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
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Date
2010
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Massey University
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Abstract
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.
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Kleinian groups, Hyperbolic 3-manifolds and 3-orbifolds