Expander Graphs in Pure and Applied Mathematics Alex Lubotzky Einstein Institute of Mathematics, Hebrew University Jerusalem 91904, ISRAEL • Alexander Lubotzky, Discrete groups, expanding graphs and invariant measures. Reprint of the 1994 edition. Modern Birkhäuser Classics. Birkhäuser Verlag, Basel, 2010. iii+192 • Shlomo Hoory, Nathan Linial and Avi Wigderson, Expander graphs and their applications. Bull. Amer. Math. Soc. (N.S.) 43 (2006), no. 4, 439561 • http://www.ams.org/meetings/national/jmm/ 2011_colloquium_lecture_notes_lubotzky_expanders.pdf Def: Expander Graphs For 0 < " 2 R, X = V ; E a graph is " − expander j j vertices edges if V Y V with Y j j 8 ⊆ ; j j ≤ 2 j@Y j ≥ "jY j where @Y = boundry of Y = fx 2 V jdist(x; Y ) = 1g not expander. expander ) fat and round expander ) logarithmic diameter History Barzdin & Kolmogorov (1967) (networks of nerve cells in the brain!) Pinsker (1973) - communication networks We want families of expanders (n; k;")-expanders, n = jV j ! 1 k-regular, k-xed (as small as possible) "-xed (as large as possible) Fact. Fixed k ≥ 3; 9" > 0 s.t. most random k-regular graphs are "-expanders. (Pick π1; : : : ; πk 2 Sym(n) at random). Many applications in CS: Communication networks pseudorandomness/Monte-Carlo algorithms derandomization error-correcting codes . Over 4,000,000 sites with expanders but many of them for dentists Still over 400,000 are about expander graphs ... e.g. V = f0; 1;:::; p − 1g [ f1g 1 x x 1 x ! ± & ! −x For applications one wants explicit construction Kazhdan property (T ) from representation theory Def. (1967) Let Γ be a nitely generated group, Γ = hSi S = S−1. Γ has (T ) if 9" > 0 s.t. 8(H; ρ) H − Hilbert space ρ:Γ!U(H)=unitary operators irreducible (no closed invariant subspace) and non-trivial (H; ρ) 6= (C; ρ0). 8 0 6= v 2 H; 9 s 2 S s.t. kρ(s)v − vk ≥ "kvk i.e., no almost-invariant vectors Explicit construction (Margulis 1973) Assume Γ = hSi has (T ), L = fN / Γj[Γ : N] < 1g: Then fCay(Γ=N; S)jN 2 Lg is a family of expanders. Remainder. G = hSi group, Cayley graph Cay(G; S): V = jGj and g1 ∼ g2 if 9s 2 S with sg1 = g2: Proof V X Cay N S Y V X N Y j j = (Γ= ; ); ⊆ ( ) = Γ= ; j j ≤ 2 need to prove j@Y j ≥ "0jY j Γ acts on Γ=N by left translations and hence on L2(Γ=N). Take 1 y 2 Y 1Y = char. function of Y = 0 y 2= Y So some s 2 S moves 1Y by ", ρ(s)(1Y ) = 1sY so 1sY is far from 1Y , i.e. many vertices in sY are not in Y ; but sY n Y ⊂ @Y and we are done. An important observation We use (T ) only for the rep's L2(Γ=N), in particular, nite dimensional! Def: Γ = hSi nitely generated groups. L = fNi g family of nite index normal subgroups of Γ. Γ has (τ ) w.r.t. L if 9" > 0 s.t. 8(H; ρ) non-trivial irr. rep. with Ker ρ ⊃ Ni for some i, 8 0 6= v 2 H 9s 2 S s.t. kρ(s)v − vk > "kvk. Cor (τ) w.r.t. L) Cay(Γ=Ni ; S) expanders! This is i !!! Thm (Kazhdan) SLn(Z) has (T ) for n ≥ 3 (n × n integral matrices, det = 1) SL2(Z) does not have (T ) nor (τ) (has a free subgroup F of nite index and F Z) but: Thm (Selberg) SL2(Z) has (τ) w.r.t. congruence subgroups fΓ(m) = Ker(SL2(Z) ! SL2(Z=mZ))g Selberg's Thm is known as: m 3 . λ1(Γ( ) n H) ≥ 16 H - upper half plane. Eigenvalues & random walks X nite k-regular graph, X = (V ; E) jV j = n: A = AX - adjancency matrix, Aij = # edges between i and j. A symmetric matrix with eigenvalues k = λ0 ≥ λ1 ≥ · · · ≥ λn−1 ≥ −k · λ0 > λ1 i X is connected · λn−1 = −k i X is bi-partite. Thm X is "-expander i 0 λ1 ≤ k − " The non-trivial eigenvalues λ 6= ±k control the rate of convergence of the random walk on X to the uniform distribution; so: Expanders , exponentially fast convergence to uniform distribution. Thm (Alon-Boppana) p For k xed, λ1(Xn k ) = 2 k − 1 + o(1) when n ! 1 ; p Ramanujan graph λ(X ) ≤ 2 k − 1 (optimal) 8k = pα + 1; p prime 9 1 many k-regular Ramanujan graphs. Open problem for other k's, e.g. k=7. Expanders & Riemannian manifolds M n-dim connected closed Riemannian manifold ∆ = −div(grad) = laplacian = Laplace − Beltrami operator. e.ν: 0 = λ0 < λ1 ≤ λ1 ≤ ::: ≤ λi ≤ ::: Fact nR kdf k2 Z o M inf M f C 1 M f 0 λ1( ) = R 2 2 ( ); = M jf j Def. The Cheeger constant h(M) Area Y h(M) = inf (@ ) Y Volume(Y ) 1 Y - Open in M with Vol(Y ) ≤ 2 Vol(M) Cheeger Inequality (1970) 1 M h2 M λ1( ) ≥ 4 ( ) Buser proved a converse: bounding h(M) by λ1(M). In summary Thm Γ = hSi nitely generated group, L = fNi g nite index normal subgroups. TFAE: Representation (i) Γ has (τ) w.r.t. L i.e. 9"1 s.t. 8(H; ρ) ··· Combinatorics (ii) 9"2 > 0 s.t. Cay(Γ=Ni ; S) are "2-expanders Random walks (iii) 9"3 > 0 s.t. λ1(Cay(Γ=Ni ; S)) ≤ k − "3 where k = jSj Measure theoretic (iv) The Haar measure on Γ^L = lim Γ=Ni is the 1 only Γ-invariant mean on L (Γ^L) If Γ = Π1(M); M-closed Riemannian manifold and fMi g the corresponding covers: Geometric (v) 9"5 > 0; h(Mi ) ≥ "5 Analytic (vi) 9"6 > 0; λ1(Mi ) ≥ "6 Back to Selberg & Kazhdan Selberg Thm M 3 λ1(Γ( ) n H) ≥ 16 Cor Cay SL 1 1 0 1 are expanders. ( 2(Fp); ( 0 1 ) ; −1 0 ) (Proof uses Weil's Riemann hypothesis for curves and Riemann surfaces). Cor p 1 1 p−1 1 1 − 2 2 can be written as a word of length 0 1 = ( 0 1 ) O log p using 1 1 and 0 1 . ( ) ( 0 1 ) −1 0 Open problem How? Algorithm? (Partial; Larsen). (New proof by Bourgain-Gamburd (Helfgott) but also without algorithm). Thm For a xed n, Cay(SLn(Fp); fA; Bg) are expanders (A; B generators for SLn(Z)). Can they all be made into a family of expanders together - all n all p? and even all q = pe ? Conj (Babai-Kantor-Lubotzky (1989)) All non abelian nite simple groups are expanders in a uniform way (same k, same "). This was indeed proved as an accumulation of several works and several methods Kassabov - Lubotzky - Nikolov (2006): Groups of Lie type except Suzuki. Kassabov (2006): Aln(n) and Sym(n). Breuillard - Green - Tao (2010): Suzuki Groups. Other generators What happened if we slightly change the set of generators? 