BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 49, Number 1, January 2012, Pages 113–162 S 0273-0979(2011)01359-3 Article electronically published on November 2, 2011
EXPANDER GRAPHS IN PURE AND APPLIED MATHEMATICS
ALEXANDER LUBOTZKY Dedicated to the memory of Jonathan Rogawski
Abstract. Expander graphs are highly connected sparse finite graphs. They play an important role in computer science as basic building blocks for network constructions, error correcting codes, algorithms, and more. In recent years they have started to play an increasing role also in pure mathematics: number theory, group theory, geometry, and more. This expository article describes their constructions and various applications in pure and applied mathematics.
Contents Introduction 113 1. Expander graphs 115 2. Examples of expanders 121 3. Applications to computer science 131 4. Expanders in number theory 135 5. Applications to group theory 142 6. Expanders and geometry 147 7. Miscellaneous 153 Acknowledgments 155 About the author 155 References 155
Introduction Expander graphs are highly connected sparse finite graphs which play a basic role in various areas of computer science. A huge amount of research has been devoted to them in the computer science literature in the last four decades. (An excellent survey of these directions is [HLW].) But they also attracted the attention of mathematicians: their existence follows easily by random considerations (`ala Erd˝os), but explicit constructions, which are very desirable for applications, are much more difficult. Various deep mathematical theories have been used to give
Received by the editors May 12, 2011, and, in revised form, June 7, 2011. 2010 Mathematics Subject Classification. Primary 01-02, 05C99. This paper is based on notes prepared for the Colloquium Lectures at the Joint Annual Meet- ing of the American Mathematical Society (AMS) and the Mathematical Association of America (MAA), New Orleans, LA, January 6–9, 2011. The author is grateful to the AMS for the oppor- tunity to present this material for a wide audience. He has benefited by responses and remarks which followed his lectures.