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Two generator discrete groups of isometries and their representation : a thesis presented in partial fulfillment of the requirements for the degree of Master of Science in Mathematics at Massey University, Albany, New Zealand
Let M Φ and Mψ be elements of PSL(2,C) representing orientation preserving isometries on
the upper half-space model of hyperbolic 3-space Φ and ψ respectively. The parameters
β = tr2(M Φ) - 4, β1 = tr2(Mψ) - 4,
γ = tr[M Φ,Mψ] - 2,
determine the discrete group (Φ ,ψ) uniquely up to conjugacy whenever
γ ≠ 0. This thesis
is concerned with explicitly lifting this parameterisation of (Φ , ψ) to PSO(1, 3) realised as a
discrete 2 generator subgroup of orientation preserving isometries on the hyperboloid model
of hyperbolic 3-space. We particularly focus on the case where both Φ and ψ are elliptic.