Some Applications of Diophantine Geometry and Model Theory to Group Theory

Some applications of diophantine geometry and model theory to group theory.

Emmanuel Breuillard

Universit´eParis-Sud, Orsay, France

Ol´eron,June 6th, 2011

Emmanuel Breuillard Diophantine geometry and group theory 1 Eﬀective Burnside-Schur theorems and the compactness theorem of ﬁrst order logic.

2 Diophantine Geometry on character varieties and a height gap theorem.

3 A uniform Tits alternative.

4 Diameter of ﬁnite simple groups.

5 Eﬀective versions of Hrushovski’s theorems on approximate groups.

Plan of the talk:

Emmanuel Breuillard Diophantine geometry and group theory 2 Diophantine Geometry on character varieties and a height gap theorem.

3 A uniform Tits alternative.

4 Diameter of ﬁnite simple groups.

5 Eﬀective versions of Hrushovski’s theorems on approximate groups.

Plan of the talk:

1 Eﬀective Burnside-Schur theorems and the compactness theorem of ﬁrst order logic.

Emmanuel Breuillard Diophantine geometry and group theory 3 A uniform Tits alternative.

4 Diameter of ﬁnite simple groups.

5 Eﬀective versions of Hrushovski’s theorems on approximate groups.

Plan of the talk:

1 Eﬀective Burnside-Schur theorems and the compactness theorem of ﬁrst order logic.

2 Diophantine Geometry on character varieties and a height gap theorem.

Emmanuel Breuillard Diophantine geometry and group theory 4 Diameter of ﬁnite simple groups.

5 Eﬀective versions of Hrushovski’s theorems on approximate groups.

Plan of the talk:

1 Eﬀective Burnside-Schur theorems and the compactness theorem of ﬁrst order logic.

2 Diophantine Geometry on character varieties and a height gap theorem.

3 A uniform Tits alternative.

Emmanuel Breuillard Diophantine geometry and group theory 5 Eﬀective versions of Hrushovski’s theorems on approximate groups.

Plan of the talk:

1 Eﬀective Burnside-Schur theorems and the compactness theorem of ﬁrst order logic.

2 Diophantine Geometry on character varieties and a height gap theorem.

3 A uniform Tits alternative.

4 Diameter of ﬁnite simple groups.

Emmanuel Breuillard Diophantine geometry and group theory Plan of the talk:

1 Eﬀective Burnside-Schur theorems and the compactness theorem of ﬁrst order logic.

2 Diophantine Geometry on character varieties and a height gap theorem.

3 A uniform Tits alternative.

4 Diameter of ﬁnite simple groups.

5 Eﬀective versions of Hrushovski’s theorems on approximate groups.

Emmanuel Breuillard Diophantine geometry and group theory ±1 ±1 For a symmetric set of generators S = {1, s1 , ..., sr } of a group G = hSi, we denote by Sk := S · ... · S the “ball of radius k” in the Cayley graph of G generated by S. Conjecture (Effective restricted Burnside; Olshanskii) Given r, n, does there exists k(r, n) ∈ N such that there are only finitely many r-generated finite groups G = hSi such that all elements in Sk have order dividing n ?

Remark (Olshanskii). The conjecture would solve a celebrated open problem of Gromov, i.e. show the existence of a non-residually ﬁnite Gromov hyperbolic group.

Eﬀective Burnside-Schur

Theorem (Restricted Burnside Problem; Kostrikin, Zelmanov) Given natural integers r, n, there are only ﬁnitely many r-generated ﬁnite groups all of whose elements have order dividing n.

Emmanuel Breuillard Diophantine geometry and group theory Conjecture (Effective restricted Burnside; Olshanskii) Given r, n, does there exists k(r, n) ∈ N such that there are only finitely many r-generated finite groups G = hSi such that all elements in Sk have order dividing n ?

Remark (Olshanskii). The conjecture would solve a celebrated open problem of Gromov, i.e. show the existence of a non-residually ﬁnite Gromov hyperbolic group.

Eﬀective Burnside-Schur

Theorem (Restricted Burnside Problem; Kostrikin, Zelmanov) Given natural integers r, n, there are only ﬁnitely many r-generated ﬁnite groups all of whose elements have order dividing n.

±1 ±1 For a symmetric set of generators S = {1, s1 , ..., sr } of a group G = hSi, we denote by Sk := S · ... · S the “ball of radius k” in the Cayley graph of G generated by S.

Emmanuel Breuillard Diophantine geometry and group theory Remark (Olshanskii). The conjecture would solve a celebrated open problem of Gromov, i.e. show the existence of a non-residually ﬁnite Gromov hyperbolic group.

Eﬀective Burnside-Schur

Theorem (Restricted Burnside Problem; Kostrikin, Zelmanov) Given natural integers r, n, there are only ﬁnitely many r-generated ﬁnite groups all of whose elements have order dividing n.

±1 ±1 For a symmetric set of generators S = {1, s1 , ..., sr } of a group G = hSi, we denote by Sk := S · ... · S the “ball of radius k” in the Cayley graph of G generated by S. Conjecture (Effective restricted Burnside; Olshanskii) Given r, n, does there exists k(r, n) ∈ N such that there are only finitely many r-generated finite groups G = hSi such that all elements in Sk have order dividing n ?

Emmanuel Breuillard Diophantine geometry and group theory Eﬀective Burnside-Schur

Theorem (Restricted Burnside Problem; Kostrikin, Zelmanov) Given natural integers r, n, there are only ﬁnitely many r-generated ﬁnite groups all of whose elements have order dividing n.

Remark (Olshanskii). The conjecture would solve a celebrated open problem of Gromov, i.e. show the existence of a non-residually ﬁnite Gromov hyperbolic group.

Emmanuel Breuillard Diophantine geometry and group theory Corollary (1st eﬀective version)

Olshanski’s conjecture holds for subgroups of GLd (d fixed). That is: given r, n, d, there exists k(r, n, d) ∈ N such that there are only finitely many r-generated finite groups G = hSi admitting an k embedding in GLd (over some field) such that all elements in S have order dividing n.

