The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group...
Transcript of The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group...
![Page 1: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/1.jpg)
The width ofa finite simple
group
Nick Gill(OU)
The width of a finite simple group
Nick Gill (OU)
September 4, 2012
![Page 2: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/2.jpg)
The width ofa finite simple
group
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.
![Page 3: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/3.jpg)
The width ofa finite simple
group
Nick Gill(OU)
Growth
Let A be a finite subset of a group G .Define
A2 = 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.
![Page 4: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/4.jpg)
The width ofa finite simple
group
Nick Gill(OU)
Growth
Let A be a finite subset of a group G .Define
A2 = 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.
![Page 5: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/5.jpg)
The width ofa finite simple
group
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].
![Page 6: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/6.jpg)
The width ofa finite simple
group
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].
![Page 7: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/7.jpg)
The width ofa finite simple
group
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].
![Page 8: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/8.jpg)
The width ofa finite simple
group
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].
![Page 9: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/9.jpg)
The width ofa finite simple
group
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].
![Page 10: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/10.jpg)
The width ofa finite simple
group
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...
![Page 11: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/11.jpg)
The width ofa finite simple
group
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...
![Page 12: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/12.jpg)
The width ofa finite simple
group
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...
![Page 13: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/13.jpg)
The width ofa finite simple
group
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...
. . .
0 0 1
3 The following n-cycle monomial matrix s
a. . .
cd
Now A3 6= G and, if q = 3, |A3| ≤ 17 · |A| < |A|1+ 5
n−1 .
![Page 14: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/14.jpg)
The width ofa finite simple
group
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...
. . .
0 0 1
3 The following n-cycle monomial matrix s
a. . .
cd
Now A3 6= G and, if q = 3, |A3| ≤ 17 · |A| < |A|1+ 5
n−1 .
![Page 15: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/15.jpg)
The width ofa finite simple
group
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...
. . .
0 0 1
3 The following n-cycle monomial matrix sa
. . .
cd
Now A3 6= G and, if q = 3, |A3| ≤ 17 · |A| < |A|1+ 5
n−1 .
![Page 16: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/16.jpg)
The width ofa finite simple
group
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...
. . .
0 0 1
3 The following n-cycle monomial matrix s
a. . .
cd
Now A3 6= G and, if q = 3, |A3| ≤ 17 · |A| < |A|1+ 5n−1 .
![Page 17: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/17.jpg)
The width ofa finite simple
group
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...
. . .
0 0 1
3 The following n-cycle monomial matrix s
a. . .
cd
Now A3 6= G and, if q = 3, |A3| ≤ 17 · |A| < |A|1+ 5
n−1 .
![Page 18: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/18.jpg)
The width ofa finite simple
group
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.
![Page 19: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/19.jpg)
The width ofa finite simple
group
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.
![Page 20: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/20.jpg)
The width ofa finite simple
group
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.
![Page 21: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/21.jpg)
The width ofa finite simple
group
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.
![Page 22: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/22.jpg)
The width ofa finite simple
group
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.
![Page 23: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/23.jpg)
The width ofa finite simple
group
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.
![Page 24: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/24.jpg)
The width ofa finite simple
group
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.
![Page 25: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/25.jpg)
The width ofa finite simple
group
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.
![Page 26: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/26.jpg)
The width ofa finite simple
group
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.
![Page 27: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/27.jpg)
The width ofa finite simple
group
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.
![Page 28: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/28.jpg)
The width ofa finite simple
group
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.
![Page 29: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/29.jpg)
The width ofa finite simple
group
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|.
![Page 30: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/30.jpg)
The width ofa finite simple
group
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|.
![Page 31: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/31.jpg)
The width ofa finite simple
group
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|.
![Page 32: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/32.jpg)
The width ofa finite simple
group
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|.
![Page 33: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/33.jpg)
The width ofa finite simple
group
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|.
![Page 34: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/34.jpg)
The width ofa finite simple
group
Nick Gill(OU)
Width
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).
![Page 35: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/35.jpg)
The width ofa finite simple
group
Nick Gill(OU)
Width
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).
![Page 36: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/36.jpg)
The width ofa finite simple
group
Nick Gill(OU)
Width
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).
![Page 37: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/37.jpg)
The width ofa finite simple
group
Nick Gill(OU)
Width
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).
![Page 38: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/38.jpg)
The width ofa finite simple
group
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.
![Page 39: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/39.jpg)
The width ofa finite simple
group
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.
![Page 40: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/40.jpg)
The width ofa finite simple
group
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.
![Page 41: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/41.jpg)
The width ofa finite simple
group
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|.
![Page 42: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/42.jpg)
The width ofa finite simple
group
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.
![Page 43: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/43.jpg)
The width ofa finite simple
group
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.
![Page 44: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/44.jpg)
The width ofa finite simple
group
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.
![Page 45: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/45.jpg)
The width ofa finite simple
group
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?
![Page 46: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/46.jpg)
The width ofa finite simple
group
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?
![Page 47: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/47.jpg)
The width ofa finite simple
group
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?
![Page 48: The width of a finite simple group - Nick Gill · 2020-04-27 · The width of a nite simple group Nick Gill (OU) Simple groups of Lie type Let A be a subset of G = G r(q). How does](https://reader030.fdocuments.us/reader030/viewer/2022041106/5f0877997e708231d42227d1/html5/thumbnails/48.jpg)
The width ofa finite simple
group
Nick Gill(OU)
Thanks for coming!