Discrete groups and computational geometry : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Albany, New Zealand

Abstract

Let f and g be Möbius transformations with finite-orders p and q respectively. Further, let γ = tr[f; g] - 2, where tr[f; g] is the trace of the commutator of f and g in the standard SL(2;C) representation of Möbius transformations. The group G = hf; gi is then defined, up to conjugacy, by the parameter set (p; q; γ), whenever γ≠ 0. If the group G is discrete and non-elementary, then it is a Kleinian group. Kleinian groups are intimately related to hyperbolic 3-orbifolds. Here we develop a computer program that constructs a fundamental domain for such Kleinian groups. These constructions are undertaken directly from the parameters given above. We use this program to investigate, and add to, recent work on the classification of arithmetic Kleinian groups generated by two (finite-order) elliptic transformations.