The Full Group of a Countable Measurable Equivalence Relation

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.