Unique amenability of topological groups

Dana Bartosova

Carnegie Mellon University

BLASTUniversity of Denver

August 6, 2018

Dana Bartosova Unique amenability of topological groups

G-flow

G × X // X - a continuous action↑ ↑topological compactgroup Hausdorff space

g(hx) = (gh)x

ex = x

(g, ·) : X //X- a homeomorphism

G //Homeo(X) - a continuous homomorphism

We call X a G-flow.

Dana Bartosova Unique amenability of topological groups

G-flow

G × X // X - a continuous action

↑ ↑topological compactgroup Hausdorff space

g(hx) = (gh)x

ex = x

(g, ·) : X //X- a homeomorphism

G //Homeo(X) - a continuous homomorphism

We call X a G-flow.

Dana Bartosova Unique amenability of topological groups

G-flow

G × X // X - a continuous action↑ ↑topological compactgroup Hausdorff space

g(hx) = (gh)x

ex = x

(g, ·) : X //X- a homeomorphism

G //Homeo(X) - a continuous homomorphism

We call X a G-flow.

Dana Bartosova Unique amenability of topological groups

G-flow

G × X // X - a continuous action↑ ↑topological compactgroup Hausdorff space

g(hx) = (gh)x

ex = x

(g, ·) : X //X- a homeomorphism

G //Homeo(X) - a continuous homomorphism

We call X a G-flow.

Dana Bartosova Unique amenability of topological groups

G-flow

G × X // X - a continuous action↑ ↑topological compactgroup Hausdorff space

g(hx) = (gh)x

ex = x

(g, ·) : X //X- a homeomorphism

G //Homeo(X) - a continuous homomorphism

We call X a G-flow.

Dana Bartosova Unique amenability of topological groups

G-flow

G × X // X - a continuous action↑ ↑topological compactgroup Hausdorff space

g(hx) = (gh)x

ex = x

(g, ·) : X //X- a homeomorphism

G //Homeo(X) - a continuous homomorphism

We call X a G-flow.

Dana Bartosova Unique amenability of topological groups

G-flow

G × X // X - a continuous action↑ ↑topological compactgroup Hausdorff space

g(hx) = (gh)x

ex = x

(g, ·) : X //X- a homeomorphism

G //Homeo(X) - a continuous homomorphism

We call X a G-flow.

Dana Bartosova Unique amenability of topological groups

Examples

f : X //X homeomorphism, X compact Hausdorff

gives rise to a Z-flowZ×X //X, (n, x) 7→ fz(x)

1 rotation of a circle ρα : R/Z // R/Z, x 7→ x+ α

2 Bernoulli shift σ : 2Z // 2Z, σ(f)(n) = f(n+ 1)

3 1+ : βZ // βZ, 1 + u = {1 + S, S ∈ u}

βZ is the Cech-Stone compactification on Z, i.e., the space ofall ultrafilters on Z

Dana Bartosova Unique amenability of topological groups

Examples

f : X //X homeomorphism, X compact Hausdorff

gives rise to a Z-flowZ×X //X, (n, x) 7→ fz(x)

1 rotation of a circle ρα : R/Z // R/Z, x 7→ x+ α

2 Bernoulli shift σ : 2Z // 2Z, σ(f)(n) = f(n+ 1)

3 1+ : βZ // βZ, 1 + u = {1 + S, S ∈ u}

βZ is the Cech-Stone compactification on Z, i.e., the space ofall ultrafilters on Z

Dana Bartosova Unique amenability of topological groups

Examples

f : X //X homeomorphism, X compact Hausdorff

gives rise to a Z-flowZ×X //X, (n, x) 7→ fz(x)

1 rotation of a circle ρα : R/Z // R/Z, x 7→ x+ α

2 Bernoulli shift σ : 2Z // 2Z, σ(f)(n) = f(n+ 1)

3 1+ : βZ // βZ, 1 + u = {1 + S, S ∈ u}

βZ is the Cech-Stone compactification on Z, i.e., the space ofall ultrafilters on Z

Dana Bartosova Unique amenability of topological groups

Examples

f : X //X homeomorphism, X compact Hausdorff

gives rise to a Z-flowZ×X //X, (n, x) 7→ fz(x)

1 rotation of a circle ρα : R/Z // R/Z, x 7→ x+ α

2 Bernoulli shift σ : 2Z // 2Z, σ(f)(n) = f(n+ 1)

3 1+ : βZ // βZ, 1 + u = {1 + S, S ∈ u}

βZ is the Cech-Stone compactification on Z, i.e., the space ofall ultrafilters on Z

Dana Bartosova Unique amenability of topological groups

Examples

f : X //X homeomorphism, X compact Hausdorff

gives rise to a Z-flowZ×X //X, (n, x) 7→ fz(x)

