#### J. Amer. Math. Soc. 22 (2009), 1157-1215.

**David Gabai**, *Department of Mathematics, Princeton University*;
**Robert Meyerhoff**,* Department of Mathematics, Boston College*; **Peter Milley**, *Department of Mathematics and Statistics, University of Melbourne*

In the 1970's, Thurston and Jørgensen showed that the volumes of orientable finite-volume hyperbolic 3-manifolds form a well-ordered subset of the real numbers. In particular, there is a smallest such volume. Around 1985, Weeks and Matveev-Fomenko independently discovered a candidate for a least-volume orientable hyperbolic 3-manifold, which is now commonly termed the Weeks manifold. Thurston, Hodgson-Weeks and Matveev-Fomenko conjectured that low volume implies low combinatorial/topological complexity. It has been a challenge to make this statement precise and to determine the hyperbolic manifolds of smallest volume.

The paper under review is a milestone in the study of orientable hyperbolic 3-manifolds of low volume. The authors depart from traditional measures of topological/combinatorial complexity, which arise from triangulations, Heegaard splittings or spines. They not only show that “Mom technology” allows a precise formulation of how low volume implies low complexity, but they also demonstrate that this platform allows topological, geometric, combinatorial and computational methods to be entwined in order to produce significant theoretical results. The paper under review is, for example, a key step in the proof of the fact that the Weeks manifold is indeed the unique orientable hyperbolic 3-manifold of least volume. It is beyond doubt that the methods and results of this paper will have major impact and lead to further crucial applications.