On amenability and groups of measurable maps

Pestov, Vladimir; Schneider, Friedrich Martin

doi:10.1016/j.jfa.2017.09.011

Cited by 10 publications

(5 citation statements)

References 23 publications

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“…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%

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: Introductionmentioning

confidence: 99%

Ordered structures and large conjugacy classes

Kwiatkowska

Malicki

2020

Journal of Algebra

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“…In Milman's seminal joint work with Misha Gromov [16], concentration of measure was identified as a source of extreme amenability: if a topological group G contains a directed family of compact subgroups whose union is dense in G and whose normalized Haar measures concentrate in G, then G is extremely amenable. Following the examples of extremely amenable groups discovered in [16], this method has since found numerous further applications [13,12,40,7,5] and extensions [37,9,41,42,48,49].…”

Section: Introductionmentioning

confidence: 99%

“…Before exploring further applications, one may wonder about potential improvements and extensions of Theorem 1.1. Indeed, the present work has been crucially inspired by the following problem by Vladimir Pestov, which was communicated to the author in 2016 and published in [42,Question 4.5].…”

Section: Introductionmentioning

confidence: 99%

Concentration of invariant means and dynamics of chain stabilizers in continuous geometries

Schneider¹

2022

Preprint

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We prove a concentration inequality for invariant means on topological groups, namely for such adapted to a chain of amenable topological subgroups. The result is based on an application of Azuma's martingale inequality and provides a method for establishing extreme amenability. Building on this technique, we exhibit new examples of extremely amenable groups arising from von Neumann's continuous geometries. Along the way, we also answer a question by Pestov on dynamical concentration in direct products of amenable topological groups.

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“…2. Also note that if we reduce the class of smoothness even further and consider groups of measurable maps with the topology of convergence in measure, we get a very strong version of amenability called extreme amenability [13,20].…”

Section: Introductionmentioning

confidence: 99%

An amenability-like property of finite energy path and loop groups

Pestov¹

2021

Comptes Rendus. Mathématique

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We show that the groups of finite energy loops and paths (that is, those of Sobolev class H 1 ) with values in a compact connected Lie group, as well as their central extensions, satisfy an amenability-like property: they admit a left-invariant mean on the space of bounded functions uniformly continuous with regard to a left-invariant metric. Every strongly continuous unitary representation π of such a group (which we call skew-amenable) has a conjugation-invariant state on B (H π ).Résumé. Nous montrons que les groupes de lacets et de chemins à énergie finie (c.à.d. de classe H 1 de Sobolev) à valeurs dans un groupe de Lie compact et connexe, ainsi que leurs extensions centrales, satisfont une version de la moyennabilité: ils admettent une moyenne invariante à gauche sur l'espace de fonctions bornées uniformément continues par rapport a une métrique invariante à gauche. Chaque représentation unitaire continue, π, d'un tel groupe (que nous disons d'être "moyennable en biais") possède un état sur B (H π ) invariant sous conjugaison.

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On amenability and groups of measurable maps

Cited by 10 publications

References 23 publications

Ordered structures and large conjugacy classes

Ordered structures and large conjugacy classes

Concentration of invariant means and dynamics of chain stabilizers in continuous geometries

An amenability-like property of finite energy path and loop groups

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