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Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Automorphism groups of spaces with manysymmetries
Aleksandra Kwiatkowska
University of Bonn
September 23, 2016
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Ultrahomogeneous structures
Definition
A countable structure M is ultrahomogeneous if everyautomorphism between finite substructures of M can be extendedto 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 structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Ultrahomogeneous structures
Definition
A countable structure M is ultrahomogeneous if everyautomorphism between finite substructures of M can be extendedto 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 structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Ultrahomogeneous structures
Definition
A countable structure M is ultrahomogeneous if everyautomorphism between finite substructures of M can be extendedto 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 structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Setup
Let F be a family of finite structures(a structure is a set A equipped with relations RA
1 ,RA2 , . . . and
functions f A1 , fA
2 , . . .).
Maps between structures in F are structure preservingmonomorphisms.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
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 structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Fraısse family-definition
A countable family F of finite structures is a Fraısse family if:
1 (F1) (joint embedding property: JEP) for any A,B ∈ F thereis C ∈ F and monomorphisms from A into C and from Bonto C ;
2 (F2) (amalgamation property: AP) for A,B1,B2 ∈ F and anymonomorphisms φ1 : A→ B1 and φ2 : A→ B2, there exist C ,φ3 : B1 → C and φ4 : B2 → C such that φ3 ◦ φ1 = φ4 ◦ φ2;
3 (F3) (hereditary property: HP) if A ∈ F and B ⊆ A, thenB ∈ F .
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Fraısse limit-definition
A countable structure L is a Fraısse limit of F if the followingtwo conditions hold:
1 (L1) (universality) for any A ∈ F there is an monomorphismfrom A into L;
2 (L2) (ultrahomogeneity) for any A ∈ F and anymonomorphisms φ1 : A→ L and φ2 : A→ L there exists anisomorphism h : L→ L such that φ2 = h ◦ φ1;
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Fraısse limit-existence and uniqueness
Theorem (Fraısse)
Let F be a countable Fraısse family of finite structures. Then:
1 there exists a Fraısse limit of F ;
2 any two Fraısse limits are isomorphic.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Examples
Example
1 If F=finite linear orders, then L=rational numbers with theorder
2 If F=finite graphs, then L=random graph
3 If F=finite Boolean algebras, then L=countable atomlessBoolean algebra
4 F=finite metric spaces with rational distances, thenL=rational Urysohn metric space
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
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 fanthat 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 ofendpoints in L
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
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 fanthat 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 ofendpoints in L
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
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 fanthat 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 ofendpoints in L
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
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 fanthat 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 ofendpoints in L
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
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 fanthat 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 ofendpoints in L
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Lelek fan
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
About the Lelek fan
Lelek fan was constructed by Lelek in 1960
Lelek fan is unique: Any two subfans of the Cantor fan withdense set of endpoints are homeomorphic (Bula-Oversteegen1990 and Charatonik 1989)
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
About the Lelek fan
Lelek fan was constructed by Lelek in 1960
Lelek fan is unique: Any two subfans of the Cantor fan withdense set of endpoints are homeomorphic (Bula-Oversteegen1990 and Charatonik 1989)
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Endpoints of the Lelek fan
The set of endpoints of the Lelek fan L is a dense Gδ set in L,it is a 1-dimensional space.
It is homeomorphic to: the complete Erdos space, the set ofendpoints of the Julia set of the exponential map, the set ofendpoints of the separable universal R-tree. (Kawamura,Oversteegen, Tymchatyn)
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Endpoints of the Lelek fan
The set of endpoints of the Lelek fan L is a dense Gδ set in L,it is a 1-dimensional space.
It is homeomorphic to: the complete Erdos space, the set ofendpoints of the Julia set of the exponential map, the set ofendpoints of the separable universal R-tree. (Kawamura,Oversteegen, Tymchatyn)
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
The pseudo-arc
Definition
The pseudo-arc is the unique hereditarily indecomposable chainablecontinuum.
continuum = compact and connected metric space;
indecomposable = not a union of two proper subcontinua;
chainable = each open cover is refined by an open coverU1,U2, . . . ,Un such that for i , j , Ui ∩ Uj 6= ∅ if and only if|j − i | ≤ 1
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
The pseudo-arc
Definition
The pseudo-arc is the unique hereditarily indecomposable chainablecontinuum.
