On a measurable analogue of small topological full groups II

Le Maître, François

Annales de l'Institut Fourier, Tome 71 (2021) no. 5, pp. 1885-1927.

Nous poursuivons l’étude des groupes pleins ${\mathrm{L}}^1$ de graphages ainsi que des adhérences de leurs groupes dérivés, que nous appelons groupes pleins ${\mathrm{L}}^1$ 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 ${\mathrm{L}}^1$ dérivé est de rang topologique fini. Nous montrons également que tout graphage est moyennable si et seulement si son groupe plein ${\mathrm{L}}^1$ l’est, et présentons des exemples variés de groupes pleins ${\mathrm{L}}^1$ (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 ${\mathrm{L}}^1$ de graphages ergodiques moyennables doit être une quasi-isométrie pour leurs distances ${\mathrm{L}}^1$ respectives. Enfin, on montre que les groupes pleins ${\mathrm{L}}^1$ des transformations de rang 1 sont de rang topologique 2.

We pursue the study of ${\mathrm{L}}^1$ full groups of graphings and of the closures of their derived groups, which we call derived ${\mathrm{L}}^1$ 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 ${\mathrm{L}}^1$ full group has finite topological rank. We further show that a graphing is amenable if and only if its ${\mathrm{L}}^1$ full group is, and explain why various examples of (derived) ${\mathrm{L}}^1$ full groups fit very well into Rosendal’s geometric framework for Polish groups. As an application, we obtain that every abstract group isomorphism between ${\mathrm{L}}^1$ full groups of amenable ergodic graphings must be a quasi-isometry for their respective ${\mathrm{L}}^1$ metrics. We finally show that ${\mathrm{L}}^1$ full groups of rank one transformations have topological rank 2.

Keywords: groupe plein $\mathrm{L}^1$, groupes dérivés, entropie de Rokhlin
Mot clés : $\mathrm{L}^1$ full groups, derived groups, Rokhlin entrop

[1] Austin, Tim Behaviour of Entropy Under Bounded and Integrable Orbit Equivalence, Geom. Funct. Anal., Volume 26 (2016) no. 6, pp. 1483-1525 | DOI | MR | Zbl

[2] Baxter, John R. A class of Ergodic Transformations Having Simple Spectrum, Proc. Am. Math. Soc., Volume 27 (1971), pp. 275-279 | DOI | MR | Zbl

[3] Becker, Howard; Kechris, Alexander S. The Descriptive Set Theory of Polish Group Actions, London Mathematical Society Lecture Note Series, 232, Cambridge University Press, 1996 | DOI

[4] Belinskaya, R. M. Partitions of Lebesgue Space in Trajectories Defined by Ergodic Automorphisms, Funct. Anal. Appl., Volume 2 (1968) no. 3, pp. 190-199 | DOI | MR | Zbl

[5] Carderi, Alessandro; Le Maître, François More Polish Full Groups, Topology Appl., Volume 202 (2016), pp. 80-105 | DOI | MR | Zbl

[6] Connes, Alain; Feldman, Jacob; Weiss, Benjamin An Amenable Equivalence Relation Is Generated by a Single Transformation, Ergodic Theory Dyn. Syst., Volume 1 (1981) no. 04, pp. 431-450 | DOI | MR | Zbl

[7] Diaconis, Persi; Graham, Ronald L. Spearman’s Footrule as a Measure of Disarray, J. R. Stat. Soc., Ser. B, Stat. Methodol., Volume 39 (1977) no. 2, pp. 262-268 | MR | Zbl

[8] Dowerk, Philip A.; Le Maître, François Bounded Normal Generation Is Not Equivalent to Topological Bounded Normal Generation, Extr. Math., Volume 34 (2019) no. 1, pp. 85-97 | DOI | MR | Zbl

[9] Dye, Henry A. On Groups of Measure Preserving Transformations. I, Am. J. Math., Volume 81 (1959) no. 1, pp. 119-159 | DOI | MR | Zbl

[10] Gaboriau, Damien Orbit Equivalence and Measured Group Theory, Proceedings of the International Congress of Mathematicians 2010 (ICM 2010), Hindustan Book Agency, 2011, pp. 1501-1527 | DOI | Zbl

[11] Giordano, Thierry; Pestov, Vladimir Some Extremely Amenable Groups Related to Operator Algebras and Ergodic Theory, J. Inst. Math. Jussieu, Volume 6 (2007) no. 2, pp. 279-315 | DOI | MR | Zbl

