Topological simplicity of automorphism groups of some generic - - PowerPoint PPT Presentation

Review of Generic structures Review of automorphism group of a countable structure Topological simplicity

Topological simplicity of automorphism groups of some generic ab-initio structures

Zaniar Ghadernezhad

Institut f¨ ur Mathematische Logik und Grundlagenforschung WWU-M¨ unster

19 Jan, 2011

Zaniar Ghadernezhad British PG Model Theory Conference - Leeds

Review of Generic structures Review of automorphism group of a countable structure Topological simplicity

Table of contents

1

Review of Generic structures Prehistory Ab-initio Dimension and independence

2

Review of automorphism group of a countable structure Automorphism group as a Polish Group Bounded automorphisms

3

Topological simplicity Existence of Bounded automorphisms Topological Simplicity

Zaniar Ghadernezhad British PG Model Theory Conference - Leeds

Review of Generic structures Review of automorphism group of a countable structure Topological simplicity Prehistory Ab-initio Dimension and independence

1

Review of Generic structures Prehistory Ab-initio Dimension and independence

2

Review of automorphism group of a countable structure Automorphism group as a Polish Group Bounded automorphisms

3

Topological simplicity Existence of Bounded automorphisms Topological Simplicity

Zaniar Ghadernezhad British PG Model Theory Conference - Leeds

Review of Generic structures Review of automorphism group of a countable structure Topological simplicity Prehistory Ab-initio Dimension and independence

Fra¨ ıss´ e-Limit

Theorem Let L be a countable first order language and let K be non-empty finite

• r countable set of finitely generated L-structure which has HP, JEP and
• AP. Then there is an L-structure D, unique up to isomorphism, such that

D has cardinality ≤ ω K is the age of D, and D is ultrahomogeneous.

Zaniar Ghadernezhad British PG Model Theory Conference - Leeds

Review of Generic structures Review of automorphism group of a countable structure Topological simplicity Prehistory Ab-initio Dimension and independence

Smooth Class

Definition A class (K, ) of finite L-structures, together with a relation on K × K, is called smooth if:

1

is reflexive.

2

is transitive.

3

B C implies B ⊆ C.

4

if A C, B ⊆ C and A ⊂ B then A B.

5

∅ A for all A ∈ K.

Zaniar Ghadernezhad British PG Model Theory Conference - Leeds

Review of Generic structures Review of automorphism group of a countable structure Topological simplicity Prehistory Ab-initio Dimension and independence

Smooth Class

Let L be finite relational language. Definition - Genericity Suppose that (K, ) is a smooth class of finite L-structures. A countable structure M is (K, )-generic if:

1

If A′ M , A′ B ∈ K, then there exists B′ M such that B ∼ =A′ B′.

2

M is a union of < Ai : i ∈ ω > such that Ai Ai+1. Theorem A smooth class (K, ) has AP if and only if there is a unique (K, )-generic structure.

Zaniar Ghadernezhad British PG Model Theory Conference - Leeds

Review of Generic structures Review of automorphism group of a countable structure Topological simplicity Prehistory Ab-initio Dimension and independence

• closure and finite closure property

Definition Let A ⊂ M then clM(A) is the smallest B ⊇ A such that B M. A structure M has finite closure property (FCP) with respect to smooth class (K, ) if for every finite A′ ⊆ M there is a finite B M in K with A′ ⊆ B. Theorem If a smooth class (K, ) has free-AP then clM and acl in generic model are equal.

Zaniar Ghadernezhad British PG Model Theory Conference - Leeds

Review of Generic structures Review of automorphism group of a countable structure Topological simplicity Prehistory Ab-initio Dimension and independence

Ab Initio

Predimension Function Let L = {R} and 0 < α ≤ 1. Let Kα denotes the class of all L-structures A such that for all substructures A′ of A δα(A′) =| A′ | −α|R(A′)| ≥ 0 . Smoothness notion Define -notion in Kα, A B if and only if δα(B′/A) ≥ 0 for all A ⊆ B′ ⊆ B. Which δα(B/A) = δα(AB) − δα(A).

Zaniar Ghadernezhad British PG Model Theory Conference - Leeds

Review of Generic structures Review of automorphism group of a countable structure Topological simplicity Prehistory Ab-initio Dimension and independence

Ab Initio

Fact (Kα, ) is smooth class. (Kα, ) has free-AP. M the (Kα, )-generic is saturated. M is stable.

Zaniar Ghadernezhad British PG Model Theory Conference - Leeds

Review of Generic structures Review of automorphism group of a countable structure Topological simplicity Prehistory Ab-initio Dimension and independence

Dimension Function

For δ define dM(X) = d(X, M) = min{δ(X ′) : X ⊆ X ′ ⊆f M} The function d(., M) is a dimension function. Definition We call d : {X : X ⊆f M} → N a dimension function if satisfies the following: d(X) ≤| X |. d(XY ) + d(X ∩ Y ) ≤ d(X) + d(Y ). X ⊆ Y ⇒ d(X) ≤ d(Y ).

