the width of a finite simple group - nick gill · 2020-04-27 · the width of a nite simple group...

The width ofa finite simple

Nick Gill(OU)

The width of a finite simple group

Nick Gill (OU)

September 4, 2012

Nick Gill(OU)

Growth

Let A be a finite subset of a group G .

DefineA2 = A · A := {a1 · a2 | a1, a2 ∈ A}.

We are interested in studying the size of A2,A3,A4, . . . we callthis the study of the growth of A.

Nick Gill(OU)

Growth

Let A be a finite subset of a group G .Define

A2 = A · A := {a1 · a2 | a1, a2 ∈ A}.

Nick Gill(OU)

Growth

Let A be a finite subset of a group G .Define

A2 = A · A := {a1 · a2 | a1, a2 ∈ A}.

Nick Gill(OU)

Doubling and tripling

Suppose first that G is abelian (the classical setting for additivecombinatorics).

If |AA| ≤ K |A| then |A`| ≤ K `|A| [P-R].

Now drop the condition that G is abelian.

If |AAA| ≤ K |A| then |A`| ≤ K 2`−5|A| [H-T].

Nick Gill(OU)

Simple groups of Lie type

Let A be a subset of G = Gr (q). How does the set A grow?This question is partially answered, with a strong value of K ,by a theorem of Pyber-Szabo, Breuillard-Green-Tao building onwork of Helfgott, Dinai, Helfgott-G.

Theorem

Fix r > 0. There exists a positive number ε such that for anygenerating set A in Gr (q) either

|AAA| ≥ |A|1+ε, or

AAA = G .

Applications are manifold: diameter bounds, expansion,sieving...

Nick Gill(OU)

Theorem

|AAA| ≥ |A|1+ε, or

AAA = G .

Nick Gill(OU)

Theorem

|AAA| ≥ |A|1+ε, or

AAA = G .

Nick Gill(OU)

Dependence on the rank [P-S]

Let A be a generating set of SLn(q) containing:

1 T , the set of diagonal matrices; |T | = (q − 1)n−1;

2 a, b, two elements generating this copy of SL2(q):A 0 · · · 00 1 0...

3 The following n-cycle monomial matrix s

a. . .

Now A3 6= G and, if q = 3, |A3| ≤ 17 · |A| < |A|1+ 5

n−1 .

Nick Gill(OU)

a. . .

Now A3 6= G and, if q = 3, |A3| ≤ 17 · |A| < |A|1+ 5

n−1 .

Nick Gill(OU)

3 The following n-cycle monomial matrix sa

Now A3 6= G and, if q = 3, |A3| ≤ 17 · |A| < |A|1+ 5

n−1 .

Nick Gill(OU)

a. . .

Now A3 6= G and, if q = 3, |A3| ≤ 17 · |A| < |A|1+ 5n−1 .

Nick Gill(OU)

a. . .

Now A3 6= G and, if q = 3, |A3| ≤ 17 · |A| < |A|1+ 5

n−1 .

Nick Gill(OU)

Babai’s conjecture

Let G be a finite simple group and A a generating set in G .Note that

1 A` = G for some integer ` (take it as small as possible).

2 ` ≥ log |G |log |A|

3 ` ≥ log |G | if |A| = 2.

Conjecture (Babai)

There exists c > 0 such that, for any finite simple group G andany generating set A ⊂ G , A` = G for some ` ≤ (log |G |)c .

Babai’s conjecture is often stated in terms of the diameter ofthe Cayley graph.

Nick Gill(OU)

2 ` ≥ log |G |log |A|

3 ` ≥ log |G | if |A| = 2.

Conjecture (Babai)

Nick Gill(OU)

2 ` ≥ log |G |log |A|

3 ` ≥ log |G | if |A| = 2.

Conjecture (Babai)

Nick Gill(OU)

2 ` ≥ log |G |log |A|

3 ` ≥ log |G | if |A| = 2.

Conjecture (Babai)

Nick Gill(OU)

2 ` ≥ log |G |log |A|

3 ` ≥ log |G | if |A| = 2.

Conjecture (Babai)

Nick Gill(OU)

2 ` ≥ log |G |log |A|

3 ` ≥ log |G | if |A| = 2.

Conjecture (Babai)

Nick Gill(OU)

A partial proof of Babai’s conjecture

Corollary

Fix r > 0. There exists c > 0 such that for any generating setA in G = Gr (q) we have A` = G for some ` ≤ (log |G |)c .

Proof.

1 The product theorem implies that either |A3| ≥ |A|1+ε orA3 = G .

2 Iterating we obtain that either |A3k | ≥ |A|(1+ε)k or

A3k = G .

3 If |A|(1+ε)k ≥ |G | we must have A3k = G .

4 Thus A` = G where ` = (log |G |)dlog1+ε 3e+1.

Nick Gill(OU)

Corollary

Proof.

A3k = G .

Nick Gill(OU)

Corollary

Proof.

A3k = G .

Nick Gill(OU)

Corollary

Proof.

A3k = G .

Nick Gill(OU)

Corollary

Proof.

A3k = G .

Nick Gill(OU)

Normal subsets

We say that A is a normal subset of G if, for all g ∈ G ,

gAg−1 := {gag−1 | a ∈ A} = A.

