Hyperbolic 3-Manifolds with Geodesic Boundary: Enumeration and Volume Calculation

We describe a natural strategy to enumerate compact hyperbolic 3-manifolds with geodesic boundary in increasing order of complexity. We show that the same strategy can be applied in order to analyze simultaneously compact manifolds and finite-volume manifolds with toric cusps. In contrast, we show that if one allows annular cusps, the number of manifolds grows very rapidly and our strategy cannot be employed to obtain a complete list. We also carefully describe how to compute the volume of our manifolds, discussing formulas for the volume of a tetrahedron with generic dihedral angles in a hyperbolic space.