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 RA

1 , RA 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

• nto C;

2 (F2) (amalgamation property: AP) for A, B1, B2 ∈ F and any

monomorphisms 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, 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 φ1 : A → L and φ2 : A → L there exists an isomorphism h: L → L such that φ2 = h ◦ φ1;

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

• rder

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]

• r 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]

• r 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 Gδ 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 Gδ set in L, it is a 1-dimensional space. It is homeomorphic to: the complete Erd˝

• s 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

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; chainable = each open cover is refined by an open cover U1, U2, . . . , Un such that for i, j, Ui ∩ Uj = ∅ if and only if |j − i| ≤ 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

A few properties of the pseudo-arc

Theorem (Bing) The pseudo-arc is unique up to homeomorphism.

Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications

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 the Hilbert space, equipped with the Hausdorff metric, homeomorphic copies of the pseudo-arc form a dense Gδ 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

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 the Hilbert space, equipped with the Hausdorff metric, homeomorphic copies 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 structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications

Projective Fra¨ ıss´ e theory – setup

1 Let L = {Ri}i∈I ∪ {fj}j∈J be a language. Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications

Projective Fra¨ ıss´ e theory – setup

1 Let L = {Ri}i∈I ∪ {fj}j∈J be a language. 2 A topological L-structure is a compact zero-dimensional

second-countable space A equipped with closed relations RA

i , i ∈ I and continuous functions f A j , j ∈ J.

Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications

Projective Fra¨ ıss´ e theory – setup

1 Let L = {Ri}i∈I ∪ {fj}j∈J be a language. 2 A topological L-structure is a compact zero-dimensional

second-countable space A equipped with closed relations RA

i , i ∈ I and continuous functions f A j , j ∈ J.

3 Epimorphisms are continuous surjections preserving the

structure.

Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications

Projective Fra¨ ıss´ e family – definition

A family F of finite topological L-structure is a projective Fra¨ ıss´ e family if:

1 (F1) (joint projection property: JPP) for any A, B ∈ F there

is C ∈ F and epimorphisms from C onto A and from C onto B;

2 (F2) (amalgamation property: AP) for A, B1, B2 ∈ F and any

epimorphisms 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 structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications

B1 φ3 C B2 φ4 A φ1 φ2 amalgamation property

Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications

Projective Fra¨ ıss´ e limit – definition

A topological L-structure L is a projective Fra¨ ıss´ e limit of F if the following three conditions hold:

1 (L1) (projective universality) for any A ∈ F there is an

epimorphism from L onto A;

2 (L2) (projective ultrahomogeneity) for any A ∈ F and any

epimorphisms φ1 : L → A and φ2 : L → A there exists an isomorphism h: L → L such that φ2 = φ1 ◦ h;

3 (L3) for any finite discrete topological space X and any

continuous function f : L → X there is an A ∈ F, an epimorphism φ: L → A, and a function f0 : A → X such that f = f0 ◦ φ.

Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications

Projective Fra¨ ıss´ e limit – existence and uniqueness

Theorem (Irwin-Solecki) Let F be a countable projective Fra¨ ıss´ e family of finite structures. Then:

1 there exists a projective Fra¨

ıss´ e limit of F;

2 any two projective 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

Example

Let F be the family of all finite sets.

Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications

Example

Let F be the family of all finite sets. The projective Fra¨ ıss´ e limit is 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

Pseudo-arc from a projective Fra¨ ıss´ e limit, part 1

Let r be a binary relation symbol. Let G be the family of all finite linear reflexive graphs.

Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications

Pseudo-arc from a projective Fra¨ ıss´ e limit, part 1

Let r be a binary relation symbol. Let G be the family of all finite linear reflexive graphs.

Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications

Pseudo-arc from a projective Fra¨ ıss´ e limit, part 1

Let r be a binary relation symbol. Let G be the family of all finite linear reflexive graphs. Theorem (Irwin-Solecki) G is a projective Fra¨ ıss´ e family.

Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications

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 structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications

An example of an epimorphism

S b a b a b c b b T a b c φ

Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications

Pseudo-arc from a projective Fra¨ ıss´ e limit, part 2

Lemma (Irwin-Solecki) Let P be the projective Fra¨ ıss´ e limit of G. Then rP is an equivalence relation such that each equivalence class has at most two elements.

Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications

Pseudo-arc from a projective Fra¨ ıss´ e limit, part 2

Lemma (Irwin-Solecki) Let P be the projective Fra¨ ıss´ e limit of G. Then rP is an equivalence relation such that each equivalence class has at most two elements. Theorem (Irwin-Solecki) P/rP is the pseudo-arc.

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 from a projective Fra¨ ıss´ e limit, part 1

Let R be a binary relation symbol. Let F be the family of all finite reflexive fans.

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 from a projective Fra¨ ıss´ e limit, part 1

Let R be a binary relation symbol. Let F be the family of all finite reflexive fans. Theorem F is a projective Fra¨ ıss´ e family.

Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications

An example of an epimorphism

x y x x r x b b a r r x r r y b x a φ S T

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 from a projective Fra¨ ıss´ e limit, part 2

Lemma Let L be the projective Fra¨ ıss´ e limit of F. Then RL

S , where

RL

S (x, y) iff RL(x, y) or RL(y, x), is an equivalence relation such

that each equivalence class has at most two elements.

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 from a projective Fra¨ ıss´ e limit, part 2

Lemma Let L be the projective Fra¨ ıss´ e limit of F. Then RL

S , where

RL

S (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/RL

S is the 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

Non-triviality of H(L)

Remark The group H(L) is non-trivial, that is, there is f ∈ H(L) such that f = Id.

Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications

Aut(L) as a subgroup of H(L)

Each automorphism h ∈ Aut(L) can be identified in a natural way with a homeomorphism h∗ ∈ H(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

Aut(L) as a subgroup of H(L)

Each automorphism h ∈ Aut(L) can be identified in a natural way with a homeomorphism h∗ ∈ H(L). Aut(L) is equipped with the compact-open topology.

Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications

Aut(L) as a subgroup of H(L)

Each automorphism h ∈ Aut(L) can be identified in a natural way 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-open topology on H(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

Projective universality and Projective Ultrahomogeneity

smooth fan = subfan of the Cantor 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

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 via

a map that takes the root to the root and is monotone on segments.

2 Let X be a smooth fan with a metric d. If f1, f2 : L → X are

two continuous surjections that take the root to the root and are monotone on segments, then for any ǫ > 0 there exists h ∈ Aut(L) such that for all x ∈ L, d(f1(x), f2 ◦ h∗(x)) < ǫ.

Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications

Corollary

Corollary The group Aut(L) is dense in H(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

Homeomorphism group of the Lelek fan–totally disconnected

A topological space X is totally disconnected if for any x, y ∈ X there is a clopen set C ⊆ X such that x ∈ C and y ∈ (X \ C).

Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications

Homeomorphism group of the Lelek fan–totally disconnected

A topological space X is totally disconnected if for any x, y ∈ X there 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 structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications

Homeomorphism group of the Lelek fan–‘locally generated’

A homeomorphism h ∈ H(L) is called an ǫ-homeomorphism if dsup(h, Id) < ǫ.

Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications

Homeomorphism group of the Lelek fan–‘locally generated’

A homeomorphism h ∈ H(L) is called an ǫ-homeomorphism if dsup(h, Id) < ǫ. Theorem For every ǫ > 0 and h ∈ H(L) there are ǫ-homeomorphisms h1, . . . , hn ∈ H(L) such that h = h1 ◦ . . . ◦ hn.

Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications

Homeomorphism group of the Lelek fan–‘locally generated’

A homeomorphism h ∈ H(L) is called an ǫ-homeomorphism if dsup(h, Id) < ǫ. Theorem For every ǫ > 0 and h ∈ H(L) there are ǫ-homeomorphisms h1, . . . , hn ∈ H(L) such that h = h1 ◦ . . . ◦ hn. Moreover, if h ∈ Aut(L), then we can choose required h1, . . . , hn in Aut(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

H(L) is a ‘large’ group

Corollary The group H(L) is not locally compact.

Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications

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 the identity that consists of compact open subgroups.

Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications

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 the identity 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 structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications

Conjugacy classes of H(L)

Theorem The group of all homeomorphisms of the Lelek fan, H(L), has a dense conjugacy class, i.e. there is g ∈ H(L) such that {hgh−1 : h ∈ H(L)} is dense.

Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications

Conjugacy classes of H(L)

Theorem The group of all homeomorphisms of the Lelek fan, H(L), has a dense 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 dense conjugacy class.

Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications

H(L) is simple

Recall that a group is simple if it has no proper normal subgroups.

Aleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra¨ ıss´ e theory Applications

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