[12] Giordano, Thierry; Putnam, Ian F.; Skau, Christian F. Full Groups of Cantor Minimal Systems, Isr. J. Math., Volume 111 (1999) no. 1, pp. 285-320 | DOI | MR | Zbl

[13] Glasner, Eli Ergodic Theory via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, 2003 | DOI

[14] Glasner, Eli; Tsirelson, Boris; Weiss, Benjamin The Automorphism Group of the Gaussian Measure Cannot Act Pointwise, Isr. J. Math., Volume 148 (2005) no. 1, pp. 305-329 | DOI | MR | Zbl

[15] Jackson, Stephen; Kechris, Alexander S.; Louveau, Alain Countable Borel Equivalence Relations, J. Math. Log., Volume 02 (2002) no. 01, pp. 1-80 | DOI | MR | Zbl

[16] Kac, Mark On the Notion of Recurrence in Discrete Stochastic Processes, Bull. Am. Math. Soc., Volume 53 (1947) no. 10, pp. 1002-1010 | MR | Zbl

[17] Kechris, Alexander S. Classical Descriptive Set Theory, Graduate Texts in Mathematics, 156, Springer, 1995 | DOI

[18] Kechris, Alexander S. Global Aspects of Ergodic Group Actions, Mathematical Surveys and Monographs, 160, American Mathematical Society, 2010 | DOI

[19] Kittrell, John; Tsankov, Todor Topological Properties of Full Groups, Ergodic Theory Dyn. Syst., Volume 30 (2010) no. 2, pp. 525-545 | DOI | MR | Zbl

[20] Le Maître, François Sur les groupes pleins préservant une mesure de probabilité, Ph. D. Thesis, ENS Lyon, France (2014)

[21] Le Maître, François On Full Groups of Non-Ergodic Probability-Measure-Preserving Equivalence Relations, Ergodic Theory Dyn. Syst., Volume 36 (2016) no. 7, pp. 2218-2245 | DOI | MR | Zbl

[22] Le Maître, François On a Measurable Analogue of Small Topological Full Groups, Adv. Math., Volume 332 (2018), pp. 235-286 | DOI | MR | Zbl

[23] Marks, Andrew S. Topological Generators for Full Groups of Hyperfinite Pmp Equivalence Relations, 2016 (https://arxiv.org/abs/1606.08080)

[24] Matui, Hiroki Some Remarks on Topological Full Groups of Cantor Minimal Systems, Int. J. Math., Volume 17 (2006) no. 02, pp. 231-251 | DOI | MR | Zbl

[25] Matui, Hiroki Topological Full Groups of One-Sided Shifts of Finite Type, J. Reine Angew. Math., Volume 2015 (2013) no. 705, pp. 35-84 | DOI | Zbl

[26] Nekrashevych, Volodymyr Simple Groups of Dynamical Origin, Ergodic Theory Dyn. Syst. (2017), pp. 1-26 | DOI

[27] Nekrashevych, Volodymyr Palindromic Subshifts and Simple Periodic Groups of Intermediate Growth, Ann. Math., Volume 187 (2018) no. 3, pp. 667-719 | DOI | MR | Zbl

[28] Pestov, Vladimir Concentration of Measure and Whirly Actions of Polish Groups, Probabilistic Approach to Geometry (Kyoto, 2008) (Advanced Studies in Pure Mathematics), Volume 57, Mathematical Society of Japan (2009), pp. 383-403 | Zbl

[29] Rokhlin, Vladimir A. Lectures on the Entropy Theory of Measure-Preserving Transformations, Russ. Math. Surv., Volume 22 (1967) no. 5, p. 1 | DOI | Zbl

[30] Rosendal, Christian Coarse Geometry of Topological Groups, Cambridge Tracts in Mathematics, 223, CUP, 2021 | DOI | MR | Zbl

[31] Seward, Brandon Ergodic Actions of Countable Groups and Finite Generating Partitions, Groups Geom. Dyn., Volume 9 (2015) no. 3, pp. 793-810 | DOI | MR | Zbl

[32] Seward, Brandon Krieger’s Finite Generator Theorem for Actions of Countable Groups I, Invent. Math., Volume 215 (2019) no. 1, pp. 265-310 | DOI | MR | Zbl