UID:
almafu_9960117277202883
Umfang:
1 online resource (xviii, 515 pages) :
,
digital, PDF file(s).
Ausgabe:
First edition.
ISBN:
1-316-43039-1
,
1-316-43536-9
,
1-316-33777-4
Inhalt:
Over the past three decades there has been a total revolution in the classic branch of mathematics called 3-dimensional topology, namely the discovery that most solid 3-dimensional shapes are hyperbolic 3-manifolds. This book introduces and explains hyperbolic geometry and hyperbolic 3- and 2-dimensional manifolds in the first two chapters and then goes on to develop the subject. The author discusses the profound discoveries of the astonishing features of these 3-manifolds, helping the reader to understand them without going into long, detailed formal proofs. The book is heavily illustrated with pictures, mostly in color, that help explain the manifold properties described in the text. Each chapter ends with a set of exercises and explorations that both challenge the reader to prove assertions made in the text, and suggest further topics to explore that bring additional insight. There is an extensive index and bibliography.
Anmerkung:
Title from publisher's bibliographic system (viewed on 05 Jan 2016).
,
Cover -- Half-title page -- Frontis piece -- Title page -- Copyright page -- Dedication -- Contents -- List of Illustrations -- Preface -- 1 Hyperbolic space and its isometries -- 1.1 Möbius transformations -- 1.2 Hyperbolic geometry -- 1.2.1 The hyperbolic plane -- 1.2.2 Hyperbolic space -- 1.3 The circle or sphere at infinity -- 1.4 Gaussian curvature -- 1.5 Further properties of Möbius transformations -- 1.5.1 Commutativity -- 1.5.2 Isometric circles and planes -- 1.5.3 Trace identities -- 1.6 Exercises and explorations -- 2 Discrete groups -- 2.1 Convergence of Möbius transformations -- 2.1.1 Some group terminology -- 2.2 Discreteness -- 2.3 Elementary discrete groups -- 2.4 Kleinian groups -- 2.4.1 The limit set Λ (G) -- 2.4.2 The ordinary (regular, discontinuity) set Ω (G) -- 2.5 Quotient manifolds and orbifolds -- 2.5.1 Covering surfaces and manifolds -- 2.5.2 Orbifolds -- 2.5.3 The conformal boundary -- 2.6 Two fundamental algebraic theorems -- 2.7 Introduction to Riemann surfaces and their uniformization -- 2.8 Fuchsian and Schottky groups -- 2.8.1 Handlebodies -- 2.9 Riemannian metrics and quasiconformal mappings -- 2.10 Teichmüller spaces of Riemann surfaces -- 2.10.1 Teichmüller mappings -- 2.11 The mapping class group MCG(R) -- 2.11.1 Dehn twists -- 2.11.2 The action of MCG(R) on R and Teich(R) -- 2.11.3 The complex structure of Teich(R) -- 2.12 Exercises and explorations -- 2.12.1 Summary of group properties -- 3 Properties of hyperbolic manifolds -- 3.1 The Ahlfors Finiteness Theorem -- 3.2 Tubes and horoballs -- 3.3 Universal properties in hyperbolic 3-manifolds and orbifolds -- 3.4 The thick/thin decomposition of a manifold -- 3.5 Fundamental polyhedra -- 3.5.1 The Ford fundamental region and polyhedron -- 3.5.2 Poincaré's Theorem -- 3.5.3 The Cayley graph dual to tessellation -- 3.6 Geometric finiteness -- 3.6.1 Finite volume.
