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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.

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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