The Arithmetic of Hyperbolic 3-ManifoldsFor the past 25 years, the Geometrization Program of Thurston has been a driving force for research in 3-manifold topology. This has inspired a surge of activity investigating hyperbolic 3-manifolds (and Kleinian groups), as these manifolds form the largest and least well-understood class of compact 3-manifolds. Familiar and new tools from diverse areas of mathematics have been utilized in these investigations, from topology, geometry, analysis, group theory, and from the point of view of this book, algebra and number theory. This book is aimed at readers already familiar with the basics of hyperbolic 3-manifolds or Kleinian groups, and it is intended to introduce them to the interesting connections with number theory and the tools that will be required to pursue them. While there are a number of texts which cover the topological, geometric and analytical aspects of hyperbolic 3-manifolds, this book is unique in that it deals exclusively with the arithmetic aspects, which are not covered in other texts. Colin Maclachlan is a Reader in the Department of Mathematical Sciences at the University of Aberdeen in Scotland where he has served since 1968. He is a former President of the Edinburgh Mathematical Society. Alan Reid is a Professor in the Department of Mathematics at The University of Texas at Austin. He is a former Royal Society University Research Fellow, Alfred P. Sloan Fellow and winner of the Sir Edmund Whittaker Prize from The Edinburgh Mathematical Society. Both authors have published extensively in the general area of discrete groups, hyperbolic manifolds and low-dimensional topology. |
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This book gives a fantastic explanation of the state of the art (as of 2003) on the study of hyperbolic 3-manifolds using arithmetic invariants. It is an essential tool for anyone working in that branch of math, either as a reference book or as a way of learning the theory. Some critical typos hold me back from the full 5 stars though, it would benefit from a 2nd edition with that fixed.
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apply arithmetic Fuchsian arithmetic Kleinian groups assume basis Bianchi groups Chapter choose closed commensurable compact complement complex place conjugate consider construction contains Corollary cusp Dedekind domain deduce defined Definition denote described determine discrete discriminant discussed elements embedding equivalent examples Exercise exists Figure finite finite covolume fixed Fuchsian groups fundamental further Furthermore geodesic geometric given gives hence holds hyperbolic 3-manifold infinitely integers invariant trace field isomorphic knot Lemma manifold matrix maximal order measure Minimum Polynomial norm Note number field obtain orbifold particular places prime ideal Proof Prove quadratic extension quaternion algebra ramified real places Recall representation respectively result ring Root satisfies Show shown Signature simple splits structure subgroup Suppose surface tetrahedral Theorem Theory unique unit valuation volume
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Bibliographic information
| Title | The Arithmetic of Hyperbolic 3-Manifolds Volume 219 of Graduate Texts in Mathematics, ISSN 0072-5285 |
| Authors | Colin Maclachlan, Colin M. MacLachlan, Alan W. Reid, Emeritus John Christy Barr Distinguished Professor Colin MacLachlan, Alan W.. Reid |
| Edition | illustrated, annotated |
| Publisher | Springer Science & Business Media, 2003 |
| ISBN | 0387983864, 9780387983868 |
| Length | 463 pages |
| Subjects | › › Mathematics / Geometry / Analytic Mathematics / Geometry / Differential Mathematics / Geometry / General Mathematics / Number Theory Mathematics / Topology Medical / General |
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