Randomness in group theoryFrom the finite to the infinite

Alex Lubotzky

Hebrew University, Jerusalem, Israel

IAS, Princeton

June 2010

Early stage

• Erdos-Turan ( 7 papers 1965-72 , “On some problems of statistical group theory”)

A typical result: A random permutation in Sym(n) has aprox. log(n) cycles and its order is n(1/2+o(1))log(n).

(but it is really combinatorics & number theory- not group theory).

• Even earlier Netto conjecture (1882): Almost any pair of elements of Sym(n) generates either Alt(n) or Sym(n)

Proved in 1969 by Dixon.

Dixon conjecture: the same true for all finite simple groups

Dixon conjecture was proved by

Kantor-Lubotzky (1990)- for classical groups

Liebeck-Shalev (1995) – for exceptional groups.

Dixon’s proof-elementary. The others (as well as Babai’s better estimate for

Alt(n)) needs CFSG.)

Denote So

the theorem says that

Proof based on Aschbacher’s (1984) classification of the maximal subgroups.

kk 1 2 k

max max

P (G)= Prob ( (x ,x , ...., x ) G generate G)

1 1 [ : ]k

kk

M G M G

MG M

G

max

( ) [ : ] sG

M G

s G M 2 GP (G) 1- (2)

| |(2) 0G G

Applications

• Magnus problem: The free group is residually any infinite family of non-abelian finite simple groups.

Many partial results- till a full proof in 1999 by Wiegel.

A probablistic much easier proof by Dixon-Pyber-Serress-Shalev.

Fix w in F2 :2Prob( (x,y) G : w(x,y) 1& , ) 1

Gx y G

A result of Liebeck and Shalev : Almost every pair (x,y) of G2 where x of order 2

and y of order 3 generate G unless

G=PSp4)pk) in which case ½.

Led to a complete solution (Lubeck-Malle 1997) of the problem which finite simple groups are quotients of the modular group PSL(2, ).

More probablistic results ( e.g. 1 ½ -

generation…) by Guralnick, Kantor,…..

Z

• Hurwitz groups :

Which finite simple groups are quotients of the triangle (2,3,7)- group ?

(i.e. appears as Iso(Sg) of order 84(g-1)

for a Riemann surface of genus g.)

Also more general triangle and Fuchsian groups (Higman’s conj. Alt(n) proved by Liebeck-Shalev etc.)

• Beauville surface is a rigid complex surface of the form C1×C2 / G , where C1 and C2 are non-singular, projective, higher genus curves and G a finite group acting freely on the product.

• Conj ( Bauer-Catanese-Grunewald) All non-abelian finite simple groups

except of Alt(5) gives such surfaces.

• proved in 2010 by Garion-Larsen-LubotzkyFor almost all finite simple groups.

• Ore conjecture (1951)

G non-abelian finite simple group. Every element g in G is a commutator, i.e., there

Exist x and y in G s.t. g=x-1y-1xy

Proved in 2009 by Liebeck, O’Brain, Shalev, Tiep

Computational group theory gave new life to “statistical group theory”.

Random algorithms need pseudo-random elements of G.

Random walks on Cayley graphs

Product replacement algorithm.

Erdos-Turan type of results for general finite simple groups: “How typical element of looks like?”

( )n qSL F

Profinite Groups

• G profinite ≡ Compact,  Hausdorff, totally disconnected

≡ inverse limit of finite groups.

Haar measure μ, μ(G)=1

Ex: (1) Γ fin. gen. discrete group

(2)

(3) Fd = free group of d generators

= free in the category of profinite groups

lim iG

ˆ limG Profinite completion N

ˆ lim

( )

pp

p

np adic integeres

ZZ ZZZ

dF

Let

Jarden (1980) For

Pf: A Borel-Canteli lemma using .

Jarden-Lubotzky (1999)

(2b) needs CFSG

1

1

{( ) : ( : ,..., }

{( ) : ( : ,..., }

dd n

dd n

S x G G x x

Q x G G x x

ˆG Z

(1) 1, ( ( )) 0

(2) 2, ( ( )) 1d

d

If d S G

If d S G

1d

p p

1ˆˆ ˆ, 2 ( )

nG F n note F Zx

(1) , ( ( )) 0

(2) ( ) , ( ( )) 0

( ) ,0 ( ( )) 1

d

d

d

d S G

a d n Q G

b d n Q G

Def (Mann 1996) G is positively finitely generated (PFG) if

for some

So is PFG but are not !

But free pro solvable are !

Theorem (Mann-Shalev 1996)

A profinite G is PFG iff G has polynomial maximal subgroup growth.

