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Introduction to Arithmetic Groups

Armand Borel
Publisher: 
AMS
Publication Date: 
2019
Number of Pages: 
118
Format: 
Paperback
Series: 
University Lecture Series
Price: 
59,99
ISBN: 
978-1-4704-5231-5
Category: 
Monograph
[Reviewed by
Benjamin Linowitz
, on
12/14/2019
]
Introduction to Arithmetic Groups by Borel is an English translation of the author's 1969 book Introduction aux groupes arithmétiques. This book was based on a series of lectures given by the author at the Institut Henri-Poincaré in 1964 and focuses on the construction of arithmetic groups with an emphasis on the problem of finding nice fundamental domains in reductive algebraic groups \(G\) defined over \(\mathbb Q\) relative to arithmetic subgroups \(\Gamma \subset G_\mathbb Q.\)
 
In his MathSciNet review of Introduction aux groupes arithmétiques, reviewer James Humphreys commented that the author's "style is concise and the proofs (in later sections) are often demanding of the reader." In order to remedy this, translation editor Dave Witte Morris has included a large number of very helpful footnotes which fix certain typographical errors present in the original manuscript, comment on situations in which the book's terminology differs from the standard terminology used nowadays, and in general, provide details to make Borel's arguments easier to follow.
 
Borel's Introduction aux groupes arithmétiques is a classic that has served generations of graduate students and researchers interested in arithmetic subgroups of algebraic groups. The AMS has done a huge service to this community by making an English translation of the book available. That said, it is really a no-nonsense account of the theory and is not appropriate for someone without prior exposure to the subject. For the absolute beginner, I'd strongly recommend they look at translation editor Dave Witte Morris' own book, also titled Introduction to Arithmetic Groups.
 
Benjamin Linowitz (benjamin.linowitz@oberlin.edu) is an Assistant Professor of Mathematics at Oberlin College.

See the publisher's web page.

Tags: 
Arithmetic Groups