The Arithmetic of Hyperbolic 3-Manifolds

Colin Maclachlan, Alan W. Reid (auth.)9780387983868, 0387983864

For 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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Table of contents :
Front Matter….Pages i-xiii
Number-Theoretic Menagerie….Pages 1-45
Kleinian Groups and Hyperbolic Manifolds….Pages 47-75
Quaternion Algebras I….Pages 77-109
Invariant Trace Fields….Pages 111-131
Examples….Pages 133-163
Applications….Pages 165-195
Orders in Quaternion Algebras….Pages 197-224
Quaternion Algebras II….Pages 225-251
Arithmetic Kleinian Groups….Pages 253-274
Arithmetic Hyperbolic 3-Manifolds and Orbifolds….Pages 275-304
Discrete Arithmetic Groups….Pages 305-330
Commensurable Arithmetic Groups and Volumes….Pages 331-370
Length and Torsion in Arithmetic Hyperbolic Orbifolds….Pages 371-414
Appendices….Pages 415-442
Back Matter….Pages 443-467