1 rotation of a circle ρα : R/Z // R/Z, x 7→ x+ α

2 Bernoulli shift σ : 2Z // 2Z, σ(f)(n) = f(n+ 1)

3 1+ : βZ // βZ, 1 + u = {1 + S, S ∈ u}

βZ is the Cech-Stone compactification on Z, i.e., the space ofall ultrafilters on Z

Dana Bartosova Unique amenability of topological groups

Examples

f : X //X homeomorphism, X compact Hausdorff

gives rise to a Z-flowZ×X //X, (n, x) 7→ fz(x)

1 rotation of a circle ρα : R/Z // R/Z, x 7→ x+ α

2 Bernoulli shift σ : 2Z // 2Z, σ(f)(n) = f(n+ 1)

3 1+ : βZ // βZ, 1 + u = {1 + S, S ∈ u}

βZ is the Cech-Stone compactification on Z, i.e., the space ofall ultrafilters on Z

Dana Bartosova Unique amenability of topological groups

Amenability

A topological group G is amenable if every G-flow admits aninvariant probability measure

1 Z2 locally compact groups

3 extremely amenable groups

G is extremely amenable if every G-flow has a fixed point

Dana Bartosova Unique amenability of topological groups

Amenability

A topological group G is amenable if every G-flow admits aninvariant probability measure

1 Z

2 locally compact groups

3 extremely amenable groups

G is extremely amenable if every G-flow has a fixed point

Dana Bartosova Unique amenability of topological groups

Amenability

A topological group G is amenable if every G-flow admits aninvariant probability measure

1 Z2 locally compact groups

3 extremely amenable groups

G is extremely amenable if every G-flow has a fixed point

Dana Bartosova Unique amenability of topological groups

Amenability

A topological group G is amenable if every G-flow admits aninvariant probability measure

1 Z2 locally compact groups

3 extremely amenable groups

G is extremely amenable if every G-flow has a fixed point

Dana Bartosova Unique amenability of topological groups

Amenability

A topological group G is amenable if every G-flow admits aninvariant probability measure

1 Z2 locally compact groups

3 extremely amenable groups

G is extremely amenable if every G-flow has a fixed point

Dana Bartosova Unique amenability of topological groups

Unique amenability

A topological group G is uniquely amenable if every G-flow witha dense orbit admits a unique invariant probability measure.

1 precompact groups

2 ???

A group is precompact if it is a subgroup of a compact group.

Dana Bartosova Unique amenability of topological groups

Unique amenability

A topological group G is uniquely amenable if every G-flow witha dense orbit admits a unique invariant probability measure.

1 precompact groups

2 ???

A group is precompact if it is a subgroup of a compact group.

Dana Bartosova Unique amenability of topological groups

Unique amenability

A topological group G is uniquely amenable if every G-flow witha dense orbit admits a unique invariant probability measure.

1 precompact groups

2 ???

A group is precompact if it is a subgroup of a compact group.

Dana Bartosova Unique amenability of topological groups

Unique amenability

A topological group G is uniquely amenable if every G-flow witha dense orbit admits a unique invariant probability measure.

1 precompact groups

2 ???

A group is precompact if it is a subgroup of a compact group.

Dana Bartosova Unique amenability of topological groups

All non-examples thus far

Given a (amenable) topological group find a flow with a denseorbit that has two disjoint subflows.

1 locally compact groups (Lau and Paterson, 1986)

2 separable topological vector spaces (Ferri and Strauss,2001)

3 groups of density < ℵω, automorphism groups, . . . (B.,2013)

Dana Bartosova Unique amenability of topological groups

All non-examples thus far

Given a (amenable) topological group find a flow with a denseorbit that has two disjoint subflows.

1 locally compact groups (Lau and Paterson, 1986)

2 separable topological vector spaces (Ferri and Strauss,2001)

3 groups of density < ℵω, automorphism groups, . . . (B.,2013)

Dana Bartosova Unique amenability of topological groups

All non-examples thus far

Given a (amenable) topological group find a flow with a denseorbit that has two disjoint subflows.

1 locally compact groups (Lau and Paterson, 1986)

2 separable topological vector spaces (Ferri and Strauss,2001)

3 groups of density < ℵω, automorphism groups, . . . (B.,2013)

Dana Bartosova Unique amenability of topological groups

All non-examples thus far

Given a (amenable) topological group find a flow with a denseorbit that has two disjoint subflows.

1 locally compact groups (Lau and Paterson, 1986)

2 separable topological vector spaces (Ferri and Strauss,2001)

3 groups of density < ℵω, automorphism groups, . . . (B.,2013)

Dana Bartosova Unique amenability of topological groups

Universal flow with a dense orbit

A pointed G-flow (X,x0) is a G-ambit if the orbit Gx0 is densein X.

