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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.

Inhaltsverzeichnis

Frontmatter

Chapter A. Hyperbolic Space

Abstract
This chapter is devoted to the definition of a Riemannian n-manifold ℍ n called hyperbolic n-space and to the determination of its geometric properties (isometries, geodesies, curvature, etc.). This space is the local model for the class of manifolds we shall deal with in the whole book. The results we are going to prove may be found in several texts (e.g. [Bea], [Co], [Ep2], [Fe], [Fo], [Greenb2], [Mag], [Mask2], [Th1, ch. 3] and [Wol]) so we shall omit precise references. The line of the present chapter is partially inspired by [Ep2], though we shall be dealing with a less general situation. For a wide list of references about hyperbolic geometry from ancient times to 1980 we address the reader to [Mi3].
Riccardo Benedetti, Carlo Petronio

Chapter B. Hyperbolic Manifolds and the Compact Two-dimensional Case

Abstract
In this chapter we are going to introduce the notion of hyperbolic manifold (i.e. a manifold modeled on hyperbolic space) via the introduction of a much more general class of manifolds. We shall prove the first essential properties of such manifolds (namely, the fact that if a hyperbolic manifold is complete then it can be obtained as a quotient of hyperbolic space). Afterwards we shall consider the special case of compact surfaces and we shall give a complete classification of the hyperbolic structures on a surface of fixed genus (that is we shall give a parametrization of the so-called Teichmüller space).
Riccardo Benedetti, Carlo Petronio

Chapter C. The Rigidity Theorem (Compact Case)

Abstract
In this chapter we are going to prove that there is a very sharp difference between 2-dimensional hyperbolic geometry and higher dimensions (at least for the compact case, but the results we shall prove generalize to the case of finite volume). Namely, we shall prove that for n ≥ 3 a connected, compact, oriented n-manifold supports at most one (equivalence class of) hyperbolic structure (while it was proved in Chapt. B that a compact surface of genus at least 2 supports uncountably many non-equivalent hyperbolic structures). This is the famous Mostow rigidity theorem: the original proof can be found in [Mos], and others (generalizing the first one) in [Mar] and [Pr]; we shall refer mostly to [Gro3], [Th1, ch. 6] and [Mu]. The core of the proof we present resides in Theorem C.4.2, relating the Gromov norm (introduced in C.3) to the volume of a compact hyperbolic manifold; this result has a deep importance independently of the rigidity theorem: in Chapters E and F we shall meet interesting applications and related ideas.
Riccardo Benedetti, Carlo Petronio

Chapter D. Margulis’ Lemma and its Applications

Abstract
In this chapter we begin the study of complete hyperbolic manifolds which are not necessarily compact. The essential tool of our investigations is Margulis’ lemma, which is proved in the first section, while the second section is devoted to the basic properties of the thin-thick decomposition of a hyperbolic manifold. In the third section several facts are deduced about the shape of the ends, in particular in the case of finite volume.
Riccardo Benedetti, Carlo Petronio

Chapter E. The Space of Hyperbolic Manifolds and the Volume Function

Abstract
In the whole of this chapter we shall always suppose manifolds are connected and oriented. It follows from the Gauss-Bonnet formula B.3.3 (for n = 2) and from the Gromov-Thurston theorem C.4.2 (for n ≥ 3) that the volume of a hyperbolic manifold is a topological invariant. Moreover B.3.3 implies that such an invariant is (topologically) complete for n = 2 in the compact case, and it may be proved that in the finite-volume case it becomes complete together with the number of cusp ends (“punctures”). Hence the problem of studying the volume function arises quite naturally: this is the aim of the present chapter.
Riccardo Benedetti, Carlo Petronio

Chapter F. Bounded Cohomology, a Rough Outline

Abstract
In this chapter we point out some of the basic ideas of the theory of bounded cohomology we first met during the proof of the rigidity theorem (Sect. C.3: compare F.2.2 below). In particular we define the groups of singular bounded cohomology and we consider the natural class of cohomology arising from the problem of the existence of a global non-vanishing section on a flat fiber bundle (known as Euler class of the bundle). In connection with the notion of Euler class we introduce and develop the definition of amenable group.
Riccardo Benedetti, Carlo Petronio

Backmatter

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