Lectures on Hyperbolic Geometry

Front Cover
Springer Science & Business Media, 1992 - Mathematics - 330 pages
In recent years hyperbolic geometry has been the object and the preparation for extensive study that has produced important and often amazing results and also opened up new questions. The book concerns the geometry of manifolds and in particular hyperbolic manifolds; its aim is to provide an exposition of some fundamental results, and to be as far as possible self-contained, complete, detailed and unified. Since it starts from the basics and it reaches recent developments of the theory, the book is mainly addressed to graduate-level students approaching research, but it will also be a helpful and ready-to-use tool to the mature researcher. After collecting some classical material about the geometry of the hyperbolic space and the Teichmüller space, the book centers on the two fundamental results: Mostow's rigidity theorem (of which a complete proof is given following Gromov and Thurston) and Margulis' lemma. These results form the basis for the study of the space of the hyperbolic manifolds in all dimensions (Chabauty and geometric topology); a unified exposition is given of Wang's theorem and the Jorgensen-Thurston theory. A large part is devoted to the three-dimensional case: a complete and elementary proof of the hyperbolic surgery theorem is given based on the possibility of representing three manifolds as glued ideal tetrahedra. The last chapter deals with some related ideas and generalizations (bounded cohomology, flat fiber bundles, amenable groups). This is the first book to collect this material together from numerous scattered sources to give a detailed presentation at a unified level accessible to novice readers.
 

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Contents

II
1
IV
3
V
7
VI
22
VII
25
VIII
37
IX
45
X
55
XXVI
190
XXVII
196
XXVIII
198
XXIX
201
XXX
207
XXXI
210
XXXII
223
XXXIII
224

XI
58
XII
61
XIII
83
XIV
84
XV
94
XVI
103
XVII
105
XVIII
121
XIX
133
XX
140
XXI
143
XXII
159
XXIII
160
XXIV
174
XXV
184
XXXIV
234
XXXV
251
XXXVI
256
XXXVII
267
XXXVIII
273
XXXIX
277
XL
280
XLI
287
XLII
294
XLIII
303
XLIV
321
XLV
324
XLVI
326
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Page 326 - An embedding theorem for connected 3-manifolds with boundary, Proc. Amer. Math. Soc. 16 (1965), 559—566.

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