On the Classification of Hyperbolic Triangle Groups and Non-Arithmetic Lattices of Hyperbolic Orbifolds

The study of hyperbolic manifolds, and more generally hyperbolic orbifolds, is inti-mately bound to the study of discrete subgroups of the isometry group of hyperbolic n-space. In the wake of certain rigidity theorems due to Mostow et al., a new program of study has developed in recent decades for the characterization of hyperbolic mani- folds by investigating certain invariants arising from the theory of numbers. Critical to the arithmetic study of hyperbolic manifolds are those discrete subgroups of the isometry group which have finite co-volume under the Haar metric, sometimes called lattices. These correlate to a particular tiling of hyperbolic space with a certain fun- damental domain. The simplest non-trivial example of these for hyperbolic orbifolds are triangle groups. These triangle groups, or more properly arithmetic Fuchsian tri- angle groups, were first classified by Takeuchi in 1983. In the proceeding manuscript, a concise introduction to the geometry of hyperbolic manifolds and orbifolds is put forth. The two primary invariants used in the study of the hyperbolic lattices, the invariant trace field and the invariant quaternion algebra, are then defined. There- after, a hyperbolic triangle group is constructed from the tessellation of the hyperbolic plane by hyperbolic triangles. A version of the classification theorem of arithmetic Fuchsian triangle groups is stated and proved. The paper concludes with a brief discussion regarding non-arithmetic lattices.