The dynamical hierarchy for Roelcke precompact Polish groups

Ibarlucía, Tomás

doi:10.1007/s11856-016-1399-1

Cited by 25 publications

(39 citation statements)

References 29 publications

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“…As explained in Subsection 2.3, the WAP compactification of a pro-oligomorphic group G is the space of types of pairs of embeddings x, y ∈ E(M) restricted to stable formulas. Dually, this can be stated by saying that WAP(G) is the closed algebra generated by the functions of the form ϕ a,b (g) = ϕ(a, gb), where ϕ(u, v) is a stable formula and a, b are tuples from M. For a more detailed explanation of this duality see [BT16,§5] or [Iba16,§4]. Hence it is natural to ask which formulas ϕ(u, v) give rise to functions in the subalgebra Hilb(G).…”

Section: 2mentioning

confidence: 99%

“…The correspondence between model-theoretic properties of ℵ 0 -categorical structures and dynamical properties of their automorphism groups is not restricted to the non-archimedean case. The correct model-theoretic setting for dealing with general Roelcke precompact Polish groups is that of continuous logic and in both [BT16] and [Iba16], the results are proved in full generality. However, the two most important tools used in this paper are currently only available in the non-archimedean setting: namely, the classification of the unitary representations on the dynamical side and the notion of one-basedness on the model-theoretic side.…”

Section: Introductionmentioning

confidence: 99%

“…We remark that every countable compactification of a Roelcke precompact Polish group G is a factor of W (G). Indeed, since countable compact systems are Asplund-representable (see, for instance, [GM06], Corollary 10.2), this follows from [Iba16], Theorem 2.9.…”

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confidence: 95%

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Eberlein oligomorphic groups

Yaacov

Ibarlucía

Tsankov

2017

Trans. Amer. Math. Soc.

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We study the Fourier-Stieltjes algebra of Roelcke precompact, non-archimedean, Polish groups and give a model-theoretic description of the Hilbert compactification of these groups. We characterize the family of such groups whose Fourier-Stieltjes algebra is dense in the algebra of weakly almost periodic functions: those are exactly the automorphism groups of ℵ 0 -stable, ℵ 0categorical structures. This analysis is then extended to all semitopological semigroup compactifica-2010 Mathematics Subject Classification. Primary: 22F50. Secondary: 03C45, 22A25, 03C98, 22A15.

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Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Eberlein oligomorphic groups

Yaacov

Ibarlucía

Tsankov

2017

Trans. Amer. Math. Soc.

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“…Finally, we point out that Rosenthal's dichotomy was imported in dynamics through the work of Köhler [9] and Glasner [5]. From there, the relationship with NIP was noticed independently by Chernikov and myself in the work [4] mentioned above and by Ibarlucía [8] in the context of ℵ 0categorical structures and automorphism groups. This paper is organized as follows: In the first section, we present the relevant part of the work of Bourgain, Fremlin and Talagrand.…”

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confidence: 76%

Rosenthal compacta and NIP formulas

Simon

2015

Fund. Math.

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We apply the work of Bourgain, Fremlin and Talagrand on compact subsets of the first Baire class to show new results about φ-types for φ NIP. In particular, we show that if M is a countable model, then an M -invariant φ-type is Borel definable. Also the space of Minvariant φ-types is a Rosenthal compactum, which implies a number of topological tameness properties.Shelah introduced the independence property (IP) for first order formulas in 1971 [13]. Some ten years later, Poizat [10] proved that a countable theory T does not have the independence property (is NIP) if and only if for any model M of T and type p ∈ S(M), p has at most 2 |M | coheirs (the bound a priori being 2 2 |M | ). Another way to state this result is to say that for any model M, the closure in S(M) of a subset of size at most κ has cardinality at most 2 κ . Thus NIP is equivalent to a topological tameness condition on the space of types.At about the same time, Rosenthal [11] studied Banach spaces not embedding l 1 . He showed that a separable Banach space B does not contain a closed subspace isomorphic to l 1 if and only if the unit ball of B is relatively sequentially compact in the bidual B * * , if and only if B * * has the same cardinality as B. Note that an element of B * * is by definition a function on B * , the topology on B * * is that of pointwise convergence, and B, identified with a subset of B * * , is dense. Shortly after this work, Rosenthal [12] and then Bourgain, Fremlin and Talagrand [2] extended the ideas of this theorem and studied systematically the pointwise closure of subsets A of continuous functions on a Polish space. It turns out that there is a sharp dichotomy: either the closureĀ contains non-measurable functions or all functions in the * Partially supported by ValCoMo (ANR-13-BS01-0006) and by MSRI, Berkeley.

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“…As the class of Roelcke flows does not seem to be of particular interest, let us simply mention that it is quite closely related to the class of strongly continuous flows as defined by , which is much better behaved. However, in the case of Roelcke precompact groups, Ibarlucía has shown in [Iba16a] that the corresponding subalgebra of RUC b (G) corresponds to the weakly almost periodic algebra (see Section 5.2). The study of the fixed point property on strongly continuous flows therefore reduces to that of equicontinuous and distal flows, which are treated in Section 5.…”

Section: Roelcke Flowsmentioning

confidence: 99%

Fixed points in compactifications and combinatorial counterparts

Thé¹

2019

Annales Henri Lebesgue

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The Kechris-Pestov-Todorcevic correspondence connects extreme amenability of topological groups with Ramsey properties of classes of finite structures. The purpose of the present paper is to recast it as one of the instances of a more general construction, allowing to show that Ramsey-type statements actually appear as natural combinatorial expressions of the existence of fixed points in certain compactifications of groups, and that similar correspondences in fact exist in various dynamical contexts. Résumé.-La correspondance de Kechris-Pestov-Todorcevic établit une relation entre la moyennabilité extrême des groupes topologiques et les propriétés de type Ramsey de certaines classes de structures finies. Le but de cet article est de la resituer comme une instance particulière d'une construction plus générale, permettant ainsi de montrer que des énoncés de type Ramsey apparaissent en fait comme l'expression combinatoire naturelle de l'existence de points fixes dans certaines compactifications de groupes, et que des correspondances similaires sont en réalité présentes dans toute une variété de contextes dynamiques.

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The dynamical hierarchy for Roelcke precompact Polish groups

Cited by 25 publications

References 29 publications

Eberlein oligomorphic groups

Eberlein oligomorphic groups

Rosenthal compacta and NIP formulas

Fixed points in compactifications and combinatorial counterparts

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