Pseudofinite groups and VC-dimension Gabriel Conant Notre Dame 10 - - PowerPoint PPT Presentation

Pseudofinite groups and VC-dimension

Gabriel Conant

Notre Dame

10 April 2018 Workshop on model theory of finite and pseudofinite structures University of Leeds

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Stable arithmetic regularity

Theorem (Terry-Wolf 2017)

Given a prime p, an integer k ≥ 1, and ǫ > 0, there is N = N(p, k, ǫ) such that the following holds. Suppose A ⊆ G = (Z/pZ)n is such that x + y ∈ A is k-stable. Then there is a subgroup H ≤ G, of index at most N, such that for all g ∈ G, |H ∩ (g + A)| ≤ ǫ|H|

• r

|H\(g + A)| ≤ ǫ|H|.

• (C.-Pillay-Terry 2017) Generalization to stable sets in arbitrary finite

groups; start working on the NIP case.

• Terry and Wolf obtain effective and very strong bounds on the index
• f H. Our methods to not give effective bounds of any kind.

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NIP arithmetic regularity

Theorem (C.-Pillay-Terry 2018)

Given k ≥ 1 and ǫ > 0, there is N = N(k, ǫ) such that the following

• holds. Let G be a finite group and fix A ⊆ G such that y · x ∈ A is

k-NIP. Then there are:

• a finite-index normal subgroup H ≤ G, of index at most N,
• a (δ, Tn)-Bohr set B ⊆ H, with n ≤ N and δ ≥ 1

N ,

• a set Z ⊆ G, with |Z| < ǫ|G|,

such that for all g ∈ G\Z |B ∩ gA| ≤ ǫ|B|

• r

|B\gA| ≤ ǫ|B|.

• If A is k-stable then we may take B = H and Z = ∅.
• If we restrict to abelian groups then we may take H = G.
• If we fix r > 0 and restrict to groups of exponent r, then we may take

B = H. This generalizes the abelian case: Alon-Fox-Zhao (2018).

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Setting

T is a complete theory, M is a monster model, G = G(M) is a ∅-definable group. Definitions: Fix a formula θ(x; ¯ y) (with x of sort G).

• A θ-formula is a finite Boolean combination of instances of θ(x; ¯

y).

• A subset of G is θ-definable if it is defined by a θ-formula.
• A subset of G is θ-type-definable if it is an intersection of boundedly

many θ-definable sets.

• G0

θ = {H : H ≤ G is θ-definable of finite index}

• G00

θ = {H : H ≤ G is θ-type-definable of bounded index}

• θ(x; ¯

y) is invariant any left translate of an instance of θ(x; ¯ y) is equivalent to an instance of θ(x; ¯ y).

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Local stable group theory

Fact (Hrushovski-Pillay 1994)

Fix θ(x; ¯ y) invariant and stable. Then G0

θ = G00 θ , and G0 θ is θ-definable

• f finite index. Given a θ-definable set X ⊆ G, let

µ(X) = |{aG0

θ : X ∩ aG0 θ is generic}|

[G : G0

θ]

. Then: (a) µ is the unique left-invariant Keisler measure on θ-definable subsets of G; and (b) if X ⊆ G is θ-definable, and D ⊆ G is the union of left cosets of G0

θ

whose intersection with X is generic, then µ(X △D) = 0.

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NIP and finitely satisfiable generics

Examples of groups definable in NIP theories:

• T = Th(R, +, ·) and G = SL2(M).

There is no left-invariant Keisler measure on G.

• T = Th(Z, +, <) and G = M.

For any 0 ≤ r ≤ 1 there is a left-invariant Keisler measure µ on G such that µ(N) = r.

Definition (T NIP)

G has finitely satisfiable generics (fsg) if there is a left-invariant Keisler measure µ on G, which is generically stable over some small M0 ≺ M (i.e. definable over M0 and finitely satisfiable in M0).

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NIP groups with fsg

Fact (HPP 2008; HP 2011; HPS 2013)

Assume T is NIP and G is fsg, witnessed by µ. Then: (a) µ is the unique left-invariant Keisler measure on G, and the unique right-invariant Keisler measure on G. (b) A definable set X ⊆ G is generic if and only if µ(X) > 0. (c) There are generic types p ∈ SG(M). (d) G00 = Stab(p) for any generic p ∈ SG(M). (e) Fix a definable set X ⊆ G and a generic type p ∈ SG(M). Define Up

X = {aG00 : X ∈ ap}.

