Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-17T17:10:02.025Z Has data issue: false hasContentIssue false
  • English
  • Français

On full groups of non-ergodic probability-measure-preserving equivalence relations

Published online by Cambridge University Press:  15 June 2015

FRANÇOIS LE MAÎTRE Open the ORCID record for FRANÇOIS LE MAÎTRE [Opens in a new window]
FRANÇOIS LE MAÎTRE*
Affiliation:
Université catholique de Louvain, Institut de Recherche en Mathématiques et Physique (IRMP), Chemin du Cyclotron 2, Box L7.01.02, 1348 Louvain-la-Neuve, Belgium email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This article generalizes our previous results [Le Maître. The number of topological generators for full groups of ergodic equivalence relations. Invent. Math. 198 (2014), 261–268] to the non-ergodic case by giving a formula relating the topological rank of the full group of an aperiodic probability-measure-preserving (pmp) equivalence relation to the cost of its ergodic components. Furthermore, we obtain examples of full groups that have a dense free subgroup whose rank is equal to the topological rank of the full group, using a Baire category argument. We then study the automatic continuity property for full groups of aperiodic equivalence relations, and find a connected metric for which they have the automatic continuity property. This allows us to provide an algebraic characterization of aperiodicity for pmp equivalence relations, namely the non-existence of homomorphisms from their full groups into totally disconnected separable groups. A simple proof of the extreme amenability of full groups of hyperfinite pmp equivalence relations is also given, generalizing a result of Giordano and Pestov to the non-ergodic case [Giordano and Pestov. Some extremely amenable groups related to operator algebras and ergodic theory. J. Inst. Math. Jussieu6(2) (2007), 279–315, Theorem 5.7].

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 7 , October 2016 , pp. 2218 - 2245
Copyright
© Cambridge University Press, 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Connes, A., Feldman, J. and Weiss, B.. An amenable equivalence relation is generated by a single transformation. Ergod. Th. & Dynam. Sys. 1(4) (1981), 431–450.CrossRefGoogle Scholar
Dye, H. A.. On groups of measure preserving transformation. I. Amer. J. Math. 81 (1959), 119–159.CrossRefGoogle Scholar
Dye, H. A.. On groups of measure preserving transformations. II. Amer. J. Math. 85 (1963), 551–576.CrossRefGoogle Scholar
Eigen, S. J.. On the simplicity of the full group of ergodic transformations. Israel J. Math. 40(3–4) (1981), 345–349.CrossRefGoogle Scholar
Epstein, I.. Some results on orbit inequivalent actions of non-amenable groups. PhD Thesis, University of California, Los Angeles, ProQuest LLC, Ann Arbor, MI, 2008.Google Scholar
Gaboriau, D.. Coût des relations d’équivalence et des groupes. Invent. Math. 139(1) (2000), 41–98.CrossRefGoogle Scholar
Glasner, E.. On minimal actions of Polish groups. Topology Appl. 85(1–3) (1998), 119–125, 8th Prague Topological Symposium on General Topology and Its Relations to Modern Analysis and Algebra, 1996.CrossRefGoogle Scholar
Giordano, T. and Pestov, V.. Some extremely amenable groups related to operator algebras and ergodic theory. J. Inst. Math. Jussieu 6(2) (2007), 279–315.CrossRefGoogle Scholar
Giordano, T., Putnam, I. F. and Skau, C. F.. Full groups of Cantor minimal systems. Israel J. Math. 111 (1999), 285–320.CrossRefGoogle Scholar
Kaimanovich, V. A.. Amenability, hyperfiniteness, and isoperimetric inequalities. C. R. Math. Acad. Sci. Paris 325(9) (1997), 999–1004.CrossRefGoogle Scholar
Kechris, A. S.. Global Aspects of Ergodic Group Actions (Mathematical Surveys and Monographs, 160) . American Mathematical Society, Providence, RI, 2010.CrossRefGoogle Scholar
Kechris, A. S. and Miller, B. D.. Topics in Orbit Equivalence (Lecture Notes in Mathematics, 1852) . Springer, Berlin, 2004.CrossRefGoogle Scholar
Kittrell, J. and Tsankov, T.. Topological properties of full groups. Ergod. Th. & Dynam. Sys. 30(2) (2010), 525–545.CrossRefGoogle Scholar
Levitt, G.. On the cost of generating an equivalence relation. Ergod. Th. & Dynam. Sys. 15(6) (1995), 1173–1181.CrossRefGoogle Scholar
Le Maître, F.. The number of topological generators for full groups of ergodic equivalence relations. Invent. Math. 198 (2014), 261–268.CrossRefGoogle Scholar
Le Maître, F.. Sur les groupes pleins préservant une mesure de probabilité. PhD Thesis, ENS Lyon, 2014ENSL0892, 2014.Google Scholar
Matui, H.. Some remarks on topological full groups of Cantor minimal systems II. Ergod. Th. & Dynam. Sys. 33(5) (2013), 1542–1549.CrossRefGoogle Scholar
Ornstein, D. S. and Weiss, B.. Ergodic theory of amenable group actions. I. The Rohlin lemma. Bull. Amer. Math. Soc. (N.S.) 2(1) (1980), 161–164.CrossRefGoogle Scholar
Rosendal, C.. Automatic continuity of group homomorphisms. Bull. Symbolic Logic 15(2) (2009), 184–214.CrossRefGoogle Scholar
Rosendal, C. and Solecki, S.. Automatic continuity of homomorphisms and fixed points on metric compacta. Israel J. Math. 162 (2007), 349–371.CrossRefGoogle Scholar
Ryzhikov, V. V.. Factorization of an automorphism of a full Boolean algebra into the product of three involutions. Mat. Zametki 54(2) (1993), 79–84, 159.Google Scholar
Veech, W. A.. Topological dynamics. Bull. Amer. Math. Soc. (N.S.) 83(5) (1977), 775–830.CrossRefGoogle Scholar