双曲几何讲义（英文版）
出版时间：2012
内容简介
One of the main themes of this book is the conflict between the "flexibility' and the "rigidity properties of the hyperbolic manifolds： the first radical difference arises between the case of dimension 2 and the case of higher dimensions （as proved in chapters B and C）， an elementary feature of thus phenomenon being the difference between the Riemann mapping theorem and Liouville's theorem， as pointed out in chapter A. Thus chapter is rather clementary and most of its material may' be the object of an undergraduate course.Together with the rigidity theorem， a basic tool for the study of hyperbolic manifolds is Margulis' lemma， a detailed proof of which we give in chapter D; as a consequence of this result in the same chapter we also give a rather accurate description， in all dimensions， of the thin-thick decomposition of a hyperbolic manifold （especially in case of finite volume）.
目录
preface
chapter a.hyperbolic space
a.1 models for hyperbolic space
a.2 isometries of hyperbolic space: hyperboloid model
a.3 conformal geometry
a.4 isometries of hyperbolic space: disc and half-spacemodels
a.5 geodesics, hyperbolic subspaces and misceuaneo,s facts
a.6 curvature of hyperbolic space
chapter b.hyperbolic manifolds and the compact two-dimensionalcase
b.1 hyperbolic, elliptic and flat manifolds
b.2 topology of compact oriented surfaces
b.3 hyperbolic, elliptic and flat surfaces
b.4 teichmiiller space
chapter c.the rigidity theorem (compact case)
c.1 first step of the proof: extension of pseudo-isometrics
c.2 second step of the proof: volume of ideal simplices
c.3 gromov norm of a compact manifold
c.4 third step of the proof:the gromov norm and the volume areproportional
c.5 conclusion of the proof, corollaries and generalizations
chapter d.margulis' lemma and its applications
d.1 margnlis' lemma
d.2 local geometry of a hyperbolic manifold
d.3 ends of a hyperbolic manifold
chapter e.the space of hyperbolic manifolds and the volumefunction
e.1 the chahauty and the geometric topology
e.2 convergence in the geometric topology: opening cusps.the caseof dimension at least three
e.3 the case of dimension different from three.conclusions andexamples
e.4 the three-dimensional case: jorgensen's part of theso-calledjorgensen-tlmrston theory
e.5 the three-dimensional case. thurston's hyperbolic surgerytheorem: statement and preliminaries
e.5-i definition and first properties of ts (non-compactthree-manifolds with \triangulation\ without vertices)
e.5-ii hyperbolic structures on an element of ts and realizationof the complete structure
e.5-iii elements of ts and standard spines
e.5-iv some links whose complements are realized a.s elements ofts
e.6 proof of thurston's hyperbolic surgery theorem
e.6-i algebraic equations of h(m)(hyperbolic structures supportedby m∈t3)
e.6-ii dimension of h(m): general ca.se
e.6-iii the case m is complete hyperbolic:the space ofdeformations
e.6-iv completion of the deformed hyperbolic structures andconclusion of the proof
e.7 applications to the study of the volume f, mction andcomplements about three-dimensional hyperbolic geometry
chapter f.bounded cohomology, a rough outline
f.1 singular cohomology
f.2 bounded singular cohomology
f.3 flat fiber bundles
f.4 euler class of a flat vector bundle
f.5 flat vector bundles on surfaces and the milner-sullivantheorem
f.6 suuivan's conjecture and amenable groups
subject index
notation index
references
