Metrizable universal minimal flows of Polish groups have a comeagre orbit

Yaacov, Itaï Ben; Melleray, Julien; Tsankov, Todor

doi:10.1007/s00039-017-0398-7

Cited by 30 publications

(61 citation statements)

References 14 publications

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“…H(W P ) has a non-metrizable universal minimal flow with a comeager orbit. Ben Yaacov-Melleray-Tsankov [3], generalizing a result of Zucker [22], showed:…”

Section: 2mentioning

confidence: 72%

“…Then note that if for some a, b ∈ S and c, the meet of a and b, we had that f (c) is strictly lower with respect to ≤ T than the meet of f (a) and f (b), then ¬C S (a, b, c) and C T (f (a), f (b), f (c)), which is impossible. Therefore we get (3). Now if we assume (3), then f also preserves ≤ S and ≤ T .…”

Section: The Universal Minimal Flow -Constructionmentioning

confidence: 92%

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Universal Minimal Flows of Generalized Ważewski Dendrites

Kwiatkowska¹

2018

J. symb. log.

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We study universal minimal flows of the homeomorphism groups of generalized Ważewski dendrites W P , P ⊆ {3, 4, . . . , ω}. If P is finite, we prove that the universal minimal flow of the homeomorphism group H(W P ) is metrizable and we compute it explicitly. This answers a question of Duchesne. If P is infinite, we show that the universal minimal flow of H(W P ) is not metrizable. This provides examples of topological groups which are Roelcke precompact and have a non-metrizable universal minimal flow with a comeager orbit.2010 Mathematics Subject Classification. 05D10, 37B05, 54F15, 03C98.

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“…H(W P ) has a non-metrizable universal minimal flow with a comeager orbit. Ben Yaacov-Melleray-Tsankov [3], generalizing a result of Zucker [22], showed:…”

Section: 2mentioning

confidence: 72%

Section: The Universal Minimal Flow -Constructionmentioning

confidence: 92%

Universal Minimal Flows of Generalized Ważewski Dendrites

Kwiatkowska¹

2018

J. symb. log.

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“…In fact, the converse is also true: if

M (G)

is metrisable, then there is a comeagre

G

‐orbit on

M (G)

and the stabiliser of a point in this orbit is extremely amenable and co‐precompact in

G

. This is proved by Zucker in and, independently, by Ben Yaacov, Melleray and Tsankov in . (The latter builds on work in and proves the result for arbitrary Polish groups

G

.…”

Section: Introductionmentioning

confidence: 85%

“…Using , we then have the following corollary, answering Question 1.5 in : Corollary There is a closed, oligomorphic permutation group

G

on a countable set whose universal minimal flow

M (G)

is not metrisable.…”

Section: Introductionmentioning

confidence: 99%

Automorphism groups and Ramsey properties of sparse graphs

Evans

Hubička

Nešetřil

2019

Proc. London Math. Soc.

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We study automorphism groups of sparse graphs from the viewpoint of topological dynamics and the Kechris, Pestov, Todorčević correspondence. We investigate amenable and extremely amenable subgroups of these groups using the space of orientations of the graph and results from structural Ramsey theory. Resolving one of the open questions in the area, we show that Hrushovski's example of an ω‐categorical sparse graph has no ω‐categorical expansion with extremely amenable automorphism group.

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“…(4) for every X, Y ⊂ E that belong to A, ε > 0, every finite coloring of Emb C (X, E) has an ε-monochromatic set of the form α • Emb C (X, Y ) for some α ∈ Aut C (E); (5) A satisfies the (ARP); (6) A satisfies the (ARP); (7) For every X, Y ∈ A, every r ∈ N and every ε > 0 there is Z ∈ A such that every r-coloring of Emb C (X, Z) has an ε-monochromatic set of the form γ…”

Section: Fraïssé Classes and Limits; Topological Dynamics Of Automorpmentioning

confidence: 99%

The Ramsey property for Banach spaces and Choquet simplices, and applications

Bartošová

Lopez-Abad

Lupini

et al. 2017

Comptes Rendus. Mathématique

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Abstract. We show that the Gurarij space G and its noncommutative analog NG both have extremely amenable automorphism group. We also compute the universal minimal flows of the automorphism groups of the Poulsen simplex P and its noncommutative analogue NP. The former is P itself, and the latter is the state space of the operator system associated with NP. This answers a question of Conley and Törnquist. We also show that the pointwise stabilizer of any closed proper face of P is extremely amenable. Similarly, the pointwise stabilizer of any closed proper biface of the unit ball of the dual of the Gurarij space (the Lusky simplex) is extremely amenable.These results are obtained via the Kechris-Pestov-Todorcevic correspondence, by establishing the approximate Ramsey property for several classes of finite-dimensional operator spaces and operator systems (with distinguished linear functionals), including: Banach spaces, exact operator spaces, function systems with a distinguished state, and exact operator systems with a distinguished state. This is the first direct application of the Kechris-Pestov-Todorcevic correspondence in the setting of metric structures. The fundamental combinatorial principle that underpins the proofs is the Dual Ramsey Theorem of Graham and Rothschild.In the second part of the paper, we obtain factorization theorems for colorings of matrices and Grassmannians over R and C, which can be considered as continuous versions of the Dual Ramsey Theorem for Boolean matrices and of the Graham-Leeb-Rothschild Theorem for Grassmannians over a finite field.Recall that given a topological group G, a compact G-space or G-flow is a compact Hausdorff space X endowed with a continuous action of G. Such a G-flow X is called minimal when every orbit is dense. There is a natural notion of morphism between G-flows, given by a (necessarily surjective) G-equivariant continuous map (factor). A minimal G-flow is universal if it factors onto any other minimal G-flow. It is a classical fact that any topological group G admits a unique (up to isomorphism of G-flows) universal minimal flow, usually denoted by M (G) [18,33]. For any locally compact non compact Polish group G, the universal minimal G-flow is nonmetrizable. At the opposite end, non locally compact topological groups often have metrizable universal minimal flows, or even reduced to a single point. A topological group for which M (G) is a singleton is called extremely amenable. (Amenability of G is equivalent to the assertion that every compact G-space has an invariant Borel measure. Thus any extremely amenable group is, in particular, amenable.)The universal minimal flow has been explicitly computed for many topological groups, typically given as automorphism groups of naturally arising mathematical structures. Examples of extremely amenable Polish groups include the group of order automorphisms of Q [61], the group of unitary operators on the separable infinite-dimensional Hilbert space [30], the automorphism group of the hyperfinite II 1 factor and of in...

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Metrizable universal minimal flows of Polish groups have a comeagre orbit

Abstract: Abstract. We prove that, whenever G is a Polish group with metrizable universal minimal flow M(G), there exists a comeagre orbit in M(G). It then follows that there exists an extremely amenable, closed, co-precompact subgroup G

Cited by 30 publications

References 14 publications

Universal Minimal Flows of Generalized Ważewski Dendrites

Universal Minimal Flows of Generalized Ważewski Dendrites

Automorphism groups and Ramsey properties of sparse graphs

The Ramsey property for Banach spaces and Choquet simplices, and applications

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