There is a compactification G ↪→ S(G) so that (S(G), e) is thegreatest ambit, that is, for every G-ambit (X,x0) there is aquotient map q : S(G) //X so that

G× (X,x0) X//

G× (S(G), e)

G× (X,x0)

id×q

��

G× (S(G), e) S(G)// S(G)

X

id×q

��

commutes and q(e) = x0.

Dana Bartosova Unique amenability of topological groups

Universal flow with a dense orbit

A pointed G-flow (X,x0) is a G-ambit if the orbit Gx0 is densein X.

There is a compactification G ↪→ S(G) so that (S(G), e) is thegreatest ambit, that is,

for every G-ambit (X,x0) there is aquotient map q : S(G) //X so that

G× (X,x0) X//

G× (S(G), e)

G× (X,x0)

id×q

��

G× (S(G), e) S(G)// S(G)

X

id×q

��

commutes and q(e) = x0.

Dana Bartosova Unique amenability of topological groups

Universal flow with a dense orbit

A pointed G-flow (X,x0) is a G-ambit if the orbit Gx0 is densein X.

There is a compactification G ↪→ S(G) so that (S(G), e) is thegreatest ambit, that is, for every G-ambit (X,x0) there is aquotient map q : S(G) //X so that

G× (X,x0) X//

G× (S(G), e)

G× (X,x0)

id×q

��

G× (S(G), e) S(G)// S(G)

X

id×q

��

commutes and q(e) = x0.

Dana Bartosova Unique amenability of topological groups

Universal flow with a dense orbit

A pointed G-flow (X,x0) is a G-ambit if the orbit Gx0 is densein X.

There is a compactification G ↪→ S(G) so that (S(G), e) is thegreatest ambit, that is, for every G-ambit (X,x0) there is aquotient map q : S(G) //X so that

G× (X,x0) X//

G× (S(G), e)

G× (X,x0)

id×q

��

G× (S(G), e) S(G)// S(G)

X

id×q

��

commutes and q(e) = x0.

Dana Bartosova Unique amenability of topological groups

Universal flow with a dense orbit

A pointed G-flow (X,x0) is a G-ambit if the orbit Gx0 is densein X.

There is a compactification G ↪→ S(G) so that (S(G), e) is thegreatest ambit, that is, for every G-ambit (X,x0) there is aquotient map q : S(G) //X so that

G× (X,x0) X//

G× (S(G), e)

G× (X,x0)

id×q

��

G× (S(G), e) S(G)// S(G)

X

id×q

��

commutes and q(e) = x0.

Dana Bartosova Unique amenability of topological groups

S(G) must be a witness

If X is a G-flow with a dense orbit, S(G) maps onto it.

In particular, if X has disjoint subflows, so does S(G).

Note, if X has disjoint flows than it has disjoint minimal flows,since every flow has a minimal subflow.

A G-flow is minimal if it has no proper non-empty closedinvariant subset iff every orbit is dense.

Every minimal subflow of S(G) is the universal minimal flow.

Dana Bartosova Unique amenability of topological groups

S(G) must be a witness

If X is a G-flow with a dense orbit, S(G) maps onto it.

In particular, if X has disjoint subflows, so does S(G).

Note, if X has disjoint flows than it has disjoint minimal flows,since every flow has a minimal subflow.

A G-flow is minimal if it has no proper non-empty closedinvariant subset iff every orbit is dense.

Every minimal subflow of S(G) is the universal minimal flow.

Dana Bartosova Unique amenability of topological groups

S(G) must be a witness

If X is a G-flow with a dense orbit, S(G) maps onto it.

In particular, if X has disjoint subflows, so does S(G).

Note, if X has disjoint flows than it has disjoint minimal flows,since

every flow has a minimal subflow.

A G-flow is minimal if it has no proper non-empty closedinvariant subset iff every orbit is dense.

Every minimal subflow of S(G) is the universal minimal flow.

Dana Bartosova Unique amenability of topological groups

S(G) must be a witness

If X is a G-flow with a dense orbit, S(G) maps onto it.

In particular, if X has disjoint subflows, so does S(G).

Note, if X has disjoint flows than it has disjoint minimal flows,since every flow has a minimal subflow.

A G-flow is minimal if it has no proper non-empty closedinvariant subset iff every orbit is dense.

Every minimal subflow of S(G) is the universal minimal flow.

Dana Bartosova Unique amenability of topological groups

S(G) must be a witness

If X is a G-flow with a dense orbit, S(G) maps onto it.

In particular, if X has disjoint subflows, so does S(G).

