Discrete Groups, Expanding Graphs and Invariant Measures

In the last—fifteen 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 difficult task. Since expanders serve as basic building blocks f...