This thesis investigates the structure and properties of hyperbolic 3-manifold groups (particularly knot and link groups) and arithmetic Kleinian groups. In Chapter 2, we establish a stronger version of a conjecture of A. Reid and others in the arithmetic case: if two elements of equal trace (e.g., conjugate elements) generate an arithmetic two-bridge knot or link group, then the elements are parabolic (and hence peripheral). In Chapter 3, we identify all Kleinian groups that can be generated by two elements for which equality holds in Jørgensen’s Inequality in two cases: torsion-free Kleinian groups and non-cocompact arithmetic Kleinian groups. / text

<p>This project aims to give an overview of knots, orderability of knot groups, and to construct knots for which the knot groups enjoy some nice properties.</p> <p>To accomplish this, we first present some preliminary results concerning knots and knot groups. We then introduce the Alexander polynomial, and explain the idea of a special polynomial originally introduced by Linnell, Rhemtulla and Rolfsen. By investigating the conditions on a special polynomial, we classify all the special Alexander polynomial of fibred knots of degree less than 10. Finally we construct examples of fibred knots which have a special Alexander polynomial.</p> / Master of Science (MS)

We study two topics on surface diffeomorphisms, their mapping classes and dynamics. For the mapping classes of a punctured disc, we study the $\ZxZ$ subgroups of the fundamental groups of the corresponding mapping tori. An application is the proof of the fact that a satellite knot with braid pattern is prime. For the mapping classes of the disc minus a Cantor set, we study a special type of reducible mapping class. This has direct application on the embeddings of solenoids in $\mathbb{S}^3$. We also give some examples of other types of mapping classes of the disc minus a Cantor set. For the dynamics of surface diffeomorphisms, we prove three formulas for computing the topological pressure of a $C^1$-generic conservative diffeomorphism with no dominated splitting and show the continuity of topological pressure with respect to these diffeomorphisms. We prove for these generic diffeomorphisms that there is no equilibrium states with positive measure theoretic entropy. In particular, for hyperbolic potentials, there are no equilibrium states. For $C^1$ generic conservative diffeomorphisms on compact surfaces with no dominated splitting and $\phi_m(x):=-\frac{1}{m}\log \Vert D_x f^m\Vert, m \in \mathbb{N}$, we show that there exist equilibrium states with zero entropy and there exists a transition point $t_0$ for the one parameter family $\lbrace t \phi_m\rbrace_{t\geq 0}$, such that there is no equilibrium states for $ t \in [0, t_0)$ and there is an equilibrium state for $t \in [t_0,+\infty)$.