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Published April 2010 | Published
Journal Article Open

Topological properties of full groups

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 F_m and F_n 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.

Additional Information

© 2009 Cambridge University Press. Received 4 September 2007 and accepted in revised form 9 January 2009. We would like to thank our respective advisors G. Hjorth and A. S. Kechris for encouragement, support and guidance as well as B. Miller for the proof of Lemma 4.8 and useful discussions. We are also grateful to the referee for making useful comments and providing some references. The second author was partially supported by NSF grant DMS-0455285.

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