Hyperbolic manifolds and Kleinian groups. (English) Zbl 0892.30035

Oxford Mathematical Monographs. Oxford: Clarendon Press. ix, 253 p. (1998).
The recent work of Sullivan and Thurston has clearly established the fact that hyperbolic manifolds and Kleinian groups must be studied together, that they are complementary aspects of the same phenomenon, that either without the other is but half a theory. The authors of the present book have pulled together many of the most magical and difficult moments of that full theory in a manner which gives each aspect its due. Some of the highlights include the Margulis constant, the Jorgensen inequality, the Selberg lemma, the Maskit-Klein combination theorems, geometric finiteness, Ahlfors measure zero theorem, Mostow rigidity theorem, Sullivan rigidity theorem, Teichmüller space, quasiconformal deformation space, the Bers embedding, pleated surfaces, the Thurston compactification, the Thurston double limit theorem, geometrically tame Kleinian groups, the Bonahon theorem, geometric and algebraic convergence of Kleinian groups.
Reviewer: J.W.Cannon (Provo)


30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable
57M25 Knots and links in the \(3\)-sphere (MSC2010)