Note, if X has disjoint flows than it has disjoint minimal flows,since every flow has a minimal subflow.

A G-flow is minimal if it has no proper non-empty closedinvariant subset iff every orbit is dense.

Every minimal subflow of S(G) is the universal minimal flow.

Dana Bartosova Unique amenability of topological groups

S(G) must be a witness

If X is a G-flow with a dense orbit, S(G) maps onto it.

In particular, if X has disjoint subflows, so does S(G).

Note, if X has disjoint flows than it has disjoint minimal flows,since every flow has a minimal subflow.

A G-flow is minimal if it has no proper non-empty closedinvariant subset iff every orbit is dense.

Every minimal subflow of S(G) is the universal minimal flow.

Dana Bartosova Unique amenability of topological groups

Why do we like S(G)?

Algebra ♥ topology.

(S(G), ·) is a right topological semigroup, i.e.,· : S(G)× S(G) // S(G) is a semigroup operation and·s : x 7→ xs is continuous.

It extends the continuous action G× S(G) // S(G),

which extends the continuous multiplication · : G×G //G.

Dana Bartosova Unique amenability of topological groups

Why do we like S(G)?

Algebra ♥ topology.

(S(G), ·) is a right topological semigroup, i.e.,· : S(G)× S(G) // S(G) is a semigroup operation and·s : x 7→ xs is continuous.

It extends the continuous action G× S(G) // S(G),

which extends the continuous multiplication · : G×G //G.

Dana Bartosova Unique amenability of topological groups

Why do we like S(G)?

Algebra ♥ topology.

(S(G), ·) is a right topological semigroup, i.e.,· : S(G)× S(G) // S(G) is a semigroup operation and·s : x 7→ xs is continuous.

It extends the continuous action G× S(G) // S(G),

which extends the continuous multiplication · : G×G //G.

Dana Bartosova Unique amenability of topological groups

Why do we like S(G)?

Algebra ♥ topology.

(S(G), ·) is a right topological semigroup, i.e.,· : S(G)× S(G) // S(G) is a semigroup operation and·s : x 7→ xs is continuous.

It extends the continuous action G× S(G) // S(G),

which extends the continuous multiplication · : G×G //G.

Dana Bartosova Unique amenability of topological groups

Minimal subflows of S(G)

Let M be a minimal subflow of S(G) and m ∈M. ThenS(G)m = M , i.e., M is a (minimal) left ideal in S(G).

Conversely, if N is a minimal left ideal of S(G) N is a subflow.

Dana Bartosova Unique amenability of topological groups

Minimal subflows of S(G)

Let M be a minimal subflow of S(G) and m ∈M. ThenS(G)m = M , i.e., M is a (minimal) left ideal in S(G).

Conversely, if N is a minimal left ideal of S(G) N is a subflow.

Dana Bartosova Unique amenability of topological groups

S(G) for G discrete

βG - Cech-Stone compactification of G

gU = {gA : A ∈ U}

defines a continuous action

G× βG // βG.

Right multiplication

(·,U) : G // βG

can be continuously extended

(·,U) : βG // βG.

UV = U − lim{gV : g ∈ G} = {A ⊂ G : {g ∈ G : g−1A ∈ V} ∈ U}

Dana Bartosova Unique amenability of topological groups

S(G) for G discrete

βG - Cech-Stone compactification of G

gU = {gA : A ∈ U}

defines a continuous action

G× βG // βG.

Right multiplication

(·,U) : G // βG

can be continuously extended

(·,U) : βG // βG.

UV = U − lim{gV : g ∈ G} = {A ⊂ G : {g ∈ G : g−1A ∈ V} ∈ U}

Dana Bartosova Unique amenability of topological groups

S(G) for G discrete

βG - Cech-Stone compactification of G

gU = {gA : A ∈ U}

defines a continuous action

G× βG // βG.

Right multiplication

(·,U) : G // βG

can be continuously extended

(·,U) : βG // βG.

UV = U − lim{gV : g ∈ G} = {A ⊂ G : {g ∈ G : g−1A ∈ V} ∈ U}

Dana Bartosova Unique amenability of topological groups

S(G) for G discrete

βG - Cech-Stone compactification of G

gU = {gA : A ∈ U}

defines a continuous action

G× βG // βG.

Right multiplication

(·,U) : G // βG

can be continuously extended

(·,U) : βG // βG.

UV = U − lim{gV : g ∈ G} = {A ⊂ G : {g ∈ G : g−1A ∈ V} ∈ U}

Dana Bartosova Unique amenability of topological groups

S(G) for G discrete

βG - Cech-Stone compactification of G

gU = {gA : A ∈ U}

defines a continuous action

G× βG // βG.

