Lectures on hyperbolic geometry. (English) Zbl 0768.51018

Benedetti, Riccardo; Petronio, Carlo

Universitext. Berlin etc.: Springer-Verlag. xiv, 330 p. (1992).

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 \((n\geq 4)\) 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)

MSC:

51M10	Hyperbolic and elliptic geometries (general) and generalizations
51-02	Research exposition (monographs, survey articles) pertaining to geometry
32Q45	Hyperbolic and Kobayashi hyperbolic manifolds
32-02	Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
32G15	Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
58B20	Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds

Keywords:

hyperbolic geometry; Thurston’s hyperbolic surgery theorem