult to distinguish topologically in their free homotopy class.

The computer programs SnapPea by Weeks and Geo by Casson have proven to be powerful tools in the study of hyperbolic 3-manifolds. Manifolds are special examples of spaces called orbifolds, which are modelled locally on R^n modulo finite groups of symmetries. SnapPea can also be used to study orbifolds but it is restricted to those whose singular set is a link.One goal of this thesis is to lay down the theory for a computer program that can work on a much larger class of 3-orbifolds. The work of Casson is generalized and implemented in a computer program Orb which should provide new insight into hyperbolic 3-orbifolds.The other main focus of this work is the study of 2-handle additions. Given a compact 3-manifold M and an essential simple closed curve α on ∂M, then we define M[α] to be the manifold obtained by gluing a 2-handle to ∂M along α. If α lies on a torus boundary component, we cap off the spherical boundary component created and the result is just Dehn filling.The case when α lies on a boundary surface of genus ≥ 2 is examined and conditions on α guaranteeing that M[α] is hyperbolic are found. This uses a lemma of Scharlemann and Wu, an argument of Lackenby, and a theorem of Marshall and Martin on the density of strip packings. A method for performing 2-handle additions is then described and employed to study two examples in detail.This thesis concludes by illustrating applications of Orb in studying orbifoldsand in the classification of knotted graphs. Hyperbolic invariants are used to distinguish the graphs in Litherland’s table of 90 prime θ-curves and provide access to new topological information including symmetry groups. Then by prescribing cone angles along the edges of knotted graphs, tables of low volume orbifolds are produced.

Thesis (M.Sc.) - Carleton University, 2006. / Includes bibliographical references (p. 162-164). Also available in electronic format on the Internet.

Thesis (M.S.)--University of Texas at El Paso, 2009. / Title from title screen. Vita. CD-ROM. Includes bibliographical references. Also available online.

Thesis (doctoral)--Universität Bonn, 1993. / Includes bibliographical references (p. 80-82).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1981. / Typescript. Vita. Description based on print version record. Includes bibliographical references (leaves 164-166).

Thesis--Wisconsin. / Vita. Includes bibliographical references (leaves 239-241).

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 1999. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.