Zaniar Ghadernezhad British PG Model Theory Conference - Leeds

Review of Generic structures Review of automorphism group of a countable structure Topological simplicity Prehistory Ab-initio Dimension and independence

d-independence

Suppose d is a dimension function. Define B

d

| ⌣

A

C if cl(AB) ∩ cl(AC) = cl(A) and d(B/AC) = d(B/A). Fact Let M be (Kα, )-generic model, then | ⌣ ≡

d

| ⌣

Zaniar Ghadernezhad British PG Model Theory Conference - Leeds

Review of Generic structures Review of automorphism group of a countable structure Topological simplicity Automorphism group as a Polish Group Bounded automorphisms

1

Review of Generic structures Prehistory Ab-initio Dimension and independence

2

Review of automorphism group of a countable structure Automorphism group as a Polish Group Bounded automorphisms

3

Topological simplicity Existence of Bounded automorphisms Topological Simplicity

Zaniar Ghadernezhad British PG Model Theory Conference - Leeds

Review of Generic structures Review of automorphism group of a countable structure Topological simplicity Automorphism group as a Polish Group Bounded automorphisms

Automorphism group

Definition A topological group G, is group with topology on G such that the group operation and the inverse function are continuous. Definition A Polish space is a separable completely metrizable topological space : a space homeomorphic to a complete metric space with a countable dense subset. A Polish group is a topological group which the topological space is Polish.

Zaniar Ghadernezhad British PG Model Theory Conference - Leeds

Review of Generic structures Review of automorphism group of a countable structure Topological simplicity Automorphism group as a Polish Group Bounded automorphisms

Defining a topology on automorphism group

Let M be countable structure and G = Aut(M). Then consider GA = {g ∈ G : ag = a for all a ∈ A}. The GA’s, for A ⊆f M, form a neighbourhood basis for the identity element for a topology on G . Fact G is a Polish group.

Zaniar Ghadernezhad British PG Model Theory Conference - Leeds

Review of Generic structures Review of automorphism group of a countable structure Topological simplicity Automorphism group as a Polish Group Bounded automorphisms

Bounded automorphisms

Definition β ∈ Aut(M) is a bounded automorphism if there exists algebraically closed finite subset A ⊆ M such that for every m ∈ M, mβ ∈ acl(mA). Example In a strongly minimal strucuters, acl is pregeometry and then Bdd form a normal subgroup. Suppose V is infinite-dimensional vector space. Consider the base B for V . Let B0 ⊆ B and β0 ∈ Sym(B0) then there is an automorphism extending β0 and fixing acl(B\B0). β is bounded.

Zaniar Ghadernezhad British PG Model Theory Conference - Leeds

Review of Generic structures Review of automorphism group of a countable structure Topological simplicity Existence of Bounded automorphisms Topological Simplicity

1

Review of Generic structures Prehistory Ab-initio Dimension and independence

2

Review of automorphism group of a countable structure Automorphism group as a Polish Group Bounded automorphisms

3

Topological simplicity Existence of Bounded automorphisms Topological Simplicity

Zaniar Ghadernezhad British PG Model Theory Conference - Leeds

Review of Generic structures Review of automorphism group of a countable structure Topological simplicity Existence of Bounded automorphisms Topological Simplicity

Setting

Let L = {R} and δα, where α is of the form 1/m. Kα = {A : L − structure with δα(A′) ≥ 0 for every A′ ⊆ A}. Let M be (Kα, )-generic and G = Aut(M). Theorem G is toplogically simple i.e. there is no non-trivial closed normal subgroup

• f G.

Zaniar Ghadernezhad British PG Model Theory Conference - Leeds

Review of Generic structures Review of automorphism group of a countable structure Topological simplicity Existence of Bounded automorphisms Topological Simplicity

If {id} = N ⊳ G then N is dense. N and G have same orbit on Mn. If A ⊆ M then NA is transitive on realization of type p over A (algebraic and non-algebraic). There is no bounded automorphism. Suppose A is closed set. If b | = p, then there exists g ∈ NA such that b | ⌣A bg.

Zaniar Ghadernezhad British PG Model Theory Conference - Leeds

Review of Generic structures Review of automorphism group of a countable structure Topological simplicity Existence of Bounded automorphisms Topological Simplicity

Suppose β ∈ Bdd, so there exists finite closed A such that mβ ∈ acl(mA). Fact If b is a point and δα(b/A) = 0, then b is fixed by β. Theorem There is no no-trivial bounded automorphism in G.

Zaniar Ghadernezhad British PG Model Theory Conference - Leeds

Review of Generic structures Review of automorphism group of a countable structure Topological simplicity Existence of Bounded automorphisms Topological Simplicity

Main Idea

If b | = p, then there exists g ∈ NA such that b | ⌣A bg. Idea If g ∈ NA such that cg / ∈ acl(cA) and c | ⌣A cg, then by conjugation of g with some elements GA, we can find f ∈ NA such that d(ccf /A) > d(ccg/A).

Zaniar Ghadernezhad British PG Model Theory Conference - Leeds