These polished lecture notes provide a readable account of recent results in hyperbolic geometry, assuming only some facility with Riemannian geometry and algebraic topology. They are organized into six chapters, labeled $A$ through $F$. The first two chapters treat the basic properties of $n$-dimensional hyperbolic manifolds, with occasional specialization to the case $n=2$, culminating in the Fenchel-Nielsen parametrization of Teichmüller space. The third chapter gives a singular chains version of the Gromov-Thurston proof for the Mostow rigidity theorem in the compact case. Chapter $D$ contains a proof of the Margulis lemma and some applications. Chapter $E$ accounts for over a third of the book; it deals with the volume function on the space of $n$- dimensional hyperbolic manifolds; included are a proof of Wang’s theorem $(n\ge 4)$ and an account of the Jorgensen-Thurston theory $(n=3)$. Here the authors provide a reorganized proof of Thurston’s hyperbolic surgery theorem, avoiding apparent gaps in previous expositions. The final chapter sketches the theory of bounded cohomology, concluding with a section on Sullivan’s conjecture and amenable groups. The text includes a spare but useful 2-page subject index, a notation index, and 175 very helpful line drawings.
Reviewer: T.J.Barth (Riverside)