Then Up

X is Borel in G/G00 and µ(X) is the Haar measure of Up X.

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Toward “local fsg”

Definition

Suppose X ⊆ Mn is ∅-definable, and let µ be a Keisler measure on X. Let φ(x; ¯ y) be a formula (with x of sort X). Then µ is generically stable with respect to φ(x; ¯ y) if there is a small model M0 ≺ M such that: (i) (definability) for any closed C ⊆ [0, 1], {¯ b ∈ M¯

y : µ(φ(x; ¯

b)) ∈ C} is φopp-type-definable over M0; (ii) (finite satisfiability) for any ¯ b ∈ M¯

y, if µ(φ(x; ¯

b)) > 0 then φ(x; ¯ b) is realized in X(M0). Concrete motivating example: NIP formulas on pseudofinite sets.

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Generic stability and pseudofiniteness

Let X be a ∅-definable set in M. Assume X is pseudofinite with pseudofinite counting measure µ.

Corollary (of the VC-theorem)

Suppose φ(x; ¯ y) is NIP (with x in sort X). Then, for any ǫ > 0, there is some n ≥ 1 and (a1, . . . , an) ∈ X n such that, for any ¯ b ∈ M¯

y,

|µ(φ(x; ¯ b)) − 1

n|{1 ≤ t ≤ n : M |

= φ(at; ¯ b)}| < ǫ. Moreover n depends only on ǫ and the VC-dimension of φ(x; ¯ y).

Corollary

If φ(x; ¯ y) is NIP (with x of sort X), then µ is generically stable with respect to φ(x; ¯ y).

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Local fsg

Fix an invariant formula θ(x; ¯ y) (with x of sort G). Let θr(x; ¯ y, z) be the formula θ(x · z; ¯ y).

Definition

θ(x; ¯ y) is fsg if there is a left and right invariant Keisler measure µ on the Boolean algebra of θr-formulas, which is generically stable with respect any formula of the form: (i) φ(u · x), (ii) φ(x · u), or (iii) φ(u1 · x) △φ(u2 · x) for some θr-formula φ(x). Note: If θ(x; ¯ y) is NIP then any ψ(x; ¯ u) as in (i), (ii), or (iii) is NIP .

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Examples of local fsg

Fix an invariant formula θ(x; ¯ y). Then θ(x; ¯ y) is fsg when:

• T is NIP and G is fsg,
• θ(x; ¯

y) is stable, or

• G is pseudofinite and θ(x; ¯

y) is NIP . Caution: The definition of fsg for θ(x; ¯ y) involves a Keisler measure on θr-formulas. It is possible that θ(x; y) is stable, while θr(x; ¯ y, z) has the independence property.

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First main result

Theorem (C.-Pillay 2018)

Let θ(x; ¯ y) be invariant, NIP , and fsg, witnessed by the measure µ. (a) (Generic types) Global generic θr-types exist. Given a θr-formula φ(x), the following are equivalent:

(i) φ(x) is left generic; (ii) φ(x) is right generic; (iii) µ(φ(x)) > 0.

(b) (Local G00)

(i) G00

θr is normal and θr-type-definable of bounded index.

(ii) G00

θr = Stab(p) for any global generic θr-type p.

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First main result, continued

Theorem (C.-Pillay 2018)

Let θ(x; ¯ y) be invariant, NIP , and fsg, witnessed by the measure µ. (c) (Local G0) G0

θr is normal and θr-type-definable of bounded index.

Moreover, G0

θr /G00 θr is the connected component of the identity in

G/G00

θr .

(d) (Uniqueness of measure) µ is the unique left-invariant Keisler measure on θr-formulas. Given a θr-formula φ(x) and a global generic θr-type p, the set Up

φ(x) = {aG00 θr : φ(x) ∈ ap}

is a Borel set in G/G00

θr , and µ(φ(x)) is the Haar measure of Up φ(x).

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First main result, continued

Theorem (C.-Pillay 2018)

Let θ(x; ¯ y) be invariant, NIP , and fsg, witnessed by the measure µ. (e) (Generic compact domination) Fix a θr-formula φ(x) and define Eφ(x) ⊆ G/G00

θr to be the set of C ∈ G/G00 θr such that

C ∩ φ(x) and C ∩ ¬φ(x) each extend to a global generic θr-type. Then Eφ(x) has Haar measure 0. The proof uses the work of many people, including:

• NIP 1, 2: Hrushovski-Peterzil-Pillay (2008); Hrushovski-Pillay (2011).
• Chernikov-Simon (2015): Definably amenable NIP groups.
• Simon (2015): Rosenthal compacta and NIP formulas.
• Simon (2017): VC-sets and generic compact domination.

