Lectures on Hyperbolic Geometry
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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algebraic assume bijective boundary canonically centred Chabauty topology closed cohomology component construction contains converges Corollary corresponding cusp define definition Dehn surgery denote diffeomorphism dimension disc easily checked edges element of T3 endowed endpoint equivalent Euclidean exists fact fiber bundle finite number finite-volume fixed point flat fiber bundle fundamental group geodesic geodesic line geometric geometric topology given glueing Gromov norm half-space model hence holonomy homeomorphic homotopy horoball hyperbolic Dehn surgery hyperbolic manifold hyperbolic structure hyperbolic surgery hyperplane identity implies induces integer intersection inversion isometry isomorphism J(Hn Lemma Let us remark loop mapping metric Moreover n-manifold neighborhood non-trivial obtained oriented orthogonal proof Proposition prove quotient quotient set realization recall represented respect result Riemannian Riemannian manifold rigidity theorem sequence simplex simplices sphere subgroup subset subspace surface tetrahedra topological space topology torus triangle trivial universal covering vertex vertices volume function well-defined