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3.7 Three-manifold surgery -- 3.7.1 Compressible and incompressible boundary -- 3.7.2 Extensions ∂M→M -- 3.8 Quasifuchsian groups -- 3.8.1 Simultaneous uniformization -- 3.9 Geodesic and measured geodesic laminations -- 3.9.1 Geodesic laminations -- 3.9.2 Measured geodesic laminations -- 3.9.3 Geometric intersection numbers -- 3.9.4 Length of measured laminations -- 3.10 The convex hull of the limit set -- 3.10.1 The bending measure -- 3.10.2 Pleated surfaces -- 3.11 The convex core -- 3.11.1 Length estimates for the convex core boundary -- 3.11.2 Bending measures on convex core boundary -- 3.12 The compact and relative compact core -- 3.13 Rigidity of hyperbolic 3-manifolds -- 3.14 Exercises and explorations -- 4 Algebraic and geometric convergence -- 4.1 Algebraic convergence -- 4.2 Geometric convergence -- 4.3 Polyhedral convergence -- 4.4 The geometric limit -- 4.5 Sequences of limit sets and regions of discontinuity -- 4.5.1 Hausdorff and Carathéodory convergence -- 4.5.2 Convergence of groups and regular sets -- 4.6 New parabolics -- 4.7 Acylindrical manifolds -- 4.8 Dehn filling and Dehn surgery -- 4.9 The prototypical example -- 4.10 Manifolds of finite volume -- 4.10.1 The Dehn Surgery Theorem -- 4.10.2 Sequences of volumes -- 4.10.3 Well ordering of volumes -- 4.10.4 Minimum volumes -- 4.11 Exercises and explorations -- 5 Deformation spaces and the ends of manifolds -- 5.1 The representation variety -- 5.1.1 The discreteness locus -- 5.1.2 The quasiconformal deformation space T(G) -- 5.2 Homotopy equivalence -- 5.2.1 Components of the discreteness locus -- 5.3 The quasiconformal deformation space boundary -- 5.3.1 Bumping and self-bumping -- 5.4 The three conjectures for geometrically infinite manifolds -- 5.5 Ends of hyperbolic manifolds -- 5.6 Tame manifolds -- 5.7 The Ending Lamination Theorem -- 5.8 The Double Limit Theorem.
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5.9 The Density Theorem -- 5.10 Bers slices -- 5.11 The quasifuchsian space boundary -- 5.11.1 The Bers (analytic) boundary -- 5.11.2 The Thurston (geometric) boundary -- 5.12 Examples of geometric limits at the Bers boundary -- 5.13 Classification of the geometric limits -- 5.14 Cannon-Thurston mappings -- 5.14.1 The Cannon-Thurston Theorem -- 5.14.2 Cannon-Thurston mappings and local connectivity -- 5.15 Exercises and explorations -- 6 Hyperbolization -- 6.1 Hyperbolic manifolds that fiber over a circle -- 6.1.1 Automorphisms of surfaces -- 6.1.2 Pseudo-Anosov mappings -- 6.1.3 The space of hyperbolic metrics -- 6.1.4 Fibering -- 6.2 Hyperbolic gluing boundary components -- 6.2.1 Skinning a bordered manifold -- 6.2.2 Totally geodesic boundary -- 6.2.3 Gluing boundary components -- 6.2.4 The Bounded Image Theorem -- 6.3 Hyperbolization of 3-manifolds -- 6.3.1 Review of definitions in 3-manifold topology -- 6.3.2 Hyperbolization -- 6.4 The three big conjectures, now theorems, for closed manifolds -- 6.4.1 Surface subgroups of π[sub(1)](M(G)) =G -- 6.4.2 Remarks on the proof of VHT and VFT: Cubulation -- 6.4.3 Prior computational evidence -- 6.5 Geometrization -- 6.6 Hyperbolic knots and links -- 6.6.1 Knot complements -- 6.6.2 Link complements -- 6.7 Computation of hyperbolic manifolds -- 6.8 The Orbifold Theorem -- 6.9 Exercises and explorations -- 7 Line geometry -- 7.1 Half-rotations -- 7.2 The Lie product -- 7.3 Square roots -- 7.4 Complex distance -- 7.5 Complex distance and line geometry -- 7.6 Exercises and explorations -- 8 Right hexagons and hyperbolic trigonometry -- 8.1 Generic right hexagons -- 8.2 The sine and cosine laws -- 8.3 Degenerate right hexagons -- 8.4 Formulas for triangles, quadrilaterals, and pentagons -- 8.5 Exercises and explorations -- Bibliography -- Index.
,
English
Weitere Ausg.:
ISBN 1-107-11674-0
Sprache:
Englisch
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