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Avi Wigderson IAS, Princeton. Expanders, groups and representations. Happy Birthday Laci !. Avi Wigderson IAS, Princeton. Expanders, groups and representations. Expanding Graphs - Properties. K regular undirected. Combinatorial: no small cuts, high connectivity.

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expanding graphs properties
Expanding Graphs - Properties




  • 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]. KExpander: (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 ?

Expansion of Finite Groups
representations of finite groups
G finite group. A representation of G is a

homomorphismρ: 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=Fdor 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

Representations of Finite Groups
cayley graphs and representations
Cayley graphs and representations


Cayley matrix (f)




Indep of f










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 thenw.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

Expansion in every group [ n=|G|]
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 Zxdiagonal (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

Concentration for matrix valued RV’s
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?
is expansion a group property1
[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=Cnacts 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)

Is expansion a group property?
expansion in near abelian groups
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.


k times

near constant degree expanders for near abelian groups meshulam w 04
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 ,… = Gk |Sk+1|

-|Sk|  O(log(k/2)|Gk|) deg “approaching” constant

-Cay(Gk, Sk) expanding

dimensions of representations in expanding groups meshuam w 04
FqGexpands with constant many orbits

Thm 1

Ghas at most exp(d) irreducible reps of dimension d.

Thm 2

Gis expanding and monomial.

Dimensions of Representations in Expanding Groups [Meshuam-W’04]

Gnaturally acts on FqG (|G|,q)=1

Assume: G is expanding Want: G  FqG expanding

Lemma. If Gis monomial, so is G  FqG

dimensions of representations in expanding groups
Ghas at most exp(d) irreducible reps of dimension d.

Ghas at most exp(d2) irreducible reps of dimension d.

Thm 2

Gis expanding and monomial.

Dimensions of Representations in Expanding Groups


Thm [de la Harpe-Robertson-Valette]

G Abelian. Conjecture fails (as it should)

G simple nonAbelian Conjecture holds(as it should)

G = SL2(p)  F2p+1 Conjectureholds& tight!

expansion in solvable groups
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

d=3, n=2

i  A3





dimension expanders

T1,T2, …,Tk: Fd Fdare (d,F)-dimension expanderif subspace VFdwith 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.

Dimension Expanders
monotone expanders
f: [n]  [n] partial monotone map:


f1,f2, …,fk: [n]  [n] are a k-monotone expander

if fipartial 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!

Monotone Expanders
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 Miin an-ball around I.

Real Monotone Expanders [Bourgain’09]

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