Hostname: page-component-7f64f4797f-d87pz Total loading time: 0 Render date: 2025-11-07T00:33:05.383Z Has data issue: false hasContentIssue false
  • English
  • Français

Full groups of minimal homeomorphisms and Baire category methods

Published online by Cambridge University Press:  02 October 2014

TOMÁS IBARLUCÍA  and
JULIEN MELLERAY
TOMÁS IBARLUCÍA
Affiliation:
Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne Cedex, France email melleray@math.univ-lyon1.fr
JULIEN MELLERAY
Affiliation:
Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne Cedex, France email melleray@math.univ-lyon1.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

We study full groups of minimal actions of countable groups by homeomorphisms on a Cantor space $X$, showing that these groups do not admit a compatible Polish group topology and, in the case of $\mathbb{Z}$-actions, are coanalytic non-Borel inside $\text{Homeo}(X)$. We point out that the full group of a minimal homeomorphism is topologically simple. We also study some properties of the closure of the full group of a minimal homeomorphism inside $\text{Homeo}(X)$.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 2 , April 2016 , pp. 550 - 573
Copyright
© Cambridge University Press, 2014 

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.)

Article purchase

Temporarily unavailable

References

Akin, E.. Good measures on Cantor space. Trans. Amer. Math. Soc. 357(7) (2005), 2681–2722.CrossRefGoogle Scholar
Anderson, R. D.. The algebraic simplicity of certain groups of homeomorphisms. Amer. J. Math. 80 (1958), 955–963.CrossRefGoogle Scholar
Angel, O., Kechris, A. S. and Lyons, R.. Random orderings and unique ergodicity of automorphism groups. Preprint, 2012, J. Eur. Math. Soc., to appear.Google Scholar
Bezuglyi, S. and Kwiatkowski, J.. Topologies on full groups and normalizers of Cantor minimal systems. Mat. Fiz. Anal. Geom. 9(3) (2002), 455–464.Google Scholar
Bezuglyi, S. and Medynets, K.. Full groups, flip conjugacy, and orbit equivalence of Cantor minimal systems. Colloq. Math. 110(2) (2008), 409–429.CrossRefGoogle Scholar
Boyle, M. and Tomiyama, J.. Bounded topological orbit equivalence and C∗ -algebras. J. Math. Soc. Japan 50(2) (1998), 317–329.CrossRefGoogle Scholar
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
Y. de Cornulier. Groupe pleins-topologiques [d’après Matui, Juschenko, Monod, . . .]. Séminaire Nicolas Bourbaki: Volume 2012-2013, exposés 1063 066 (Société Mathématique de France, Paris, 2013).Google Scholar
Dye, H. A.. On groups of measure preserving transformations. 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
Gamarnik, D.. Minimality of the group Autohomeom(C). Serdica 17(4) (1991), 197–201.Google Scholar
Grigorchuk, R. and Medynets, K.. On algebraic properties of topological full groups. Preprint, 2012,arXiv:1105.0719, Mat. Sbornik (2014), to appear.CrossRefGoogle Scholar
Giordano, T., Matui, H., Putnam, I. F. and Skau, C. F.. Orbit equivalence for Cantor minimal ℤd-systems. Invent. Math. 179(1) (2010), 119–158.CrossRefGoogle Scholar
Giordano, T., Putnam, I. F. and Skau, C. F.. Topological orbit equivalence and C ∗ -crossed products. J. Reine Angew. Math. 469 (1995), 51–111.Google 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
Glasner, E. and Weiss, B.. Weak orbit equivalence of Cantor minimal systems. Internat. J. Math. 6(4) (1995), 559–579.CrossRefGoogle Scholar
Juschenko, K. and Monod, N.. Cantor systems, piecewise translations and simple amenable groups. Ann. of Math. (2) 178(2) (2013), 775–787.CrossRefGoogle Scholar
Kechris, A. S.. Classical Descriptive Set Theory (Graduate Texts in Mathematics, 156). Springer, New York, 1995.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 Rosendal, C.. Turbulence, amalgamation and generic automorphisms of homogeneous structures. Proc. Lond. Math. Soc. 94(2) (2007), 302–350.CrossRefGoogle Scholar
Kechris, A. S., Sokic, M. and Todorcevic, S.. Ramsey properties of finite measure algebras and topological dynamics of the group of measure preserving automorphisms: some results and an open problem. Preprint, 2012.Google Scholar
Matui, H.. Some remarks on topological full groups of Cantor minimal systems. Internat. J. Math. 17(2) (2006), 231–251.CrossRefGoogle Scholar
Medynets, K.. Reconstruction of orbits of Cantor systems from full groups. Bull. Lond. Math. Soc. 43(6) (2011), 1104–1110.CrossRefGoogle Scholar
Rosendal, C.. On the non-existence of certain group topologies. Fund. Math. 187(3) (2005), 213–228.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
Wei, T.-J. Descriptive properties of measure preserving actions and the associated unitary representations. PhD Thesis, California Institute of Technology, 2005.Google Scholar