continuum = compact and connected metric space;
indecomposable = not a union of two proper subcontinua;
chainable = each open cover is refined by an open coverU1,U2, . . . ,Un such that for i , j , Ui ∩ Uj 6= ∅ if and only if|j − i | ≤ 1
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
The pseudo-arc
Definition
The pseudo-arc is the unique hereditarily indecomposable chainablecontinuum.
continuum = compact and connected metric space;
indecomposable = not a union of two proper subcontinua;
chainable = each open cover is refined by an open coverU1,U2, . . . ,Un such that for i , j , Ui ∩ Uj 6= ∅ if and only if|j − i | ≤ 1
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
The pseudo-arc
Definition
The pseudo-arc is the unique hereditarily indecomposable chainablecontinuum.
continuum = compact and connected metric space;
indecomposable = not a union of two proper subcontinua;
chainable = each open cover is refined by an open coverU1,U2, . . . ,Un such that for i , j , Ui ∩ Uj 6= ∅ if and only if|j − i | ≤ 1
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
A few properties of the pseudo-arc
Theorem (Bing)
The pseudo-arc is unique up to homeomorphism.
Theorem (Bing)
In the space of all subcontinua of either [0, 1]n, n > 1, or theHilbert space, equipped with the Hausdorff metric, homeomorphiccopies of the pseudo-arc form a dense Gδ set.
Theorem (Bing, Moise)
The pseudo-arc is homogeneous.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
A few properties of the pseudo-arc
Theorem (Bing)
The pseudo-arc is unique up to homeomorphism.
Theorem (Bing)
In the space of all subcontinua of either [0, 1]n, n > 1, or theHilbert space, equipped with the Hausdorff metric, homeomorphiccopies of the pseudo-arc form a dense Gδ set.
Theorem (Bing, Moise)
The pseudo-arc is homogeneous.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
A few properties of the pseudo-arc
Theorem (Bing)
The pseudo-arc is unique up to homeomorphism.
Theorem (Bing)
In the space of all subcontinua of either [0, 1]n, n > 1, or theHilbert space, equipped with the Hausdorff metric, homeomorphiccopies of the pseudo-arc form a dense Gδ set.
Theorem (Bing, Moise)
The pseudo-arc is homogeneous.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Projective Fraısse theory – setup
1 Let L = {Ri}i∈I ∪ {fj}j∈J be a language.
2 A topological L-structure is a compact zero-dimensionalsecond-countable space A equipped with closed relationsRAi , i ∈ I and continuous functions f Aj , j ∈ J.
3 Epimorphisms are continuous surjections preserving thestructure.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Projective Fraısse theory – setup
1 Let L = {Ri}i∈I ∪ {fj}j∈J be a language.
2 A topological L-structure is a compact zero-dimensionalsecond-countable space A equipped with closed relationsRAi , i ∈ I and continuous functions f Aj , j ∈ J.
3 Epimorphisms are continuous surjections preserving thestructure.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Projective Fraısse theory – setup
1 Let L = {Ri}i∈I ∪ {fj}j∈J be a language.
2 A topological L-structure is a compact zero-dimensionalsecond-countable space A equipped with closed relationsRAi , i ∈ I and continuous functions f Aj , j ∈ J.
3 Epimorphisms are continuous surjections preserving thestructure.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Projective Fraısse family – definition
A family F of finite topological L-structure is a projective Fraıssefamily if:
1 (F1) (joint projection property: JPP) for any A,B ∈ F thereis C ∈ F and epimorphisms from C onto A and from C ontoB;
2 (F2) (amalgamation property: AP) for A,B1,B2 ∈ F and anyepimorphisms φ1 : B1 → A and φ2 : B2 → A, there exist C ,φ3 : C → B1 and φ4 : C → B2 such that φ1 ◦ φ3 = φ2 ◦ φ4.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
B1
φ3
C
B2
φ4
A
φ1 φ2
amalgamation propertyAleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Projective Fraısse limit – definition
A topological L-structure L is a projective Fraısse limit of F ifthe following three conditions hold:
1 (L1) (projective universality) for any A ∈ F there is anepimorphism from L onto A;
2 (L2) (projective ultrahomogeneity) for any A ∈ F and anyepimorphisms φ1 : L→ A and φ2 : L→ A there exists anisomorphism h : L→ L such that φ2 = φ1 ◦ h;
3 (L3) for any finite discrete topological space X and anycontinuous function f : L→ X there is an A ∈ F , anepimorphism φ : L→ A, and a function f0 : A→ X such thatf = f0 ◦ φ.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Projective Fraısse limit – existence and uniqueness
Theorem (Irwin-Solecki)
Let F be a countable projective Fraısse family of finite structures.Then:
1 there exists a projective Fraısse limit of F ;
2 any two projective Fraısse limits are isomorphic.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Example
Let F be the family of all finite sets.
