Hyperbolic manifolds and discrete groups
type: Item
instance of: scholarly article
title: Hyperbolic manifolds and discrete groups
zbMATH Open document ID: 0958.57001
author: Michael Kapovich
published in: Progress in Mathematics
publication date: 2001-01-07T00:00:00Z
review text: The main goal of the book is to present a proof of Thurston's Hyperbolization Theorem (``The Big Monster''): Suppose that \(M\) is a compact atoroidal Haken 3-manifold that has zero Euler characteristic. Then the interior of \(M\) admits a complete hyperbolic structure of finite volume. (This is exactly the beginning of the book). The hyperbolization theorem has been announced by \textit{W. P. Thurston} around 1976. Thurston himself never published a complete proof of his theorem (see the interesting comments in the last section of his paper [Bull. Am. Math. Soc., New Ser. 30, No. 2, 161-177 (1994; Zbl 0817.01031)], but prepared the ground for its proof by explaining the general scene and the basic concepts involved in various papers and preprints, and in particular in his Princeton lecture notes which were widely distributed and which revolutionized the whole field of low-dimensional topology, hyperbolic geometry, Kleinian groups and geometric group theory (part of these notes have appeared in his book ``Three-dimensional geometry and topology'', Princeton Math. Ser. 35 (1997; Zbl 0873.57001)). An early and careful account of the proof has been given by Morgan in the book [\textit{J. W. Morgan} (ed.) and \textit{H. Bass} (ed.), The Smith conjecture, Pure Appl. Math., Academic Press, 122 (1984; Zbl 0599.57001)] whose positive solution was a first major success of the new theory (using the still incomplete proof of the hyperbolization theorem). Then, in the next twenty years, many mathematicians worked on various aspects of the new theory, formalizing and extending the ideas of Thurston, often from different points of view. We recommend the excellent introduction of the present book for the history of the various contributions, and also for a sketch of the proof itself which we will not repeat here in detail (there are too many different aspects and persons involved in the story). The proof of the hyperbolization theorem naturally divides into two cases: the ``generi…
reviewed by: Bruno Zimmermann
Mathematics Subject Classification ID: 57-02
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 30F40
Mathematics Subject Classification ID: 20F65
zbMATH DE Number: 1548759
zbMATH Keywords: hyperbolization of 3-manifolds
zbMATH Keywords: hyperbolization theorem
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1548759