Eﬀective Burnside-Schur

Theorem (Burnside 1904, Schur 1914)

Let K be a field and S ⊂ GLd (K) a finite symmetric set. If every element of the subgroup hSi has finite order, then hSi is finite.

Emmanuel Breuillard Diophantine geometry and group theory Eﬀective Burnside-Schur

Theorem (Burnside 1904, Schur 1914)

Let K be a field and S ⊂ GLd (K) a finite symmetric set. If every element of the subgroup hSi has finite order, then hSi is finite.

Corollary (1st eﬀective version)

Emmanuel Breuillard Diophantine geometry and group theory Equivalent way to see it: use ultraproducts as follows. If conclusion fails, one can ﬁnd a sequence of ﬁelds {K } , a sequence of k k>0 k symmetric sets Sk of size r, such that every element in Sk has order dividing n, and yet |hSk i| → +∞. Then consider the ultraproduct Q Q U hSk i ⊂ U GLd (Kk ) = GLd (K), where Y K = Kk . U

Eﬀective Burnside-Schur

Corollary (1st eﬀective version)

Proof. Arguing by contradiction, this follows easily from the Compactness Theorem and the Burnside-Schur theorem.

Emmanuel Breuillard Diophantine geometry and group theory If conclusion fails, one can ﬁnd a sequence of ﬁelds {K } , a sequence of k k>0 k symmetric sets Sk of size r, such that every element in Sk has order dividing n, and yet |hSk i| → +∞. Then consider the ultraproduct Q Q U hSk i ⊂ U GLd (Kk ) = GLd (K), where Y K = Kk . U

Eﬀective Burnside-Schur

Corollary (1st eﬀective version)

Proof. Arguing by contradiction, this follows easily from the Compactness Theorem and the Burnside-Schur theorem. Equivalent way to see it: use ultraproducts as follows.

Emmanuel Breuillard Diophantine geometry and group theory Then consider the ultraproduct Q Q U hSk i ⊂ U GLd (Kk ) = GLd (K), where Y K = Kk . U

Eﬀective Burnside-Schur

Corollary (1st eﬀective version)

Emmanuel Breuillard Diophantine geometry and group theory Eﬀective Burnside-Schur

Corollary (1st eﬀective version)

Emmanuel Breuillard U Diophantine geometry and group theory Q The set S := U Sk has size r and yet for all k > 1, every element of Sk has order divisible by n. But K is a ﬁeld! so by the Burnside-Schur theorem, hSi is ﬁnite. In turn, this forces hSk i to be bounded independently of k, contrary to the standing assumption. QED.

Eﬀective Burnside-Schur

Emmanuel Breuillard Diophantine geometry and group theory But K is a ﬁeld! so by the Burnside-Schur theorem, hSi is ﬁnite. In turn, this forces hSk i to be bounded independently of k, contrary to the standing assumption. QED.

Eﬀective Burnside-Schur

Proof. Arguing by contradiction, this follows easily from the Compactness Theorem and the Burnside-Schur theorem. Equivalent way to see it: use ultraproducts as follows. If conclusion fails, one can ﬁnd a sequence of ﬁelds {K } , a sequence of k k>0 k symmetric sets Sk of size r, such that every element in Sk has order dividing n, and yet |hSk i| → +∞. Then consider the ultraproduct Q Q U hSk i ⊂ U GLd (Kk ) = GLd (K), where Y K = Kk . U Q The set S := U Sk has size r and yet for all k > 1, every element of Sk has order divisible by n.

Emmanuel Breuillard Diophantine geometry and group theory In turn, this forces hSk i to be bounded independently of k, contrary to the standing assumption. QED.

Eﬀective Burnside-Schur

Emmanuel Breuillard Diophantine geometry and group theory Eﬀective Burnside-Schur

Emmanuel Breuillard Diophantine geometry and group theory We had 1st effective version: ∀n, ∃k, K s.t. every element in Sk has order 6 n ⇒ |hSi| 6 K. we prove the following stronger version: 2nd effective version: ∃k = k(d) s.t. ∀n, ∃K(n) s.t. every k element in S has order 6 n ⇒ |hSi| 6 K(n). In particular: Theorem (2nd effective version; B. ’08)

There exists k = k(d) ∈ N such that if S ⊂ GLd over some field, and every element of Sk has finite order, then hSi is finite.

Open pb: ﬁnd optimal upper bounds for K(n) (proof shows nC(d) K(n) 6 e ).

Eﬀective Burnside-Schur

... in fact one can do better and switch two more quantiﬁers.

Emmanuel Breuillard Diophantine geometry and group theory we prove the following stronger version: 2nd eﬀective version: ∃k = k(d) s.t. ∀n, ∃K(n) s.t. every k element in S has order 6 n ⇒ |hSi| 6 K(n). In particular: Theorem (2nd eﬀective version; B. ’08)

There exists k = k(d) ∈ N such that if S ⊂ GLd over some field, and every element of Sk has finite order, then hSi is finite.

Open pb: ﬁnd optimal upper bounds for K(n) (proof shows nC(d) K(n) 6 e ).

Eﬀective Burnside-Schur

... in fact one can do better and switch two more quantiﬁers. We had 1st eﬀective version: ∀n, ∃k, K s.t. every element in Sk has order 6 n ⇒ |hSi| 6 K.

Emmanuel Breuillard Diophantine geometry and group theory In particular: Theorem (2nd eﬀective version; B. ’08)

There exists k = k(d) ∈ N such that if S ⊂ GLd over some field, and every element of Sk has finite order, then hSi is finite.

Open pb: ﬁnd optimal upper bounds for K(n) (proof shows nC(d) K(n) 6 e ).

Eﬀective Burnside-Schur

... in fact one can do better and switch two more quantifiers. We had 1st effective version: ∀n, ∃k, K s.t. every element in Sk has order 6 n ⇒ |hSi| 6 K. we prove the following stronger version: 2nd effective version: ∃k = k(d) s.t. ∀n, ∃K(n) s.t. every k element in S has order 6 n ⇒ |hSi| 6 K(n).

Emmanuel Breuillard Diophantine geometry and group theory Open pb: ﬁnd optimal upper bounds for K(n) (proof shows nC(d) K(n) 6 e ).

