Expanders, groups and representations Avi Wigderson IAS, Princeton

Happy Birthday Laci !

Expanders, groups and representations Avi Wigderson IAS, Princeton

Expanding Graphs - Properties K regular undirected Combinatorial: no small cuts, high connectivity Probabilistic: rapid convergence of random walk Algebraic: small second eigenvalue Theorem. [Cheeger, Buser, Tanner, Alon-Milman, Alon, Jerrum-Sinclair,…]: All properties are equivalent! (K)= max {|| P – J/n ||}. P random walk on K (K)  [0,1]. K Expander: (K)<.999 (1-(K)>.001)

Expansion of Finite Groups G finite group, SG, symmetric. The Cayley graph Cay(G;S) has xsx for all xG, sS. Cay(Cn ; {-1,1}) Cay(F2n ; {e1,e2,…,en}) Basic Q: for which G,S is Cay(G;S) expanding ?

Representations of Finite Groups G finite group. A representation of G is a homomorphism ρ: G  GLd(F) ρ(x)ρ(y)=ρ(xy) for all x,yG ρ irreducible if it has no nontrivial invariant subspace: ρ(x)VV for all xG  V=Fd or V=ϕ. Irrep(G): { ρ1, ρ2, …, ρt } di=dim ρi 1=d1≤ d2≤ …≤ dt i di2 =n Independent of F if (char F, |G|)=1 [Babai-Ronyai] Polytime alg for Irrep(G) over C

Cayley graphs and representations ρ2(f) ρ1(f) ρt(f) Fourier Transform Indep of f . Cayley matrix (f) f(xy-1) x y f: G  F e.g. f = pS = ρ(f) = xG ρ(x) 1/|S| if xS 0 otherwise (Cay(G;S)) = ||Ps –J/n|| = maxρ≠1 || ρ(pS) ||

Expansion in every group [ n=|G|] [Kassabov-Lubotzky-Nikolov’06] G simple nonAbelian Then  |S|=O(1) such that Cay(G;S) expands. Fact: G Abelian, Cay(G;S) expands  |S|>log n [Alon-Roichman’94] G finite group. SG random, |S|=k=100*log n then w.h.p Cay(G;S) expands. Proof [Loh-Schulman’04, Landau-Russell’04, Xiao-W’06] Random walk matrix: PS(x,y)=1/k iff xy-1S Claim: (Cay(G;S)) = || Z || where Z=PS–J/n

Concentration for matrix valued RV’s Claim: (Cay(G;S)) = || Z || where Z=PS–J/n Z = (1/k) xSZx where Zx =P{x,x-1} –J/n Claim: xG Zx =0, Zx symm., ||Zx|| ≤1 xG. [Ahlswede-Winter’02] : generalizes Chernoff (n=1) PrS[ || xS Zx || > k/2 ] < n exp(-k) Comment: Tight when Zx diagonal (Abelian case) Conjecture: G finite, ρ  Irrep(G) (dim ρ = n) then PrS[ || xS ρ(x) || > k/2 ] < exp(-k) Comment: Holds for Abelian & some simple gps

Is expansion a group property? [Lubotzky-Weiss’93] Is there a group G, and two generating subsets |S1|,|S2|=O(1) such that Cay(G;S1) expands but Cay(G;S2) doesn’t ? (call such G schizophrenic) nonEx1: Cn - no S expands nonEx2: SL2(p)-every S expands [Bruillard-Gamburd’09] [Alon-Lubotzky-W’01] SL2(p)(F2)p+1 schizophrenic [Kassabov’05] Symn schizophrenic

Is expansion a group property? [Alon-Lubotzky-W’01] SL2(p)  F2p+1 schizophrenic [Reingold-Vadhan-W’00] zig-zag product theorem. [Alon-Lubotzky-W’01] G, H groups. G acts on H. Cay(G;S) expands with |S|=O(1) Cay(H; tT tG) expands with |T|=O(1) Then Cay(GH; STS) expands with |STS|=O(1) Ex: G=Cn acts on H=F2n by cyclic shifts Cay(H,e1G) not expanding e1G = {e1,e2,…,en} Cay(H,vGuG) expanding for random u,v in F2n Problem: Explicit u,v. (vGuG gen. good code)

