More Polish full groups

Carderi, Alessandro; Maître, François Le

doi:10.1016/j.topol.2015.12.065

Cited by 3 publications

(13 citation statements)

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“…Quasi-isometric compatible norms will result in the same L 1 full groups, so actions of Polish boundedly generated groups have canonical L 1 full groups associated with them based on to the work of C. Rosendal [Ros18]. Our study also parallels the generalization of the full group construction introduced by A. Carderi and the first named author in [CLM16], where full groups were defined for Borel measure-preserving actions of Polish groups.…”

Section: Introductionmentioning

confidence: 83%

“…The only part where certain care needs to be exercised in this regard appears in Section 2. We define L 1 full groups for Borel measurepreserving actions of Polish normed groups, and we need a genuine action on the space X for these to make sense just as in [CLM16]. Boolean actions (also called near actions) of general Polish groups do not admit realizations in general [GTW05], and even when they do, it could happen that different realizations yield different full groups.…”

Section: Theorem 2 Let Gmentioning

confidence: 99%

“…Our goal is to prove that the topology of the norm • 1 on [G X ] 1 is a Polish topology. Following the approach taken in [CLM16], we provide a different definition of the L 1 full group, where Polishness of the topology will be readily obtainable, and then argue that the two constructions are isometrically isomorphic. Note that our construction encompasses that from [CLM16] since the latter corresponds to the case when G is equipped with a compatible bounded norm.…”

Section: Full Groups Of Polish Group Actionsmentioning

confidence: 99%

“…D(x 1 , x 2 ) = inf g ∈G { g : g x 1 = x 2 }. Following [CLM16], a full group of the action is denoted by [R G ] and is defined as the collection of all measure-preserving T ∈ Aut(X , µ) that satisfy xR G T x for all x ∈ X . The L 1 full group [G X ] 1 is given by those T ∈ [R G ] for which the map X ∋ x → D(x, T x) is integrable.…”

Section: Introductionmentioning

confidence: 99%

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$\mathrm{L}^1$ full groups of flows

Maître,

Slutsky

2021

Preprint

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We introduce the concept of an L 1 full group associated with a measure-preserving action of a Polish normed group on a standard probability space. Such groups are shown to carry a natural separable complete metric, and are thus Polish. Our construction generalizes L 1 full groups of actions of discrete groups, which have been studied recently by the first author.We show that under minor assumptions on the actions, topological derived subgroups of L 1 full groups are topologically simple and -when the acting group is locally compact and amenable -are whirly amenable and generically two-generated.For measure-preserving actions of the real line (also known as measure-preserving flows), the topological derived subgroup of an L 1 full groups is shown to coincide with the kernel of the index map, which implies that L 1 full groups of free measure-preserving flows are topologically finitely generated if and only if the flow admits finitely many ergodic components. The latter is in a striking contrast to the case of Z-actions, where the number of topological generators is controlled by the entropy of the action.

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Section: Introductionmentioning

confidence: 83%

Section: Theorem 2 Let Gmentioning

confidence: 99%

Section: Full Groups Of Polish Group Actionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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$\mathrm{L}^1$ full groups of flows

Maître,

Slutsky

2021

Preprint

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“…We recall now the definition of orbit full groups and their Polish topology (see [CM16] for more details and proofs). Definition 1.10.…”

Section: Orbit Full Groupsmentioning

confidence: 99%

Orbit full groups for locally compact groups

Carderi

Maître

2017

Trans. Amer. Math. Soc.

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We show that the topological rank of an orbit full group generated by an ergodic, probability measure-preserving free action of a non-discrete unimodular locally compact Polish group is two. For this, we use the existence of a cross section and show that for a locally compact Polish group, the full group generated by any dense subgroup is dense in the orbit full group of the action of the group.We prove that the orbit full group of a free action of a locally compact Polish group is extremely amenable if and only if the acting group is amenable, using the fact that the full group generates the von Neumann algebra of the action.

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On a measurable analogue of small topological full groups II

Maître¹

2022

Annales De l'Institut Fourier

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We pursue the study of L 1 full groups of graphings and of the closures of their derived groups, which we call derived L 1 full groups. Our main result shows that aperiodic probability measure-preserving actions of finitely generated groups have finite Rokhlin entropy if and only if their derived L 1 full group has finite topological rank. We further show that a graphing is amenable if and only if its L 1 full group is, and explain why various examples of (derived) L 1 full groups fit very well into Rosendal's geometric framework for Polish groups. As an application, we obtain that every abstract group isomorphism between L 1 full groups of amenable ergodic graphings must be a quasi-isometry for their respective L 1 metrics. We finally show that L 1 full groups of rank one transformations have topological rank 2.Résumé. -Nous poursuivons l'étude des groupes pleins L 1 de graphages ainsi que des adhérences de leurs groupes dérivés, que nous appelons groupes pleins L 1 dérivés. Notre résultat principal montre que toute action préservant la mesure de probabilité d'un groupe de type fini est d'entropie de Rokhlin finie si et seulement si son groupe plein L 1 dérivé est de rang topologique fini. Nous montrons également que tout graphage est moyennable si et seulement si son groupe plein L 1 l'est, et présentons des exemples variés de groupes pleins L 1 (parfois dérivés) qui rentrent dans le cadre de la géométrie des groupes polonais construits par Rosendal. On en déduit que tout isomorphisme abstrait entre des groupes pleins L 1 de graphages ergodiques moyennables doit être une quasi-isométrie pour leurs distances L 1 respectives. Enfin, on montre que les groupes pleins L 1 des transformations de rang 1 sont de rang topologique 2.

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More Polish full groups

Cited by 3 publications

References 54 publications

$\mathrm{L}^1$ full groups of flows

$\mathrm{L}^1$ full groups of flows

Orbit full groups for locally compact groups

On a measurable analogue of small topological full groups II

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