  Our goal is to better understand the relationship between the polyhedron and
the group associated with a fundamental domain in H^3. In this paper, we will
study torsion-free groups and determine a formula for how many edge classes a
given abstract polyhedron must have. We will use that result to classify all
fundamental domains on the cube with torsion-free groups, including a
discussion of the explicit groups associated to those domains. We will then
turn to more general fundamental domains and prove a series of results about
how properties of the group place restrictions on the edge classes in the
quotient manifold. These results give insight into how the polyhedron and the
group associated to a fundamental domain interact, as well as offering concrete
tools to find fundamental domains.
