Connected Polish groups with ample generics

Kaïchouh, Adriane; Maître, François Le

doi:10.1112/blms/bdv078

Cited by 9 publications

(18 citation statements)

References 28 publications

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“…Remark 3. 9. Using a similar approach, and a construction as in the proof of Lemma 5.6, one can also prove that the class of partial automorphisms of ordered K n -free graphs does not have WAP, for every n ≥ 3.…”

Section: Corollary 37mentioning

confidence: 93%

“…It is known that there exist Polish groups sharing both of these features. Pestov-Schneider [15] proved that, for any Polish group G, the group L 0 (G), i.e., the group of measurable functions with values in G, is extremely amenable, provided that G is amenable, and Kaïchouh-Le Maître [9] proved that L 0 (G) has ample generics whenever G has. As S ∞ , i.e., the group of all permutations of natural numbers, is amenable, and has ample generics, L 0 (S ∞ ) is extremely amenable and it has ample generics.…”

Section: Introductionmentioning

confidence: 99%

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Ordered structures and large conjugacy classes

Kwiatkowska

Malicki

2020

Journal of Algebra

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This article is a contribution to the following problem: does there exist a Polish non-archimedean group (equivalently: automorphism group of a Fraïssé limit) that is extremely amenable, and has ample generics. As Fraïssé limits whose automorphism groups are extremely amenable must be ordered, i.e., equipped with a linear ordering, we focus on ordered Fraïssé limits. We prove that automorphism groups of the universal ordered boron tree, and the universal ordered poset have a comeager conjugacy class but no comeager 2-dimensional diagonal conjugacy class. We also provide general conditions implying that there is no comeager conjugacy class, comeager 2-dimensional diagonal conjugacy class or non-meager 2-dimensional topological similarity class in the automorphism group of an ordered Fraïssé limit. We provide a number of applications of these results.2010 Mathematics Subject Classification. 03E15, 54H11.

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Section: Corollary 37mentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

Ordered structures and large conjugacy classes

Kwiatkowska

Malicki

2020

Journal of Algebra

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“…Since (G, d, d u ) has metric generics, there is g ∈ G such that S = Orb(g) du is comeager in G (here we take the closure with respect to d u ). Then by [KLM15] Orb…”

Section: Notationmentioning

confidence: 99%

“…The work of Kaïchouh and Le Maître [KLM15] shows how to build connected examples. Let L 0 ([0, 1], G) denote the space of measurable functions from [0, 1] to G, which is a group with pointwise multiplication and Polish with the topology of convergence in measure.…”

Section: Introductionmentioning

confidence: 99%

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Isometry Groups of Borel Randomizations

Berenstein¹,

Zamora²

2020

Notre Dame J. Formal Logic

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Automorphism groups of countable structures and groups of measurable functions

Kwiatkowska

Malicki

2019

Isr. J. Math.

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Let G be a topological group and let µ be the Lebesgue measure on the interval [0, 1]. We let L 0 (G) to be the topological group of all µ-equivalence classes of µ-measurable functions defined on [0,1] with values in G, taken with the pointwise multiplication and the topology of convergence in measure. We show that for a Polish group G, if L 0 (G) has ample generics, then G has ample generics, thus the converse to a result of Kaïchouh and Le Maître.We further study topological similarity classes and conjugacy classes for many groups Aut(M ) and L 0 (Aut(M )), where M is a countable structure. We make a connection between the structure of groups generated by tuples, the Hrushovski property, and the structure of their topological similarity classes. In particular, we prove the trichotomy that for every tuplef of Aut(M ), where M is a countable structure such that algebraic closures of finite sets are finite, either the countable group f is precompact, or it is discrete, or the similarity class off is meager, in particular the conjugacy class off is meager. We prove an analogous trichotomy for groups L 0 (Aut(M )).2010 Mathematics Subject Classification. 03E15, 54H11.

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Connected Polish groups with ample generics

Cited by 9 publications

References 28 publications

Ordered structures and large conjugacy classes

Ordered structures and large conjugacy classes

Isometry Groups of Borel Randomizations

Automorphism groups of countable structures and groups of measurable functions

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