Eﬀective Burnside-Schur

There exists k = k(d) ∈ N such that if S ⊂ GLd over some field, and every element of Sk has finite order, then hSi is finite.

Emmanuel Breuillard Diophantine geometry and group theory (proof shows nC(d) K(n) 6 e ).

Eﬀective Burnside-Schur

There exists k = k(d) ∈ N such that if S ⊂ GLd over some field, and every element of Sk has finite order, then hSi is finite.

Open pb: ﬁnd optimal upper bounds for K(n)

Emmanuel Breuillard Diophantine geometry and group theory Eﬀective Burnside-Schur

There exists k = k(d) ∈ N such that if S ⊂ GLd over some field, and every element of Sk has finite order, then hSi is finite.

Open pb: ﬁnd optimal upper bounds for K(n) (proof shows nC(d) K(n) 6 e ).

Emmanuel Breuillard Diophantine geometry and group theory we need some diophantine geometry. Let G be a reductive algebraic group over Q. Say G = GLd . We are going to build a height function bh on the “character variety” Gr //G = “Gr modulo the diagonal action by conjugation”. × Recall the deﬁnition of the logarithmic Weil height on . Let × Q K 6 Q be a number ﬁeld.

1 X + For x ∈ K, set h(x) = nv log |x|v , [K : Q] v∈VK

where, as usual, VK =set of places of K, nv = [Kv : Qv ], and log+ = max{log, 0}.

Heights on character varieties

This time, the compactness theorem is not enough...

Emmanuel Breuillard Diophantine geometry and group theory Let G be a reductive algebraic group over Q. Say G = GLd . We are going to build a height function bh on the “character variety” Gr //G = “Gr modulo the diagonal action by conjugation”. × Recall the deﬁnition of the logarithmic Weil height on . Let × Q K 6 Q be a number ﬁeld.

1 X + For x ∈ K, set h(x) = nv log |x|v , [K : Q] v∈VK

where, as usual, VK =set of places of K, nv = [Kv : Qv ], and log+ = max{log, 0}.

Heights on character varieties

This time, the compactness theorem is not enough... we need some diophantine geometry.

Emmanuel Breuillard Diophantine geometry and group theory × Recall the deﬁnition of the logarithmic Weil height on . Let × Q K 6 Q be a number ﬁeld.

1 X + For x ∈ K, set h(x) = nv log |x|v , [K : Q] v∈VK

where, as usual, VK =set of places of K, nv = [Kv : Qv ], and log+ = max{log, 0}.

Heights on character varieties

This time, the compactness theorem is not enough... we need some diophantine geometry. Let G be a reductive algebraic group over Q. Say G = GLd . We are going to build a height function bh on the “character variety” Gr //G = “Gr modulo the diagonal action by conjugation”.

Emmanuel Breuillard Diophantine geometry and group theory Let × K 6 Q be a number ﬁeld.

1 X + For x ∈ K, set h(x) = nv log |x|v , [K : Q] v∈VK

where, as usual, VK =set of places of K, nv = [Kv : Qv ], and log+ = max{log, 0}.

Heights on character varieties

Emmanuel Breuillard Diophantine geometry and group theory Heights on character varieties

1 X + For x ∈ K, set h(x) = nv log |x|v , [K : Q] v∈VK

where, as usual, VK =set of places of K, nv = [Kv : Qv ], and log+ = max{log, 0}.

Emmanuel Breuillard Diophantine geometry and group theory 1 X + h(A) = nv log ||A||v , [K : Q] v∈VK

where ||A||v is the operator norm associated to the standard norm d 2 ∞ on Kv (i.e. ` if v is archimedean, ` is v is non-archimedean). r Given S ∈ GLd (K) , we set

1 X + h(S) = nv log ||S||v , [K : Q] v∈VK where

||S||v := max{||s||v , s ∈ S}.

Heights on character varieties

× For a d × d matrix A ∈ Md,d (Q ), we set

Emmanuel Breuillard Diophantine geometry and group theory where ||A||v is the operator norm associated to the standard norm d 2 ∞ on Kv (i.e. ` if v is archimedean, ` is v is non-archimedean). r Given S ∈ GLd (K) , we set

1 X + h(S) = nv log ||S||v , [K : Q] v∈VK where

||S||v := max{||s||v , s ∈ S}.

Heights on character varieties

× For a d × d matrix A ∈ Md,d (Q ), we set

1 X + h(A) = nv log ||A||v , [K : Q] v∈VK

Emmanuel Breuillard Diophantine geometry and group theory 1 X + h(S) = nv log ||S||v , [K : Q] v∈VK where

||S||v := max{||s||v , s ∈ S}.

Heights on character varieties

× For a d × d matrix A ∈ Md,d (Q ), we set

1 X + h(A) = nv log ||A||v , [K : Q] v∈VK

where ||A||v is the operator norm associated to the standard norm d 2 ∞ on Kv (i.e. ` if v is archimedean, ` is v is non-archimedean). r Given S ∈ GLd (K) , we set

Emmanuel Breuillard Diophantine geometry and group theory Heights on character varieties

× For a d × d matrix A ∈ Md,d (Q ), we set

1 X + h(A) = nv log ||A||v , [K : Q] v∈VK

where ||A||v is the operator norm associated to the standard norm d 2 ∞ on Kv (i.e. ` if v is archimedean, ` is v is non-archimedean). r Given S ∈ GLd (K) , we set

1 X + h(S) = nv log ||S||v , [K : Q] v∈VK where

||S||v := max{||s||v , s ∈ S}.

Emmanuel Breuillard Diophantine geometry and group theory Deﬁnition We call normalized height the quantity 1 bh(S) := lim h(Sn), n→+∞ n where Sn = S · ... · S is the n-th fold product set.

Heights on character varieties

r Given S ∈ GLd (K) , we set

1 X + h(S) = nv log ||S||v , [K : Q] v∈VK where

||S||v := max{||s||v , s ∈ S}.

Emmanuel Breuillard Diophantine geometry and group theory Heights on character varieties

r Given S ∈ GLd (K) , we set

1 X + h(S) = nv log ||S||v , [K : Q] v∈VK where

||S||v := max{||s||v , s ∈ S}.

