Subgroups of isometries of Urysohn-Katetov metric spaces...

Subgroups of isometries of Urysohn-Katetovmetric spaces of uncountable density

Brice Rodrigue Mbombo

IME-USP, BrazilJoint work with Vladimir Pestov

May 24, 2014

Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces

Universal topological group

Let C be a class of topologicals groups.

G ∈ C is universal for C if for every H ∈ C there is anisomorphism between H and a subgroup of G


Ulam’s Question, 1935

Ulam

Does there exist a universal topological group with a countablebase?


Remind

Teleman

Every topological group G has a topologically faithfulrepresentation on a Banach space B: embeddingG −→ Iso(B)

Every topological group G has a topologically faithfulrepresentation on a compact space X : embeddingG −→ Homeo(X ).


Proofs of Teleman results

1st Proof

B = RUCB(G ), X = the unit ball in B? with w?-topology.

RUCB(G )=Right Uniformly Continuous Bounded functionsf : G −→ C.

2nd Proof

X = S(G ) and B = C (X )

S(G )= the maximal ideal space of the abelian unital C ?-algebraRUCB(G ) = the Samuel compactification of (G ,UR)


Precise Teleman theorem

Uspenskij

For every topological group G there exists a metric space M suchthat w(G ) = w(M) and G is isomorphic (as a topological group)to a subgroup of Iso(M).


Solutions of Ulam Question

1st solution: Uspenskij, 1986

Homeo([0, 1]ℵ0) contains all groups with a countable base.

G ↪→ Homeo(X ) ↪→ Homeo(P(X )) = Homeo([0, 1]ℵ0)Note: Each infinite-dimensional convex compact set lying in ametrizable locally convex space is homeomorphic to the Hilbertcube. (Keller).



2nd solution: Uspenkij, 1990

Iso(U) contains all groups with a countable base, where U is theuniversal polish Urysohn space.

U = the Urysohn universal metric space = the complete separablespace, contains isometric copies of all separable spaces, and isultrahomogeneous =that is every isometry between two finitemetric subspace of U extends to a global isometry of U onto itself.



Let X be a polish space

Katetov

Construct X = X0 ↪→ X1 ↪→ ..., withw(X ) = w(X0) = w(X1) = ... and Xω =

⋃Xn = U is the Urysohn

space.

Uspenskij

Iso(X ) = Iso(X0) ↪→ Iso(X1) ↪→ ..., whenceG ↪→ Iso(X ) ↪→ Iso(Xω) = Iso(U).



Let X be a polish space

Katetov

Construct X = X0 ↪→ X1 ↪→ ..., withw(X ) = w(X0) = w(X1) = ... and Xω =

⋃Xn = U is the Urysohn

space.

Uspenskij

Iso(X ) = Iso(X0) ↪→ Iso(X1) ↪→ ..., whenceG ↪→ Iso(X ) ↪→ Iso(Xω) = Iso(U).



3rd solution: Ben Yaacov, 2012

Iso(G) contains all groups with a countable base, where G is theGurarij space.

A Gurarij space is a Banach space G having the property that forany ε > 0, finite-dimensional Banach space E ⊆ F and anisometric embedding ϕ : E −→ G there is a linear mapψ : F −→ G extending ϕ such that in addition, for allx ∈ F , (1− ε)‖x‖ ≤ ‖ψ(x)‖ ≤ (1 + ε)‖x‖



3rd solution: Ben Yaacov, 2012

Iso(G) contains all groups with a countable base, where G is theGurarij space.

A Gurarij space is a Banach space G having the property that forany ε > 0, finite-dimensional Banach space E ⊆ F and anisometric embedding ϕ : E −→ G there is a linear mapψ : F −→ G extending ϕ such that in addition, for allx ∈ F , (1− ε)‖x‖ ≤ ‖ψ(x)‖ ≤ (1 + ε)‖x‖



Ben Yaacov

Let E be a separable Banach space,

Construct E = E0 ↪→ E1 ↪→ ..., withw(E ) = w(E0) = w(E1) = ... and Eω =

⋃En = G is the

Gurarij space.

Iso(E ) = Iso(E0) ↪→ Iso(E1) ↪→ ..., whenceG ↪→ Iso(E ) ↪→ Iso(Eω) = Iso(G).



Ben Yaacov

Let E be a separable Banach space,

Construct E = E0 ↪→ E1 ↪→ ..., withw(E ) = w(E0) = w(E1) = ... and Eω =

⋃En = G is the

Gurarij space.

Iso(E ) = Iso(E0) ↪→ Iso(E1) ↪→ ..., whenceG ↪→ Iso(E ) ↪→ Iso(Eω) = Iso(G).


Comparison

Pestov

Iso(U) is extremely amenable or has the fixed point on compactaproperty.