Right multiplication

(·,U) : G // βG

can be continuously extended

(·,U) : βG // βG.

UV = U − lim{gV : g ∈ G} = {A ⊂ G : {g ∈ G : g−1A ∈ V} ∈ U}

Dana Bartosova Unique amenability of topological groups

S(G) for G discrete

βG - Cech-Stone compactification of G

gU = {gA : A ∈ U}

defines a continuous action

G× βG // βG.

Right multiplication

(·,U) : G // βG

can be continuously extended

(·,U) : βG // βG.

UV = U − lim{gV : g ∈ G} = {A ⊂ G : {g ∈ G : g−1A ∈ V} ∈ U}

Dana Bartosova Unique amenability of topological groups

S(G) for G discrete

βG - Cech-Stone compactification of G

gU = {gA : A ∈ U}

defines a continuous action

G× βG // βG.

Right multiplication

(·,U) : G // βG

can be continuously extended

(·,U) : βG // βG.

UV = U − lim{gV : g ∈ G}

= {A ⊂ G : {g ∈ G : g−1A ∈ V} ∈ U}

Dana Bartosova Unique amenability of topological groups

S(G) for G discrete

βG - Cech-Stone compactification of G

gU = {gA : A ∈ U}

defines a continuous action

G× βG // βG.

Right multiplication

(·,U) : G // βG

can be continuously extended

(·,U) : βG // βG.

UV = U − lim{gV : g ∈ G} = {A ⊂ G : {g ∈ G : g−1A ∈ V} ∈ U}

Dana Bartosova Unique amenability of topological groups

Near (ultra)filters

Let G be a topological group and Ne(G) a base of openneighbourhoods of the identity.

A family F of subsets of G has the finite near intersectionproperty (fnip) if for every F1, F2, . . . , Fn ∈ F and everyV ∈ Ne(G)

n⋂i=1

V Fi 6= ∅.

F is a near filter if it has fnip and is closed upwards.F is a near ultrafilter if it a maximal near filter.

Dana Bartosova Unique amenability of topological groups

Near (ultra)filters

Let G be a topological group and Ne(G) a base of openneighbourhoods of the identity.

A family F of subsets of G has the finite near intersectionproperty (fnip) if

for every F1, F2, . . . , Fn ∈ F and everyV ∈ Ne(G)

n⋂i=1

V Fi 6= ∅.

F is a near filter if it has fnip and is closed upwards.F is a near ultrafilter if it a maximal near filter.

Dana Bartosova Unique amenability of topological groups

Near (ultra)filters

Let G be a topological group and Ne(G) a base of openneighbourhoods of the identity.

A family F of subsets of G has the finite near intersectionproperty (fnip) if for every F1, F2, . . . , Fn ∈ F and everyV ∈ Ne(G)

n⋂i=1

V Fi 6= ∅.

F is a near filter if it has fnip and is closed upwards.F is a near ultrafilter if it a maximal near filter.

Dana Bartosova Unique amenability of topological groups

Near (ultra)filters

Let G be a topological group and Ne(G) a base of openneighbourhoods of the identity.

A family F of subsets of G has the finite near intersectionproperty (fnip) if for every F1, F2, . . . , Fn ∈ F and everyV ∈ Ne(G)

n⋂i=1

V Fi 6= ∅.

F is a near filter if it has fnip and is closed upwards.

F is a near ultrafilter if it a maximal near filter.

Dana Bartosova Unique amenability of topological groups

Near (ultra)filters

Let G be a topological group and Ne(G) a base of openneighbourhoods of the identity.

A family F of subsets of G has the finite near intersectionproperty (fnip) if for every F1, F2, . . . , Fn ∈ F and everyV ∈ Ne(G)

n⋂i=1

V Fi 6= ∅.

F is a near filter if it has fnip and is closed upwards.F is a near ultrafilter if it a maximal near filter.

Dana Bartosova Unique amenability of topological groups

S(G) for G topological

S(G) = the space of near ultrafilters with base for closed sets

A = {U ∈ S(G) : A ∈ U}

for A ⊂ G.

The action G× S(G) // S(G), gU = {gU : U ∈ U}

Extends to S(G)× S(G) // S(G)

UV = U − lim{gV : g ∈ G}

= {A ⊂ G : V {g ∈ G : g−1A ∈ V} ∈ U , V ∈ Ne(G)}

Dana Bartosova Unique amenability of topological groups

S(G) for G topological

S(G) = the space of near ultrafilters with base for closed sets

A = {U ∈ S(G) : A ∈ U}

for A ⊂ G.

