On a measurable analogue of small topological full groups II
Annales de l'Institut Fourier, Volume 71 (2021) no. 5, pp. 1885-1927.

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.

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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DOI: 10.5802/aif.3443
Keywords: groupe plein L 1 , groupes dérivés, entropie de Rokhlin
Le Maître, François 1

1 Université de Paris, Sorbonne Université, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, 75013 Paris (France)
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Le Maître, François. On a measurable analogue of small topological full groups II. Annales de l'Institut Fourier, Volume 71 (2021) no. 5, pp. 1885-1927. doi : 10.5802/aif.3443. https://aif.centre-mersenne.org/articles/10.5802/aif.3443/

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