Pseudofinite groups and VC-dimension Gabriel Conant Notre Dame 10 April 2018 Workshop on model theory of finite and pseudofinite structures University of Leeds 1 / 21
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 / p Z ) 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 | | H \ ( g + A ) | ≤ ǫ | H | . or • (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 of H . Our methods to not give effective bounds of any kind. 2 / 21
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, T n ) -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 | or | 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). 3 / 21
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. • G 0 θ = � { H : H ≤ G is θ -definable of finite index } • G 00 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 ) . 4 / 21
Local stable group theory Fact (Hrushovski-Pillay 1994) Fix θ ( x ; ¯ y ) invariant and stable. Then G 0 θ = G 00 θ , and G 0 θ is θ -definable of finite index. Given a θ -definable set X ⊆ G , let µ ( X ) = |{ aG 0 θ : X ∩ aG 0 θ is generic }| . [ G : G 0 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 G 0 θ whose intersection with X is generic, then µ ( X △ D ) = 0. 5 / 21
NIP and finitely satisfiable generics Examples of groups definable in NIP theories : • T = Th ( R , + , · ) and G = SL 2 ( 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 M 0 ≺ M (i.e. definable over M 0 and finitely satisfiable in M 0 ). 6 / 21
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 ∈ S G ( M ) . ( d ) G 00 = Stab ( p ) for any generic p ∈ S G ( M ) . ( e ) Fix a definable set X ⊆ G and a generic type p ∈ S G ( M ) . Define X = { aG 00 : X ∈ ap } . U p X is Borel in G / G 00 and µ ( X ) is the Haar measure of U p Then U p X . 7 / 21
Toward “local fsg” Definition Suppose X ⊆ M n 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 M 0 ≺ M such that: ( i ) (definability) for any closed C ⊆ [ 0 , 1 ] , y : µ ( φ ( x ; ¯ { ¯ b ∈ M ¯ b )) ∈ C } is φ opp -type-definable over M 0 ; ( ii ) (finite satisfiability) for any ¯ b ∈ M ¯ y , if µ ( φ ( x ; ¯ b )) > 0 then φ ( x ; ¯ b ) is realized in X ( M 0 ) . Concrete motivating example: NIP formulas on pseudofinite sets. 8 / 21
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 ( a 1 , . . . , a n ) ∈ X n such that, for any ¯ b ∈ M ¯ y , | µ ( φ ( x ; ¯ = φ ( a t ; ¯ b )) − 1 n |{ 1 ≤ t ≤ n : M | 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 ) . 9 / 21
Local fsg Fix an invariant formula θ ( x ; ¯ y ) (with x of sort G ). Let θ r ( 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 θ r -formulas, which is generically stable with respect any formula of the form: ( i ) φ ( u · x ) , ( ii ) φ ( x · u ) , or ( iii ) φ ( u 1 · x ) △ φ ( u 2 · x ) for some θ r -formula φ ( x ) . Note : If θ ( x ; ¯ y ) is NIP then any ψ ( x ; ¯ u ) as in ( i ) , ( ii ) , or ( iii ) is NIP . 10 / 21
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 θ r -formulas. It is possible that θ ( x ; y ) is stable, while θ r ( x ; ¯ y , z ) has the independence property. 11 / 21
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 θ r -types exist. Given a θ r -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 G 00 ) ( i ) G 00 θ r is normal and θ r -type-definable of bounded index. ( ii ) G 00 θ r = Stab ( p ) for any global generic θ r -type p. 12 / 21
First main result, continued Theorem (C.-Pillay 2018) Let θ ( x ; ¯ y ) be invariant, NIP , and fsg, witnessed by the measure µ . ( c ) (Local G 0 ) G 0 θ r is normal and θ r -type-definable of bounded index. Moreover, G 0 θ r / G 00 θ r is the connected component of the identity in G / G 00 θ r . ( d ) (Uniqueness of measure) µ is the unique left-invariant Keisler measure on θ r -formulas. Given a θ r -formula φ ( x ) and a global generic θ r -type p, the set U p φ ( x ) = { aG 00 θ r : φ ( x ) ∈ ap } θ r , and µ ( φ ( x )) is the Haar measure of U p is a Borel set in G / G 00 φ ( x ) . 13 / 21
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 θ r -formula φ ( x ) and define E φ ( x ) ⊆ G / G 00 θ r to be the set of C ∈ G / G 00 θ r such that C ∩ φ ( x ) and C ∩ ¬ φ ( x ) each extend to a global generic θ r -type. Then E φ ( 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 . 14 / 21
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 θ r -formulas. Main example: G is pseudofinite, θ ( x ; ¯ y ) is NIP , and µ is the pseudofinite counting measure. 15 / 21
Structure of θ r -definable sets: the profinite case Theorem (C.-Pillay-Terry 2018) Suppose G / G 00 θ r is profinite (i.e. G 00 θ r = G 0 θ r ), and fix a θ r -definable set A ⊆ G. For any ǫ > 0 there are: • a θ r -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 / G 00 θ r is profinite : • θ ( x ; ¯ y ) is stable. For ǫ small enough, we obtain Z = ∅ . • G has finite exponent (any compact torsion group is profinite). 16 / 21
A counterexample Example : (language of groups with unary predicate A ) • Let M p = ( Z / p Z , + , A p ) where p is prime and A p = { 0 , . . . , p − 1 2 } , • T = Th ( � U M p ) 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. 17 / 21
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