Arithmetic Kleinian groups and their Fuchsian subgroups
The aim of the thesis is to study in depth a certain class of hyperbolic 3-manifolds; namely those which are the quotient of hyperbolic 3-space by an arithmetic Kleinian group. In particular we consider the distribution and characterization of arithmetic Kleinian groups in the class of all Kleinian groups of finite covolume, the Fuchsian subgroup structure and the relationship between the Fuchsian subgroups (when they exist) and the arithmetic Kleinian group. In chapter 2 a characterization of arithmetic Kleinian groups via the traces of the elements in the group is given and, appealing directly to this, in chapter 3, a set of necessary and sufficient algebraic conditions for the existence of non-elementary Fuchsian subgroups is deduced. These conditions are given an equivalent alternative description in chapter 5 from which a technique is developed making identification of the field of definition a relatively simple algebraic operation. The technique is illustrated, taking as examples the eight arithmetic tetrahedral groups of Lanner. This enables an investigation of covolumes in the commensurability class of each group. The final chapter (chapter 6) investigates geometric and topological analogues for the manifolds associated to torsion-free arithmetic Kleinian groups which contain non-elementary Fuchsian subgroups. For such manifolds we answer in the affirmative conjectures of Thurston and Waldhausen on existence of haken covers and the first betti number.