The action G× S(G) // S(G), gU = {gU : U ∈ U}

Extends to S(G)× S(G) // S(G)

UV = U − lim{gV : g ∈ G}

= {A ⊂ G : V {g ∈ G : g−1A ∈ V} ∈ U , V ∈ Ne(G)}

Dana Bartosova Unique amenability of topological groups

S(G) for G topological

S(G) = the space of near ultrafilters with base for closed sets

A = {U ∈ S(G) : A ∈ U}

for A ⊂ G.

The action G× S(G) // S(G), gU = {gU : U ∈ U}

Extends to S(G)× S(G) // S(G)

UV = U − lim{gV : g ∈ G}

= {A ⊂ G : V {g ∈ G : g−1A ∈ V} ∈ U , V ∈ Ne(G)}

Dana Bartosova Unique amenability of topological groups

S(G) for G topological

S(G) = the space of near ultrafilters with base for closed sets

A = {U ∈ S(G) : A ∈ U}

for A ⊂ G.

The action G× S(G) // S(G), gU = {gU : U ∈ U}

Extends to S(G)× S(G) // S(G)

UV = U − lim{gV : g ∈ G}

= {A ⊂ G : V {g ∈ G : g−1A ∈ V} ∈ U , V ∈ Ne(G)}

Dana Bartosova Unique amenability of topological groups

S(G) for G topological

S(G) = the space of near ultrafilters with base for closed sets

A = {U ∈ S(G) : A ∈ U}

for A ⊂ G.

The action G× S(G) // S(G), gU = {gU : U ∈ U}

Extends to S(G)× S(G) // S(G)

UV = U − lim{gV : g ∈ G}

= {A ⊂ G : V {g ∈ G : g−1A ∈ V} ∈ U , V ∈ Ne(G)}

Dana Bartosova Unique amenability of topological groups

S(G) for G topological

S(G) = the space of near ultrafilters with base for closed sets

A = {U ∈ S(G) : A ∈ U}

for A ⊂ G.

The action G× S(G) // S(G), gU = {gU : U ∈ U}

Extends to S(G)× S(G) // S(G)

UV = U − lim{gV : g ∈ G}

= {A ⊂ G : V {g ∈ G : g−1A ∈ V} ∈ U , V ∈ Ne(G)}

Dana Bartosova Unique amenability of topological groups

Thick sets

A subset T ⊆ G is thick if TβG

contains a minimal left ideal ofβG.

Equivalently, {gT : g ∈ G} has the finite instersection property.

Equivalently, for every finite F ⊂ G there is x ∈ G such thatFx ⊂ G.

Minimal left ideals ≡ maximal filters of thick sets.

Dana Bartosova Unique amenability of topological groups

Thick sets

A subset T ⊆ G is thick if TβG

contains a minimal left ideal ofβG.

Equivalently, {gT : g ∈ G} has the finite instersection property.

Equivalently, for every finite F ⊂ G there is x ∈ G such thatFx ⊂ G.

Minimal left ideals ≡ maximal filters of thick sets.

Dana Bartosova Unique amenability of topological groups

Thick sets

A subset T ⊆ G is thick if TβG

contains a minimal left ideal ofβG.

Equivalently, {gT : g ∈ G} has the finite instersection property.

Equivalently, for every finite F ⊂ G there is x ∈ G such thatFx ⊂ G.

Minimal left ideals ≡ maximal filters of thick sets.

Dana Bartosova Unique amenability of topological groups

Thick sets

A subset T ⊆ G is thick if TβG

contains a minimal left ideal ofβG.

Equivalently, {gT : g ∈ G} has the finite instersection property.

Equivalently, for every finite F ⊂ G there is x ∈ G such thatFx ⊂ G.

Minimal left ideals ≡ maximal filters of thick sets.

Dana Bartosova Unique amenability of topological groups

Prethick sets

A subset T ⊆ G is prethick if TS(G)

contains a minimal leftideal in S(G).

Equivalently, V T is thick for every open V ∈ Ne(G).

Minimal left ideals ≡ maximal near filters of prethick sets.

Dana Bartosova Unique amenability of topological groups

Prethick sets

A subset T ⊆ G is prethick if TS(G)

contains a minimal leftideal in S(G).

Equivalently, V T is thick for every open V ∈ Ne(G).

Minimal left ideals ≡ maximal near filters of prethick sets.

Dana Bartosova Unique amenability of topological groups

Prethick sets

A subset T ⊆ G is prethick if TS(G)

contains a minimal leftideal in S(G).

Equivalently, V T is thick for every open V ∈ Ne(G).

Minimal left ideals ≡ maximal near filters of prethick sets.

Dana Bartosova Unique amenability of topological groups

Disjoint thick sets

Theorem (Carlson, Hindman, McLeod, Strauss (2008))

Discrete G of cardinality κ can be split into κ disjoint thick sets.

