Hyperbolic Coxeter Polytopes with Multiple Infinity Edges

An ongoing problem for the study of Coxeter polyhedra is the classification and construction of such polyhedra in a hyperbolic setting be they finite volume, compact, or otherwise. Within this setting, Coxeter groups may be represented as isometry groups acting on n-dimensional Lobachevsky space. This understanding differs significantly from the purely algebraic study of hyperbolicity for a Coxeter group, where the more varied conditions for parallel hyperplanes are suppressed in the group's presentation. Yet, in many cases, multiple polyhedra can be realized as the fundamental domain for a single Coxeter group up to isometry. This dissertation specifically investigates the formation of Coxeter polytopes with multiple parallel facets, relating the polytope's hyperbolicity to the eigen-structures and index of the polytope's gram matrix. This relation may be further computed through basic computational tools such as MATLAB.