Two elliptic generator Kleinian groups : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Albany, New Zealand - Zhang, Qingxiang (Massey University, 2010)
This thesis studies the discreteness of Kleinian groups and the geometry of their associated orbit spaces: hyperbolic 3-manifolds and 3-orbifolds. Thurston's geometrization theorem states that the interior of every compact 3-manifold can be decomposed into pieces which have geometric structures. Most of these pieces have hyperbolic structures. Every hyperbolic 3-manifold can be described as H3=G, where H3 is the 3-dimensional hyperbolic space and G is a torsion free Kleinian group. The compact 3-manifolds other than hyperbolic 3-manifolds have been classified. Therefore the study of Kleinian groups holds the key to understanding 3-manifolds in general. We investigate various conditions for determining the discreteness of a Kleinian group. Such a group is discrete if and only if all its two generator subgroups are discrete, thus we study the question of discreteness through two generator groups. Here we consider Kleinian groups generated by two elements of finite order. Once the order of these generators is fixed, these groups lie in a one complex-dimensional parameter space with a highly fractal boundary. Computational investigations of various types into the size and structure of this parameter space form an integral part of this thesis. In particular an analysis of Dehn surgery on 2-bridge knots and links gives us data indicating the boundary of the parameter space "internally" analogous to the boundary in the Riley slice (the parabolic case). We then investigate this boundary "externally" and numerically using the rational pleating rays and rational words, which were developed by Keen and Series to study the Riley slice boundary. We are then able to identify some boundary points via the interplay between algebraic and geometric convergence of sequences of Kleinian groups. We find an interesting connection between Conway's notation for 2-bridge knots and links and rational words. We then apply these descriptions of parameter spaces to advance the identification problem for two generator arithmetic Kleinian groups, in fact we give a conjecturally complete list of certain families of such groups that were previously identified as being the most difficult cases.
Keywords: Kleinian groups; Hyperbolic 3-manifolds and 3-orbifolds; Fields of Research::230000 Mathematical Sciences::230100 Mathematics::230105 Group theory and generalisations (incl. topological groups and Lie groups)