ar X iv : m at h . G T / 0 60 60 72 v 2 3 1 Ju l 2 00 6 1 MOM TECHNOLOGY AND VOLUMES OF HYPERBOLIC 3 - MANIFOLDS

This paper is the first in a series whose goal is to understand the structure of low-volume complete orientable hyperbolic 3-manifolds. Here we introduce Mom technology and enumerate the hyperbolic Mom-n manifolds for n ≤ 4. Our longterm goal is to show that all low-volume closed and cusped hyperbolic 3-manifolds are obtained by filling a hyperbolic Mom-n manifold, n ≤ 4 and to enumerate the low-volume manifolds obtained by filling such a Mom-n. William Thurston has long promoted the idea that volume is a good measure of the complexity of a hyperbolic 3-manifold (see, for example, [Th1] page 6.48). Among known low-volume manifolds, Jeff Weeks ([We]) and independently Sergei Matveev and Anatoly Fomenko ([MF]) have observed that there is a close connection between the volume of closed hyperbolic 3-manifolds and combinatorial complexity. One goal of this project is to explain this phenomenon, which is summarized by the following: