Discrete Groups, Expanding Graphs and Invariant MeasuresIn 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. |
Contents
II | 1 |
III | 5 |
IV | 7 |
V | 13 |
VI | 18 |
VII | 19 |
VIII | 27 |
IX | 30 |
XXX | 86 |
XXXI | 88 |
XXXII | 94 |
XXXIII | 99 |
XXXIV | 101 |
XXXV | 106 |
XXXVI | 112 |
XXXVII | 115 |
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amenable bi-partite bounded Cayley graph Chapter character cocompact complementary series compute congruence subgroup conjugacy class contain the trivial Corollary cuspidal deduce defined definition denote dense discrete group eigenvalues elements equivalent exists expanding graphs explicit construction family of expanders finite index finitely additive fixed free group group G Hamiltonian quaternions Hecke operator hence implies infinite integer invariant mean irreducible unitary representation isomorphic K-invariant k-regular graph Kazhdan group Laplacian lattice in G Lebesgue measure Lemma Let G Lie groups Margulis Math matrix modulo non-trivial norm one-dimensional p-adic PGL2 PGL2(Qp prime proof Proposition proved PSL2 quaternion algebra quotient Ramanujan conjecture Ramanujan graphs representation of G respect Ruziewicz problem Sarnak satisfies Selberg's Theorem SLn(p SO(n space spherical function subrepresentation subset superconcentrators symmetric topology tree trivial representation unique vector vertices weakly contain