Discrete Groups, Expanding Graphs and Invariant Measures

This text presents the solution to two problems - the construction of expanding graphs (which are of fundamental importance for communication networks and computer science) and the Ruziewicz problem concerning the finitely added invariant measures on spheres. Both problems were partially solved using the Kazhdan property (T) from representation theory of semi-simple Lie groups. Later, complete solutions were obtained for both problems using the Ramanjuan 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. The book has been made accessible to graduate students in mathematics and computer science. A number of problems and suggestions for further research are included.