Overview
- Brings together much of the existing literature on arithmetic Kleinan groups in a clear and concise way
- No such presentation currently exists
- Contains many examples and lots of problems
- Includes supplementary material: sn.pub/extras
Part of the book series: Graduate Texts in Mathematics (GTM, volume 219)
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Table of contents (14 chapters)
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Front Matter
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Back Matter
Keywords
About this book
Reviews
From the reviews:
"In this book Machlachlan and Reid give a comprehensive treatment of hyperbolic 3-manifolds and Kleinian groups from the viewpoint of algebraic number theory. … Throughout the book, Machlachlan and Reid use examples to motivate and illustrate the ideas they develop. … This book is a welcome addition to the literature on Kleinian groups and hyperbolic geometry. It is both an accessible introduction to the number theoretic side of the field and a convenient source of reference material for the expert." (J. R. Parker, Proceedings of the Edinburgh Mathematical Society, Issue 48, 2005)
"This book is aimed at … exposing readers … to the specific techniques from algebra and number theory needed to effectively study arithmetic manifolds and orbifolds. … The list of references is quite extensive, but even more useful are the Further Reading sections … which comprise a carefully annotated bibliography of the field. This book fills a real void in the literature, providing working topologists and graduate students with an accessible introduction to the useful and beautiful world of arithmetic hyperbolic 3-manifolds and orbifolds." (Kerry N. Jones, Mathematical Reviews, 2004 i)
"The book gives a comprehensive introduction into the theory of quaternion algebras and its orders, deals with trace fields for Kleinian groups, a notion which allows us to determine the associated number field and quaternion algebra. … The book is well written and is a substantial addition to the literature. It provides a suitable introduction into a deep area of research still under development." (J. Schwermer, Monatshefte für Mathematik, Vol. 145 (4), 2005)
"This is a book of great importance on the theory of hyperbolic manifolds (and Kleinian groups) since it is the first to provide a complete, precise, clearly-written and self-contained exposition of the arithmetic aspects of the theory. For this, the book fills a void inthe mathematics literature concerning hyperbolic geometry. The authors are two of the most fine mathematicians in the subject, and have made fundamental and beautiful contributions to the material included. … the reviewer highly recommends this beautiful book … ." (Alberto Cavicchioli, Zentralblatt MATH, Vol. 1025, 2003)
"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 that 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." (L’ENSEIGNEMENT MATHEMATIQUE, Vol. 49, (1-2), 2003)
Authors and Affiliations
Bibliographic Information
Book Title: The Arithmetic of Hyperbolic 3-Manifolds
Authors: Colin Maclachlan, Alan W. Reid
Series Title: Graduate Texts in Mathematics
DOI: https://doi.org/10.1007/978-1-4757-6720-9
Publisher: Springer New York, NY
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eBook Packages: Springer Book Archive
Copyright Information: Springer Science+Business Media New York 2003
Hardcover ISBN: 978-0-387-98386-8Published: 14 November 2002
Softcover ISBN: 978-1-4419-3122-1Published: 29 July 2011
eBook ISBN: 978-1-4757-6720-9Published: 17 April 2013
Series ISSN: 0072-5285
Series E-ISSN: 2197-5612
Edition Number: 1
Number of Pages: XIII, 467
Topics: Manifolds and Cell Complexes (incl. Diff.Topology), Geometry, Number Theory