Deﬁnition We call normalized height the quantity 1 bh(S) := lim h(Sn), n→+∞ n where Sn = S · ... · S is the n-th fold product set.

Emmanuel Breuillard Diophantine geometry and group theory Theorem (B. ’08) (i) (height zero points) bh(S) = 0 ⇐⇒ hSi is virtually unipotent. (ii) (Bogomolov-type Height Gap Theorem) ∃ε = ε(d) > 0 such that, unless hSi is virtually solvable, we have

bh(S) > ε.

(iii) (Comparison with heights of eigenvalues) if hSi is Zariski-dense in G, then for some C = C(d), c = c(d) > 0,

1 hd(S) max{h(λ); λ eigenvalue of some g ∈ Sc } Cbh(S). C 6 6

Heights on character varieties

properties of bh(S) ↔ group theoretic properties of hSi.

Emmanuel Breuillard Diophantine geometry and group theory (ii) (Bogomolov-type Height Gap Theorem) ∃ε = ε(d) > 0 such that, unless hSi is virtually solvable, we have

bh(S) > ε.

(iii) (Comparison with heights of eigenvalues) if hSi is Zariski-dense in G, then for some C = C(d), c = c(d) > 0,

1 hd(S) max{h(λ); λ eigenvalue of some g ∈ Sc } Cbh(S). C 6 6

Heights on character varieties

properties of bh(S) ↔ group theoretic properties of hSi. Theorem (B. ’08) (i) (height zero points) bh(S) = 0 ⇐⇒ hSi is virtually unipotent.

Emmanuel Breuillard Diophantine geometry and group theory (iii) (Comparison with heights of eigenvalues) if hSi is Zariski-dense in G, then for some C = C(d), c = c(d) > 0,

1 hd(S) max{h(λ); λ eigenvalue of some g ∈ Sc } Cbh(S). C 6 6

Heights on character varieties

properties of bh(S) ↔ group theoretic properties of hSi. Theorem (B. ’08) (i) (height zero points) bh(S) = 0 ⇐⇒ hSi is virtually unipotent. (ii) (Bogomolov-type Height Gap Theorem) ∃ε = ε(d) > 0 such that, unless hSi is virtually solvable, we have

bh(S) > ε.

Emmanuel Breuillard Diophantine geometry and group theory Heights on character varieties

bh(S) > ε.

(iii) (Comparison with heights of eigenvalues) if hSi is Zariski-dense in G, then for some C = C(d), c = c(d) > 0,

1 hd(S) max{h(λ); λ eigenvalue of some g ∈ Sc } Cbh(S). C 6 6

Emmanuel Breuillard Diophantine geometry and group theory The 2nd eﬀective version of Burnside-Schur follows easily from Properties (i) and (iii) of bh(S).

Heights on character varieties

Theorem (B. ’08) (i) (height zero points) bh(S) = 0 ⇐⇒ hSi is virtually unipotent. (ii) (Bogomolov-type Height Gap Theorem) ∃ε = ε(d) > 0 such that unless hSi is virtually solvable, we have

bh(S) > ε.

(iii) (Comparison with heights of eigenvalues) if hSi is Zariski-dense in G, then for some C = C(d), c = c(d) > 0,

1 hd(S) max{h(λ); λ eigenvalue of some g ∈ Sc } Cbh(S). C 6 6

Emmanuel Breuillard Diophantine geometry and group theory Heights on character varieties

Theorem (B. ’08) (i) (height zero points) bh(S) = 0 ⇐⇒ hSi is virtually unipotent. (ii) (Bogomolov-type Height Gap Theorem) ∃ε = ε(d) > 0 such that unless hSi is virtually solvable, we have

bh(S) > ε.

(iii) (Comparison with heights of eigenvalues) if hSi is Zariski-dense in G, then for some C = C(d), c = c(d) > 0,

1 hd(S) max{h(λ); λ eigenvalue of some g ∈ Sc } Cbh(S). C 6 6

The 2nd eﬀective version of Burnside-Schur follows easily from Properties (i) and (iii) of bh(S).

Emmanuel Breuillard Diophantine geometry and group theory Heights on character varieties

Theorem (B. ’08) (i) (height zero points) bh(S) = 0 ⇐⇒ hSi is virtually unipotent. (ii) (Bogomolov-type Height Gap Theorem) ∃ε = ε(d) > 0 such that unless hSi is virtually solvable, we have

bh(S) > ε.

(iii) (Comparison with heights of eigenvalues) if hSi is Zariski-dense in G, then for some C = C(d), c = c(d) > 0,

1 hd(S) max{h(λ); λ eigenvalue of some g ∈ Sc } Cbh(S). C 6 6

The 2nd eﬀective version of Burnside-Schur follows easily from Properties (i) and (iii) of bh(S).

Emmanuel Breuillard Diophantine geometry and group theory 2) some geometry of symmetric spaces and buildings, in particular we make use of non-positive curvature. 3) a “spectral radius formula” for several matrices that relates the n n growth of ||S ||v to that of eigenvalues of S . 4) Bilu’s theorem on equidistribution of Galois orbits of small points on tori. 5) The Bogomolov conjecture for tori (Zhang’s theorem).

Heights on character varieties

Some ingredients of the proof: 1) a geometric reformulation of the problem in terms of minimal displacement of S on each symmetric space or Bruhat-Tits building associated to G(Kv ).

Emmanuel Breuillard Diophantine geometry and group theory 3) a “spectral radius formula” for several matrices that relates the n n growth of ||S ||v to that of eigenvalues of S . 4) Bilu’s theorem on equidistribution of Galois orbits of small points on tori. 5) The Bogomolov conjecture for tori (Zhang’s theorem).

Heights on character varieties

Emmanuel Breuillard Diophantine geometry and group theory 4) Bilu’s theorem on equidistribution of Galois orbits of small points on tori. 5) The Bogomolov conjecture for tori (Zhang’s theorem).

Heights on character varieties

Some ingredients of the proof: 1) a geometric reformulation of the problem in terms of minimal displacement of S on each symmetric space or Bruhat-Tits building associated to G(Kv ). 2) some geometry of symmetric spaces and buildings, in particular we make use of non-positive curvature. 3) a “spectral radius formula” for several matrices that relates the n n growth of ||S ||v to that of eigenvalues of S .

