The number one story of 2006 in Science, the weekly magazine of the AAAS, was the solution of the Poincaré conjecture by Grigori Perelman. His solution was based on the completion of the geometrization conjecture of Thurston — that a compact three-dimensional manifold is composed of pieces, each endowed with a geometric structure coming from a list of eight possible geometries on three-space. This is a subtle generalization of what happens for surfaces — the tori with genus greater than one can be formed from a regular polygon in the hyperbolic plane with identifications determined by the fundamental group. This endows the surface with a hyperbolic geometry.
This book consists of papers, submitted after a conference at Warwick in 1984, and collected first into two books, Analytic and Geometric Aspects of Hyperbolic Space and Low-Dimensional Topology and Kleinian Groups, published in 1987. The new edition has put these two volumes together and provided an expansive foreward bringing the reader up to date.
The contributions are in four parts: "Notes on Notes of Thurston" by Canary, Epstein and Green which presents an exposition of parts of the lecture notes of Thurston, that later (1997) appeared in a polished version as Three-dimensional geometry and topology. The goal in to fix the foundations needed to understand Thurston's proof of the existence of the existence of hyperbolic structures on Haken atoroidal 3-manifolds.
The second part, by Epstein and Marden, is "Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces." The three-fold title explains the focus of this portion of the book, with the theorem of Sullivan giving a universal bound on the distortion of the boundary of the core of a hyperbolic manifold, presented as H3/Γ and its visible boundary S2∞/Γ. These results play a key role in Thurston's hyperbolization theorem for fiber bundles over the circle with fiber a surface.
Part three of the book is by Thurston, "Earthquakes in two-dimensional hyperbolic geometry." An earthquake is way to deform hyperbolic structures and the goal is a proof that any two hyperbolic structures on a surface can be deformed to each other via a (left) earthquake. The analogue for the complex plane of an earthquake is a quasi-conformal mapping. The final paper is "Lectures on measures on limit sets of Kleinian group" by Patterson, and the editors deem it the most important paper of the collection. The author develops from scratch the theory of measures on limit sets of Kleinian groups acting on any dimensional hyperbolic space (for nice choices of group). The results here have been developed further, especially by Sullivan, as a tool in differential geometry and geometric group theory.
This collection is aimed at the researcher beginning work on hyperbolic manifolds. Together with Thurston's notes (both versions), the reader will find a remarkable set of tools with which to plumb the mysteries of these manifolds.
John McCleary is Professor of Mathematics at Vassar College.