1 1 1 0 Ex 1 Cay(SL2(Fp); f( 0 1 ) ; ( 1 1 )g are expanders (Selberg) 1 2 1 0 Ex 2 Cay(SL2(Fp); f( 0 1 ) ; ( 2 1 )g are expanders (Pf: 1 2 1 0 ( 0 1 ) ; ( 2 1 ) is of nite index in SL2(Z) and use Selberg.) 1 3 1 0 What about Ex 3 Cay(SL2; f( 0 1 ) ; ( 3 1 )g)? 1 3 1 0 ( 0 1 ) ; ( 3 1 ) is of innite index in SL2(Z) (but Zariski dense?) Lubotzky 1-2-3 problem. Answer: Yes! (Bourgain-Gamburd/Helfgott) with Far reaching generalizations; Breuillard-Green-Tao, Pyber-Szabo, Salehi-Golsedy-Varju. These generalizations have dramatic number theoretic applications. This will be the topic of lecture II. Thm 9 1 many primes Proof. Put a topology on Z by declaring the arithmetic progressions Y a dn n to be a basis for the topology d 0 a;d = f + = 2 Zg ( 6= ) For every p 2 Z; pZ = Yo;p is open and closed. S Z n p prime pZ = {±1g is not open so 91-many primes. Homework: Let Z^ = completion of Z w.r.t. this topology. Then Q 1. Z^ = Z^p (Z^p − p-adic integers). p 2. The invertible elements of Z^ is equal to P n P (where P = fp 2 Zjp primeg 3. (2) is exactly Dirichlet primes on arithmetic progressions. Expander Graphs in Number Theory Alex Lubotzky Einstein Institute of Mathematics, Hebrew University Jerusalem 91904, ISRAEL • Alexander Lubotzky, Discrete groups, expanding graphs and invariant measures. Reprint of the 1994 edition. Modern Birkhäuser Classics. Birkhäuser Verlag, Basel, 2010. iii+192 • Shlomo Hoory, Nathan Linial and Avi Wigderson, Expander graphs and their applications. Bull. Amer. Math. Soc. (N.S.) 43 (2006), no. 4, 439561 • http://www.ams.org/meetings/national/jmm/ 2011_colloquium_lecture_notes_lubotzky_expanders.pdf Thm (Dirichlet) b; q 2 Z with (b; q) = 1, 9 1 many primes in b + qZ or: x 2 Z, ν(x) = # prime factors of x, then for every 0 b; q 2 Z; 9 1 x s in b + qZ with ν(x) ≤ 1 + ν ((b; q)). Twin Prime Conjecture 9 1 many p with p + 2 also a prime, or: 91 many x 2 Z with ν (x(x + 2)) ≤ 2. a stronger version TPC on arithmetic progressions. A far reaching generalization (Schinzel): • f0g= 6 Λ ≤ Z a subgroup, i.e. Λ = qZ; q 6= 0 and b 2 Z • θ = orbit of b under Λ = b + qZ • f (x) 2 Q[x] a poly, integral on θ Say: (θ; f ) primitive if 8 2 ≤ k 2 Z, 9x 2 θ s:t: (f (x); k) = 1: Conjecture If f (x) 2 Q[x] is a product of t irreducible factors &(θ; f ) primitive then 91 x 2 θ with ν(f (x)) ≤ t Higher dimensional generalization Conjecture (Hardy-Littlewood) n • Λ ≤ Z • 8j, the j-th coordinate is non-constant on Λ n • b 2 Z ; θ = b + Λ • f (x) = x1 · ::: · xn; (θ; f )-primitive.