1. Expanders, groups and representations
Avi Wigderson
IAS, Princeton

2. Happy Birthday
Laci !

3. Expanders, groups and representations
Avi Wigderson
IAS, Princeton

4. Expanding Graphs - PropertiesK
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)< (1-(K)>.001)

5. 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 ?

6. 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

7. 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) ||

8. 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

9. 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

10. 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

11. 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)

12. 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 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

13. 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

14. 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 
G has at most exp(d) irreducible reps of dimension d.
G is expanding and monomial.
Thm 
Lemma. If G is monomial, so is G  FqG

15. Dimensions of Representations in Expanding Groups
G has at most exp(d) irreducible reps of dimension d.
G is expanding and monomial.
Thm 
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 Conjecture holds & tight!

16. 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

17. 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.

18. 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!

19. 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

20. 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