Note that A is normal if and only if A is a union of conjugacyclasses of G .Liebeck and Shalev proved a (much) stronger version ofBabai’s conjecture for normal subsets of simple groups:

Theorem

There exists a constant c > 0 such that, for A a non-trivialnormal subset of a simple group G , we have G = A` where` ≤ c log |G |/ log |A|.

Nick Gill(OU)

Normal subsets

gAg−1 := {gag−1 | a ∈ A} = A.

Theorem

Nick Gill(OU)

Normal subsets

gAg−1 := {gag−1 | a ∈ A} = A.

Note that A is normal if and only if A is a union of conjugacyclasses of G .

Liebeck and Shalev proved a (much) stronger version ofBabai’s conjecture for normal subsets of simple groups:

Theorem

Nick Gill(OU)

Normal subsets

gAg−1 := {gag−1 | a ∈ A} = A.

Theorem

Nick Gill(OU)

Normal subsets

gAg−1 := {gag−1 | a ∈ A} = A.

Theorem

Nick Gill(OU)

We define the width of G with respect to A to be the minimumnumber ` such that

G = A1A2 · · ·A`

and A1, . . . ,A` are all conjugates of A in G . Write w(G ,A).

Examples for a simple group G .

1 If G is of Lie type, A is a Sylow p-subgroup thenw(G ,A) ≤ 25 [LP01].In fact w(G ,A) ≤ 5 [BNP08].

2 If G = Gr (q), an untwisted simple group of Lie type ofrank r > 1, and A is a particular subgroup isomorphic toSL2(q), then w(G ,A) ≤ 5|Φ+| [LNS11].

In particular, if G = PSLn(q) and A as above thenw(G ,A) ≤ 5

2 n(n + 1).

Nick Gill(OU)

G = A1A2 · · ·A`

and A1, . . . ,A` are all conjugates of A in G . Write w(G ,A).Examples for a simple group G .

2 n(n + 1).

Nick Gill(OU)

G = A1A2 · · ·A`

2 n(n + 1).

Nick Gill(OU)

G = A1A2 · · ·A`

2 If G = Gr (q), an untwisted simple group of Lie type ofrank r > 1, and A is a particular subgroup isomorphic toSL2(q), then w(G ,A) ≤ 5|Φ+| [LNS11].In particular, if G = PSLn(q) and A as above thenw(G ,A) ≤ 5

2 n(n + 1).

Nick Gill(OU)

The Product Decomposition Conjecture

Liebeck, Nikolov and Shalev conjectured the following:

Conjecture

There exists a constant c > 0 such that, for A any subset of afinite simple group G of size at least 2, we have G = A1 · · ·A`

where A1, . . . ,A` are conjugates of A and ` ≤ c log |G |/ log |A|.

Note the similarity to Babai’s conjecture - but both theassumptions and the conclusion are much stronger.

Nick Gill(OU)

Conjecture

Nick Gill(OU)

Conjecture

Nick Gill(OU)

Some results

We start with a result of Gill, Pyber, Short, Szabo:

Theorem

Fix r > 0. There exists c > 0 such that, for A any subset ofG = Gr (q) of size at least 2, we have G = A1A2 · · ·A` where` ≤ c log |G |/ log |A|.

Nick Gill(OU)

A product theorem for conjugates

On the way to proving this result we came across somethinglike a product theorem for conjugates:

Theorem

Fix r > 0. There exists ε > 0 such that, for A any subset ofG = Gr (q), there exists g ∈ G such that |A · Ag | ≥ |A|1+ε orA3 = G .

We conjecture that the constant ε should be independent of r ,indeed it should be uniform across all simple groups.Note too that, when we’re allowed to take conjugates, weachieve growth in two steps, not three.

Nick Gill(OU)

Theorem

We conjecture that the constant ε should be independent of r ,indeed it should be uniform across all simple groups.

Note too that, when we’re allowed to take conjugates, weachieve growth in two steps, not three.

Nick Gill(OU)

Theorem

We conjecture that the constant ε should be independent of r ,indeed it should be uniform across all simple groups.Note too that, when we’re allowed to take conjugates, weachieve growth in two steps, not three.

Nick Gill(OU)

The Skew Doubling Lemma

An explanation for the two step growth is found in thefollowing result:

Theorem

Let A be a non-empty finite set of a group G such that, forsome K > 0, |AA′| ≤ K |A| for every conjugate A′ of A. Then

|A1 · · ·A`| ≤ K 14(`−1)|A|

where A1, . . . ,A` are conjugates of A or A−1.

Note that, if A is normal, we effectively regain the doublinglemma for abelian groups.Could it be that classical additive combinatorics for sets inabelian groups is really about normal subsets of arbitrarygroups?

Nick Gill(OU)

Theorem

|A1 · · ·A`| ≤ K 14(`−1)|A|

Note that, if A is normal, we effectively regain the doublinglemma for abelian groups.

Could it be that classical additive combinatorics for sets inabelian groups is really about normal subsets of arbitrarygroups?

Nick Gill(OU)

Theorem

|A1 · · ·A`| ≤ K 14(`−1)|A|

Note that, if A is normal, we effectively regain the doublinglemma for abelian groups.Could it be that classical additive combinatorics for sets inabelian groups is really about normal subsets of arbitrarygroups?

Nick Gill(OU)

Thanks for coming!

the width of a finite simple group - nick gill · 2020-04-27 · the width of a nite simple group...

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