Emmanuel Breuillard Diophantine geometry and group theory 5) The Bogomolov conjecture for tori (Zhang’s theorem).

Heights on character varieties

Emmanuel Breuillard Diophantine geometry and group theory Heights on character varieties

Emmanuel Breuillard Diophantine geometry and group theory we get an uniform Tits alternative: Theorem (Uniform Tits Alternative, B. ’08) There is N = N(d) ∈ N such that if S is a finite symmetric set in GLd (over any field), then (i) either hSi has a solvable subgroup of finite index, (ii) or SN contains two elements a, b, such that ha, bi is a non-abelian free group.

Remark: The proof uses the Tits “ping-pong method” (extending earlier work of Eskin-Mozes-Oh and Breuillard-Gelander) and relies crucially on the Bogomolov-type result for bh(S) presented above.

Uniform Tits Alternative

... in fact the above theorem allows to show more than the eﬀective Burnside-Schur result...

Emmanuel Breuillard Diophantine geometry and group theory Theorem (Uniform Tits Alternative, B. ’08) There is N = N(d) ∈ N such that if S is a finite symmetric set in GLd (over any field), then (i) either hSi has a solvable subgroup of finite index, (ii) or SN contains two elements a, b, such that ha, bi is a non-abelian free group.

Remark: The proof uses the Tits “ping-pong method” (extending earlier work of Eskin-Mozes-Oh and Breuillard-Gelander) and relies crucially on the Bogomolov-type result for bh(S) presented above.

Uniform Tits Alternative

... in fact the above theorem allows to show more than the eﬀective Burnside-Schur result... we get an uniform Tits alternative:

Emmanuel Breuillard Diophantine geometry and group theory (i) either hSi has a solvable subgroup of ﬁnite index, (ii) or SN contains two elements a, b, such that ha, bi is a non-abelian free group.

Remark: The proof uses the Tits “ping-pong method” (extending earlier work of Eskin-Mozes-Oh and Breuillard-Gelander) and relies crucially on the Bogomolov-type result for bh(S) presented above.

Uniform Tits Alternative

Emmanuel Breuillard Diophantine geometry and group theory (ii) or SN contains two elements a, b, such that ha, bi is a non-abelian free group.

Remark: The proof uses the Tits “ping-pong method” (extending earlier work of Eskin-Mozes-Oh and Breuillard-Gelander) and relies crucially on the Bogomolov-type result for bh(S) presented above.

Uniform Tits Alternative

... in fact the above theorem allows to show more than the effective Burnside-Schur result... we get an uniform Tits alternative: Theorem (Uniform Tits Alternative, B. ’08) There is N = N(d) ∈ N such that if S is a finite symmetric set in GLd (over any field), then (i) either hSi has a solvable subgroup of finite index,

Emmanuel Breuillard Diophantine geometry and group theory Remark: The proof uses the Tits “ping-pong method” (extending earlier work of Eskin-Mozes-Oh and Breuillard-Gelander) and relies crucially on the Bogomolov-type result for bh(S) presented above.

Uniform Tits Alternative

Emmanuel Breuillard Diophantine geometry and group theory Uniform Tits Alternative

Remark: The proof uses the Tits “ping-pong method” (extending earlier work of Eskin-Mozes-Oh and Breuillard-Gelander) and relies crucially on the Bogomolov-type result for bh(S) presented above.

Emmanuel Breuillard Diophantine geometry and group theory (ii) is a countable union of algebraic conditions, each equivalent to N 00 S ∈ Wn := “every two words in S have a relation of length 6 n Then Uniform Tits reads: [ V = Wn n

Theorem (Uniform Tits Alternative, B. ’08) There are N = N(d), M = M(d) ∈ N such that if S is a ﬁnite symmetric set in GLd (C), then (i) either hSi has a solvable subgroup of index at most M, (ii) or SN contains two elements a, b, such that ha, bi is a non-abelian free group.

Note: r (i) is an algebraic condition on S in GLd , equivalent to S ∈ V := “hSi has a solvable sugroup of ﬁnite index”

Emmanuel Breuillard Diophantine geometry and group theory Then Uniform Tits reads: [ V = Wn n

Note: r (i) is an algebraic condition on S in GLd , equivalent to S ∈ V := “hSi has a solvable sugroup of ﬁnite index”

(ii) is a countable union of algebraic conditions, each equivalent to N 00 S ∈ Wn := “every two words in S have a relation of length 6 n

Emmanuel Breuillard Diophantine geometry and group theory [ V = Wn n

Note: r (i) is an algebraic condition on S in GLd , equivalent to S ∈ V := “hSi has a solvable sugroup of ﬁnite index”

(ii) is a countable union of algebraic conditions, each equivalent to N 00 S ∈ Wn := “every two words in S have a relation of length 6 n Then Uniform Tits reads:

Emmanuel Breuillard Diophantine geometry and group theory Theorem (Uniform Tits Alternative, B. ’08) There are N = N(d), M = M(d) ∈ N such that if S is a ﬁnite symmetric set in GLd (C), then (i) either hSi has a solvable subgroup of index at most M, (ii) or SN contains two elements a, b, such that ha, bi is a non-abelian free group.

Note: r (i) is an algebraic condition on S in GLd , equivalent to S ∈ V := “hSi has a solvable sugroup of ﬁnite index”

(ii) is a countable union of algebraic conditions, each equivalent to N 00 S ∈ Wn := “every two words in S have a relation of length 6 n Then Uniform Tits reads: [ V = Wn n

Emmanuel Breuillard Diophantine geometry and group theory Hence in fact:

V(C) = Wn(C),

for all n large enough.

Since V and Wn are deﬁned over Z, this equality holds also for algebraically closed ﬁelds of characteristic p if p is large enough, say p > p(n).

Problem: Find the best bound on p(n).

Classical eﬀective versions of the Hilbert Nullstellensatz give A p(n) 6 exp(n ) for some A > 1.

Then Uniform Tits reads: [ V = Wn n

|S| But V and Wn are algebraic subvarieties in G .

Emmanuel Breuillard Diophantine geometry and group theory Since V and Wn are deﬁned over Z, this equality holds also for algebraically closed ﬁelds of characteristic p if p is large enough, say p > p(n).

