Avi wigderson ias princeton
1 / 20

Expanders, groups and representations - PowerPoint PPT Presentation

  • Uploaded on

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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
Download Presentation

PowerPoint Slideshow about ' Expanders, groups and representations' - linda-jacobs

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Avi wigderson ias princeton

Avi Wigderson

IAS, Princeton

Expanders, groups and representations

Avi wigderson ias princeton1

Avi Wigderson

IAS, Princeton

Expanders, groups and representations

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 ,… <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

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:

x<y and f(x),f(y) defined, then f(x)<f(y).

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