Expansion in Near-Abelian Groups G group. [G;G] commutator subgroup of G [G;G] = <{ xyx-1y-1 : x,y G }> G= G0 > G1> … > Gk = Gk+1 Gi+1=[Gi;Gi] G is k-step solvable if Gk=1. Abelian groups are 1-step solvable [Lubotzky-Weiss’93] If G is k-step solvable, Cay(G;S) expanding, then |S| ≥ O(log(k)|G|) [Meshulam-W’04] There exists k-step solvable Gk, |Sk| ≤ O(log(k/2)|Gk|), and Cay(Gk;Sk) expanding. loglog….log k times

Near-constant degree expanders for near Abelian groups [Meshulam-W’04] Iterate: G’ = G  FqG Start with G1 = Z2 Get G1 , G2,…, Gk ,… |Gk+1|>exp (|Gk|) S1 , S2,…, Sk ,… <Sk > = Gk |Sk+1|<poly (|Sk|) - |Sk|  O(log(k/2)|Gk|) deg “approaching” constant - Cay(Gk, Sk) expanding

Dimensions of Representations in Expanding Groups [Meshuam-W’04] G naturally acts on FqG (|G|,q)=1 Assume: G is expanding Want: G  FqG expanding FqG expands with constant many orbits Thm 1  G has at most exp(d) irreducible reps of dimension d. G is expanding and monomial. Thm 2  Lemma. If G is monomial, so is G  FqG

Dimensions of Representations in Expanding Groups G has at most exp(d) irreducible reps of dimension d. G is expanding and monomial. Thm 2  Conjecture Thm [de la Harpe-Robertson-Valette]  G has at most exp(d2) irreducible reps of dimension d. G Abelian. Conjecture fails (as it should) G simple nonAbelian Conjecture holds (as it should) G = SL2(p)  F2p+1 Conjecture holds & tight!

Expansion in solvable groups G is solvable if it is k-step solvable for some k= k(n). Can G expand with O(1) generators? [Lubotzky-Weiss’93] p fixed. Gn = (p) / (pn) (pm) = Ker SL2(Z)  SL2(pm) [Rozenman-Shalev-W’04] (not solvable) d fixed. Gk = Aut*(Tkn) Iterative: Gk+1 = Gk  Ad zig-zag thm, perfect groups,… Challenge: Beat k=loglog n YES! k > loglog n 0 1 3 2 d=3, n=2 i  A3

Dimension Expanders [Barak-Impagliazzo-Shpilka-W’01] T1,T2, …,Tk: Fd  Fd are (d,F)-dimension expander if subspace VFd with dim(V) < d/2  i[k] s.t. dim(TiVV) < (1-) dim(V) Fact: k=O(1) random Ti’s suffice for every F,d. Conjecture [W’04]: Cay(G;{x1,x2,…,xk}) expander, ρIrred(G) of dim d over F, then ρ(x1),ρ(x2),…,ρ(xk) are (d,F)-dimension expander. [Lubotzky-Zelmanov’04] True for F=C.

Monotone Expanders f: [n]  [n] partial monotone map: x<y and f(x),f(y) defined, then f(x)<f(y). f1,f2, …,fk: [n]  [n] are a k-monotone expander if fi partial monotone and the (undirected) graph on [n] with edges (x,fi(x)) for all x,i, is an expander. [Dvir-Shpilka] k-monotone exp  2k-dimension exp F,d Explicit (log n)-monotone expander [Dvir-W’09] Explicit (log*n)-monotone expander (zig-zag) [Bourgain’09] Explicit O(1)-monotone expander [Dvir-W’09] Existence  Explicit reduction Open: Prove that O(1)-mon exp exist!

Real Monotone Expanders [Bourgain’09] Explicitly constructs f1,f2, …,fk: [0,1]  [0,1] continuous, Lipshitz, monotone maps, such that for every S  [0,1] with (S)< ½, there exists i[k] such that (Sf2(S)) < (1-) (S) Monotone expanders on [n] – by discretization M=( )SL2(R), xR, let fM(x) = (ax+b)/(cx+d) Take sufficiently many such Mi in an -ball around I. a b c d

Open Problems Conjecture [B ‘yesterday] Cay(G;S) with |S|=O(1). Assume 99% of the vertices are reached by length d path. Then diameter < 1.99 d Conjecture [W ‘today] SL2(p)(F2)p+1 is a counterexample