Problem: Find the best bound on p(n).

Classical eﬀective versions of the Hilbert Nullstellensatz give A p(n) 6 exp(n ) for some A > 1.

Then Uniform Tits reads: [ V = Wn n

|S| But V and Wn are algebraic subvarieties in G . Hence in fact:

V(C) = Wn(C), for all n large enough.

Emmanuel Breuillard Diophantine geometry and group theory Problem: Find the best bound on p(n).

Classical eﬀective versions of the Hilbert Nullstellensatz give A p(n) 6 exp(n ) for some A > 1.

Then Uniform Tits reads: [ V = Wn n

|S| But V and Wn are algebraic subvarieties in G . Hence in fact:

V(C) = Wn(C), for all n large enough.

Since V and Wn are deﬁned over Z, this equality holds also for algebraically closed ﬁelds of characteristic p if p is large enough, say p > p(n).

Emmanuel Breuillard Diophantine geometry and group theory Classical eﬀective versions of the Hilbert Nullstellensatz give A p(n) 6 exp(n ) for some A > 1.

Then Uniform Tits reads: [ V = Wn n

|S| But V and Wn are algebraic subvarieties in G . Hence in fact:

V(C) = Wn(C), for all n large enough.

Since V and Wn are deﬁned over Z, this equality holds also for algebraically closed ﬁelds of characteristic p if p is large enough, say p > p(n).

Problem: Find the best bound on p(n).

Emmanuel Breuillard Diophantine geometry and group theory Then Uniform Tits reads: [ V = Wn n

|S| But V and Wn are algebraic subvarieties in G . Hence in fact:

V(C) = Wn(C), for all n large enough.

Since V and Wn are deﬁned over Z, this equality holds also for algebraically closed ﬁelds of characteristic p if p is large enough, say p > p(n).

Problem: Find the best bound on p(n).

Classical eﬀective versions of the Hilbert Nullstellensatz give A p(n) 6 exp(n ) for some A > 1.

Emmanuel Breuillard Diophantine geometry and group theory In particular we have proved: Corollary (uniform exponential growth)

There are e, c, N, M > 0 depending on d only such that if S ⊂ GLd (Fp) and hSi has no solvable subgroup of index at most M, then (i) ∃a, b ∈ SN with no relation up to length (log p)e . n e (ii) |S | > exp(cn) for every n 6 (log p) .

Classical eﬀective versions of the Hilbert Nullstellensatz give A p(n) 6 exp(n ) for some A > 1. A So V = Wn in char p for p > p(n) ' exp(n ).

Emmanuel Breuillard Diophantine geometry and group theory Classical eﬀective versions of the Hilbert Nullstellensatz give A p(n) 6 exp(n ) for some A > 1. A So V = Wn in char p for p > p(n) ' exp(n ). In particular we have proved: Corollary (uniform exponential growth)

Emmanuel Breuillard Diophantine geometry and group theory Let diam(G, S) be the diameter of Cay(G, S) and

diam(G) := sup diam(G, S). S

Conjecture (Uniform logarithmic diameter for FSG’s of Lie type, folklore)

Let r ∈ N. There is a constant Cr > 0 such that

diam(G) 6 Cr log |G|,

for an arbitrary ﬁnite simple group of Lie type G with rank 6 r.

Applications: diameter of ﬁnite simple groups

Let G be a ﬁnite simple group, S a ﬁnite symmetric generating set, and Cay(G, S) its Cayley graph.

Emmanuel Breuillard Diophantine geometry and group theory Conjecture (Uniform logarithmic diameter for FSG’s of Lie type, folklore)

Let r ∈ N. There is a constant Cr > 0 such that

diam(G) 6 Cr log |G|,

for an arbitrary ﬁnite simple group of Lie type G with rank 6 r.

Applications: diameter of ﬁnite simple groups

Let G be a ﬁnite simple group, S a ﬁnite symmetric generating set, and Cay(G, S) its Cayley graph.

Let diam(G, S) be the diameter of Cay(G, S) and

diam(G) := sup diam(G, S). S

Emmanuel Breuillard Diophantine geometry and group theory Applications: diameter of ﬁnite simple groups

Let G be a ﬁnite simple group, S a ﬁnite symmetric generating set, and Cay(G, S) its Cayley graph.

Let diam(G, S) be the diameter of Cay(G, S) and

diam(G) := sup diam(G, S). S

Conjecture (Uniform logarithmic diameter for FSG’s of Lie type, folklore)

Let r ∈ N. There is a constant Cr > 0 such that

diam(G) 6 Cr log |G|,

for an arbitrary ﬁnite simple group of Lie type G with rank 6 r.

Emmanuel Breuillard Diophantine geometry and group theory In fact, it is not obvious to find even one infinite family {Gn} of finite groups for which diam(Gn) log |Gn| (a question Lubotzky’s)

Conjecture (Uniform logarithmic diameter for FSG’s of Lie type, folklore)

Let r ∈ N. There is a constant Cr > 0 such that

diam(G) 6 Cr log |G|, for an arbitrary finite simple group of Lie type G with rank 6 r. In Breuillard-Gamburd (2010), using the uniform Tits alternative, we gave the first example of such a family among finite simple groups: we showed that Gn = PSL2(Fpn ) for some infinite family of primes pn satisfies the conjecture and indeed are uniform expanders.

Emmanuel Breuillard Diophantine geometry and group theory Conjecture (Uniform logarithmic diameter for FSG’s of Lie type, folklore)

Let r ∈ N. There is a constant Cr > 0 such that

In fact, it is not obvious to find even one infinite family {Gn} of finite groups for which diam(Gn) log |Gn| (a question Lubotzky’s)

Emmanuel Breuillard Diophantine geometry and group theory Theorem (There can be only few exceptions to Conjecture 3)

Given r, k, ε > 0 there is an explicit C = C(r, k, ε) > 0 such that if Pgood denotes the set of prime numbers p such that

diam(G) 6 C log |G|

for all ﬁnite simple groups of Lie type of the form G = G(Fps ), where s 6 k, G is a simple algebraic group of rank at most r, ε Then for all X > 1, |{p ∈/ Pgood , p 6 X }| 6 X .

