UNIVERSAL CONSTRAINTS OF KLEINIAN GROUPS AND HYPERBOLIC GEOMETRY

Recent advances in geometry have shown the wide application of hyperbolic geometry not only in Mathematics but also in real-world applications. As in two dimensions, it is now clear that most three-dimensional objects (configuration spaces and manifolds) are modelled on hyperbolic geometry. This point of view explains a great many things from large-scale cosmological phenomena, such as the shape of the universe, right down to the symmetries of groups and geometric objects, and various physical theories. Kleinian groups are basically discrete groups of isometries associated with tessellations of hyperbolic space. They form the fundamental groups of hyperbolic manifolds. Over the last few decades, the theory of Kleinian groups has. flourished because of its intimate connections with low-dimensional topology and geometry, especially with three-manifold theory. The universal constraints for Kleinian groups in part arise from a novel description of the moduli spaces of discrete groups and generalize known universal constraints for Fuchsian groups - discrete subgroups of isometries of the hyperbolic plane. These generalizations will underpin a new understanding of the geometry and topology of hyperbolic three-manifolds and their associated singular spaces, hyperbolic three-orbifolds. The novel approach in this dissertation is to use a fundamental result concerning spaces of finitely generated Kleinian groups: they are closed in the topology of algebraic convergence. Indeed, this is also true in higher dimensions when minor additional and necessary conditions are imposed. For instance, giving a uniform bound on the torsion in a sequence, or asking that the limit set is in geometric position. In fact, this property holds more generally for groups of isometries of negatively curved metrics because of the Margulis-Gromov lemma. In particular, new polynomial trace identities in the Lie group SL(2; C) are discovered to expose various quantifiable inequalities (including Jørgensen’s inequality) in a more general setting for Kleinian groups and the geometry of associated three-manifolds. This approach offers further substantive advances to address the quite complicated analytic and topological properties of hyperbolic orbifolds, thereby advancing the solutions to important unsolved problems