Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T01:12:23.648Z Has data issue: false hasContentIssue false
  • English
  • Français

Topological properties of full groups

Published online by Cambridge University Press:  23 June 2009

JOHN KITTRELL  and
TODOR TSANKOV
JOHN KITTRELL*
Affiliation:
Mathematical Sciences Building 6363, University of California, Los Angeles, CA 90095, USA (email: [email protected])
TODOR TSANKOV*
Affiliation:
Department of Mathematics 253–37, California Institute of Technology, Pasadena, CA 91125, USA (email: [email protected])
*
Current address: Knightsbridge Asset Management, LLC, 660 Newport Center Drive, Suite 460, Newport Beach, CA 92660, USA.
Current address: Equipe d’Analyse, boîte 186, Université Pierre et Marie Curie, 4, Place Jussieu, 75252 Paris Cedex 05, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study full groups of countable, measure-preserving equivalence relations. Our main results include that they are all homeomorphic to the separable Hilbert space and that every homomorphism from an ergodic full group to a separable group is continuous. We also find bounds for the minimal number of topological generators (elements generating a dense subgroup) of full groups allowing us to distinguish between full groups of equivalence relations generated by free, ergodic actions of the free groups Fm and Fn if m and n are sufficiently far apart. We also show that an ergodic equivalence relation is generated by an action of a finitely generated group if an only if its full group is topologically finitely generated.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 2 , April 2010 , pp. 525 - 545
Copyright
Copyright © Cambridge University Press 2009

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

[1]Bessaga, C. and Pełczyński, A.. Selected Topics in Infinite-dimensional Topology. PWN—Polish Scientific Publishers, Warsaw, 1975.Google Scholar
[2]Danilenko, A. I.. The topological structure of Polish groups and groupoids of measure space transformations. Publ. Res. Inst. Math. Sci. 31(5) (1995), 913–940.Google Scholar
[3]Dobrowolski, T. and Toruńczyk, H.. Separable complete ANRs admitting a group structure are Hilbert manifolds. Topology Appl. 12(3) (1981), 229–235.CrossRefGoogle Scholar
[4]Dye, H. A.. On groups of measure preserving transformations. II. Amer. J. Math. 85 (1963), 551–576.CrossRefGoogle Scholar
[5]Fathi, A.. Le groupe des transformations de [0,1] qui préservent la mesure de Lebesgue est un groupe simple. Israel J. Math. 29(2–3) (1978), 302–308.CrossRefGoogle Scholar
[6]Feldman, J. and Moore, C. C.. Ergodic equivalence relations, cohomology, and von Neumann algebras. I. Trans. Amer. Math. Soc. 234(2) (1977), 289–324.Google Scholar
[7]Gaboriau, D.. Coût des relations d’équivalence et des groupes. Invent. Math. 139(1) (2000), 41–98.CrossRefGoogle Scholar
[8]Gaboriau, D. and Popa, S.. An uncountable family of nonorbit equivalent actions of 𝔽n. J. Amer. Math. Soc. 18(3) (2005), 547–559 (electronic).CrossRefGoogle Scholar
[9]Giordano, T. and Pestov, V.. Some extremely amenable groups. C. R. Math. Acad. Sci. Paris 334(4) (2002), 273–278.CrossRefGoogle Scholar
[10]Giordano, T., Putnam, I. F. and Skau, C. F.. Full groups of Cantor minimal systems. Israel J. Math. 111 (1999), 285–320.CrossRefGoogle Scholar
[11]Herman, R. H., Putnam, I. F. and Skau, C. F.. Ordered Bratteli diagrams, dimension groups and topological dynamics. Internat. J. Math. 3(6) (1992), 827–864.CrossRefGoogle Scholar
[12]Hjorth, G. and Molberg, M.. Free continuous actions on zero-dimensional spaces. Topology Appl. 153(7) (2006), 1116–1131.Google Scholar
[13]Ioana, A.. Orbit inequivalent actions for groups containing a copy of 𝔽2. Preprint, available at arXiv:math/0701027, 2007.Google Scholar
[14]Keane, M.. Contractibility of the automorphism group of a nonatomic measure space. Proc. Amer. Math. Soc. 26 (1970), 420–422.Google Scholar
[15]Kechris, A. S.. Global aspects of ergodic group actions and equivalence relations. Available athttp://www.math.caltech.edu/people/kechris.html.Google Scholar
[16]Kechris, A. S. and Miller, B. D.. Topics in Orbit Equivalence (Lecture Notes in Mathematics, 1852). Springer, Berlin, 2004.CrossRefGoogle Scholar
[17]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
[18]Kittrell, J.. Full groups and hyperfiniteness. PhD Thesis, UCLA, 2007.Google Scholar
[19]Kleppner, A.. Measurable homomorphisms of locally compact groups. Proc. Amer. Math. Soc. 106(2) (1989), 391–395.CrossRefGoogle Scholar
[20]Matui, H.. Some remarks on topological full groups of Cantor minimal systems. Internat. J. Math. 17(2) (2006), 231–251.CrossRefGoogle Scholar
[21]Medynets, K.. On approximation of homeomorphisms of a Cantor set. Fund. Math. 194(1) (2007), 1–13.CrossRefGoogle Scholar
[22]Melleray, J.. Topology of the isometry group of the Urysohn space. Unpublished note available athttp://math.univ-lyon1.fr/∼melleray/.Google Scholar
[23]Miller, B. D.. Density of topological full groups. Unpublished note available athttp://glimmeffros.googlepages.com/, 2005.Google Scholar
[24]Nhu, N. T.. The group of measure preserving transformations of the unit interval is an absolute retract. Proc. Amer. Math. Soc. 110(2) (1990), 515–522.CrossRefGoogle Scholar
[25]Pestov, V.. Dynamics of Infinite-dimensional Groups (University Lecture Series, 40). American Mathematical Society, Providence, RI, 2006 (The Ramsey–Dvoretzky–Milman phenomenon, Revised edition of Dynamics of infinite-dimensional groups and Ramsey-type phenomena, Inst. Mat. Pura. Apl. (IMPA), Rio de Janeiro, 2005).CrossRefGoogle Scholar
[26]Rosendal, C.. Automatic continuity in homeomorphism groups of compact 2-manifolds. Israel J. Math. 166 (2008), 349–367.CrossRefGoogle Scholar
[27]Rosendal, Christian and Solecki, S.. Automatic continuity of homomorphisms and fixed points on metric compacta. Israel J. Math. 162 (2007), 349–371.CrossRefGoogle Scholar
[28]Ryzhikov, V. V.. Representation of transformations preserving the Lebesgue measure, in the form of a product of periodic transformations. Math. Z. 38(6) (1985), 860–865, 957.Google Scholar
[29]Törnquist, A.. Orbit equivalence and actions of . J. Symbolic Logic 71(1) (2006), 265–282.CrossRefGoogle Scholar
[30]van Mill, J.. Infinite-dimensional Topology (North-Holland Mathematical Library, 43). North-Holland, Amsterdam, 1989.Google Scholar