Their idea of proof was used to prove a strong form of Pak’s conjecture

Theorem (Lubotzky 2002) G a finite group generated by d elements. The expected numbers of random generators for G is at most ed(G)+2e log(log(G))+11

1(( ) | ,..., ) 0dnx G x x G d N

Z ˆ ( 2)nF n

Finitely generated groups(discrete groups)

• Random groups a’ la Gromov

free group on

= ball of radius n in Cay .

Choose randomly l elements r1 ,…rl of Bn and look at

, Ask for “typical” properties of Γ,

Warning :

(i) Γ is not typical finitely generated group. It is finitely presented !

(ii) The issue of repetition is not clear!

1( ,...., )d dF F x x 1{ ,...., }dS x x( )s

nB ( ; )S

1,....d

l

Fr r

Various models

1. d fixed, l fixed and n∞.

2. d fixed, 0<δ<1 fixed, and n∞.

Theory by Gromov, Olshanski, Ollivier, Zuk, Arzhantseva,…

See: Ollivier / January 2005 Inritation of Random Groups.

theorem:

(1) If δ < ½ , Γ is hyperbolic (with prob 1 as n∞.)

(2) If δ> ½ , Γ is trivial.

(3) If δ> 1/3 , Γ has Kazhdan (T).

Cor: hyperbolic groups with (T)

(Explicit examples are known )

| ( ) |n dl B F

( ,1)( )Sp n R

Warning: The importance of the model

(I) Results for ‘typical’ can be different (Stallings graphs- see Bassimo-Martino-Nicaud-Ventura-Weil 2010)

(II) Let’s see a “ proof” for Gromov problem:

Is every hyperbolic group residually finite?

Everyone is sure the answer is no!

“Moral Proof” Take a random Γ, say, in the model “fixed d, fixed l” and look at .

By Gromov, Γ is hyperbolic. G is “morally” a random profinite group on d gen’s, l relations. By Jarden-Lubotzky, G is finite with positive probability. Hence Γ is not residually finite with pos. probability. “QED”

ˆG

Expanders, group sieve and random elements in infinite discrete groups

joint work with Chen MeiriInspired by affine group sieve a’ la Sarnak, Bourgain, Gamburd,

Igor Rivin, Emanuel Kowalski,…

Let’s start with applications then method:

• For ,let

“Thm” (Breuillard, Lubotzky, Meiri, Cornulier)

Extenion of Eskin, Mozes, Oh, Breuillard, Gelander….

S( ) # ( )

log ( )( ) lim

S S

n n

S n

n

C conj classes of represented in B

Cand

n

( ) . . ( )

( ) ( ).S

dC C d s t if GL F and not virtually solvable

then C d

(II)

Thm Malcev (60’s)

Thm (Hrushovski – Kropholler –Lubotzky-Shalev 1995)

(1) If Γ solvable and finite index subgroup, then Γ is virtually nilpotent (but it is possible

coset of fin. Index subgroup).

(2) If Γ is linear and coset of a f.i.

subgp then Γ is virt. solvable.

2

. . 1 ,

{ | }

( )

m m

m

m

fin gen group m

g g The set of m powers

and P The set of powers

, , mnilpotent m contains a finite index subgroup

m

m))(( FGLd

m

“Thm” (Lubotzky-Meiri 2011)

Note (1) quantitative; “exponential small”

(not just “linearly small” as in [HKLS])

(2) All powers together and not just one m at a time

(1) is needed for (2) !

( ) . . . ,

0

| ( ) ( ) |

( )

0

GL F fin gen not virt solvabledS then

SP B nn eSBn

for every n

(III) The mapping class grouptheorem (Maher ?; Rivin 2006; Kowalski

2008) The set of non pseudo-anasov elements of the mapping class group is exponentialy small.

(Conjectured by Thurston)

A similar result for the Torreli subgroup

(Lubotzky – Meiri).

Analogous result for Aut(Fn).

fixed m (can assume m is prime).

Lemma 1

(ε indep of m, but Q depends on m).

Γ is mapped onto ,

By property ( ) (expanders),

maps aprox. uniformly onto ,

for all primes (some c>1).

Sieving over these primes, shows is exponentially small.

Sketch of proof of application II for

)(SnB

3

3

0, . .

| ( ) |, 1| ( ) |

m

positively dense set of primes Q s t

SL pp Q

SL p

3( )SL p

ncp ( )m S

nB

p Q

3( )SL p

3( )SL Z

The results of Breuillard-Green-Tau,Pyber-Szabo, Varju-Salehi-Golsefidy, Helfgott,….are used for general Γ (assuming its zariski closure

is ‘nice’).

To get that is also exp small we need –(i) uniformity in m.(ii) For small n, is only virt. unipotent elements (sub variety & very small) unless and then use uniformity in m.

m

mP )(

mnB

)(nOm

The general Γ extends this method using strong Aprox, all the above resuls + some alg gps & numbers theoretic methods to handle non- connected or non-simply connected cases.