The projective Fraısse limit is the Cantor set.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Example
Let F be the family of all finite sets.
The projective Fraısse limit is the Cantor set.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Pseudo-arc from a projective Fraısse limit, part 1
Let r be a binary relation symbol. Let G be the family of all finitelinear reflexive graphs.
Theorem (Irwin-Solecki)
G is a projective Fraısse family.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Pseudo-arc from a projective Fraısse limit, part 1
Let r be a binary relation symbol. Let G be the family of all finitelinear reflexive graphs.
Theorem (Irwin-Solecki)
G is a projective Fraısse family.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Pseudo-arc from a projective Fraısse limit, part 1
Let r be a binary relation symbol. Let G be the family of all finitelinear reflexive graphs.
Theorem (Irwin-Solecki)
G is a projective Fraısse family.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Epimorphisms
A continuous surjection φ : S → T is an epimorphism iff
rT (a, b)
⇐⇒ ∃c , d ∈ S(φ(c) = a, φ(d) = b, and rS(c , d)
).
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
An example of an epimorphism
Sb a b a b c b b
Ta b c
φ
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Pseudo-arc from a projective Fraısse limit, part 2
Lemma (Irwin-Solecki)
Let P be the projective Fraısse limit of G. Then rP is anequivalence relation such that each equivalence class has at mosttwo elements.
Theorem (Irwin-Solecki)
P/rP is the pseudo-arc.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Pseudo-arc from a projective Fraısse limit, part 2
Lemma (Irwin-Solecki)
Let P be the projective Fraısse limit of G. Then rP is anequivalence relation such that each equivalence class has at mosttwo elements.
Theorem (Irwin-Solecki)
P/rP is the pseudo-arc.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Lelek fan from a projective Fraısse limit, part 1
Let R be a binary relation symbol. Let F be the family of all finitereflexive fans.
Theorem
F is a projective Fraısse family.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Lelek fan from a projective Fraısse limit, part 1
Let R be a binary relation symbol. Let F be the family of all finitereflexive fans.
Theorem
F is a projective Fraısse family.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
An example of an epimorphism
xy
xx
rx
b
b
a
r
rx
r
r
yb
xa φ
ST
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Lelek fan from a projective Fraısse limit, part 2
Lemma
Let L be the projective Fraısse limit of F . Then RLS , where
RLS (x , y) iff RL(x , y) or RL(y , x), is an equivalence relation such
that each equivalence class has at most two elements.
Theorem
L/RLS is the Lelek fan.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Lelek fan from a projective Fraısse limit, part 2
Lemma
Let L be the projective Fraısse limit of F . Then RLS , where
RLS (x , y) iff RL(x , y) or RL(y , x), is an equivalence relation such
that each equivalence class has at most two elements.
Theorem
L/RLS is the Lelek fan.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Non-triviality of H(L)
Remark
The group H(L) is non-trivial, that is, there is f ∈ H(L) such thatf 6= Id.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Aut(L) as a subgroup of H(L)
Each automorphism h ∈ Aut(L) can be identified in a naturalway with a homeomorphism h∗ ∈ H(L).
Aut(L) is equipped with the compact-open topology.
The topology on Aut(L) is finer than the compact-opentopology on H(L).
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Aut(L) as a subgroup of H(L)
Each automorphism h ∈ Aut(L) can be identified in a naturalway with a homeomorphism h∗ ∈ H(L).
Aut(L) is equipped with the compact-open topology.
The topology on Aut(L) is finer than the compact-opentopology on H(L).
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Aut(L) as a subgroup of H(L)
Each automorphism h ∈ Aut(L) can be identified in a naturalway with a homeomorphism h∗ ∈ H(L).
Aut(L) is equipped with the compact-open topology.
The topology on Aut(L) is finer than the compact-opentopology on H(L).
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Projective universality and Projective Ultrahomogeneity
smooth fan = subfan of the Cantor fan
Theorem
1 Each smooth fan is a continuous image of the Lelek fan L viaa map that takes the root to the root and is monotone onsegments.
2 Let X be a smooth fan with a metric d . If f1, f2 : L→ X aretwo continuous surjections that take the root to the root andare monotone on segments, then for any ε > 0 there existsh ∈ Aut(L) such that for all x ∈ L, d(f1(x), f2 ◦ h∗(x)) < ε.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Projective universality and Projective Ultrahomogeneity
smooth fan = subfan of the Cantor fan
Theorem
1 Each smooth fan is a continuous image of the Lelek fan L viaa map that takes the root to the root and is monotone onsegments.
