Modern Birkhäuser Classics

Discrete Groups, Expanding Graphs and Invariant Measures

Authors: Lubotzky, Alex

  • Award-winning monograph of the Ferran Sunyer i Balaguer Prize 1993
  • Exciting tour through several areas of mathematics
  • Recent mathematical research
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About this book

In the last ?fteen years two seemingly unrelated problems, one in computer science and the other in measure theory, were solved by amazingly similar techniques from representation theory and from analytic number theory. One problem is the - plicit construction of expanding graphs («expanders»). These are highly connected sparse graphs whose existence can be easily demonstrated but whose explicit c- struction turns out to be a dif?cult task. Since expanders serve as basic building blocks for various distributed networks, an explicit construction is highly des- able. The other problem is one posed by Ruziewicz about seventy years ago and studied by Banach [Ba]. It asks whether the Lebesgue measure is the only ?nitely additive measure of total measure one, de?ned on the Lebesgue subsets of the n-dimensional sphere and invariant under all rotations. The two problems seem, at ?rst glance, totally unrelated. It is therefore so- what surprising that both problems were solved using similar methods: initially, Kazhdan’s property (T) from representation theory of semi-simple Lie groups was applied in both cases to achieve partial results, and later on, both problems were solved using the (proved) Ramanujan conjecture from the theory of automorphic forms. The fact that representation theory and automorphic forms have anything to do with these problems is a surprise and a hint as well that the two questions are strongly related.

Reviews

From reviews:

"This exciting book marks the genesis of a new field. It is a field in which one passes back and forth at will through the looking glass dividing the discrete from the continuous. (...) The book is a charming combination of topics from group theory (finite and infinite), combinatorics, number theory, harmonic analysis." - Zentralblatt MATH

"The Appendix, written by J. Rogawski, explains the Jacquet-Langlands theory and indicates Deligne’s proof of the Petersson-Ramanujan conjecture. It would merit its own review. (...) In conclusion, this is a wonderful way of transmitting recent mathematical research directly "from the producer to the consumer". - MathSciNet

"The book is accessible to mature graduate students in mathematics and theoretical computer science. It is a nice presentation of a gem at the border of analysis, geometry, algebra and combinatorics. Those who take the effort to glance what happens behind the scene won’t regret it." - Acta Scientiarum Mathematicarum


Table of contents (10 chapters)

  • Expanding Graphs

    Lubotzky, Alexander

    Pages 1-6

  • The Banach-Ruziewicz Problem

    Lubotzky, Alexander

    Pages 7-18

  • Kazhdan Property (T) and its Applications

    Lubotzky, Alexander

    Pages 19-40

  • The Laplacian and its Eigenvalues

    Lubotzky, Alexander

    Pages 41-60

  • The Representation Theory of PGL 2

    Lubotzky, Alexander

    Pages 61-76

Buy this book

eBook $69.99
price for USA (gross)
  • ISBN 978-3-0346-0332-4
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $89.99
price for USA
  • ISBN 978-3-0346-0331-7
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Discrete Groups, Expanding Graphs and Invariant Measures
Authors
Series Title
Modern Birkhäuser Classics
Copyright
1994
Publisher
Birkhäuser Basel
Copyright Holder
Springer Basel AG
eBook ISBN
978-3-0346-0332-4
DOI
10.1007/978-3-0346-0332-4
Softcover ISBN
978-3-0346-0331-7
Series ISSN
2197-1803
Edition Number
1
Number of Pages
XI, 196
Additional Information
Originally published in the series: Progress in Mathematics Vol 125
Topics