Automorphism groups of spaces with many symmetries Aleksandra - PowerPoint PPT Presentation

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications Automorphism groups of spaces with many symmetries Aleksandra Kwiatkowska University of Bonn September 23, 2016 Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications Ultrahomogeneous structures Definition A countable structure M is ultrahomogeneous if every automorphism between finite substructures of M can be extended to an automorphism of the whole M . Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications Ultrahomogeneous structures Definition A countable structure M is ultrahomogeneous if every automorphism between finite substructures of M can be extended to an automorphism of the whole M . Examples: rationals with the ordering, the Rado graph Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications Ultrahomogeneous structures Definition A countable structure M is ultrahomogeneous if every automorphism between finite substructures of M can be extended to an automorphism of the whole M . Examples: rationals with the ordering, the Rado graph How to construct ultrahomogeneous structures? Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications Setup Let F be a family of finite structures (a structure is a set A equipped with relations R A 1 , R A 2 , . . . and functions f A 1 , f A 2 , . . . ). Maps between structures in F are structure preserving monomorphisms. Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications Examples Example 1 F =finite linear orders 2 F =finite graphs 3 F =finite Boolean algebras 4 F =finite metric spaces with rational distances Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications Fra¨ ıss´ e family-definition A countable family F of finite structures is a Fra¨ ıss´ e family if: 1 (F1) (joint embedding property: JEP) for any A , B ∈ F there is C ∈ F and monomorphisms from A into C and from B onto C ; 2 (F2) (amalgamation property: AP) for A , B 1 , B 2 ∈ F and any monomorphisms φ 1 : A → B 1 and φ 2 : A → B 2 , there exist C , φ 3 : B 1 → C and φ 4 : B 2 → C such that φ 3 ◦ φ 1 = φ 4 ◦ φ 2 ; 3 (F3) (hereditary property: HP) if A ∈ F and B ⊆ A , then B ∈ F . Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications Fra¨ ıss´ e limit-definition A countable structure L is a Fra¨ ıss´ e limit of F if the following two conditions hold: 1 (L1) (universality) for any A ∈ F there is an monomorphism from A into L ; 2 (L2) (ultrahomogeneity) for any A ∈ F and any monomorphisms φ 1 : A → L and φ 2 : A → L there exists an isomorphism h : L → L such that φ 2 = h ◦ φ 1 ; Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications Fra¨ ıss´ e limit-existence and uniqueness Theorem (Fra¨ ıss´ e) Let F be a countable Fra¨ ıss´ e family of finite structures. Then: 1 there exists a Fra¨ ıss´ e limit of F ; 2 any two Fra¨ ıss´ e limits are isomorphic. Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications Examples Example 1 If F =finite linear orders, then L =rational numbers with the order 2 If F =finite graphs, then L =random graph 3 If F =finite Boolean algebras, then L =countable atomless Boolean algebra 4 F =finite metric spaces with rational distances, then L =rational Urysohn metric space Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications Lelek fan C – the Cantor set Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications Lelek fan C – the Cantor set continuum - compact and connected metric space Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications Lelek fan C – the Cantor set continuum - compact and connected metric space Cantor fan F is the cone over the Cantor set: C × [0 , 1] / C × { 0 } Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications Lelek fan C – the Cantor set continuum - compact and connected metric space Cantor fan F is the cone over the Cantor set: C × [0 , 1] / C × { 0 } subfan of the Cantor fan - subcontinuum of the Cantor fan that contains the top point, and is not homeomorphic to [0,1] or to a point Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications Lelek fan C – the Cantor set continuum - compact and connected metric space Cantor fan F is the cone over the Cantor set: C × [0 , 1] / C × { 0 } subfan of the Cantor fan - subcontinuum of the Cantor fan that contains the top point, and is not homeomorphic to [0,1] or to a point Lelek fan L is a subfan of the Cantor fan with a dense set of endpoints in L Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications Lelek fan Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications About the Lelek fan Lelek fan was constructed by Lelek in 1960 Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications About the Lelek fan Lelek fan was constructed by Lelek in 1960 Lelek fan is unique: Any two subfans of the Cantor fan with dense set of endpoints are homeomorphic (Bula-Oversteegen 1990 and Charatonik 1989) Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications Endpoints of the Lelek fan The set of endpoints of the Lelek fan L is a dense G δ set in L , it is a 1-dimensional space. Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications Endpoints of the Lelek fan The set of endpoints of the Lelek fan L is a dense G δ set in L , it is a 1-dimensional space. It is homeomorphic to: the complete Erd˝ os space, the set of endpoints of the Julia set of the exponential map, the set of endpoints of the separable universal R -tree. (Kawamura, Oversteegen, Tymchatyn) Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications The pseudo-arc Definition The pseudo-arc is the unique hereditarily indecomposable chainable continuum. Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications The pseudo-arc Definition The pseudo-arc is the unique hereditarily indecomposable chainable continuum. continuum = compact and connected metric space; Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications The pseudo-arc Definition The pseudo-arc is the unique hereditarily indecomposable chainable continuum. continuum = compact and connected metric space; indecomposable = not a union of two proper subcontinua; Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries