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Lectures on hyperbolic geometry. (English) Zbl 0768.51018
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.

##### 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