Automorphism groups of spaces with many symmetriesmtm.ufsc.br/~coloquio/20162/slides/kwiatkowska.pdf · Automorphism groups of spaces with many symmetries Aleksandra Kwiatkowska University

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 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?

Definition

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.

Examples

Example

1 F=finite linear orders

2 F=finite graphs

3 F=finite Boolean algebras

4 F=finite metric spaces with rational distances

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 .

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;

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.

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

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

Lelek fan

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)

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)

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)

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)

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

The pseudo-arc

Definition

The pseudo-arc

Definition

The pseudo-arc

Definition

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.

Theorem (Bing)

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.

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.

φ1 φ2

amalgamation propertyAleksandra Kwiatkowska Automorphism groups of spaces with many symmetries

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.

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.

Example

Let F be the family of all finite sets.

The projective Fraısse limit is the Cantor set.

Example

Let F be the family of all finite sets.

The projective Fraısse limit is the Cantor set.

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.

G is a projective Fraısse family.

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)

An example of an epimorphism

Sb a b a b c b b

Ta b c

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.

P/rP is the pseudo-arc.

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.

P/rP is the pseudo-arc.

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.

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

Theorem

F is a projective Fraısse family.

An example of an epimorphism

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.

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.

Non-triviality of H(L)

Remark

The group H(L) is non-trivial, that is, there is f ∈ H(L) such thatf 6= Id.

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).

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.

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.

Corollary

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

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.

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.

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).

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).

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).

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.

Corollary

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.

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.

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.

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.

Automorphism groups of spaces with many symmetriesmtm.ufsc.br/~coloquio/20162/slides/kwiatkowska.pdf · Automorphism groups of spaces with many symmetries Aleksandra Kwiatkowska University

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