Folklore

Homeo(Iℵ0) act transitively on Iℵ0

Question

What about Iso(U) and Iso(G) or Iso(G) and Homeo(Iℵ0)


Precise question

Question

Is the group Iso(G) extremely amenable?

Same question

is the class KB of all finite dimensional Banach space has theapproximate Ramsey property?

Melleray and Tsankov

A metric Fraısse class K have approximate Ramsey property ifthe following happens: for all A � B ∈ K and for all ε > 0, there isC ∈ K such that for any coloring γ of AC there is β ∈ BC suchthat |γ(a)− γ(b)| < ε for all a, b ∈ AC(β)


Kechris, Pestov and Todorcevic

Extreme amenability of closed subgroups of S∞

Melleray and Tsankov

Extreme amenability of all polish groups.


Non-separable case

The question of existence of a universal topological group of agiven uncountable weight m remains open. In fact, it is open forany given cardinal m > ℵ0.


The Urysohn-Katetov space

Katetov, 1986

Let m be an infinite cardinal such that

sup {mn : n < m} = m, (1)

there exists a unique up to an isometry complete metric space Um

of weight m, such that

Um contains an isometric copy of every other metric space ofweight ≤ m

Um is < m-homogeneous, that is, an isometry between anytwo metric subspaces of density < m extends to a globalself-isometry of Um.

In particular, Uℵ0 is just the classical Urysohn space U.


The ideal candidate

The topological group Iso(Um), equipped with the topology ofsimple convergence, has weight m, and was a candidate for auniversal topological group of weight m.


SIN and FSIN groups

A topological group G is call SIN (Small InvariantNeighbourhoods)if it admits a base at the identity consistingof invariant neighborhoods.

A topological group G is called functionally balanced, orsometimes FSIN (“Functionally SIN”) if every right uniformlycontinuous bounded function on G is left uniformlycontinuous.

Every SIN group is FSIN. The converse implication has beenestablished for:

locally compact groups (Itzkowitz),metrizable groups (Protasov),locally connected groups (Megreslishvili, Nickolas and Pestov),

among others

It remains an open problem in the general case known as theItzkowitz problem


SIN and FSIN groups





among others



SIN and FSIN groups





among others



Property (OB): Rosendal

A topological group G has property (OB) if whenever G actsby isometries on a metric space (X ; d) every orbit is bounded.

Examples of such groups include, among others:

the infinite permutation group S∞ (Bergman),the unitary group U(`2) with the strong opererator topology(Atkin),the isometry group of the Urysohn sphere (that is, a sphere inthe Urysohn space) (Rosendal)Homeo([0, 1]ℵ0 ) (Rosendal).


Theorem

If G is a topological subgroup of Iso(Um) of density < m, havingproperty (OB), then G is FSIN.

Corollary

If G is a topological subgroup of Iso(Um) of density < m havingproperty (OB) which is either metrizable or locally connected, thenG is a SIN group.


Theorem

If G is a topological subgroup of Iso(Um) of density < m, havingproperty (OB), then G is FSIN.

Corollary

If G is a topological subgroup of Iso(Um) of density < m havingproperty (OB) which is either metrizable or locally connected, thenG is a SIN group.


Some groups with no-embeddings in Iso(Um)

1 Denote U© the unit sphere in the Urysohn metric space. Thegroup Iso(U©) is both metrizable and locally connected(Melleray), has property (OB) (Rosendal ) and is not SIN.

2 The group S∞ is Polish and has the property (OB)(Bergman).

3 The group Homeo([0, 1]ℵ0) is a non-SIN Polish group withproperty (OB)(Rosendal).

4 In particular the group Iso(U) admits no embedding intoIso(Um) as a topological subgroup.


Questions

Let κ be an uncountable cardinal. Does there exist a universaltopological group of weight κ?

Is Homeo([0, 1]κ) such a group?


Some universal results

Theorem

Every metrizable SIN group of weight ≤ m embeds into Iso(Um)

It is not clear for us whether every SIN group of weight ≤ membeds into Iso(Um).At the same time, not all subgroups of Iso(Um) are SIN:

Theorem

Every Pm-groups of weight m embeds into Iso(Um).

Pm-groups=the intersection of every family of strictly less than mopen subsets is open.


A Pm-groups which is not SIN

Pestov

Let K be a field of cofinality m. The linear group GL(K, n) is aPm-groups which is not SIN.


More questions

An ultrahomogeneous metric space X is oscillation stable if everyfinite partition γ of X contains an element A ∈ γ whose eachε-neighbourhood Aε, ε > 0, contains an isometric copy of X .

Odell and Schlumprecht

The unit sphere in the Hilbert space is not oscillation stable

Nguyen Van The and Sauer

The unit sphere of the separable Urysohn space U is oscillationstable.

Question

Is the same true for the unit sphere of the space Um? with m > ℵ0.


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