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Outer Circles An Introduction to Hyperbolic 3-Manifolds

ISBN-10: 0521839742

ISBN-13: 9780521839747

Edition: 2007

Authors: A. Marden

List price: $108.00
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Description:

We live in a three-dimensional space; what sort of space is it? Can we build it from simple geometric objects? The answers to such questions have been found in the last 30 years, and Outer Circles describes the basic mathematics needed for those answers as well as making clear the grand design of the subject of hyperbolic manifolds as a whole. The purpose of Outer Circles is to provide an account of the contemporary theory, accessible to those with minimal formal background in topology, hyperbolic geometry, and complex analysis. The text explains what is needed, and provides the expertise to use the primary tools to arrive at a thorough understanding of the big picture. This picture is further filled out by numerous exercises and expositions at the ends of the chapters and is complemented by a profusion of high quality illustrations. There is an extensive bibliography for further study.
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Book details

List price: $108.00
Copyright year: 2007
Publisher: Cambridge University Press
Publication date: 5/31/2007
Binding: Hardcover
Pages: 446
Size: 6.75" wide x 9.75" long x 0.85" tall
Weight: 2.288
Language: English

List of Illustrations
Preface
Hyperbolic space and its isometries
Mobius transformations
Hyperbolic geometry
The circle or sphere at infinity
Gaussian curvature
Further properties of Mobius transformations
Exercises and explorations
Discrete groups
Convergence of Mobius transformations
Discreteness
Elementary discrete groups
Kleinian groups
Quotient manifolds and orbifolds
Two fundamental algebraic theorems
Introduction to Riemann surfaces and their uniformization
Fuchsian and Schottky groups
Riemannian metrics and quasiconformal mappings
Exercises and explorations
Properties of hyperbolic manifolds
The Ahlfors Finiteness Theorem
Tubes and horoballs
Universal properties
The thick/thin decomposition of a manifold
Fundamental polyhedra
Geometric finiteness
Three-manifold surgery
Quasifuchsian groups
Geodesic and measured geodesic laminations
The convex hull of the limit set
The convex core
The compact and relative compact core
Rigidity
Exercises and explorations
Algebraic and geometric convergence
Algebraic convergence
Geometric convergence
Polyhedral convergence
The geometric limit
Convergence of limit sets and regions of discontinuity
New parabolics
Acylindrical manifolds
Dehn surgery
The prototypical example
Manifolds of finite volume
The Dehn surgery theorems for finite volume manifolds
Exercises and explorations
Deformation spaces and the ends of manifolds
The representation variety
Homotopy equivalence
The quasiconformal deformation space boundary
The three great conjectures
Ends of hyperbolic manifolds
Tame manifolds
Quasifuchsian spaces
The quasifuchsian space boundary
Geometric limits at boundary points
Exercises and explorations
Hyperbolization
Hyperbolic manifolds that fiber over a circle
Automorphisms of surfaces
The Double Limit Theorem
Manifolds fibered over the circle
The Skinning Lemma
Hyperbolic manifolds with totally geodesic boundary
Skinning the manifold (Part II)
The Hyperbolization Theorem
Knots and links
Geometrization
The Orbifold Theorem
Exercises and Explorations
Line geometry
Half-rotations
The Lie product
Square roots
Complex distance
Complex distance and line geometry
Exercises and explorations
Right hexagons and hyperbolic trigonometry
Generic right hexagons
The sine and cosine laws for generic right hexagons
Degenerate right hexagons
Formulas for triangles, quadrilaterals, and pentagons
Exercises and explorations
Bibliography
Index