Lectures on Hyperbolic GeometryIn 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. |
Contents
II | 1 |
IV | 3 |
V | 7 |
VI | 22 |
VII | 25 |
VIII | 37 |
IX | 45 |
X | 55 |
XXVIII | 190 |
XXIX | 196 |
XXX | 198 |
XXXI | 201 |
XXXII | 207 |
XXXIII | 210 |
XXXIV | 223 |
XXXV | 224 |
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Common terms and phrases
algebraic assume bijective boundary Chabauty topology compact complete component connected consider construction contains converges corresponding cusp Def(M define definition Dehn surgery denote diffeomorphism dimension easily checked edges element of T3 endowed endpoint equivalent Euclidean exists fact fiber bundle finite number fixed point flat fiber bundle fundamental group geodesic geodesic line geometric given glueing Gromov norm hence holonomy homeomorphic homotopy horoball hyperbolic Dehn surgery hyperbolic manifold hyperbolic structure identity IH³ implies infinity integer intersection isometry isomorphism Lemma Let us remark loop M₁ mapping metric Moreover n-manifold neighborhood non-trivial obtained oriented orthogonal proof Proposition prove quotient quotient set recall respect result Riemannian Riemannian manifold rigidity theorem sequence simplex space sphere subgroup subset subspace surface tetrahedra topological space topology torus triangle trivial universal covering vertex vertices vol(M volume function well-defined
Popular passages
Page 324 - An embedding theorem for connected 3-manifolds with boundary, Proc. Amer. Math. Soc. 16 (1965), 559—566.
References to this book
A Course in Metric Geometry Dmitri Burago,I︠U︡riĭ Dmitrievich Burago,Sergeĭ Ivanov No preview available - 2001 |