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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: jw.kittrell@gmail.com)
TODOR TSANKOV*
Affiliation:
Department of Mathematics 253–37, California Institute of Technology, Pasadena, CA 91125, USA (email: todor@math.jussieu.fr)
*
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.
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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.

Information

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

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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 $\Bbb F\sb n$. 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