Diameter of ﬁnite simple groups

Using recent results on approximate groups by Hrushovski, Pyber-Szabo and Breuillard-Green-Tao, one can now push the method of Breuillard-Gamburd to obtain:

Emmanuel Breuillard Diophantine geometry and group theory if Pgood denotes the set of prime numbers p such that

diam(G) 6 C log |G|

for all ﬁnite simple groups of Lie type of the form G = G(Fps ), where s 6 k, G is a simple algebraic group of rank at most r, ε Then for all X > 1, |{p ∈/ Pgood , p 6 X }| 6 X .

Diameter of ﬁnite simple groups

Using recent results on approximate groups by Hrushovski, Pyber-Szabo and Breuillard-Green-Tao, one can now push the method of Breuillard-Gamburd to obtain: Theorem (There can be only few exceptions to Conjecture 3)

Given r, k, ε > 0 there is an explicit C = C(r, k, ε) > 0 such that

Emmanuel Breuillard Diophantine geometry and group theory ε Then for all X > 1, |{p ∈/ Pgood , p 6 X }| 6 X .

Diameter of ﬁnite simple groups

Given r, k, ε > 0 there is an explicit C = C(r, k, ε) > 0 such that if Pgood denotes the set of prime numbers p such that

diam(G) 6 C log |G|

for all ﬁnite simple groups of Lie type of the form G = G(Fps ), where s 6 k, G is a simple algebraic group of rank at most r,

Emmanuel Breuillard Diophantine geometry and group theory Diameter of ﬁnite simple groups

Given r, k, ε > 0 there is an explicit C = C(r, k, ε) > 0 such that if Pgood denotes the set of prime numbers p such that

diam(G) 6 C log |G|

for all ﬁnite simple groups of Lie type of the form G = G(Fps ), where s 6 k, G is a simple algebraic group of rank at most r, ε Then for all X > 1, |{p ∈/ Pgood , p 6 X }| 6 X .

Emmanuel Breuillard Diophantine geometry and group theory There are two periods in the growth of a Cayley graph ball B(n) for G = G(Fps ).

1) an early period, when the growth is exponential: |B(n)| > exp(cn),

2) and a later period, when the growth is no longer exponential but still exp(nα) for some α < 1

then we reach B(n) = G.

Now some words about proofs.

Emmanuel Breuillard Diophantine geometry and group theory 1) an early period, when the growth is exponential: |B(n)| > exp(cn),

2) and a later period, when the growth is no longer exponential but still exp(nα) for some α < 1

then we reach B(n) = G.

Now some words about proofs.

There are two periods in the growth of a Cayley graph ball B(n) for G = G(Fps ).

Emmanuel Breuillard Diophantine geometry and group theory when the growth is exponential: |B(n)| > exp(cn),

2) and a later period, when the growth is no longer exponential but still exp(nα) for some α < 1

then we reach B(n) = G.

Now some words about proofs.

There are two periods in the growth of a Cayley graph ball B(n) for G = G(Fps ).

1) an early period,

Emmanuel Breuillard Diophantine geometry and group theory 2) and a later period, when the growth is no longer exponential but still exp(nα) for some α < 1

then we reach B(n) = G.

Now some words about proofs.

There are two periods in the growth of a Cayley graph ball B(n) for G = G(Fps ).

1) an early period, when the growth is exponential: |B(n)| > exp(cn),

Emmanuel Breuillard Diophantine geometry and group theory when the growth is no longer exponential but still exp(nα) for some α < 1

then we reach B(n) = G.

Now some words about proofs.

There are two periods in the growth of a Cayley graph ball B(n) for G = G(Fps ).

1) an early period, when the growth is exponential: |B(n)| > exp(cn),

2) and a later period,

Emmanuel Breuillard Diophantine geometry and group theory then we reach B(n) = G.

Now some words about proofs.

There are two periods in the growth of a Cayley graph ball B(n) for G = G(Fps ).

1) an early period, when the growth is exponential: |B(n)| > exp(cn),

2) and a later period, when the growth is no longer exponential but still exp(nα) for some α < 1

Emmanuel Breuillard Diophantine geometry and group theory Now some words about proofs.

There are two periods in the growth of a Cayley graph ball B(n) for G = G(Fps ).

1) an early period, when the growth is exponential: |B(n)| > exp(cn),

2) and a later period, when the growth is no longer exponential but still exp(nα) for some α < 1 then we reach B(n) = G.

Emmanuel Breuillard Diophantine geometry and group theory Emmanuel Breuillard Diophantine geometry and group theory The poor bounds on the size of first prime p = p(n) for which V = Wn holds also in characteristic p obtained in Corollary 11 (and gotten from the effective Nullstellensatz) need to be bootstrapped using a pigeonhole argument at the expense of loosing a small (but perhaps infinite) family of primes. Open problem: can one take p(n) 6 exp(Cn) ? (currently we C know p(n) 6 exp(n )).

Exponential growth in the early period follows from the uniform Tits alternative as outlined above.

Emmanuel Breuillard Diophantine geometry and group theory Open problem: can one take p(n) 6 exp(Cn) ? (currently we C know p(n) 6 exp(n )).

Exponential growth in the early period follows from the uniform Tits alternative as outlined above. The poor bounds on the size of first prime p = p(n) for which V = Wn holds also in characteristic p obtained in Corollary 11 (and gotten from the effective Nullstellensatz) need to be bootstrapped using a pigeonhole argument at the expense of loosing a small (but perhaps infinite) family of primes.

Emmanuel Breuillard Diophantine geometry and group theory Exponential growth in the early period follows from the uniform Tits alternative as outlined above. The poor bounds on the size of first prime p = p(n) for which V = Wn holds also in characteristic p obtained in Corollary 11 (and gotten from the effective Nullstellensatz) need to be bootstrapped using a pigeonhole argument at the expense of loosing a small (but perhaps infinite) family of primes. Open problem: can one take p(n) 6 exp(Cn) ? (currently we C know p(n) 6 exp(n )).

