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 employed to analyze simultaneously compact manifolds and finite-volume manifolds having toric cusps. In opposition to this we show that, if one allows annular cusps, the number of manifolds grows very rapidly, and that our strategy cannot be employed to obtain a complete list. We also carefully describe how to compute the volume of our manifolds, discussing formulae for the volume of a tetrahedron with generic dihedral angles in hyperbolic space.