Discrete Groups, Expanding Graphs and Invariant Measures
Alexander Lubotzky
The book presents the solutions to two problems: the first is the construction of expanding graphs – graphs which are of fundamental importance for communication networks and computer science; the second is the Ruziewicz problem concerning the finitely additive invariant measures on spheres. Both problems were partially solved using the Kazhdan property (T) from representation theory of semi-simple Lie groups. Later, complete soultions were obtained for both problems using the Ramanujan conjecture from analytic number theory. The author, who played an important role in these developments, explains the two problems and their solutions from a perspective which reveals why all these seemingly unrelated topics are so interconnected. The unified approach shows interrelations between different branches of mathematics such as graph theory, measure theory, Riemannian geometry, discrete subgroups of Lie groups, representation theory and analytic number theory.  Special efforts were made to make the book accessible to graduate students in mathematics and computer science. A number of problems and suggestions for further research are presented.  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