Mom technology and hyperbolic 3-manifolds

TL;DR

This paper reviews Mom-technology's role in identifying minimal-volume hyperbolic 3-manifolds and surveys key results and open problems in hyperbolic 3-manifold volume research.

Contribution

It provides an expository overview of recent advances in Mom-technology and its application to hyperbolic 3-manifolds, including new insights into minimal volumes.

Findings

Identification of the minimum-volume compact hyperbolic 3-manifold

Determination of the 10 smallest cusped hyperbolic 3-manifolds

Survey of key results on hyperbolic 3-manifold volumes

Abstract

This is an expository paper on Mom-technology, describing the recent work of the authors in this area (found in arXiv:math/0606072, arXiv:0705.4325, and arXiv:0809.0346) concerning the use of Mom-technology to find the minimum-volume compact hyperbolic 3-manifold and the 10 smallest cusped hyperbolic 3-manifolds. In addition we provide a survey of a selection of results on volumes of hyperbolic 3-manifolds, and a discussion of outstanding open problems in this area.