gt.geometric topology - Do you know how to construct a compact hyperbolic 3-manifold with three or four totally geodesic boundary components?

A combinatorial way to construct examples consists of taking any (topological) ideal triangulation with k ideal vertices where all edges have valence >=7 (i.e. each edge meets at least 7 tetrahedra, counted with multiplicity: if you take at least 6 you might find toric cusps). After removing an open star of the vertices you get a 3-manifold with k boundary components which admits such a metric.

(To prove this, see http://arxiv.org/abs/math/0402339, or prove that there cannot be any normal surfaces with non-negative Euler characteristic in such a triangulation and use geometrization.)

You can constrcut concretely the metric if all edges have the same valence v>=7: in this case you realize every tetrahedron as a regular truncated hyperbolic tetrahedron with dihedral angles at all 6 edges equal to 2pi/v (such an object exists since this angle is smaller than pi/3).

The number of tetrahedra you need for such a construction of course grows with k. For k=1, two tetrahedra suffice (see Thurston's knotted Y from his lecture notes).