Corollary

If G is as above, βG contains 22κdisjoint minimal left ideals.

CONSTRUCTION

1 enumerate finite subsets of G into (Fλ)λ<κ.

2 recursively pick (xλ)λ>κ so that Fλxλ ∩⋃µ<λ Fµxµ = ∅.

3 κ =⋃ν<κIν , so that for any ν every finite set F ⊂ G is

contained in Fλ for some λ ∈ Iν .4 Tν =

⋃λ∈Iν Fλxλ for ν < κ are disjoint thick.

Dana Bartosova Unique amenability of topological groups

Disjoint thick sets

Theorem (Carlson, Hindman, McLeod, Strauss (2008))

Discrete G of cardinality κ can be split into κ disjoint thick sets.

Corollary

If G is as above, βG contains 22κdisjoint minimal left ideals.

CONSTRUCTION

1 enumerate finite subsets of G into (Fλ)λ<κ.

2 recursively pick (xλ)λ>κ so that Fλxλ ∩⋃µ<λ Fµxµ = ∅.

3 κ =⋃ν<κIν , so that for any ν every finite set F ⊂ G is

contained in Fλ for some λ ∈ Iν .4 Tν =

⋃λ∈Iν Fλxλ for ν < κ are disjoint thick.

Dana Bartosova Unique amenability of topological groups

Disjoint thick sets

Theorem (Carlson, Hindman, McLeod, Strauss (2008))

Discrete G of cardinality κ can be split into κ disjoint thick sets.

Corollary

If G is as above, βG contains 22κdisjoint minimal left ideals.

CONSTRUCTION

1 enumerate finite subsets of G into (Fλ)λ<κ.

2 recursively pick (xλ)λ>κ so that Fλxλ ∩⋃µ<λ Fµxµ = ∅.

3 κ =⋃ν<κIν , so that for any ν every finite set F ⊂ G is

contained in Fλ for some λ ∈ Iν .4 Tν =

⋃λ∈Iν Fλxλ for ν < κ are disjoint thick.

Dana Bartosova Unique amenability of topological groups

Near disjoint thick sets

For V ∈ Ne(G), set

κ(V ) = min{|Γ| : ∀g, h ∈ Γ, gV ∩ hV = ∅ & ΓV2 = G}.

Theorem (B., 2013)

If there is V ∈ Ne(G) so that κ(V ) = κ(V 6) ≥ ℵ0 then Gcontains κ(V ) thick sets whose V -saturations are disjoint.

Corollary

If G is as above, S(G) contains 22κ(V )

disjoint minimal leftideals.

Dana Bartosova Unique amenability of topological groups

Near disjoint thick sets

For V ∈ Ne(G), set

κ(V ) = min{|Γ| : ∀g, h ∈ Γ, gV ∩ hV = ∅ & ΓV2 = G}.

Theorem (B., 2013)

If there is V ∈ Ne(G) so that κ(V ) = κ(V 6) ≥ ℵ0 then Gcontains κ(V ) thick sets whose V -saturations are disjoint.

Corollary

If G is as above, S(G) contains 22κ(V )

disjoint minimal leftideals.

Dana Bartosova Unique amenability of topological groups

Near disjoint thick sets

For V ∈ Ne(G), set

κ(V ) = min{|Γ| : ∀g, h ∈ Γ, gV ∩ hV = ∅ & ΓV2 = G}.

Theorem (B., 2013)

If there is V ∈ Ne(G) so that κ(V ) = κ(V 6) ≥ ℵ0 then Gcontains κ(V ) thick sets whose V -saturations are disjoint.

Corollary

If G is as above, S(G) contains 22κ(V )

disjoint minimal leftideals.

Dana Bartosova Unique amenability of topological groups

Near disjoint thick sets

For V ∈ Ne(G), set

κ(V ) = min{|Γ| : ∀g, h ∈ Γ, gV ∩ hV = ∅ & ΓV2 = G}.

Theorem (B., 2013)

If there is V ∈ Ne(G) so that κ(V ) = κ(V 6) ≥ ℵ0 then Gcontains κ(V ) thick sets whose V -saturations are disjoint.

Corollary

If G is as above, S(G) contains 22κ(V )

disjoint minimal leftideals.