Emmanuel Breuillard Diophantine geometry and group theory Theorem (Product Theorem, Hrushovski, Pyber-Szabo and Breuillard-Green-Tao) Let r ∈ N. There exists a constant γ = γ(r) > 0 such that

1+γ |AAA| > min{|A| , |G|}, for every generating subset A of any ﬁnite simple group of Lie type G of rank at most r.

Approximate groups

Subexponential growth in the later period follows from the aforementioned results on Approximate groups. Namely:

Emmanuel Breuillard Diophantine geometry and group theory As a consequence, for every Cayley graph of G, either 1+γ |B(3n)| > |B(n)| or B(3n) = G. Hence we get: a) subexponential lower bound on the growth in the later period. C b) that diam(G) 6 (log |G|) for some C > 0 depending only on rank(G)... but not enough for 6 C · log |G| !

Approximate groups

Subexponential growth in the later period follows from the aforementioned results on Approximate groups. Namely: Theorem (Product Theorem, Hrushovski, Pyber-Szabo and Breuillard-Green-Tao) Let r ∈ N. There exists a constant γ = γ(r) > 0 such that

1+γ |AAA| > min{|A| , |G|}, for every generating subset A of any ﬁnite simple group of Lie type G of rank at most r.

Emmanuel Breuillard Diophantine geometry and group theory a) subexponential lower bound on the growth in the later period. C b) that diam(G) 6 (log |G|) for some C > 0 depending only on rank(G)... but not enough for 6 C · log |G| !

Approximate groups

1+γ |AAA| > min{|A| , |G|}, for every generating subset A of any ﬁnite simple group of Lie type G of rank at most r.

As a consequence, for every Cayley graph of G, either 1+γ |B(3n)| > |B(n)| or B(3n) = G. Hence we get:

Emmanuel Breuillard Diophantine geometry and group theory C b) that diam(G) 6 (log |G|) for some C > 0 depending only on rank(G)... but not enough for 6 C · log |G| !

Approximate groups

1+γ |AAA| > min{|A| , |G|}, for every generating subset A of any ﬁnite simple group of Lie type G of rank at most r.

As a consequence, for every Cayley graph of G, either 1+γ |B(3n)| > |B(n)| or B(3n) = G. Hence we get: a) subexponential lower bound on the growth in the later period.

Emmanuel Breuillard Diophantine geometry and group theory but not enough for 6 C · log |G| !

Approximate groups

1+γ |AAA| > min{|A| , |G|}, for every generating subset A of any ﬁnite simple group of Lie type G of rank at most r.

Emmanuel Breuillard Diophantine geometry and group theory Approximate groups

1+γ |AAA| > min{|A| , |G|}, for every generating subset A of any ﬁnite simple group of Lie type G of rank at most r.

As a consequence, for every Cayley graph of G, either 1+γ |B(3n)| > |B(n)| or B(3n) = G. Hence we get: a) subexponential lower bound on the growth in the later period. C b) that diam(G) 6 (log |G|) for some C > 0 depending only on rank(G)... but not enough for 6 C · log |G| ! Emmanuel Breuillard Diophantine geometry and group theory In 2009, using model theory, Hrushovski obtained the ﬁrst general results on the structure of approximate groups. For approximate subgroups of simple algebraic groups, he obtained essentially complete results. Namely, approximate groups are close to genuine subgroups unless they are trapped in a proper algebraic subgroup. The aforementioned subsequent works of Pyber-Szabo and Breuillard-Green-Tao aim at giving a better bound on how close to a genuine group is a given approximate group, resulting in the above stated theorem.

Approximate groups

The above product theorem can be reformulated in terms of “approximate groups” (T. Tao), that is ﬁnite subsets A of a group G, such that AA can be covered by a small amount of translates of A.

Emmanuel Breuillard Diophantine geometry and group theory For approximate subgroups of simple algebraic groups, he obtained essentially complete results. Namely, approximate groups are close to genuine subgroups unless they are trapped in a proper algebraic subgroup. The aforementioned subsequent works of Pyber-Szabo and Breuillard-Green-Tao aim at giving a better bound on how close to a genuine group is a given approximate group, resulting in the above stated theorem.

Approximate groups

Emmanuel Breuillard Diophantine geometry and group theory The aforementioned subsequent works of Pyber-Szabo and Breuillard-Green-Tao aim at giving a better bound on how close to a genuine group is a given approximate group, resulting in the above stated theorem.

Approximate groups

The above product theorem can be reformulated in terms of “approximate groups” (T. Tao), that is ﬁnite subsets A of a group G, such that AA can be covered by a small amount of translates of A. In 2009, using model theory, Hrushovski obtained the ﬁrst general results on the structure of approximate groups. For approximate subgroups of simple algebraic groups, he obtained essentially complete results. Namely, approximate groups are close to genuine subgroups unless they are trapped in a proper algebraic subgroup.

Emmanuel Breuillard Diophantine geometry and group theory Approximate groups

Emmanuel Breuillard Diophantine geometry and group theory References: Heights on character varieties, uniform Tits

E. Breuillard, A Height Gap Theorem for ﬁnite subsets of GLd (Q) and non amenable subgroups, to appear Annals of Math (2011).

E. Breuillard, A strong Tits alternative, preprint, arXiv:0804.1395.

E. Breuillard, Heights on SL2 and free subgroups, Zimmer Volume, Chicago Univ. Press (2011).

E. Breuillard and A. Gamburd, Strong uniform expansion in SL(2, p), Geom. Anal. Func. Anal. Vol. 20-5 (2010), 1201-1209.

Emmanuel Breuillard Diophantine geometry and group theory References: Approximate groups and applications to ﬁnite groups

E. Breuillard, B. J. Green and T. C. Tao, Approximate subgroups of linear groups, to appear Geom. Anal. Func. Anal. (2011).

H. A. Helfgott, Growth and generation in SL2(Z/pZ), Ann. of Math. (2) 167 (2008), no. 2, 601–623.

E. Hrushovski, Stable group theory and approximate subgroups, preprint (2009), arXiv:0909.2190.

L. Pyber and E. Szab´o, Growth in ﬁnite simple groups of Lie type, preprint (2010), arXiv:1001.4556.

Emmanuel Breuillard Diophantine geometry and group theory Thank you!

Emmanuel Breuillard Diophantine geometry and group theory