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.