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Graduate Texts in Mathematics #149
Foundations of Hyperbolic Manifolds
1 Euclidean Geometry.- 2 Spherical Geometry.- 3 Hyperbolic Geometry.- 4 Inversive Geometry.- 5 Isometries of Hyperbolic Space.- 6 Geometry of Discrete Groups.- 7 Classical Discrete Groups.- 8 Geometric Manifolds.- 9 Geometric Surfaces.- 10 Hyperbolic 3-Manifolds.- 11 Hyperbolic n-Manifolds.- 12 Geometrically Finite n-Manifolds.- 13 Geometric Orbifolds.
- GenresMathematicsGeometry
747 pages, Hardcover
First published January 1, 1994
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August 2, 2011
Having previously failed to grasp the subject after purchasing two other books on hyperbolic manifolds, this book, now in an edition revised in 2005, finally gives me hope I may one day begin to understand hyperbolic manifolds, and, more importantly, given me faith that is worth making the necessary effort.
The author has succeeded in his aim of creating a work of reference that is largely self-contained: the "foundations" mentioned in the title are laid very deeply - roughly half the book (7 chapters out of 13) is used to give a detailed presentation of background material. For instance, there is an entire chapter on Spherical geometry, and one on the classical discrete groups, (which includes a detailed proof of Selberg's lemma).
All this background material is presented both carefully and well, so that a substantial part of the book will prove a valuable reference to non-specialists.
Chapters 8 through 11 are the chapters specifically devoted to manifold theory, presenting first surfaces, then 3-manifolds and then n-manifolds. Chapter 12 introduces Kleinian Groups, and contains several very nice proofs about limit sets, which are the first I have seen that have made me really believe manifold theory could be a way to make the subject of Kleinian groups easier to understand rather than harder.
The final chapter concludes with a proof of Poincaré's Polyhedron theorem.
The author suggests that the material in this book could be studied reasonably thoroughly in a one year course, but that to cover it in depth would take around two years. So, having only owned this book for some six weeks at the time I write this, I am probably reviewing it too soon, and freely confess that there is much in that I do not understand as yet (and I think I will require nearer two years than one to work through the book properly). But I already feel justified in recommending it warmly because I am confident most potential purchases will appreciate not being endlessly referred to other works or "well-known" results in the development of the material, and agree that this is a book that will repay careful study and strikes a very nice balance between algebraic, geometric and topological points of view.
The author has succeeded in his aim of creating a work of reference that is largely self-contained: the "foundations" mentioned in the title are laid very deeply - roughly half the book (7 chapters out of 13) is used to give a detailed presentation of background material. For instance, there is an entire chapter on Spherical geometry, and one on the classical discrete groups, (which includes a detailed proof of Selberg's lemma).
All this background material is presented both carefully and well, so that a substantial part of the book will prove a valuable reference to non-specialists.
Chapters 8 through 11 are the chapters specifically devoted to manifold theory, presenting first surfaces, then 3-manifolds and then n-manifolds. Chapter 12 introduces Kleinian Groups, and contains several very nice proofs about limit sets, which are the first I have seen that have made me really believe manifold theory could be a way to make the subject of Kleinian groups easier to understand rather than harder.
The final chapter concludes with a proof of Poincaré's Polyhedron theorem.
The author suggests that the material in this book could be studied reasonably thoroughly in a one year course, but that to cover it in depth would take around two years. So, having only owned this book for some six weeks at the time I write this, I am probably reviewing it too soon, and freely confess that there is much in that I do not understand as yet (and I think I will require nearer two years than one to work through the book properly). But I already feel justified in recommending it warmly because I am confident most potential purchases will appreciate not being endlessly referred to other works or "well-known" results in the development of the material, and agree that this is a book that will repay careful study and strikes a very nice balance between algebraic, geometric and topological points of view.
Displaying 1 of 1 review