The full group of a countable measurable equivalence relation

Mercer, Richard

doi:10.1090/S0002-9939-1993-1139480-1

The full group of a countable measurable equivalence relation
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Proc. Amer. Math. Soc. 117 (1993), 323-333 Request permission

Abstract:

We study the group of all "$R$-automorphisms" of a countable equivalence relation $R$ on a standard Borel space, special Borel automorphisms whose graphs lie in $R$. We show that such a group always contains periodic maps of each order sufficient to generate $R$. A construction based on these periodic maps leads to totally nonperiodic $R$-automorphisms all of whose powers have disjoint graphs. The presence of a large number of periodic maps allows us to present a version of the Rohlin Lemma for $R$-automorphisms. Finally we show that this group always contains copies of free groups on any countable number of generators.

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