2 Let X be a smooth fan with a metric d . If f1, f2 : L→ X aretwo continuous surjections that take the root to the root andare monotone on segments, then for any ε > 0 there existsh ∈ Aut(L) such that for all x ∈ L, d(f1(x), f2 ◦ h∗(x)) < ε.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Corollary
Corollary
The group Aut(L) is dense in H(L).
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Homeomorphism group of the Lelek fan–totallydisconnected
A topological space X is totally disconnected if for any x , y ∈ Xthere is a clopen set C ⊆ X such that x ∈ C and y ∈ (X \ C ).
Proposition
The group H(L) is totally disconnected.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Homeomorphism group of the Lelek fan–totallydisconnected
A topological space X is totally disconnected if for any x , y ∈ Xthere is a clopen set C ⊆ X such that x ∈ C and y ∈ (X \ C ).
Proposition
The group H(L) is totally disconnected.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Homeomorphism group of the Lelek fan–‘locally generated’
A homeomorphism h ∈ H(L) is called an ε-homeomorphism ifdsup(h, Id) < ε.
Theorem
For every ε > 0 and h ∈ H(L) there are ε-homeomorphismsh1, . . . , hn ∈ H(L) such that h = h1 ◦ . . . ◦ hn.Moreover, if h ∈ Aut(L), then we can choose required h1, . . . , hnin Aut(L).
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Homeomorphism group of the Lelek fan–‘locally generated’
A homeomorphism h ∈ H(L) is called an ε-homeomorphism ifdsup(h, Id) < ε.
Theorem
For every ε > 0 and h ∈ H(L) there are ε-homeomorphismsh1, . . . , hn ∈ H(L) such that h = h1 ◦ . . . ◦ hn.
Moreover, if h ∈ Aut(L), then we can choose required h1, . . . , hnin Aut(L).
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Homeomorphism group of the Lelek fan–‘locally generated’
A homeomorphism h ∈ H(L) is called an ε-homeomorphism ifdsup(h, Id) < ε.
Theorem
For every ε > 0 and h ∈ H(L) there are ε-homeomorphismsh1, . . . , hn ∈ H(L) such that h = h1 ◦ . . . ◦ hn.Moreover, if h ∈ Aut(L), then we can choose required h1, . . . , hnin Aut(L).
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
H(L) is a ‘large’ group
Corollary
The group H(L) is not locally compact.
To show the corollary above we needed:
Theorem (van Dantzig)
A totally disconnected locally compact group admits a basis at theidentity that consists of compact open subgroups.
Corollary
The group H(L) is not a non-archimedean group.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
H(L) is a ‘large’ group
Corollary
The group H(L) is not locally compact.
To show the corollary above we needed:
Theorem (van Dantzig)
A totally disconnected locally compact group admits a basis at theidentity that consists of compact open subgroups.
Corollary
The group H(L) is not a non-archimedean group.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
H(L) is a ‘large’ group
Corollary
The group H(L) is not locally compact.
To show the corollary above we needed:
Theorem (van Dantzig)
A totally disconnected locally compact group admits a basis at theidentity that consists of compact open subgroups.
Corollary
The group H(L) is not a non-archimedean group.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Conjugacy classes of H(L)
Theorem
The group of all homeomorphisms of the Lelek fan, H(L), has adense conjugacy class, i.e. there is g ∈ H(L) such that{hgh−1 : h ∈ H(L)} is dense.
Theorem
The group of all automorphisms of L, Aut(L), has a denseconjugacy class.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
Conjugacy classes of H(L)
Theorem
The group of all homeomorphisms of the Lelek fan, H(L), has adense conjugacy class, i.e. there is g ∈ H(L) such that{hgh−1 : h ∈ H(L)} is dense.
Theorem
The group of all automorphisms of L, Aut(L), has a denseconjugacy class.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
H(L) is simple
Recall that a group is simple if it has no proper normal subgroups.
Theorem
The group of all homeomorphisms of the Lelek fan, H(L), is simple.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
Ultrahomogeneous structuresThe pseudo-arc and the Lelek fan
Projective Fraısse theoryApplications
H(L) is simple
Recall that a group is simple if it has no proper normal subgroups.
Theorem
The group of all homeomorphisms of the Lelek fan, H(L), is simple.
Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries
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