Minimality in topological groups

Preliminariesπ-uniform actions

Homeomorphism GroupsSemidirect Product

Luie Polev

Bar-Ilan University, Israel

May 23, 2013

Luie Polev Minimality in topological groups

1 Preliminaries

2 π-uniform actions

3 Homeomorphism GroupsHomeomorphisms of the unit intervalHomeomorphisms of the unit circle

4 Semidirect Product

Definitions

Definition (Stephenson and Doıtchinov)

A Hausdorff topological group G is minimal if it does not admit astrictly coarser Hausdorff group topology.

Examples

I Every compact topological group is minimal.

I The subgroup Q/Z of the circle group T is minimal.

I Locally compact abelian groups are minimal if and only if theyare compact (Stephenson, 1971). So for example, the groupsZ (with the discrete topology) and R are not minimal.

Definitions

Examples

Definitions

Examples

Definitions

Examples

Definition

Let G be a topological group. A topological group X is called aG -group if G acts continuously on X by group automorphismsπ : G × X → X .

Notation

For a topological group (G , τ) and its subgroup H denote by τ/Hthe usual quotient topology on the coset space.

A useful lemma that we will use in the sequel:

Merson’s Lemma

Let (G , γ) be a not necessarily Hausdorff topological group and Hbe a not necessarily closed subgroup of G . Assume that γ1 ⊆ γ isa coarser group topology on G such that γ1|H = γ|H andγ1/H = γ/H. Then γ1 = γ.

Definition (Megrelishvili, 1985)

Let π : G × X → X be an action of a topological group G on aspace X . A uniformity µ on X (or, the action) is said to beπ-uniform at e ∈ G if

∀ε ∈ µ ∃U ∈ Ne ∃δ ∈ µ (ux , uy) ∈ ε ∀(x , y) ∈ δ ∀u ∈ U.

Motivation

Examples

I Every isometric action on a metric space is π-uniform.

I Every G -group X is π-uniform with respect to left (or right)uniformity on X .

Motivation

Examples

Motivation

Examples

Motivation

Examples

Homeomorphisms of the unit intervalHomeomorphisms of the unit circle

Definition

A compact space K is M-compact if the topological groupHomeo(K ) is minimal.

Question

Which compact spaces are M-compact?

Partial Answers

(van Mill, 2012) The n-dimensional Menger universalcontinuum is not M-compact for n > 0 .

(Gamarnik, 1991) The space [0, 1] is M-compact.

(Gamarnik) The space [0, 1]n for n > 1 is not M-compact.

(Gamarnik) The Cantor cube 2ω is M-compact (note that it isthe 0-dimensional Menger cube).

Definition

Question

Partial Answers

Definition

Question

Partial Answers

Definition

Question

Partial Answers

Definition

Question

Partial Answers

Definition

Question

Partial Answers

Homeo ([0, 1])

To prove that Homeo ([0, 1]) is minimal, Gamarnik used the notionof π-uniform topology. He first showed that the compact opentopology on Homeo ([0, 1]) is minimal within the class ofπ-uniform topologies, and then proved that it is indeed minimal.

These groups are minimal:

H+([0, 1]) - the subgroup of Homeo ([0, 1]) that consists ofincreasing homeomorphisms of [0, 1].

Let X denote the double arrow space. Then H+(X ) is thesubgroup of Homeo (X ) that consists of increasinghomeomorphisms of X .

Homeo ([0, 1])

Theorem

The group Homeo (S1) is minimal with respect to the compactopen topology.

Denote by γ the compact open topology on Homeo (S1), andlet γ1 ⊆ γ be a coarser Hausdorff group topology onHomeo (S1).

Let Homeo (S1, a) be the stabilizer subgroup.

Homeo (S1, a) is topologically isomorphic to the minimaltopological group Homeo ([0, 1]), and sinceγ1|Homeo (S1,a) ⊆ γ|Homeo (S1,a) is a Hausdorff group topology,we can conclude that γ1|Homeo (S1,a) = γ|Homeo (S1,a).

Theorem

Proof.

The quotient space Homeo (S1)/Homeo (S1, a) ishomeomorphic to S1, and sinceγ1/Homeo (S1, a) ⊆ γ/Homeo (S1, a) we conclude thatγ1/Homeo (S1, a) = γ/Homeo (S1, a).

Merson’s lemma concludes the proof.

Proof.

The quotient space Homeo (S1)/Homeo (S1, a) ishomeomorphic to S1, and sinceγ1/Homeo (S1, a) ⊆ γ/Homeo (S1, a) we conclude thatγ1/Homeo (S1, a) = γ/Homeo (S1, a).

Merson’s lemma concludes the proof.

Semidirect Product

For a G -group X , denote by X hπ G the topological semidirectproduct.

Minimality

X hπ G is not necessarily minimal even if G and X are bothminimal.

Indeed, letting π act trivially, the semidirect product is thedirect product X × G (for example, (Z, τp)× (Z, τp)).

X hπ G can be minimal even if X and G are not minimal. Forexample, the group Rh R+ is minimal (Dierolf andSchwanengel).

(Megrelishvili) For a compact abelian group G , the semidirectproduct G hAut(G ) is minimal.

Semidirect Product

Minimality

Semidirect Product

Minimality

Semidirect Product

Minimality

Semidirect Product

Minimality

Main Theorem

Theorem

Let G be a compact topological group. Then G hAut(G ) is aminimal group.

The structure of the proof

Let τ be the given topology on G , and σ the compact-opentopology on Aut(G ). Denote by γ the product topology onG hAut(G ). Assume that γ1 ⊆ γ is a coarser Hausdorffgroup topology on G hAut(G ).

Identifying G with the subgroup G × idG , the idea is to showthat γ1|G = γ|G and γ1/G = γ/G , and conclude the proofusing Merson’s lemma.

Main Theorem

Theorem

Main Theorem

Theorem

The ”crucial” steps

The compact open topology σ on Aut(G ) is minimal withinthe class of α-uniform topologies on Aut(G ).

The action α : (Aut(G ), γ1/G )× (G , γ1|G )→ (G , γ1|G ) iscontinuous at the identity (idG , eG ) and thus it is α-uniform.

The ”crucial” steps

The compact open topology σ on Aut(G ) is minimal withinthe class of α-uniform topologies on Aut(G ).

The action α : (Aut(G ), γ1/G )× (G , γ1|G )→ (G , γ1|G ) iscontinuous at the identity (idG , eG ) and thus it is α-uniform.

Minimality in topological groups

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