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Assumptions

For the rest of the talk, fix an invariant formula θ(x; ¯ y). Assume θ(x; ¯ y) is NIP and fsg. Let µ be the unique left-invariant Keisler measure on θr-formulas. Main example: G is pseudofinite, θ(x; ¯ y) is NIP , and µ is the pseudofinite counting measure.

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Structure of θr-definable sets: the profinite case

Theorem (C.-Pillay-Terry 2018)

Suppose G/G00

θr is profinite (i.e. G00 θr = G0 θr ), and fix a θr-definable set

A ⊆ G. For any ǫ > 0 there are:

• a θr-definable finite-index normal subgroup H ≤ G, and
• a set Z ⊆ G, which is a union of cosets of H, with µ(Z) < ǫ,

such that the following properties hold. (i) (regularity) For any g ∈ G\Z, µ(H ∩ gA) = 0 or µ(H\gA) = 0. (ii) (structure) There is D ⊆ G, which is a union of cosets of H, such that µ((A\Z) △D) = 0; Examples when G/G00

θr is profinite:

• θ(x; ¯

y) is stable. For ǫ small enough, we obtain Z = ∅.

• G has finite exponent (any compact torsion group is profinite).

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A counterexample

Example: (language of groups with unary predicate A)

• Let Mp = (Z/pZ, +, Ap) where p is prime and Ap = {0, . . . , p−1

2 },

• T = Th(

U Mp) for some nonprincipal ultrafilter U, and

• set θ(x; y) := x + y ∈ A.

Then:

• θ(x; y) is NIP (and thus fsg),
• A = A(M) is θ-definable with measure 1

2, and

• G = M has no proper definable subgroups.

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Bohr sets

Definition

Let L be a compact metrizable group, with an invariant metric d. Given δ > 0, a (δ, L)-Bohr set in G is a subset of G of the form Bπ

δ := π-1({λ ∈ L : d(λ, 1L) < δ}),

where π: G → L is a homomorphism. Example: Suppose H ≤ G is normal of finite index. Then H = Bπ

δ ,

where π: G → G/H is the canonical map and δ <

1 [G:H].

Exercise: Any Bohr set in G is symmetric, contains 1G, and is left and right generic. Morever, given π: G → L as above, Bπ

δ · Bπ δ ⊆ Bπ 2δ.

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Structure of θr-definable sets: the general case

Let G/G00

θr = lim

← − Li, where Li is a compact Lie group, for i ∈ N. For each i ∈ N, the canonical projection πi : G → Li is θr-definable (i.e. π-1

i (C) is θr-type-definable for any closed C ⊆ Li).

Theorem (C.-Pillay-Terry 2018)

Fix a θr-definable set A ⊆ G. For any ǫ > 0 there are:

• an i ∈ N, and
• a θr-definable set Z ⊆ G, with µ(Z) < ǫ,

such that, for arbitrarily small δ > 0, the following properties hold. (i) (regularity) For any g ∈ G\Z, µ(Bπi

δ ∩ gA) = 0 or µ(Bπi δ \gA) = 0.

(ii) (structure) There is D ⊆ G, which is a union of finitely many translates of Bπi

δ , such that µ((A △D)\Z) = 0;

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A better theorem for pseudofinite groups

Assume G is pseudofinite.

Fact (Pillay 2017)

If Γ ≤ G is type-definable of bounded index, then the connected component of the identity in G/Γ is commutative.

Theorem (C.-Pillay-Terry 2018)

Fix a θr-definable set A ⊆ G. For any ǫ > 0, there are:

• a θr-definable finite-index normal subgroup H ≤ G,
• a θr-definable homomorphism π: H → Tn, for some n ≥ 0, and
• a θr-definable Z ⊆ G, with µ(Z) < ǫ,

such that, for arbitrarily small δ > 0, the following properties hold. (i) (regularity) For any g ∈ G\Z, µ(Bπ

δ ∩ gA) = 0 or µ(Bπ δ \gA) = 0.

(ii) (structure) There is D ⊆ G, which is a union of finitely many translates of Bπ

δ , such that µ((A △D)\Z) = 0;

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thank you

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