subgroups of isometries of urysohn-katetov metric spaces...
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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
Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces
Ulam’s Question, 1935
Ulam
Does there exist a universal topological group with a countablebase?
Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces
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 ).
Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces
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)
Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces
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).
Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces
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).
Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces
Solutions of Ulam Question
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.
Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces
Solutions of Ulam Question
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).
Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces
Solutions of Ulam Question
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).
Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces
Solutions of Ulam Question
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‖
Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces
Solutions of Ulam Question
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‖
Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces
Solutions of Ulam Question
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).
Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces
Solutions of Ulam Question
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).
Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces
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)
Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces
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)
Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces
Kechris, Pestov and Todorcevic
Extreme amenability of closed subgroups of S∞
Melleray and Tsankov
Extreme amenability of all polish groups.
Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces
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.
Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces
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.
Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces
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.
Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces
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
Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces
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
Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces
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
Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces
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).
Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces
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.
Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces
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.
Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces
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.
Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces
Questions
Let κ be an uncountable cardinal. Does there exist a universaltopological group of weight κ?
Is Homeo([0, 1]κ) such a group?
Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces
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.
Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces
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.
Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces
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.
Brice Rodrigue Mbombo Subgroups of isometries of Urysohn-Katetov metric spaces