This text is based on graduate courses given by the authors. The seminal work of W. Thurston in the 1970's, in which geometric techniques were introduced to 3-dimensional topology, has made the area one of the most active areas of current research. Recently there has been considerable interest in developing techniques based on connections of hyperbolic 3-manifolds with number theory to attack problems on hyperbolic 3-manifolds. This is primarily the subject matter of the book.
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)