Dana Bartosova Unique amenability of topological groups

Open

1 Is there a topological group G such that no V ∈ Ne(G)satisfies κ(V ) = κ(V 6) ≥ ℵ0?

2 Is there a non-precompact group G with a single minimalleft ideal in S(G)?

3 Is there a non-precompact extremely amenable group Gwith a single minimal left ideal in S(G)?

4 Is there a non-precompact group G that is uniquelyamenable.

Dana Bartosova Unique amenability of topological groups

Open

1 Is there a topological group G such that no V ∈ Ne(G)satisfies κ(V ) = κ(V 6) ≥ ℵ0?

2 Is there a non-precompact group G with a single minimalleft ideal in S(G)?

3 Is there a non-precompact extremely amenable group Gwith a single minimal left ideal in S(G)?

4 Is there a non-precompact group G that is uniquelyamenable.

Dana Bartosova Unique amenability of topological groups

Open

1 Is there a topological group G such that no V ∈ Ne(G)satisfies κ(V ) = κ(V 6) ≥ ℵ0?

2 Is there a non-precompact group G with a single minimalleft ideal in S(G)?

3 Is there a non-precompact extremely amenable group Gwith a single minimal left ideal in S(G)?

4 Is there a non-precompact group G that is uniquelyamenable.

Dana Bartosova Unique amenability of topological groups

Open

1 Is there a topological group G such that no V ∈ Ne(G)satisfies κ(V ) = κ(V 6) ≥ ℵ0?

2 Is there a non-precompact group G with a single minimalleft ideal in S(G)?

3 Is there a non-precompact extremely amenable group Gwith a single minimal left ideal in S(G)?

4 Is there a non-precompact group G that is uniquelyamenable.

Dana Bartosova Unique amenability of topological groups

Another point of attack

K(S(G)) = minimal (both-sided) ideal in (S(G), ·)

=⋃

minimal left ideals of S(G)

If G is discrete than K(βG) is not closed when

1 G is countable. (Hindman and Strauss, 1998)

2 G can be algebraically embdedded into a compact group.(Zelenyuk, 2009)

It means that S(G) need to contain infinitely many minimal leftideals.

Theorem (Hindman and Strauss, 2017)

K(βN) is not Borel.

Dana Bartosova Unique amenability of topological groups

Another point of attack

K(S(G)) = minimal (both-sided) ideal in (S(G), ·)=

⋃minimal left ideals of S(G)

If G is discrete than K(βG) is not closed when

1 G is countable. (Hindman and Strauss, 1998)

2 G can be algebraically embdedded into a compact group.(Zelenyuk, 2009)

It means that S(G) need to contain infinitely many minimal leftideals.

Theorem (Hindman and Strauss, 2017)

K(βN) is not Borel.

Dana Bartosova Unique amenability of topological groups

Another point of attack

K(S(G)) = minimal (both-sided) ideal in (S(G), ·)=

⋃minimal left ideals of S(G)

If G is discrete than K(βG) is not closed when

1 G is countable. (Hindman and Strauss, 1998)

2 G can be algebraically embdedded into a compact group.(Zelenyuk, 2009)

It means that S(G) need to contain infinitely many minimal leftideals.

Theorem (Hindman and Strauss, 2017)

K(βN) is not Borel.

Dana Bartosova Unique amenability of topological groups

Another point of attack

K(S(G)) = minimal (both-sided) ideal in (S(G), ·)=

⋃minimal left ideals of S(G)

If G is discrete than K(βG) is not closed when

1 G is countable. (Hindman and Strauss, 1998)

2 G can be algebraically embdedded into a compact group.(Zelenyuk, 2009)

It means that S(G) need to contain infinitely many minimal leftideals.

Theorem (Hindman and Strauss, 2017)

K(βN) is not Borel.

Dana Bartosova Unique amenability of topological groups

Another point of attack

K(S(G)) = minimal (both-sided) ideal in (S(G), ·)=

⋃minimal left ideals of S(G)

If G is discrete than K(βG) is not closed when

1 G is countable. (Hindman and Strauss, 1998)

2 G can be algebraically embdedded into a compact group.(Zelenyuk, 2009)

It means that S(G) need to contain infinitely many minimal leftideals.

Theorem (Hindman and Strauss, 2017)

K(βN) is not Borel.

Dana Bartosova Unique amenability of topological groups

Another point of attack

K(S(G)) = minimal (both-sided) ideal in (S(G), ·)=

⋃minimal left ideals of S(G)

If G is discrete than K(βG) is not closed when

1 G is countable. (Hindman and Strauss, 1998)

2 G can be algebraically embdedded into a compact group.(Zelenyuk, 2009)

It means that S(G) need to contain infinitely many minimal leftideals.

Theorem (Hindman and Strauss, 2017)

K(βN) is not Borel.

Dana Bartosova Unique amenability of topological groups

G Polish

Theorem (B. and Zucker 2017)

K(S(G)) is closed if and only if minimal left ideals of S(G) aremetrizable.

Dana Bartosova Unique amenability of topological groups

The end

THANK YOU!

Dana Bartosova Unique amenability of topological groups