A linear bound on the tetrahedral number of manifolds of bounded volume (after Jorgensen and Thurston)

We provide a detailed proof of the following folklore theorem: Let mu > 0 be a Margulis constant for 3-dimensional hyperbolic space. Then for any d>0 there exists a constant K>0, depending on mu and d, so that for any complete finite volume hyperbolic 3-manifold M, the d-neighborhood of the mu-thick part of M can be triangulated using at most K Vol(M) tetrahedra; here Vol is the hyperbolic volume function. As a corollary, we obtain the following topological interpretation of the volume: the minimal number of tetrahedra required to triangulate a link exterior in M is linearly equivalent to Vol(M); for a precise statement see Corollary 1.3.