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 Review of Generic structures 1 Prehistory Ab-initio Dimension and independence Review of automorphism group of a countable structure 2 Automorphism group as a Polish Group Bounded automorphisms Topological simplicity 3 Existence of Bounded automorphisms Topological Simplicity Zaniar Ghadernezhad British PG Model Theory Conference - Leeds
Review of Generic structures Prehistory Review of automorphism group of a countable structure Ab-initio Topological simplicity Dimension and independence Review of Generic structures 1 Prehistory Ab-initio Dimension and independence Review of automorphism group of a countable structure 2 Automorphism group as a Polish Group Bounded automorphisms Topological simplicity 3 Existence of Bounded automorphisms Topological Simplicity Zaniar Ghadernezhad British PG Model Theory Conference - Leeds
Review of Generic structures Prehistory Review of automorphism group of a countable structure Ab-initio Topological simplicity Dimension and independence Fra¨ ıss´ e-Limit Theorem Let L be a countable first order language and let K be non-empty finite or 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 Prehistory Review of automorphism group of a countable structure Ab-initio Topological simplicity Dimension and independence Smooth Class Definition A class ( K , � ) of finite L -structures, together with a relation � on K × K , is called smooth if: � is reflexive. 1 � is transitive. 2 B � C implies B ⊆ C . 3 if A � C , B ⊆ C and A ⊂ B then A � B . 4 ∅ � A for all A ∈ K . 5 Zaniar Ghadernezhad British PG Model Theory Conference - Leeds
Review of Generic structures Prehistory Review of automorphism group of a countable structure Ab-initio Topological simplicity 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: If A ′ � M , A ′ � B ∈ K , then there exists B ′ � M such that 1 B ∼ = A ′ B ′ . M is a union of < A i : i ∈ ω > such that A i � A i +1 . 2 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 Prehistory Review of automorphism group of a countable structure Ab-initio Topological simplicity Dimension and independence � -closure and finite closure property Definition Let A ⊂ M then cl M ( 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 cl M and acl in generic model are equal. Zaniar Ghadernezhad British PG Model Theory Conference - Leeds
Review of Generic structures Prehistory Review of automorphism group of a countable structure Ab-initio Topological simplicity Dimension and independence Ab Initio Predimension Function Let L = { R } and 0 < α ≤ 1. Let K α denotes the class of all L -structures A such that for all substructures A ′ of A δ α ( A ′ ) = | A ′ | − α | R ( A ′ ) | ≥ 0 . Smoothness notion Define � -notion in K α , A � B if and only if δ α ( B ′ / A ) ≥ 0 for all A ⊆ B ′ ⊆ B . Which δ α ( B / A ) = δ α ( AB ) − δ α ( A ). Zaniar Ghadernezhad British PG Model Theory Conference - Leeds
Review of Generic structures Prehistory Review of automorphism group of a countable structure Ab-initio Topological simplicity Dimension and independence Ab Initio Fact ( K α , � ) is smooth class. ( K α , � ) has free-AP. M the ( K α , � )-generic is saturated. M is stable. Zaniar Ghadernezhad British PG Model Theory Conference - Leeds
Review of Generic structures Prehistory Review of automorphism group of a countable structure Ab-initio Topological simplicity Dimension and independence Dimension Function For δ define d M ( 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 Prehistory Review of automorphism group of a countable structure Ab-initio Topological simplicity Dimension and independence d-independence Suppose d is a dimension function. Define d B | C ⌣ A if cl ( AB ) ∩ cl ( AC ) = cl ( A ) and d ( B / AC ) = d ( B / A ). Fact Let M be ( K α , � )-generic model, then d ⌣ ≡ | | ⌣ Zaniar Ghadernezhad British PG Model Theory Conference - Leeds
Review of Generic structures Automorphism group as a Polish Group Review of automorphism group of a countable structure Bounded automorphisms Topological simplicity Review of Generic structures 1 Prehistory Ab-initio Dimension and independence Review of automorphism group of a countable structure 2 Automorphism group as a Polish Group Bounded automorphisms Topological simplicity 3 Existence of Bounded automorphisms Topological Simplicity Zaniar Ghadernezhad British PG Model Theory Conference - Leeds
Review of Generic structures Automorphism group as a Polish Group Review of automorphism group of a countable structure Bounded automorphisms Topological simplicity 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 Automorphism group as a Polish Group Review of automorphism group of a countable structure Bounded automorphisms Topological simplicity Defining a topology on automorphism group Let M be countable structure and G = Aut ( M ). Then consider G A = { g ∈ G : a g = a for all a ∈ A } . The G A ’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 Automorphism group as a Polish Group Review of automorphism group of a countable structure Bounded automorphisms Topological simplicity 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 , m β ∈ 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 B 0 ⊆ B and β 0 ∈ Sym ( B 0 ) then there is an automorphism extending β 0 and fixing acl ( B \ B 0 ). β is bounded. Zaniar Ghadernezhad British PG Model Theory Conference - Leeds
Review of Generic structures Existence of Bounded automorphisms Review of automorphism group of a countable structure Topological Simplicity Topological simplicity Review of Generic structures 1 Prehistory Ab-initio Dimension and independence Review of automorphism group of a countable structure 2 Automorphism group as a Polish Group Bounded automorphisms Topological simplicity 3 Existence of Bounded automorphisms Topological Simplicity Zaniar Ghadernezhad British PG Model Theory Conference - Leeds
Review of Generic structures Existence of Bounded automorphisms Review of automorphism group of a countable structure Topological Simplicity Topological simplicity Setting Let L = { R } and δ α , where α is of the form 1 / m . K α = { A : L − structure with δ α ( A ′ ) ≥ 0 for every A ′ ⊆ A } . Let M be ( K α , � )-generic and G = Aut ( M ). Theorem G is toplogically simple i.e. there is no non-trivial closed normal subgroup of G . Zaniar Ghadernezhad British PG Model Theory Conference - Leeds
Review of Generic structures Existence of Bounded automorphisms Review of automorphism group of a countable structure Topological Simplicity Topological simplicity If { id } � = N ⊳ G then N is dense. N and G have same orbit on M n . If A ⊆ M then N A 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 ∈ N A such that b | ⌣ A b g . Zaniar Ghadernezhad British PG Model Theory Conference - Leeds
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